Aspects of W-symmetries - ICCUBicc.ub.edu › ~gomis › documents › tesis › tesi_J_Herrero.pdf · No em vull oblidar dels companys de fatigues: Sergi, Alex, Jordi, Antonio, Paula,`

Aspects of W-symmetries

Jose Herrero Izquierdo

UNIVERSITAT de BARCELONA

Departament d’Estructura i Constituents de la Materia

Aspects of W-symmetries

Memoria de la tesi realitzada per

Jose Herrero Izquierdo

per tal d’optar al grau de

Doctor en Ciencies Fısiques.

Marc del 2000

Vull expressar el meu agraıment a en Joaquim Gomis per haver acceptat la direcciod’aquest treball i per haver fet tot el possible, cientıficament i humanament, perqueaquest arribes a bon port.

I am also indebted to Prof. Kiyoshi Kamimura, from Toho University (Japan) fora pleasant and fruitful scientific collaboration that has helped me so much in the real-ization of this work. Thank you!

Pels mateixos motius cal agrair la contribucio cientıfica d’en Jaume Roca, contribu-cio que ha estat tan decisiva com sincera la seva amistat.

Un record a la meva famılia cientıfica, pel seu recolzament en tants aspectes: ames d’en Quim i en Jaume cal esmentar en Josep Maria Pons, en Jordi Parıs, enFredy Zamora, en Joan Simon i en David Mateos. Estenc l’agraıment als amics de laUniversitat Politecnica.

Agraeixo aquelles persones que em van rebre en diverses estades cientıfiques: alProf. Steven Weinberg, de la Universitat d’Austin (Texas), aixı com, un altre cop, aen Quim i la seva famılia; al Prof. Walter Troost, de la Universitat de Leuven (Bel-gica) i, novament, a en Jordi. Tambien a Alberto Ibort, entonces en la UniversidadComplutense de Madrid.

Al Comissionat per a Universitats i Recerca, de la Generalitat de Catalunya, pelseu suport economic.

No em vull oblidar dels companys de fatigues: Sergi, Alex, Jordi, Antonio, Paula,Jose Mari, Assum, a mes d’en Fredy, Jaume, Jordi i Joan, ja esmentats. Vosaltres souels protagonistes dels bons records que m’enduc d’aquests anys d’ECM.

Y, finalmente, quiero dar las gracias a toda mi familia y, en especial, a mis padres,pues este trabajo es, en cierto modo, obra suya. Es por ello que se lo dedico con todomi carino.

Barcelona, diecinueve de marzo del ano dos mil.

. . . all novelty is but oblivion.

Francis Bacon,

Essays, LVIII.

Contents

1 Introduction 1

1.1 Conformal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Quantum W-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 Classical limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.4 Overview of W-symmetry applications . . . . . . . . . . . . . . . . . . . 141.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Classical W-algebras 19

2.1 Drinfel’d–Sokolov reductions . . . . . . . . . . . . . . . . . . . . . . . . . 192.1.1 sl(2,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1.2 sl(3,R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.1.3 Towards a classification: WG

S algebras . . . . . . . . . . . . . . . 222.2 WG

S -transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.3 Integrable systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3.1 The KP equations . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Particle Mechanics Models with W-symmetries 35

3.1 Particle mechanics model with sp(2M,R) symmetry . . . . . . . . . . . 363.1.1 Diffeomorphism invariance of the sp(2M,R) model . . . . . . . . 393.1.2 Finite gauge transformations of the sp(2M,R) model . . . . . . . 41

3.2 W2 model and finite gauge transformations . . . . . . . . . . . . . . . . 423.3 sp(4,R) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.3.1 (0, 1) embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.3.2 (1, 1) (principal) embedding . . . . . . . . . . . . . . . . . . . . . 513.3.3 (1

2 , 0) embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . 543.4 sl(3,R) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.4.1 W3 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.4.2 W2

3 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

v

vi CONTENTS

4 Relations between W-algebras 634.1 Linearizing W3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Non-local V-algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 674.3 Secondary reductions of W2

3 . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.1 From W2

3 to W3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.3.2 Non-local V(2)

2,2 algebra . . . . . . . . . . . . . . . . . . . . . . . . 724.3.3 The change of variables connecting W1+3 and W lin

3 . . . . . . . . 734.4 Secondary reductions of W(0,1)

sp(4,R) . . . . . . . . . . . . . . . . . . . . . . 754.4.1 Linearizing W(2, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . 764.4.2 Non-local V(1)

2,2 algebra . . . . . . . . . . . . . . . . . . . . . . . . 76

4.5 From W( 12,0)

sp(4,R) to W(0,1)sp(4,R) . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5 W3 Geometry 815.1 Extension of the base space . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.1.1 W2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.1.2 W3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.1.3 Super-Virasoro . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.2 W3 diffeomorphisms in the W3 space . . . . . . . . . . . . . . . . . . . . 965.2.1 Geometrical status of the W3 fields . . . . . . . . . . . . . . . . . 1015.2.2 Finite transformations and the W3-Schwarzian derivative . . . . 1025.2.3 A covariant system of differential equations . . . . . . . . . . . . 105

5.3 Generalization to WN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

Conclusions 111

A sl(2,R) embeddings 115

B Signature of the space-time for the particle models 121

Resum 125

Bibliography 135

Chapter 1

Introduction

This thesis is about W-symmetries. In this first chapter an introduction to the conceptof W-algebra will be presented. W-symmetries are extensions of the two-dimensionalconformal symmetry and, for this reason, the first section of the chapter will reviewthe basic concepts of conformal symmetry (standard references for this issue are [1]).The aim of this review is to prepare the definition of (quantum) W-algebra, whichis the subject of sect. 1.2. The forthcoming chapters of the thesis deal with the so-called classical limits of W-algebras; these limits are stated in sect. 1.3. Once the basictheoretical framework is set, sect. 1.4 is devoted to put W-symmetries in the context ofthe different —and, apparently, disconnected— fields of Theoretical Physics where theyhave been applied. Finally, a sketch of the following chapters of the thesis is presentedin sect. 1.5.

1.1 Conformal symmetry

Consider a d-dimensional manifold equipped with a metric tensor that in a coordinatesystem is expressed as gµν(x). A coordinate transformation xµ 7→ x′µ(x) is said to beconformal if the form of the metric tensor is preserved up to a factor:

g′µν(x) = Ω(x) gµν(x). (1.1.1)

Infinitesimally, this condition is equivalent to the conformal Killing equation:

gµα ∂νεα + gνα ∂µεα + εα ∂αgµν =

1d

(2 ∂αεα + εα gσρ ∂αgσρ) gµν , (1.1.2)

where

εµ(x) ≡ x′µ(x)− xµ ≡ δxµ (1.1.3)

parametrizes the infinitesimal coordinate transformation.

1

2 Introduction

In a flat manifold the metric can be chosen to be ηµν , the constant diagonal metricof signature (p, q). The conformal Killing equation is written:

ηµα ∂νεα + ηνα ∂µεα =

2d

ηµν ∂αεα. (1.1.4)

When d 6= 2 the general solution of this differential equation depends on (d+1)(d+2)/2constant parameters:

εµ(x) = aµ + ωµν xν + b xµ + cν (2xµxν − ηµνxαxα) , ωµν = ηµα ωα

ν = −ωνµ.

(1.1.5)

In this situation, the conformal transformations are a realization of a Lie group whoseLie algebra is so(p + 1, q + 1). The isometric transformations are a subset of theconformal transformations and, therefore, the Lorentz transformations are included in(1.1.5).

In any two-dimensional manifold a coordinate system can be chosen such that themetric tensor is written (at least locally) as:

gµν(x) = eφ(x)ηµν , (1.1.6)

where ηµν is a constant metric. The conformal Killing equation becomes eq. (1.1.4)with d = 2. When ηµν is minkowskian (ηµν = diag(+,−)) then the general solution ofeq. (1.1.4) is given by:

ε1(x) =12

(a+(x1 + x2) + a−(x1 − x2)

), ε2(x) =

12

(a+(x1 + x2)− a−(x1 − x2)

),

(1.1.7)

or, in terms of the light-cone coordinates:

y± ≡ x1 ± x2 ⇒ δy± = a±(y±). (1.1.8)

Here a+ and a− are two arbitrary real functions. In other words, when d = 2 theconformal transformations are the realization of an infinite-dimensional group. Theexistence of such a huge symmetry —not to be confused with a gauge symmetry—implies that one can obtain a large number of results from a d = 2 conformal-invarianttheory only by invoking symmetry arguments.

The infinitesimal transformations (1.1.8) are a realization of the algebra Vir⊕Vir,where Vir is the (classical) Virasoro algebra,

[ln, lm] = (n−m) ln+m, n, m ∈ Z, (1.1.9)

whose generators can be taken to be

l±n = − (y±

)n+1 ∂

∂y±, ∀n ∈ Z. (1.1.10)

1.1 Conformal symmetry 3

If the two-dimensional manifold is euclidean (ηµν = diag(+, +)) then the situationis similar but instead of the light-cone cooordinates one has to introduce a complexcoordinate:

z ≡ x1 + ix2 ⇔ x1 =12

(z + z) , x2 =12i

(z − z) , (1.1.11)

Then the conformal Killing equation is solved in this way:

δz = a(z), (1.1.12)

where a(z) is an arbitrary holomorphic infinitesimal complex function. The infinitesimalgenerators of these transformations are:

ln = −zn+1 ∂

∂z, ∀n ∈ Z, (1.1.13)

which span VirC, the Virasoro algebra over C. The algebra of conformal transforma-tions on the original real euclidean plane is the algebra over R generated by ln andilm (n,m ∈ Z). Note that, unlike the minkowskian case, this algebra can not be de-composed as two mutually commuting Virasoro algebras. An alternative interpretationconsists in treating z as another complex coordinate —independent of z. The (infinites-imal) conformal transformations are then viewed as the set of arbitrary holomorphicchanges of coordinates in this extended manifold of complex dimension equal to two.The algebra of transformations in the original real manifold is then the proper real formof VirC⊕VirC which corresponds to the identification of the second complex coordinateas the complex conjugate of the first one. Both approaches are equivalent.

Finite conformal transformations depend on the global properties of the two-dimen-sional manifold. When this manifold is topologically equivalent to the two-dimensionalplane R2 then finite conformal transformations associated with infinitesimal generatorsin eq. (1.1.10) or eq. (1.1.13) having n < −1 are forbidden. The whole set of infinites-imal generators produce well-defined finite transformations when the topology of themanifold is equivalent to R2 − 0. This is the case, for instance, of a cylinder. Thefinite transformations corresponding to eq. (1.1.12) are, of course, parametrized as:

z′ = f(z), (1.1.14)

for any f , holomorphic function. On the other hand, if the manifold is compactifiedas R2 ∪ ∞, which is topologically equivalent to the Riemann sphere S2, then finitetransformations are restricted to those generated by the elements in eq. (1.1.10) oreq. (1.1.13) with index −1, 0 and 1. In the minkowskian case the generators span asubalgebra of Vir ⊕ Vir which is equivalent to sl(2,R) ⊕ sl(2,R) ' so(2, 2). If themetric is euclidean then the corresponding (real) algebra is sl(2,C)R ' so(3, 1) and the

4 Introduction

associated finite transformations (in the complex language) are the so-called Mobius orprojective transformations:

z′ =a z + b

c z + d, a, b, c, d ∈ C, ad− bc = 1. (1.1.15)

These transformations form a (non-linear) realization of SO0(3, 1), the proper ortho-chronous leaf of the usual space-time Lorentz group, which is isomorphic to SL(2,C)/Z2.

When talking about a realization of the conformal group one has to guaranteethe existence of the inverse transformation. This is achieved by demanding the non-vanishing of f ′(z) in eq. (1.1.14) throughout the manifold, resulting that the onlyglobally defined transformations providing a realization of the conformal group areprecisely those defined in eq. (1.1.15).

It is worth mentioning that so far the analysis has been carried out for the finitetransformations that are connected with the identity. However, the set of conformaltransformations of a manifold may have several disconnected leaves. For instance,when d = 2 one can find confomal transformations not controlled by any holomorphiccomplex function. A trivial example of this is the transformation z′ = z. It can beproved that the conformal group in two dimensions has two leaves parametrized byholomorphic and antiholomorphic functions respectively.

From now on it will be assumed that the manifold is a two-dimensional infinitelylong euclidean cylinder. Computations will be performed on an associated manifoldof complex dimension two, which is the result of treating z and z as two indepen-dent complex coordinates as pointed out before. Therefore, the powerful tools of the(holomorphic) complex calculus will be applied.

It still remains to specify the realization of the conformal group as a group oftransformations acting on the fields φ(z, z). Among the several possibilities yielding theright algebra of transformations (VirC⊕VirC), the following one is specially important:

φ′(z, z) =(

∂f(z)∂z

)h (∂f(z)

∂z

)h

φ(f(z), f(z)), h, h ∈ R, (1.1.16)

when

z = f(z′), z = f(z′). (1.1.17)

A field transforming in such a way is called a primary field of weights (h, h). 1 In-finitesimally:

δφ = ε(z) ∂φ + ε(z) ∂φ +(h∂ε + h ∂ε

)φ, (1.1.18)

1The return to the original cylinder is performed when f(z) = f(z). Only real fields (φ = φ) are

defined on the cylinder; this implies that the only primary fields that can be defined back on the real

manifold must have h = h.


where:

δz = −ε(z), δz = −ε(z), ∂ ≡ ∂

∂z, ∂ ≡ ∂

∂z. (1.1.19)

A primary field with h = 0 is called a chiral field. It can be shown that for such afield the quantity ∂φ(z, z) is a null field in the sense that every correlator containing itvanishes. Therefore, a chiral field can be referred to as φ(z).

The action S[gµν(x), φA(x)] defines a conformal field theory if it is invariant un-der conformal transformations—the φA(x) fields being primary fields or not. Generalcovariance will be assumed to be a (gauge) symmetry of the action. Another gaugesymmetry that will typically be present is the Weyl rescaling of the metric tensor:

δgµν(x) = λ(x) gµν(x), δφA(x) = 0. (1.1.20)

The presence of quantum anomalies prevents the metric field gµν(x) to be completelygauge-fixed to a constant value. However, general covariance is usually chosen to bepreserved at the quantum level, so that it will be assumed that the so-called conformalgauge of eq. (1.1.6) can be safely reached in a local coordinate system.

The energy-momentum tensor of the theory is defined (except a global sign andother constant factors) as:

Tµν(x) ∼ 1√|g|

δS[g, φ]δgµν

, |g| ≡ |det(gµν)|. (1.1.21)

If the metric were flat this would concide with the definition of the Noether current asso-ciated with the invariance of the action under translations [2]. The energy-momentumtensor is symmetric and, when the Weyl symmetry (1.1.20) is present, it is traceless: 2

Tµν = T νµ, gµνTµν = 0. (1.1.22)

Also, when the equations of motion for the φA(x) fields are considered, the energy-momentum tensor satisfies the following equation:

∇µTµν = 0, (1.1.23)

which, in the conformal gauge (1.1.6), is written as a conservation equation:

∂µTµν = 0. (1.1.24)

By taking all this into account one realizes that the energy-momentum tensor has onlytwo independent real components. In the complex coordinate system adapted to the

2The tracelessness of T µν can also be proved by taking into account the conformal invariance of the

action and the equations of motion for the φA(x) fields.

6 Introduction

conformal gauge with an euclidean ηµν matrix, the only non-vanishing components ofTµν are:

T (z) ≡ Tzz =12

(T11(x)− iT12(x)) , and T (z) ≡ Tzz =12

(T11(x) + iT12(x)) ,

(1.1.25)

with ∂T (z) = 0, and ∂T (z) = 0.

The importance of the energy-momentum tensor is based on the fact that it can beseen as the generator of the conformal symmetry. Indeed, all the conserved Noethercurrents jµ(x) associated with the conformal symmetry can be written in terms of it:

jµ(x) = Tµν(x)εν(x). (1.1.26)

This current is conserved (i.e. ∂µjµ = 0) when εν(x) satisfies the conformal Killingequation and the equations of motion for the φA(x) fields are used.

An essential ingredient for quantizing the conformal symmetry is a time coordinate.One can interpret the infinitely long cylinder as the world-sheet of a closed stringparametrized by a (Wick-rotated) time coordinate τ ranging from −∞ to +∞, and aspace coordinate σ compactified to a radius-one circle. The following identifications:

x1 = eτ cosσ, x2 = eτ sinσ ⇒ z = eτ+iσ, (1.1.27)

define a mapping from the cylinder to R2 − 0. The equal-time surfaces are now cir-cumferences around the origin and the conserved charge associated with the conformalsymmetry becomes:

Q =∫

τdxµjµ(x) =

∮

|z|

(dz ε(z) T (z) + dz ε(z) T (z)

). (1.1.28)

In the extended complex manifold (C−0)2, i.e. when T (z) and T (z) are independentquantities, one splits Q in two independent charges:

Q =∮

|z|dz ε(z) T (z), and Q =

∮

|z|dz ε(z) T (z). (1.1.29)

Conformal transformations are generated by these charges through an appropriatePoisson-bracket structure:

δφ(w, w) = Q,φ(w, w)ET +Q,φ(w,w)

ET

, (1.1.30)

where ‘ET’ means that the Poisson bracket is defined only on equal-time surfaces, i.e.on circumferences.


Fields become operators after quantizing. Equal-time Poisson brackets are regardedas commutators with a radial-ordering (i.e. time-ordering) prescription:

φ1(z, z), φ2(w, w)

ET

→ 1i~

[φ1(z, z), φ2(w, w)

]|z|=|w| ≡

≡ 1i~

limδ→0

(φ1(z, z)φ2(w, w)

∣∣|z|=|w|+δ

− φ2(w,w)φ1(z, z)∣∣|z|=|w|−δ

). (1.1.31)

This procedure is called radial quantization. Equation (1.1.30) becomes (a 2π factorhas been introduced for convenience):

δφ(w,w) =1~

limδ→0

∮

|z|=|w|+δ

dz

2πiε(z) T (z)φ(w,w)−

∮

|z|=|w|−δ

dz

2πiε(z) φ(w, w)T (z)

,

(1.1.32)

for transformations generated by Q. An advantage of working with complex coordinatesis that the powerful complex calculus can be used. Therefore, a convenient deformationof integration contours leads to:

δφ(w, w) =∮

Cw

dz

2πiε(z)

1~R (T (z)φ(w,w)) , (1.1.33)

Here Cw is a contour surrounding w, and R symbolizes the radial ordering of the fields,namely: 3

R (A(z, z)B(w,w)) ≡

A(z, z)B(w,w) if |z| > |w|,B(w,w)A(z, z) if |z| < |w|.

(1.1.34)

The radial product is commutative and associative. It is clear that the expressionR (T (z)φ(w,w)) must be singular when z → w in order to obtain a non-vanishinginfinitesimal transformation law. For instance, if φ(w, w) is a primary field with weightsh and h, the matching of eq. (1.1.18) —assumed to be valid at the quantum level too—with eq. (1.1.33) dictates:

1~R (T (z)φ(w,w)) =

hφ(w, w)(z − w)2

+∂φ(w, w)

z − w+ · · · (1.1.35)

where the dots represent non-singular pieces when z → w. This is an example ofwhat is called an operator product expansion (OPE). The singular parts appearing inthe R (A(z, z)B(w,w)) OPE will be denoted by A(z, z)B(w,w). A normal ordered

product of operators is defined [3]:

(AB) (z) ≡∮

Cz

dw

2πi

R (A(w)B(z))w − z

, (1.1.36)

3All the fields are supposed to be bosonic. For fermionic fields commutators must be substituted

by anticommutators where necessary, and a minus sign can appear in the second row of eq. (1.1.34).

8 Introduction

where Cz is a contour enclosing the point z. An operator product algebra is a setof chiral fields Φi(z) together with their derivatives ∂kΦi(z), all composite operatorsdefined through eq. (1.1.36), and the unity operator I, which is closed under the OPEoperation:

1~A(z)B(w) =

kmax∑

k=1

ABk(w)

(z − w)k. (1.1.37)

The requirements of commutativity and associativity of the radial ordering induce anumber of properties of the ABk(z) operators appearing in eq. (1.1.37) [4]:

BAk(z) =∑

s≥k

(−1)s

(s− k)!∂s−k

z ABs(z), (1.1.38a)

ABCkl(z) = BAClk(z) +l∑

s=1

(l − 1s− 1

)ABsCk+l−s(z). (1.1.38b)

The R symbol will be omited in the following: a radial ordering will be understoodwhen necessary. Transformations generated by Q are computed in the same way butwith T (z) and h instead of T (z) and h.

The energy-momentum tensor depends on the fields φA of the theory. Thereforethe OPE corresponding to the product of T (z) with itself will be derived from thefundamental OPE’s involving φA. However, as T (z) is the generator of the conformaltransformations (see eq. (1.1.33)), the T (z)T (w) OPE has to be consistent with theconformal algebra:

[δε1 , δε2 ]φ(z, z) = δεφ(z, z), with ε(z) = ε2(z) ∂ε1(z)− ε1(z) ∂ε2(z). (1.1.39)

Further requirements of commutativity, associativity and closeness of the operator prod-uct algebra spanned by T (z) almost fix the T (z)T (w) OPE:

1~T (z) T (w) =

~c/2(z − w)4

+2T (w)

(z − w)2+

∂T (w)z − w

, (1.1.40)

Here c is a real number which is called the central charge and depends on the particulartheory—i.e. on the representation of the conformal group (see below). If c = 0 thenT (z) itself is a chiral weight-two primary field. In general, T (z) is said to be a chiralquasi-primary field. A ~ appears in the central-charge term for dimensional reasons;its presence here indicates that this central-charge term is the consequence of somequantum effect. The infinitesimal transformation law for T (z) is:

δT (z) = ε(z) ∂T (z) + 2 T (z) ∂ε(z) +~c12

∂3ε(z). (1.1.41)


A similar OPE is derived for the radial product T (z)T (w), but with a c real constantinstead of c. Also: T (z) T (w) = 0.

An operator product algebra containing an energy-momentum field T (z) satisfyingeq. (1.1.40) is called conformal. If one can assign a dimension d to every operator inthe algebra such that:

d[I] = 0, d[∂A] = d[A] + 1, d[(AB)] = d[A] + d[B],

d[ABk] = d[A] + d[B]− k, (1.1.42)

then the operator product algebra is called graded. It is clear that the dimension ofT (z) is equal to 2.

Chiral dimension-d operators in a graded conformal operator product algebra canbe decomposed in this way:

φ(z) =∑

k∈Zφk z−k−d ⇔ φk =

∮

C

dz

2πizk+d−1 φ(z), (1.1.43)

where C is a contour surrounding the origin. The operators φk are called modes andthey satisfy a set of commutation relations inherited from the corresponding OPE’s:

[φk, ψl] =∮

C0

dw

2πiwl+dψ−1

∮

Cw

dz

2πizk+dφ−1 φ(z)ψ(w),

>C0

∧Cw&%

'$p

Om (1.1.44)

where a radial ordering between the φ(z) and ψ(w) fields is understood and the contoursare arranged in the way it is indicated. For instance:

T (z) =∑

k∈ZLk z−k−2 ⇒ 1

~[Lm, Ln] = (m− n) Lm+n +

~c12

m (m2 − 1) δm+n,0.

(1.1.45)

This is precisely the Virasoro algebra eq. (1.1.9) but with an extra central term. Also,if φ(z) is a weight-h primary field, then:

φ(z) =∑

k∈Zφk z−k−h ⇒ 1

~[Lm, φn] = (m (h− 1)− n) φm+n. (1.1.46)

It can be shown [4] that the requirement of associativity in the operator product algebrainduces the Jacobi identity for the algebra of modes.

10 Introduction

Representations of the Virasoro algebra

An important conformally-invariant theory is given by the following action:

S =18π

∫

Σd2x

√|g| gµν∂µX(x) ∂νX(x). (1.1.47)

Taking the gauge (1.1.6) with ηµν euclidean, and considering the complex coordi-nates (1.1.11), this action is written as a free scalar-field action:

S =12π

∫

Σd2x ∂X ∂X. (1.1.48)

It can be shown that ∂X is a chiral field. Computation (making use of, for instance, thepath-integral formalism) of the 〈X(x)X(y)〉 propagator yields the OPE of the chiralfield ∂X with itself:

∂X(z) ∂X(w) = − 1(z − w)2

. (1.1.49)

The (holomorphic) energy-momentum tensor can be taken to be:

T (z) = −12

(∂X ∂X) (z). (1.1.50)

The OPE of T (z) with itself is then eq. (1.1.40) with c = 1.

The free scalar-field theory is just an example of representation of the Virasoroalgebra. A systematic study of the Virasoro algebra representations can be carried out.A highest-weight (holomorphic) representation is generated by a weight-h primary fieldφ(z), or, in the language of Hilbert-space states, by |h, c〉 ≡ limz→0 φ(z)|0〉. Here |0〉is the sl(2,R)-invariant vacuum, satisfying Ln|0〉 = 0, for n ≥ −1, and c is the centralcharge of the theory. The highest-weight state is such that:

L0|h, c〉 = h|h, c〉, Ln|h, c〉 = 0, n > 0. (1.1.51)

The states generated by the action of the remaining Virasoro operators, L−n (n > 0),on |h, c〉 are called descendant states, and the corresponding fields are the descendant orsecondary fields of φ(z). The set of |h, c〉 and its descendants is called a Verma module.Not every Verma module will give rise to an irreducible and unitary representation ofthe Virasoro algebra. This will happen only [5] if either c ≥ 1 and h ≥ 0, or

c = 1− 6m(m + 1)

, m ∈ N, m ≥ 2, and (1.1.52a)

h =((m + 1)r −ms)2 − 1

4m(m + 1), r, s ∈ N, 1 ≤ r ≤ m− 1, 1 ≤ s ≤ m. (1.1.52b)

1.2 Quantum W-algebras 11

1.2 Quantum W-algebras

A (quantum) W-algebra is a graded conformal operator product algebra such that alloperators but the energy-momentum field T (z) are chiral primary fields. 4

The new chiral operators in the operator product algebra can be regarded as gener-ators of new symmetries just as the chiral T (z) field is the generator of the conformalsymmetry. These are the so-called extended conformal symmetries or W-symmetries.A great number of conformal invariant theories are also W-invariant. This implies thatthe huge knowledge of the theory derived from conformal symmetry arguments couldbe even larger by considering W-symmetries. However, W-algebras are hard to han-dle: they typically contain high-order derivative terms and non-linearities. A relatedproblem is the difficulty of defining W-symmetries intrinsically, i.e., as a group of trans-formations acting on some (space-time) manifold. That is the reason why the definitionof W-symmetry has been given according to its action at the level of fields. This is alsothe way W-algebras were historically first introduced in 1985 by A.B. Zamolodchikovin his seminal article [6]. The huge amount of articles on W-algebras that appearedsince that date and until 1992 were systematized in the standard review reference [7](see also [8] for a commented reprint collection updated to 1994).

Following [7] a W-algebra containing, besides the weight-two energy-momentumfield, a set of extra primary fields having weights h1, . . . , hn, will be denoted as beingof type W(2, h1, . . . , hn). For instance, a unique W-algebra of type W(2, 3) can bedefined for every value of the central charge c except for c = −22/5. This algebra isdenoted by W3 and its OPE’s read [6]:

1~

T (z) T (w) =~c/2

(z − w)4+

2T (w)(z − w)2

+∂T (w)z − w

, (1.2.1a)

1~

T (z) W3(w) =3W3(w)(z − w)2

+∂W3(w)z − w

, (1.2.1b)

1~

W3(z) W3(w) =~c/3

(z − w)6+

2T (w)(z − w)4

+∂T (w)

(z − w)3+

+(

310

∂2T (w) +32

~ (5c + 22)Λ(w)

)1

(z − w)2+

+(

115

∂3T (w) +16

~ (5c + 22)∂Λ(w)

)1

z − w. (1.2.1c)

Here W3(z) is the new weight-three field, and

Λ(z) ≡ (TT ) (z)− 3~10

∂2T (z). (1.2.2)

4According to this simple definition the superconformal algebra, for instance, should be considered

as a W-algebra. However, the name W-algebra is usually reserved for higher-weight bosonic extensions

of the Virasoro algebra. Similarly, extensions of the super-conformal algebra are referred to as super

W-algebras.

12 Introduction

It is worth stressing that the third OPE (1.2.1c) is totally determined (except a globalconstant factor) out of the basic previous two OPE’s and the requirements of conser-vation of the conformal dimension (1.1.42) and of commutativity and associativity ofthe radial-ordered product. The corresponding algebra of modes is:

1~

[Lm, Ln] = (m− n) Lm+n +~c12

m (m2 − 1) δm+n,0, (1.2.3a)

1~

[Lm,Wn] = (2m− n) Wn+m, (1.2.3b)

1~

[Wm,Wn] =115

(m− n) (m2 + n2 − 12mn− 4)Lm+n +

16~ (5c + 22)

(m− n) Λm+n +

+~c360

m (m2 − 1) (m2 − 4) δm+n,0, (1.2.3c)

where Lm and Wm are the modes associated with T (z) and W3(z), respectively. Thenon-linearity of W3 is encoded in the Λm objects, which are the modes of the compositeoperator Λ(z) (see eq. (1.1.36) and (1.1.43)):

Λm =−2∑

r=−∞LrLm−r +

∞∑

r=−1

Lm−rLr − 3~10

(m + 2) (m + 3)Lm. (1.2.4)

Therefore, it is clear thatW3 can not be considered a (infinite-dimensional) Lie algebra.

A complete classification of (quantum) W-algebras is still lacking. Some of themonly exist for a discrete set of values of the central charge c, the so-called exotic W-algebras. For example, a W(2, 4, 4)-type algebra can only be constructed if c = 1 orc = −656

11 [9]. Non-exotic W-algebras, like W3, are also called deformable W-algebras.The W3 algebra can be generalized to W(2, 3, . . . , N)-type algebras, which are theso-called WN algebras. The first algebra of this series, W2, is the Virasoro algebra.A systematic method for finding quantum W-algebras is the direct construction bydemanding commutativity and associativity of the radial product. See, for instance,the extensive search for W-algebras with two or three generators for low values of theirweights started in ref. [9] and [10]. The algebraic computations become more and morecumbersome as the weight and/or the number of fields in the W algebra increase. Theuse of computer tools like the one of Thielemans [11] [4] is almost mandatory in thistask.

There are alternative methods for obtaining quantum W-algebras. For instance,the study of conformal symmetries of correlation functions containing higher-weightoperators was, in fact, the original method used in [6] for defining the W3 algebra(see also [12]). The study of Casimir algebras [3] or coset constructions [13] links theW-algebras with the affine or Kac–Moody algebras. This fundamental relationshipconstitutes the core of the connections of W-algebras with Wess–Zumino–Novikov–Witten and Toda models, and with the theory of integrable systems through the so-

1.3 Classical limits 13

called Drinfel’d–Sokolov reductions. The new methods for obtaining (quantum) W-algebras that provide these connections will be explained in chapter 2. However, thesenew methods (including the coset constructions [14]) do not cover all the cases ofquantum W-algebras that can be obtained from direct construction methods.

1.3 Classical limits

A first classical limit5 can be defined by setting c →∞ as well as ~→ 0, such that theproduct ~c remains a finite non-zero constant. The structure of the resulting classicalW-algebra is very similar to the corresponding quantum W-algebra, but it is expressedin terms of equal-time Poisson brackets instead of OPE’s. Infinitesimal transformationlaws in the chiral algebra are computed through eq. (1.1.30):

δφ(w) = Q,φ(w)(ET) , Q =∫

dz (ε(z) T (z) + ρα(z) Wα(z)) . (1.3.1)

Matching of these transformation laws with the classical limit of those coming fromeq. (1.1.33) is achieved once the following identifications are set:

1~A(z) B(w) → A(z), B(w), 1

(z − w)n →1

(n− 1)!∂n−1

w δ(z − w),

(AB) (z) → A(z)B(z). (1.3.2)

All fields can be supposed to live on a circumference because Poisson-brackets aredefined only on equal-time surfaces. Therefore complex coordinates as z can now beregarded to be the (real) coordinate of that circumference. It can be checked that therequirements of commutativity and associativity of the radial product (see eq. (1.1.38))yield anticommutativity of the Poisson bracket and the Jacobi identity on the classicallimit.

As an example, the classical W3 algebra reads:

T (z), T (w) =~c12

∂3wδ(z − w) + 2 T (w) ∂wδ(z − w) + ∂T (w) δ(z − w), (1.3.3a)

T (z),W3(w) = 3W3(w) ∂wδ(z − w) + ∂W3(w)δ(z − w), (1.3.3b)

W3(z),W3(w) =~c360

∂5wδ(z − w) +

13T (w) ∂3

wδ(z − w) +12∂T (w) ∂2

wδ(z − w) +

+(

310

∂2T (w) +325~c

T (w)2)

∂wδ(z − w) +

+(

115

∂3T (w) +325~c

T (w)∂T (w))

δ(z − w). (1.3.3c)

5Discussions on classical limits of W-algebras can be found in ref. [15], [16] and [17].

14 Introduction

From a physical point of view this limit can not be considered a full classical limit dueto the presence of terms exhibiting the central charge, which is known to be associatedwith a quantum effect—the anomaly. However, the resulting W-algebras do nicelyarise in a purely classical context, apparently disconnected from quantum field theory,namely the theory of integrable systems. Moreover, it should be noted that classicallythe actual value of the central charge has no meaning because it can be arbitrarilychosen by a suitable normalization of the Poisson-bracket structure.

The true classical limit of a W-algebra is obtained after retaining the leading termsin a further ~c → 0 truncation of the above-mentioned first classical limit. In the caseof the W3 algebra this second classical limit yields:

T (z), T (w) = 2 T (w) ∂wδ(z − w) + ∂T (w) δ(z − w), (1.3.4a)

T (z),W3(w) = 3 W3(w) ∂wδ(z − w) + ∂W3(w)δ(z − w), (1.3.4b)

W3(z),W3(w) =325~c

(T (w)2∂wδ(z − w) + T (w)∂T (w)δ(z − w)

). (1.3.4c)

Note that even at this level non-linearities are still present.

The class of infinitely-generated W-algebras is an important group. The simplestone is the w∞ algebra [18]. Its commutation relations are:

[w(s)m , w(t)

n ] =(

(t− 1)m− (s− 1)n)

w(s+t−2)m+n , s, t ≥ 2. (1.3.5)

The generators, w(s)m , can be identified with the modes of weight-s fields (s ≥ 2).

This (linear) algebra is related with the algebra of area-preserving diffeomorphisms ofa two-dimensional cylinder or, equivalently, with the algebra of symplectomorphismson the cylinder. (The algebra (1.3.5) arises when w

(s)m is identified with xm+s−1ps−1

through the usual Poisson bracket x, p = 1.) The w∞ algebra can be regarded as aN →∞ limit and a classical truncation of the WN algebras. There exist a great varietyof W∞-type algebras that are quantizations, deformations and extensions of this w∞algebra [19].

1.4 Overview of W-symmetry applications

The ultimate goal underlying a great part of the research efforts in modern TheoreticalPhysics is to achieve a unified description of all the interactions that are present inNature. The quantum-mechanics paradigm has proved to be a successful guideline inthis process and reliable theories have been proposed for all forces but gravity. Thesetting of a quantum theory for gravity will probably imply the abandonment of somenowadays well-established basic ideas on the objects a physical theory has to deal

1.4 Overview of W-symmetry applications 15

with. In this sense, string theory is the wide framework of almost all the theoreticaldevelopments made in treating this problem.

Two-dimensional conformal symmetry is a basic ingredient in the description of anytheory based on strings. The archetypical example is the bosonic string described bythe action (1.1.47). It has three gauge-symmetries that classically can be fixed but oneof them becomes anomalous at the quantum level. By adopting the chiral gauge thataction becomes:

S =12π

∫

Σd2x

(∂X ∂X + h(x) ∂X∂X

). (1.4.1)

The h field is the remaining gauge degree of freedom coming from the metric gµν .The algebra of gauge transformations of this action is the (classical) Virasoro algebra.Extension of this symmetry by considering more than one scalar field (X(x) → Xi(x))and the inclusion of a cubic piece

∫

Σd2x b(x) dijk∂Xi∂Xj∂Xk, (1.4.2)

with the dijk constants satisfying some conditions, makes the action invariant undera gauge symmetry that is precisely the classical W3 algebra in the form given byeq. (1.3.4) [20].

By integrating out the matter X fields in the action (1.4.1) one obtains an inducedaction for the remaining gauge variable h. This action can be covariantized by includingall degrees of freedom of the gµν metric yielding the Polyakov action [21],

Sind ∼∫

d2x√|g|R 1√

|g|¤√|g|R, (1.4.3)

which is therefore regarded as a description for two-dimensional gravity. (R is thescalar curvature of the gµν metric.) By doing the same with the action with theextra term (1.4.2) one obtains an induced action that is consequently referred to asthe (chiral) description of two-dimensional W3-gravity (see, for instance, ref. [22]).Covariantization of this action is a difficult task [23] due to the lacking of a W3-stringaction that generalizes eq. (1.1.47). The core of the problem is the question of howto endow the b field with a geometrical meaning in the framework of a would-be ‘W3

geometry’ as, for instance, a component of some ‘W3 metric’.Although an action for W-string theories can not written down, a great number of

features of these theories are known thanks to, for instance, cohomological techniques(see, for example, the articles grouped in ref. [24]). Generalization of old ideas is oneof the main tools in Theoretical Physics research. Thus, application of W-symmetriesto string theory was an obvious task to be done. Proliferation of new string theoriesshould not be necessarily seen as a diversion in the way of seeking for a physically

16 Introduction

admissible unified theory. Actually, the new trends in the field point to the directionof treating apparently disconnected string theories as different faces of the same entity.

The existence of unitary and irreducible representations of W-algebras has beendetermined. It turns out that, for values of the central charge less than a certain value,the field content of these representations is fairly well established. This is a general-ization of the minimal models appearing for c < 1 when studying the representationsof the Virasoro algebra (see eq. (1.1.52)). The range of values of c where these ‘Wminimal models’ appear extends beyond that c = 1 barrier. For example [25], the WN

minimal models appear for values of the central charge in the series:

c = (N − 1)(

1− N(N + 1)m(m + 1)

), m ∈ N, m ≥ N. (1.4.4)

Therefore, the study of W-invariant theories may contribute to the important issue ofclassifying all possible conformal field theories.

Related with this, it is well known that these Virasoro minimal models describephase transitions of two-dimensional statistical systems such as the Ising model. Ithappens, for instance, that the three-state Potts model, which can be described withone of these minimal models, can be also identified with a W3 minimal model [26].

This feature, namely the discovering of W-structures in previously known theoriesand models, has been as fruitful for the development of the study of W-algebras as theconstruction of ad hoc theories with W-symmetries. W-algebras appeared in the theoryof integrable systems of differential equations, more related with hydrodynamics andclassical mechanics than with string theory, thus providing a nice example of correlationbetween apparently disconnected parts of Physics. This relationship is explained withsome detail in sect. 2.3.

Another example of this peculiarity is the application of W-symmetries (specificallysome W∞-type algebras) to the study of the quantum Hall effects, one of the activefields of research in applied Physics [27].

1.5 Outline of the thesis

After this introduction to what a W-algebra is, chapter 2 will review a number of ba-sic procedures and notions that are necessary for the developments to be presented inthe remainder of the thesis. Specifically, this second chapter is divided in two parts.The first one presents a way for obtaining classical W-algebras: the Drinfel’d–Sokolovreduction method. This procedure avoids the cumbersome expressions arising when aquantum W-algebra is directly constructed through the fulfilment of the Jacobi iden-tities. The second part of the chapter is a short survey of the theory of the so-calledgeneralized KdV and KP hierarchies of integrable differential equations, already men-tioned in the previous section.

1.5 Outline of the thesis 17

The original results of the thesis are developed in chapters 3, 4 and 5. In chapter 3,a dynamical implementation of W-symmetries in the context of particle mechanics ispresented. The Drinfel’d–Sokolov reduction method ensures the invariance under aclassical W-algebra of any Lagrangian that can be constructed with the rules givenin this chapter. Somehow, these models are the one-dimensional counterparts of theW-gravity theories mentioned in the previous section, and serve as toy models forunderstanding several features ofW-symmetries. For instance, these particle mechanicsmodels allow the interpretation of W-algebras as symmetries of differential equations—i.e., of the equations of motion of the models— and, therefore, the link betweenclassical W-algebras and integrable systems is established.

As a consequence of this, algebraic manipulations of the equations of motion forthese particle mechanics models may detect relations between W-algebras that other-wise, with the mere inspection of their Poisson brackets, would be unnoticed or hard tofind. This idea is developed in chapter 4, where a number of links between interestinggeneralizations of the W-algebra concept will appear. In this way, the so-called linearand non-local W-algebras can be nicely bound to the standard Drinfel’d–Sokolov re-duction scheme. Also, the expressions giving the embedding of a classical W-algebrain a bigger one are easily obtained.

It has already been mentioned that a base-space implementation of W-symmetriesis lacking. This is an unsatisfactory feature of W-symmetries that hinders a unifiedtreatment with usual conformal symmetries —it certainly exists a conformal geometry,so, what about W-geometries? Chapter 5 is devoted to the presentation of a proposalfor defining a W-geometry. Again, the equations of motion for the particle mechanicsmodels will be a fruitful starting point, and the theory of integrable systems will appearas an essential ingredient for this proposal.

A summary of the original results of the thesis is presented in the Conclusions.Finally, two appendices are included: the first one deals with the theory of the sl(2,R)embeddings in a Lie algebra and the second one is a complement to chapter 3.

Chapter 2

Classical W-algebras

2.1 Drinfel’d–Sokolov reductions

The idea of reducing or constraining a Kac–Moody algebra was exploited by Drinfel’dand Sokolov in ref. [28] for obtaining integrable systems of differential equations. Thealgebraic structures underlying the integrability of these systems were computable withthe pseudo-differential operator formalism developed by Gel’fand and Dickey [29] (seesect. 2.3). After the introduction of W-algebras in 1985 it was realized that these alge-braic structures of integrable systems were precisely classical limits of W-algebras [30].This result is a generalization of the previously noted [31] relationship existing be-tween the Virasoro algebra and the simplest of those integrable systems, namely, theKorteweg–de Vries equation. Therefore, the Drinfel’d–Sokolov (DS) reduction can beregarded as a method for obtaining classical W-algebras1 out of Kac–Moody algebras.In fact, this is the most powerful method for obtaining classical W-algebras. Further-more, the linking with well-known algebraic structures (Kac–Moody algebras) will opena way for setting a classification scheme for classical W-algebras.

The starting point is an affine (Kac–Moody) algebra associated with a Lie algebra G.This Lie algebra will be sometimes referred to as the vacuum preserving algebra.

J(z) = Ja(z)Ta, Ta ∈ G, a = 1, . . . ,dimG. (2.1.1)

A Lie–Poisson bracket is defined:

Ja(z), Jb(w) = ~κ γab ∂wδ(z − w) + fabc Jc(w) δ(z − w). (2.1.2)

Here κ is a constant and f cab are the structure constants of the Lie algebra. The ~

constant has been introduced for dimensional reasons. Indices are raised and lowered1These classical W-algebras are those obtained from the quantum ones through the ~→ 0, c →∞

limit explained in sect. 1.3.

19

20 Classical W-algebras

with γab, which is related to the Cartan–Killing metric of G, gab, and its dual Coxeternumber h∨:

γab =1

2h∨gab. (2.1.3)

2.1.1 sl(2,R)

The basic idea of the DS reduction method is to introduce a second-class set of con-straints in the space of affine currents Ja(z) so that a Dirac-bracket can be defined.Consider the case G = sl(2,R) spanned by E+, E− and E0:

[E0, E±] = ±E±, [E+, E−] = E0. (2.1.4)

The introduction of a first-class constraint, namely,

Φ1(z) ≡ J+(z)− a = 0, a = constant, (2.1.5)

generates a gauge symmetry in the space of affine currents. This symmetry can befixed by considering a new constraint:

Φ2(z) ≡ J0(z) = 0. (2.1.6)

The whole set of constraints becomes second-class and a new, constraint-compatible,Lie–Poisson bracket substitutes eq. (2.1.2): the Dirac bracket. This is defined in thefollowing way:

f(z), g(w)∗ ≡(f(z), g(w) −

∫dxdy f(z),Φµ(x)∆µν(x, y)Φν(y), g(w)

)∣∣∣∣red

,

(2.1.7)

where ∆µν(x, y) ≡ Φµ(x),Φν(y) and∫

dx∆µα(z, x)∆αν(x,w) = δνµ δ(z − w). (2.1.8)

The Dirac bracket of the only non-constrained current, J−(z), with itself reads:

J−(z), J−(w)∗ =2~κa

(−(~κ)2

a∂3

wδ(z − w) + 2J−(w) ∂wδ(z − w) + ∂J− δ(z − w)

),

(2.1.9)

which is eq. (1.3.3a) with:

T (z) =a

2~κJ−(z), ~c = −6~κ. (2.1.10)

In conclusion, the Virasoro (W2) algebra arises as a DS reduction of the sl(2,R) Kac–Moody algebra.

2.1 Drinfel’d–Sokolov reductions 21

2.1.2 sl(3,R)

As sl(2,R) ⊂ sl(3,R) one would expect the existence of a DS reduction of the sl(3,R)Kac–Moody algebra such that the resulting Dirac-bracket algebra contains the Virasoroalgebra. The presence of new non-constrained fields, appart from the generator of theVirasoro algebra, would imply that this Dirac-bracket algebra is in fact a W-algebra.Intuitively, it is clear that the method depends on how sl(2,R) is embedded into sl(3,R).This can be done in two inequivalent ways and, therefore, two different W-algebras willarise.

A generic sl(3,R) affine current can be written in the following way:

J(z) = Ja(z)Ta =

Jα0 (z) Jα(z) Jα+β(z)

J−α(z) Jβ0 (z)− Jα

0 (z) Jβ(z)J−α−β(z) J−β(z) −Jβ

0 (z)

. (2.1.11)

The first-class constraints corresponding to the first reduction are:

Φ1(z) ≡ Jα+β(z) = 0, Φ2(z) ≡ Jα(z)− a = 0, Φ3(z) ≡ Jβ(z)− a = 0,

(2.1.12)

where a is a constant, and the gauge-fixed current is of the form:

J(z) =

0 a 0J−α(z) 0 a

J−α−β(z) J−α(z) 0

. (2.1.13)

The remaining currents span the W3 algebra of eq. (1.3.3) through the following iden-tifications:

T (z) =2a

~κJ−α(z), W3(z) =

√−2

5a2

(~κ)2J−α−β(z), ~c = −24~κ.

(2.1.14)

The second reduction starts by considering only two first-class constraints:

Φ1(z) ≡ Jα+β(z)− a = 0, Φ2(z) ≡ Jβ(z) = 0, a = constant, (2.1.15)

and the gauge-fixed form of the current contains four degrees of freedom:

J(z) =

Jα0 (z) 0 a

J−α(z) −2Jα0 (z) 0

J−α−β(z) J−β(z) Jα0 (z)

. (2.1.16)


The W-algebra spanned by these four generators is called W23 and was originally con-

sidered in ref. [32]. The quantum OPE’s and classical Poisson-brackets are displayed insubsect. 3.4.2 and sect. 4.3, respectively. The W2

3 generators can be taken as follows:

T (z) =a

~κ

(J−α−β(z) +

3a

(Jα0 (z))2

), (2.1.17a)

H(z) = 2Jα0 (z), B+(z) =

√2a

3~κJ−α(z), B−(z) =

√2a

3~κJ−β(z).

(2.1.17b)

Here the W23 central charge c classically is equal to:

~c = −6~κ. (2.1.18)

2.1.3 Towards a classification: WGS algebras

The key point is how to choose a suitable set of second-class constraints in a givenKac–Moody algebra in order to get a classical W-algebra. The properties of the Diracbracket guarantee that the reduced algebra is indeed antisymmetric and that it satisfiesthe Jacobi identity. Therefore, the only requirement for this reduced algebra to be a(classical) W-algebra is the existence of a basis, T (z),Wi(z), such that:

1. T (z) is the generator of a Virasoro algebra, and

2. the remaining generators Wi(z) are primary fields with respect to T (z):

T (z),Wi(w) = hiWi(w) ∂wδ(z − w) + ∂Wi(w) δ(z − w), ∀i. (2.1.19)

There exists a well-established procedure, that will be referred to as canonical DSreduction to have these conditions satisfied. The inputs are a Kac–Moody algebrabased on a maximally non-compact real Lie algebra G, 2 and a sl(2,R) subalgebra ofG, S = E0, E+, E− (see appendix A for a sketch of the theory of sl(2,R) embeddingsin a Lie algebra G). The second-class constraints are such that they put J(z) to theform:

J(z) = jred(z) + aE+, a = constant, (2.1.20)

where jred(z), which contains the remaining fields, is such that:

jred(z) ∈ ker adE−. (2.1.21)

2Given a simple complex Lie algebra, its maximally non-compact real form is spanned by its Cartan

(or Chevalley) generators over R. Maximally non-compact real forms of the classical complex Lie

algebras An, Bn, Cn and Dn are sl(n + 1,R), so(n, n + 1), sp(n,R) and so(n, n), respectively.

2.1 Drinfel’d–Sokolov reductions 23

Under these conditions it can be proved (see next section) that the reduced algebra,which is spanned by the elements in jred(z), is a W-algebra. It will be denoted by WG

S .

The relationship existing between the theory of sl(2,R) embeddings in a Lie algebraand classical W-algebras was first stressed in ref. [33]. (An extensive treatment of thisissue can be found in ref. [34].) An obvious question arised: is every classicalW-algebraincluded in the WG

S labelling system? The results of ref. [35] point to the directionof giving an affirmative answer to this question (see, however, ref. [36]). Therefore,the canonical DS reduction method would provide a first step towards a classificationscheme for classical W-algebras.

Kac–Moody algebras are the symmetry algebras of Wess–Zumino–Novikov–Witten(WZNW) models. It was shown [37] that, by constraining the sl(2,R) WZNW modelthrough the DS reduction procedure explained above, one obtains the so-called Liouvilleaction, which is therefore invariant under the Virasoro algebra. This Liouville actionplays an important role in the study of two-dimensional induced gravity because itcan be obtained from the Polyakov action (1.4.3) when the conformal gauge (1.1.6) ischosen for the metric gµν . In fact, the presence of a sl(2,R) Kac–Moody symmetry intwo-dimensional gravity (in the light-cone gauge) was already noted in ref. [38].

When the WZNW model is associated with a (maximally non-compact) Lie algebradifferent from sl(2,R), G, the DS reduction induced by the principal sl(2,R) embeddingin G (see appendix A) gives rise [37] [39] to the so-called abelian Toda models.3 Thegeneralized or non-abelian Toda models arise [41] [34] when the sl(2,R) embedding isa non-principal one. Toda models are therefore two-dimensional theories which areinvariant under classical W-algebras, a relationship that was stated in ref. [42] (seealso ref. [43]). Furthermore, the equations of motion for Toda models were alreadypresented as a example of integrable differential equations that can be obtained fromDS reduction in the original article [28].

The classification scheme that can be set for classical W-algebras can not be ex-tended for the quantum ones due to the existence of non-deformable quantum W-algebras. However, classical W-algebras can be quantized by employing a variety ofmethods. The classical WN algebras are obtained through the DS reduction of thesl(n,R) Kac–Moody current algebras induced by the principal sl(2,R) embedding insl(n,R). The generators of these algebras can be represented in terms of free fields byusing the so-called Miura transformation. Direct quantization of this representationprovides the quantum WN algebras [25]. The Toda field theory representation of clas-sical W-algebras can be also quantized [44]. More or less ad hoc constructions thatmimic the DS reduction at the quantum level were proposed for finding the quantumgenerators of some algebras out of (a representation of) Kac–Moody generators, making

3A treatment of Toda systems can be found in ref. [40].


use of the BRST quantization technology. This was the case of the Virasoro algebra inref. [45], where a relation between the central charge of the quantum Virasoro algebra c

and the sl(2,R) Kac–Moody level κ was written down:

c = 13− 6 (κ + 2)− 6κ + 2

. (2.1.22)

The corresponding classical limit is eq. (2.1.10). Similar relations with the sl(3,R)Kac–Moody level were set for the W3 central charge:

c = 50− 24 (κ + 3)− 24κ + 3

, (2.1.23)

and also for W23 [46]:

c = 25− 6 (κ + 3)− 24κ + 3

. (2.1.24)

Equations (2.1.14) and (2.1.18) are the classical limits of these expressions. The gen-eralization of eq. (2.1.23) to the WN algebras was also given in ref. [45].

Finally, the formulation of the sl(2,R) embedding technique with the BRST lan-guage yielded a general method for quantizing every classical W-algebra obtainedthrough a canonical DS reduction with a principal [47] or arbitrary [48] sl(2,R) embed-ding in a Lie algebra.

2.2 WGS -transformations

In this section a proof of the existence of a Virasoro subalgebra in every WGS algebra

is presented. The infinitesimal transformations generated by the Kac–Moody alge-bra (2.1.2), namely,

δJb(w) =∫

dz γac εc(z)Ja(z), Jb(w), (2.2.1)

can be written as a set of Yang–Mills-like gauge transformations:

δΛ = β + [β, Λ]. (2.2.2)

Here, for convenience, the complex coordinate z has been identified with a real coordi-nate t. The dot means derivative with respect to t. Also:

− 1~κ

Ja(z)Ta → Λa(t)Ta ≡ Λ(t), −εa(z)Ta → βa(t)Ta ≡ β(t). (2.2.3)

The reduction in eq. (2.1.20) is then regarded as a partial gauge-fixing of the Λ(t)matrix:

Λ(t) = W (t) + M, (2.2.4)

2.2 WGS -transformations 25

where M is a non-zero constant element of G and W (t) = W b(t)T ′b. The Lie algebraelements T ′b span GW , a subspace of G (b = 1, . . . , dimGW < dimG) and W b are theremnant fields living in GW . The remnant transformations are those preserving thisgauge-fixing:

[M, β] + δW = β − [W,β] . (2.2.5)

These are the infinitesimal transformations generated by the reduced algebra throughthe proper Dirac bracket. Within this formalism, instead of computing the Diracbracket one has to solve eq. (2.2.5). Solving this equation is tantamount to express-ing a subset of the gauge parameters βa(t) as a function of another subset (remnantparameters) and the remnant fields W b(t) in a purely algebraic way.

If the reduction is a canonical DS reduction induced by a sl(2,R) embedding, S,of the original Lie algebra G, then this algebraic process for solving eq. (2.2.5) can becarried out as it will be shown in the following. Prescriptions in eq. (2.1.20) are thenwritten (a = 1, for simplicity):

M = E+, GW = ker adE−, (2.2.6)

where E+, E− and E0 are the defining elements of the sl(2,R) embedding (2.1.4). Themapping adS given by adS : s → ad s where s ∈ S and

ad s : G −→ Gg −→ [s, g],

is a representation of S on G. This representation is completely reducible so G (as avector space) decomposes to a direct sum of invariant subspaces of spin j (integer orhalf-integer) and multiplicity nj (branching):

G =∑

j≥0

nj∑

i=1

+ G(i)j , G(i)

j =j∑

m=−j

+ G(i)j,m,

∑

j≥0

nj(2j + 1) = dimG. (2.2.7)

The G(i)j,m subspaces are one-dimensional eigenspaces of adE0 with eigenvalue equal

to m. A spin 1 subspace is always present in the branching, namely, S itself (denotedby G1). They define a gradation of G:

Gm =

∑

j≥m

nj∑

i=1

+ G(i)j,m, if m is an eigenvalue of adE0

0 , otherwise

⇒ G =∑

m + Gm, [Gm, Gn] ⊂ Gm+n. (2.2.8)


According to (2.2.6), every remnant field lives in Gm=−j and there are

N(S) =∑

j≥0

nj (2.2.9)

such fields.The presence of this gradation ensures an algebraic process for solving eq. (2.2.5).

Indeed, remnant parameters live in ker adE+, i.e., in Gm=j . Restrictions of eq. (2.2.5) tosubspaces Gj allow to express in an algebraic way parameters living in Gj−1 as functionsof parameters living in Gj (and fields), because M is in G1. So, as one goes down on thespectrum of m’s, an algebraic algorithm arises for expressing all the gauge parametersas functions of those living in ker adE+ and fields. Finally, restrictions of eq. (2.2.5) tosubspaces Gm=−j give the transformations of the remnant fields, δW b.

The existence of a Virasoro subalgebra can be shown by performing a decompositionof parameters: β → β, ε. Consider the following change:

β = β + εΛ + εH, (2.2.10)

where H =∑

α kαHα is a general element of the Cartan subalgebra H of G, with con-stant coefficients and β = βcT ′′c (c = 1, . . . ,dimG − 1). With this change Virasorotransformations appear both before and after the gauge-fixing. However, the transfor-mation laws of the remnant fields after the gauge-fixing are generally different from theoriginal ones.

In order to examine the Virasoro transformations of the gauge field Λ, the followingdecomposition is useful:

Λ =∑

γ

ΛγEγ +∑α

ΛαHα, (2.2.11)

where Eγ , Hα form a Cartan–Weyl basis of the Lie algebra G. The definition (2.2.10),together with eq. (2.2.5), produces the following Virasoro transformations before thegauge-fixing:

δΛγ = εΛγ + (1 +∑α

(α, γ)kα)εΛγ , (2.2.12a)

δΛα = εΛα + εΛα + kαε. (2.2.12b)

Therefore, fields living on the Cartan subalgebra (indices α) transform as weight-oneprimary fields, generally with an inhomogeneous extension term. Instead, the fieldsliving in the root spaces (indices γ) transform as primary fields of weight 1+

∑α(α, γ)kα.

The appearance of an inhomogeneous kαε term in the transformation law of the weight-one fields is a consequence of the inclusion of the εH piece in eq. (2.2.10). This equationwithout the εH term is equivalent to taking the usual Sugawara energy-momentum

2.2 WGS -transformations 27

tensor as the generator of the Virasoro transformations in the DS reduction framework.Then, the addition of the εH term in (2.2.10) corresponds to changing the realizationof the Virasoro group by considering an improved Sugawara energy-momentum tensoras the generator.

The transformations generated by ε remain to be Virasoro transformations afterthe gauge-fixing procedure. Indeed, the parameter ε lives in the subspace generated byE+ so it is one of the remnant parameters (one can take the T ′′c Lie algebra elementsas the generators of all the G(i)

j,m subspaces, except the one of G1,1, i.e. E+). Eq. (2.2.5)after the gauge-fixing reads:

[M, β] = ˙β − δW − [W, β] + εW + ε(M + W ) + εH + ε[H,M + W ]. (2.2.13)

The following expansion of H and W is useful for solving this equation:

H = k0E0 +∑

i

kiHi +∑

σ

kσHσ, (2.2.14a)

W =∑

i

W iHi +∑α

Wαeα +∑

ρ

W ρeρ. (2.2.14b)

In the expansion of H, E0 is the central sl(2,R)-embedding element, Hi span GW ∩Hand Hσ form a basis of the rest of H. In the expansion of W , W i are the fields livingin H, Wα are the fields living in G0 but not in H and W ρ are the rest of remnant fields.The following identities hold:

[E0, eρ] = −j(ρ)eρ, [E0, eα] = 0, (2.2.15a)

[Hi, eρ] = ri(ρ)eρ, [Hi, eα] = ri(α)eα. (2.2.15b)

One can study the propagation of the parameter ε through the relations imposedby eq. (2.2.13) at each level in the gradation of G. The result of this analysis is:

β = −(1 + k0) ε E0 − ε kσ Hσ − ε E− + (terms without ε). (2.2.16)

After introducing eq. (2.2.16) in eq. (2.2.13) one gets the residual infinitesimal trans-formations of the remnant fields, δW , under the ε sector. There are some cancellationsdue to the presence of the term εH which cut off the propagation of the k0 and kσ pa-rameters. Hence, the only surviving arbitrariness comes from the ki parameters. Theresult is summarized as:

• The field T living in the subspace generated by E−, which is one of the eρ gener-ators, transforms as a quasi-primary field of weight two:

δT = ε T + 2 ε T − ...ε . (2.2.17)


• Fields living in the subspace spanned by Hi, W i, transform as weight-one fieldsplus a ε term:

δW i = ε W i + εW i + ki ε, (i = 1, . . . , dimH ∩ GW ). (2.2.18)

• The rest of remnant fields living in Gm=−j , W ρ and Wα, are primary fields:

δW ρ = ε W ρ +

(1 + j(ρ) +

∑

i

kiri(ρ)

)εW ρ, (2.2.19a)

δWα = ε Wα +

(1 +

∑

i

kiri(α)

)εWα. (2.2.19b)

In general, the field living in Gm=−j has weight 1 + j apart from possible shifts,which exist in case the subspace H ∩ GW is non-trivial. The following relation holds:

∑weights :=

∑

j≥0

nj(1 + j) =12

(dimG + N(G)) . (2.2.20)

There is no explicit general formula for the transformations generated by the otherremnant parameters. They are precisely specific chiral W-transformations because onehas a set of infinitesimal transformations with closed algebra and containing a Virasorosector with the weight-two quasi-primary field T .

2.3 Integrable systems

Consider a dynamical system described by a set of quantities xi(t), i = 1, . . . , N (forinstance, coordinate and momenta variables) satisfying a number of evolution equations:

dxi(t)dt

= Fi(xk(t)), i = 1, . . . , N. (2.3.1)

This system is said to be Hamiltonian if it can be written in the following form:

dxi(t)dt

= Hij∂H(xk(t))

∂xj(t), i = 1, . . . , N, (2.3.2)

where Hij are the components of the so-called Hamiltonian structure, which is the ‘in-verse’ of a symplectic 2-form, and H(xk(t)) is a scalar function known as the Hamil-tonian. The Hamiltonian structure defines a Poisson bracket in the space parametrizedby xi(t):

F (xk(t)) , G(xl(t))

= Hij

∂F

∂xi(t)∂G

∂xj(t). (2.3.3)

2.3 Integrable systems 29

As an example, consider the following dynamical system:

uk(t) = uk+3 − 3uk+1 + 3uk−1 − uk−3 + 3(u2

k+1 − u2k−1

), k = 1, . . . , N,

(2.3.4)

with u0(t) ≡ uN (t), etc. It is a Hamiltonian system with:

Hij = δi,j−1 − δi,j+1, H =N∑

k=1

(u3

k −12

(uk+1 − uk−1)2

). (2.3.5)

The passage to a system with an infinite number of degrees of freedom, namely,

N →∞, uk(t) → u(x, t), uk+1(t)− uk−1(t) → ∂xu(x, t), (2.3.6)

turns eq. (2.3.4) into the celebrated Korteweg–de Vries (KdV) differential equation,originally introduced for describing solitary wave phenomena in Hydrodynamics:

∂tu(x, t) = ∂3xu(x, t) + 6u(x, t) ∂xu(x, t). (2.3.7)

One can extend all the above-mentioned concepts for a system with an infinite numberof degrees of freedom. For example, eq. (2.3.2) becomes:

∂tu(x, t) =∫

dyH(x, y)δH[u]δu(y, t)

, (2.3.8)

and, eq. (2.3.5) reads:

H(x, y) = ∂xδ(x− y), H[u] =∫

dx

(u(x, t)3 − 1

2(∂xu(x, t))2

). (2.3.9)

In fact, the KdV equation is a bihamiltonian system, because one can check that thefollowing choices also give rise to it:

H′(x, y) = −12∂3

xδ(x− y)− 2u(x, t) ∂xδ(x− y)− ∂xu(x, t) δ(x− y), (2.3.10a)

H ′[u] = −∫

dx u(x, t)2. (2.3.10b)

The important thing to note is that the Poisson bracket induced by this second Hamil-tonian structure is precisely the (classical) Virasoro algebra with non-vanishing centralcharge [31]:

u(x, t), u(y, t) =12∂3

yδ(x− y) + 2u(y, t) ∂yδ(x− y) + ∂yu(y, t) δ(x− y). (2.3.11)

An integrable system is a Hamiltonian system described by xk(t), k = 1, . . . , N(or u(x, t)) such that one can find N independent functions Ii(xk(t)) that are ininvolution:

Ii, Ij = 0. (2.3.12)


The dynamical equations of an integrable system can be solved due to the existence ofthese conserved quantities.

An indication of the integrability of a Hamiltonian system is the possibility ofreformulating it by means of a so-called Lax-pair:

∂tL = [A,L] , (2.3.13)

where L and A are some operators depending on uk(t) or u(x, t). One can show thatthe KdV equation admits this Lax-pair description with:

L = ∂2x + u(x, t), A = 4∂3

x + 6u(x, t)∂x + 3∂xu(x, t). (2.3.14)

An infinite number of conserved quantities, which are in involution, can be found. Thefirst ones are:

I1 =∫

u(x, t)dx, I2 =∫

u(x, t)2dx, I3 =∫ (

2u(x, t)3 − (∂xu(x, t))2)

dx, . . .

(2.3.15)

Therefore, the KdV equation is an integrable system.The so-called pseudo-differential operator formalism [49] generalizes in a systematic

way this Lax-pair formulation of integrable systems, and opens the possibility of find-ing new differential equations representing infinite-dimensional integrable systems in ahierarchical way. This framework, apparently disconnected from conformal theory, willbe surprisingly related to the theory of W-algebras.

From now on, any operator of the form:

L = ∂nx +

n−1∑

k=0

uk(x, . . . ) ∂kx , (2.3.16)

where uk may depend on other variables as well as of x, will be called a Lax opera-tor. The starting point for constructing the pseudo-differential operator formalism forintegrable systems is this Lax operator:

L = ∂n + un−2(x, t)∂n−2 + un−3(x, t)∂n−3 + . . . + u1(x, t)∂ + u0(x, t), (2.3.17)

where ∂ ≡ ∂x. The nth root of this operator, L1/n, can be defined as a pseudo-differential operator by introducing negative powers of ∂ and generalizing the compo-sition law to:

∂k f(x) =∞∑

l=0

(k

l

)∂lf(x)∂k−l, (2.3.18)


where now:(

k

l

)≡ k(k − 1) · · · (k − l + 1)

l!,

(k

0

)≡ 1. (2.3.19)

The mth power of L1/n, Lm/n, is also a pseudo-differential operator. The part ofLm/n with non-negative powers of ∂, L

m/n+ ≡ Pm, is such that it can form a Lax pair

with L. In other words, the following expressions encode a set of systems of differentialequations for ui(x, t) (i = 0, . . . , n− 2):

∂tmL = [Pm, L]. (2.3.20)

It can be proved that these systems are Hamiltonian for any n and m. In fact, theyare bihamiltonian. The two Hamiltonian structures, H(∞) and H(0), are given by theAdler mapping [50]:

H(∞)(X) = [X, L]+, (2.3.21a)

H(0)(X) = (LX)+L− L(XL)+, (2.3.21b)

where

X =n−1∑

i=0

∂−i−1 δf

δui≡ δf

δL, (2.3.22)

and f is an arbitrary functional depending on the ui(x, t) quantities. Here L is equal toL in eq. (2.3.17) plus a un−1(x, t)∂n−1 term. The actual expressions for the Hamiltonianstructures when acting on the space of functionals depending only on the n − 2 fieldsappearing in L are obtained after a un−1 = 0 reduction.

Then eq. (2.3.20) can be written in a Hamiltonian form (see eq. (2.3.22)):

∂tmL = H(∞)(δH

(∞)m

δL) = H(0)(

δH(0)m

δL), (2.3.23)

where the Hamiltonians are:

H(∞)m = − n

n + m

∫dx resL(m+n)/n, H(0)

m =n

m

∫dx resLm/n. (2.3.24)

Here resK refers to the coefficient of the ∂−1 term of the pseudo-differential operatorK =

∑Ni=−∞Ki∂

i.These dynamical systems are also integrable. All the differential equations in the

hiererchy labelled by n share the same conserved quantities Ik. They are preciselyproportional to the Hamiltonians H

(0)k , which are in involution with respect to the

Poisson bracket induced by the Hamiltonian structure H(0).


When n = 2 and m = 3 eq. (2.3.20) becomes (except a numerical factor) the KdVequation, and the Adler mapping (eq. (2.3.21)) provides the two Hamiltonian structuresdisplayed above (eq. (2.3.9) is the first structure, and eq. (2.3.10a) is the second one,H(0)).

The n = 3 hierarchy of differential equations is generated by the following Laxoperator:

L = ∂3 + u(x, t)∂ + v(x, t). (2.3.25)

The first non-trivial integrable system in this hierarchy is obtained when m = 2. It isdefined by the following differential equations:

∂tu(x, t) = 2∂v − ∂2u, ∂tv(x, t) = ∂2v − 23∂3u− 2

3u∂u. (2.3.26)

The so-called Boussinesq equation arises as a consistency condition for these two equa-tions:

∂2t u = −1

3∂4u− 4

3∂ (u∂u) . (2.3.27)

The second Hamiltonian structure corresponding to this integrable system is such thatthe fundamental Poisson brackets are equal to the defining relations of the classicalW3-algebra (eq. (1.3.3)) with:

T ≡ ~c24

u, W3 ≡ ~c24

√−2

5

(v − 1

2u

). (2.3.28)

This relationship between integrable systems and W-algebras can be extended toarbitrary n: the second Hamiltonian structure corresponding to the Lax operator oforder N is related with the Poisson brackets defining the classical WN algebra (see,for instance, ref. [30] and [51]). These are the integrable systems associated with thesl(n,R) algebras that were introduced in ref. [28]. In the same article, integrable systemsrelated with the Bn, Cn and Dn series of Lie algebras are studied. The correspondingclassical W-algebras are those obtained via DS reduction through the principal sl(2,R)embedding in a Lie algebra (see appendix A). However, integrable differential equationsassociated with non-principal W-algebras can be also constructed. For example, thehierarchy corresponding to the W2

3 algebra was presented in ref. [52]. A generalizationof the method of ref. [28] has been considered in ref. [53] (see also ref. [54]). Thegeneralized Drinfel’d–Sokolov hierarchies constructed in this way include systems ofdifferential equations which are invariant under a great variety of classical W-algebras(such as W2

3 ). However, it seems that not every classical W-algebra constructed viaDS reduction can be interpreted as the symmetry algebra of one of those generalizedDrinfel’d–Sokolov hierarchies [55].


2.3.1 The KP equations

According to the definition of Pm one deduces that they satisfy a kind of zero-curvaturecondition [56]:

∂tiPj − ∂tjPi + [Pj , Pi] = 0. (2.3.29)

As a consequence, all the flows generated by Pm (with fixed n) via eq. (2.3.20) arecompatible. Therefore, it makes sense to consider the fields uk appearing in the Laxoperator L as functions of all the time variables tm (m = 1, . . . ,∞) simultaneously.Due to the fact that P1 = ∂, the first time variable can be identified with x and onecan write:

uk = uk(t1, t2, t3, . . . ) = uk(ti). (2.3.30)

In fact, the generalized KdV equations presented above are a subset of a more gen-eral class of integrable systems described by the so-called generalized Kadomtsev-Petviashvili (KP) equations, which are Hamiltonian and integrable too. They aredefined in the following way:

∂tmQ = [Pm, Q], (2.3.31)

where now Q is an arbitrary pseudo-differential operator of the form

Q = ∂ +∞∑

i=1

qi(tk)∂−i, (2.3.32)

and Pm ≡ Qm+ . Therefore, the generalized KP equations are a set of differential equa-

tions for the qi variables, which, in general, depend of the infinite times tk. Equation(2.3.29) is sometimes given as an alternative definition of the generalized KP hierar-chy of differential equations. The generalized KdV equations associated with the Laxoperator L of order n arise when the qi functions are such that Qn = Qn

+ ≡ L.Solutions of the generalized KP (and KdV) equations can be found via the τ -

function formalism [57]. The basic idea is that, given a solution qi(tk) of thegeneralized KP equations, one can construct a scalar τ(tk) function satisfying:

∂t1∂tn log τ(tk) = res Qn. (2.3.33)

Note that∫

dt1 resQn are precisely the conserved quantities of the generalized KP equa-tions. Conversely, a τ function satisfying eq. (2.3.33) will provide a solution qi(tk)of the generalized KP equations. The Baker–Akhiezer function ψ(tk, λ) is the objectthat links this τ function with the corresponding KP solution. It satisfies the followingdifferential equations:

Qψ(tk, λ) = λψ(tk, λ), ⇒ Lψ(tk, λ) = λn ψ(tk, λ) for KdV,

(2.3.34a)

∂tiψ(tk, λ) = Pi ψ(tk, λ), (i = 1, . . . ,∞). (2.3.34b)

Chapter 3

Particle Mechanics Models

with W-symmetries

A method for constructing particle mechanics models exhibiting (gauge)W-symmetrieswill be presented in this chapter. This method exploits the link between affine and Walgebras: the W-invariant Lagrangians appear after a partial gauge-fixing of a simpleconstrained model which is invariant under a gauge sp(2M,R). The different partialgauge-fixings leading to W-invariant Lagrangians are determined by the set of inequiv-alent sl(2,R) embeddings in the original gauge algebra.

The resulting Lagrangians can be viewed as a one-dimensional analogue of themodels of matter coupled with two-dimensional W-gravity. Therefore, the models alsoprovide a realization ofW-symmetries on these ‘matter’ variables xi(t). Their equationsof motion are consequently invariant under the corresponding W-symmetries. Theseequations of motion have, schematically, the form:

Lx(t) = 0, (3.1)

where L is a Lax-type operator (see eq. (2.3.16)):

L =dn

dtn+

n−1∑

k=0

Uk(t)dk

dtk.

Therefore, L and eq. (3.1) will be referred to as the Lax operator and Lax equationassociated with that W-algebra. The Lax operators for the most common W-algebras,such as W2 and W3 (see sect. 2.3) will be recovered in this way. Other W-algebras willbe associated with matrix Lax operators. These differential equations will prove to bea useful tool in order to relate different W-algebras, as will be shown in next chapter.The contents of this chapter have been published in articles [58] and [59].

35

36 Particle Mechanics Models with W-symmetries

3.1 Particle mechanics model with sp(2M,R) symmetry

The starting point is a reparametrization-invariant model of M relativistic particleswith an sp(2M,R) gauge algebra living in a Minkowskian d-dimensional space-time(with flat metric ηµν). The dimension d satisfies d > 2M + 1 so the constraints do nottrivialize the model. 1 The canonical action is given by (a sum over repeated indicesis assumed):

S =∫

dt(pi,µ xµ

i − λAijφAij

), µ = 1, . . . , d, i, j = 1, . . . ,M, A = 1, 2, 3.

(3.1.1)

The variable xµi (t) is the world-line coordinate of the i-th particle and pi,µ(t) is its

corresponding momentum. The Lagrange multipliers λAij (t) implement the constraintsφAij = 0 and satisfy:

λ1ji = λ1ij , λ3ji = λ3ij . (3.1.2)

The explicit form of φAij is:

φ1ij =12

ηµνpi,µ pj,ν , φ2ij = pi,µ xµj and φ3ij =

12

ηµν xµi xν

j . (3.1.3)

These 2M2 + M constraints close under the usual Poisson bracket xµi , pj,ν = δijδ

µν

giving a realization of the sp(2M,R) algebra. 2

It is useful to introduce a matrix notation for the coordinates and momenta of theparticles

R =

(r

p

), with r =

x1...

xM

, p =

p1...

pM

. (3.1.4)

The conjugate of R is given by

R = R>J2M =(p>, −r>

), (3.1.5)

where J2M is the 2M × 2M symplectic matrix

(0 −II 0

). The Lagrange multipliers

can be written in the form of a 2M × 2M symplectic matrix

Λ =

(B A

−C −B>

), (3.1.6)

1See appendix B for a discussion on the dimension and signature of this space-time.2From now on, the space-time index µ will be dropped and a ηµν will be assumed to be inserted

whenever necessary.

3.1 Particle mechanics model with sp(2M,R) symmetry 37

where the components of the M × M matrices A,B, C are the Lagrange multipliersλ1ij , λ2ij , λ3ij respectively.

The canonical action (3.1.1) can be written in a matrix form as

S =∫

dt12

RDR, (3.1.7)

where D is the covariant derivative

D =ddt− Λ. (3.1.8)

In this formulation the gauge invariance of the action is expressed in a manifestlyinvariant form of Yang–Mills type3 with the gauge algebra sp(2M,R): 4

δΛ = β − [Λ, β], (3.1.9a)

δR = β R, (3.1.9b)

where β is the following 2M × 2M matrix of gauge parameters:

β =

(βB βA

−βC −β>B

). (3.1.10)

The components βA, βB and βC are the M×M matrices associated with the constraintsφ1ij , φ2ij and φ3ij , respectively. The equations of motion of the matter fields are:

DR = R− ΛR = 0. (3.1.11)

The infinitesimal transformation law (3.1.9a) is the compatibility condition of the pairof equations (3.1.9b) and (3.1.11):

0 = [(δ − β),D] R = −(δΛ− β + [Λ, β]

)R. (3.1.12)

A part of the equations of motion can be used for expressing the momenta pi interms of the Lagrangian variables xi and xi:

p = A−1 (r −B r) ≡ K. (3.1.13)

The action is now rewritten as:

S =∫

dt12

(K>AK − r>C r

), (3.1.14)

and the gauge transformations become:

δr = βA K + βB r, δΛ = β − [Λ, β]. (3.1.15)

3For a previous discussion of geometrical models with Yang–Mills gauge theories see ref. [60].4The supersymmetric version has been studied in ref. [61] (see also subsect. 5.1.3).


A characteristic feature of these Lagrangian transformations is that the algebra is open,except for sp(2,R):

[δ1, δ2] Λ = δβ∗Λ, (3.1.16a)

[δ1, δ2] r = δβ∗r −(β

(2)A A−1 β

(1)A − β

(1)A A−1 β

(2)A

)[L]r, (3.1.16b)

where β∗ ≡ [β(2), β(1)] and [L]r is the Euler-Lagrange equation of motion of r. Thereare two reasons for the appearance of an open algebra: (i) the transformations of themomenta at the Lagrangian and Hamiltonian level do not generally coincide, and (ii)there are more than one first-class constraints quadratic in the momenta.

This algebra can be made to close by introducing M auxiliary vectors (F1, . . . , FM )and modify the transformation law of the coordinates r as follows:

δr = βA (K + F ) + βB r. (3.1.17)

The transformation of F is determined by the condition that K + F transforms as p inthe Hamiltonian formalism. Explicitly one gets:

δF = −A−1[βA ∂t (K + F ) + βA B> (K + F ) + (δA− βB A) F + βA C r

], (3.1.18)

while the transformation of Λ remains unchanged:

δΛ = β − [Λ, β]. (3.1.19)

The new algebra closes off-shell. The invariant action under the modified gauge trans-formations is:

S =∫

dt12

(K>A K − r>C r − F>AF

). (3.1.20)

The redundancy of the auxiliary variables F is guaranteed by the action itself, whichimplies F = 0 as the equation of motion.

The invariance of this model under eq. (3.1.9a) allows the application of the formal-ism developed in section 2.2. In particular, one can show the invariance of the modelunder one-dimensional diffeomorphisms (Diff), or t-reparametrizations (see next sub-section). Furthermore, according to the general discussion in section 2.2, by performinga partial gauge-fixing of the Λ matrix induced by a sl(2,R) embedding on sp(2M,R),

Λ(t) = M + W (t), (3.1.21)

the remnant gauge transformations can be interpreted as W-transformations. As aconsequence, the corresponding partial gauge-fixed particle mechanics model will beinvariant under a W-symmetry. This gauge-fixing procedure will be explicitly shownin the next sections by considering several examples.


3.1.1 Diffeomorphism invariance of the sp(2M,R) model

According to the developments made in section 2.2, one can obtain diffeomorphismtransformations out of eq. (3.1.19) by performing the change of gauge parameters givenin eq. (2.2.10):

β = β + εΛ + ε H. (3.1.22)

Here H is an arbitrary element of the sp(2M,R) Cartan subalgebra:

H =M∑

i=1

kαiHαi , (3.1.23)

where αi are the simple roots of sp(2M,R) and kαi are constants. When G=sp(2M,R)then H is the following diagonalized matrix:

H =

(N 00 −N>

), (3.1.24a)

N =1

4(M + 1)

kα1 0kα2 − kα1

. . .kαM−1 − kαM−2

0 2kαM − kαM−1

,

(3.1.24b)

where αM is the longest root.As stated in section 2.2, the Lagrange multipliers transform as primary fields with,

eventually, ε terms under the ε sector infinitesimal transformations. For the matterand auxiliary variables, this change of parameters produces the following infinitesimaltransformations:

δεr = ε r + ε N r + εA F, (3.1.25a)

δεF = −ε F − ε (F + NF ) + ε(A−1BA F −A−1A F −B>F − K −B>K − C r

).

(3.1.25b)

These transformations are equivalent to diffeomorphism transformations. In order toshow this, one can introduce an antisymmetric combination of the equations of motion:

δεqi(t) = δεq

i(t) +∫

dt′M ij(t, t′) [L]qj (t′), (3.1.26a)


where:

qi = xi (i = 1, . . . , M)

qi = Fi (i = M + 1, . . . , 2M)

and M(t, t′) =

(0 ε(t) δ(t− t′) I

−ε(t) δ(t− t′) I M(t, t′)

),

M(t, t′) = −ε(t)(B>(t)A−1(t)−A−1(t)B(t)

)δ(t− t′) +

+ ε(t′) A−1(t′)ddt′

δ(t− t′)− ε(t) A−1(t)ddt

δ(t− t′).

(3.1.26b)

It can be shown that

M>(t′, t) = −M(t, t′), (3.1.27)

so δε is a symmetry transformation of the action too, and

δεr = ε r + ε N r, δεF = ε F − ε N F, (3.1.28)

which are diffeomorphism transformations for the matter and auxiliary variables. Theytransform as primary fields.

In summary, the infinitesimal gauge transformations of the sp(2M,R) model beforethe gauge-fixing are:

• Diffeomorphism transformations:

δΛγ = ε Λγ +

(1 +

∑α

(α, γ)kα

)εΛγ , δΛα = ε Λα + ε Λα + kα ε,

(3.1.29a)

δεr = ε r + ε N r, δεF = ε F − ε N F, (3.1.29b)

These Diff transformations may be regarded as realizations of the Virasoro algebragenerated by (improved) Sugawara energy-momentum tensors. The freedom inchoosing the different Virasoro realizations is reflected in the arbitrariness of thekα constants. When all of them are zero one obtains the usual realization withall the gauge fields having conformal weight equal to one, and the matter andauxiliary fields being scalars.

• Yang–Mills-type transformations:

δΛ = ˙β − [Λ, β], δr = βA (K + F ) + βB r, (3.1.30a)

δF = −A−1[βA

(K + F

)+ βA B>K + βA C r −B βA F + ˙

βA F]− β>B F,

(3.1.30b)


where (see section 2.2):

Λ =∑

γ

ΛγEγ +∑α

ΛαHα =

(B A

−C −B>

), β =

(βB βA

−βC −β>B

).

(3.1.31)

After performing the gauge-fixing, Lagrange multipliers still transform as primaryor quasi-primary fields (see eq. (2.2.17), (2.2.18) and (2.2.19)), whereas matter andauxiliary fields do not have, in general, a nice behavior under Diff transformations.For instance, if the E− element of the sl(2,R) embedding is taken to live in the A

or B sectors of a general sp(2M,R) matrix, then non-desired ε terms appear in theresidual ε transformations coming from the algorithm described in section 2.2 (seeeq. (2.2.16)) through the βA or βB factors of (3.1.17) and (3.1.18). In any case, theonly undetermined constants that remain in the infinitesimal transformations after thegauge-fixing from those in the decomposition of H (2.2.14a) are the ki constants as inthe residual transformations for the Lagrange multipliers.

3.1.2 Finite gauge transformations of the sp(2M,R) model

Finite transformations5 can be obtained by exponentiating the infinitesimal ones asXi′ = expθαΓαXi, where the generators Γα = Ri

α∂

∂Xi satisfy [Γα,Γβ] = f γαβ Γγ and

Xi represents any of the variables. The coefficients f γαβ are the structure functions of

the sp(2M,R) gauge algebra.It is useful to perform the integration using the matrix notation. The explicit

form of the finite gauge transformations before the gauge-fixing is considered in thefollowing four sets of transformations. Any finite transformations may be obtained bythe composition of them.

• Diffeomorphism transformations:

Λ′γ(t) = f(t)1+P

α(α,γ)kα Λγ(f(t)), Λ′α(t) = f(t) Λα(f(t)) + kαf(t)f(t)

,

(3.1.32a)

r′i(t) = f(t)Nii r(f(t)), F ′i (t) = f(t)−Nii F (f(t)), i = 1, . . . ,M,

(3.1.32b)

where N = (Nij) is the diagonalized constant matrix given in eq. (3.1.24b).

5For a discussion on finite gauge transformations, see ref. [62].


• Transformations parametrized by βA:

A′ = A + ˙βA − βAB> −BβA+ βACβA, B′ = B − βAC, C ′ = C,

(3.1.33a)

r′ = r + βA(K + F ), F ′ = A′−1[AF − βA∂t(K + F ) + B>(K + F ) + Cr

].

(3.1.33b)

• Transformations parametrized by βB:

A′ = eβBAeβ>B , B′ = eβB (B − ∂t)e−βB , C ′ = e−β>B Ce−βB ,

(3.1.34a)

r′ = eβBr, F ′ = e−β>B F. (3.1.34b)

• Transformations parametrized by βC :

A′ = A, B′ = B + AβC , C ′ = C + ˙βC + βCB + B>βC+ βCAβC ,

(3.1.35a)

r′ = r, F ′ = F. (3.1.35b)

The finite transformations of the gauge-fixed model are (excluding the diffeomor-phism transformations), in general, difficult to find. Some explicit cases are presentedin the following.

3.2 W2 model and finite gauge transformations

In this section the particle mechanics model with sp(2,R) gauge symmetry [63] is stud-ied. In this case the Lagrangian gauge transformations close off-shell and no auxiliaryvariables have to be introduced. The finite form of the residual diffeomorphism trans-formations will be constructed from the knowledge of the finite transformations beforethe gauge-fixing. This may be a useful procedure when direct integration of the in-finitesimal residual gauge transformations cannot be performed in a simple way.

The model is described by the first-order action (3.1.1) with gauge transformations(3.1.9b) and (3.1.9a), where R, Λ and β are given by:

R =

(x

p

), Λ =

(λ2 λ1

−λ3 −λ2

), β =

(β2 β1

−β3 −β2

). (3.2.1)

After the elimination of p via its equation of motion the action reads:

S =∫

dt

[1

2λ1(x− λ2 x)2 − λ3

2x2

], (3.2.2)

3.2 W2 model and finite gauge transformations 43

and the gauge transformations are:

δλ1 = 2 β2 λ1 − 2β1 λ2 + β1, (3.2.3a)

δλ2 = β3 λ1 − β1 λ3 + β2, (3.2.3b)

δλ3 = 2 β3 λ2 − 2β2 λ3 + β3, (3.2.3c)

δx = β2 x +β1

λ1(x− λ2 x). (3.2.3d)

The gauge algebra still closes off-shell. That is because, as it has been mentioned insection 3.1, Lagrangian open algebras can only occur in theories which possess morethan one constraint quadratic in the momenta.

In the following, the issues of partial gauge-fixing and remnant gauge transforma-tions are studied along the lines of subsection 3.1.1. The matrix of Lagrange multipliersΛ can be rewritten as:

Λ = λ1 E+ + 2 λ2 h− 2λ3 E−, (3.2.4)

which defines the (trivial) embedding of sl(2,R) in sp(2,R). In this simple case the spaceof remnant fields is generated by E− alone: GW = ker adE− = 〈E−〉, the remnant gaugeparameter belongs to ker adE+, and the partial gauge-fixing is given by:

λ1 = 1, λ2 = 0, λ3 ≡ 6~c

λ. (3.2.5)

The associated partially gauge-fixed action is:

Spgf =∫

dt

(x2

2− 3~c

λ x2

), (3.2.6)

which produces the matter equation of motion:

x +6~c

λ x = 0. (3.2.7)

This is precisely the Lax equation Lx = 0 where L is the Lax operator associated withthe standard KdV equation (see eq. (2.3.14)).

The existence of a diffeomorphism symmetry sector —the only remnant symmetryin this model— can be shown by changing gauge parameters according to (3.1.22). Inthe present case H ∩ GW = 0. Hence no arbitrary constants ki can be introduced.The redefinition (3.1.22) is here, in components,

β1 = ε, β2 = σ, β3 = ρ +6~c

λ ε. (3.2.8)

If the partial gauge-fixing (3.2.5) is imposed then the remnant transformations areparametrized by ε and the other parameters are written in terms of it:

σ = −12

ε, ρ =12

ε. (3.2.9)


Using (3.2.5), (3.2.8) and (3.2.9) in (3.2.3) one shows that the remnant transformationsare indeed (world-line) diffeomorphisms:

δλ = ε λ + 2 ε λ +~c12

...ε , (3.2.10a)

δx = ε x− 12

ε x. (3.2.10b)

These infinitesimal transformations can be integrated directly to give their standardfinite forms:

x′(t) = (f)−1/2 x(f(t)), (3.2.11a)

λ′(t) = (f)2 λ(f(t)) +~c12

f...f − 3

2(f)2

(f)2. (3.2.11b)

In other words, according to subsect. 2.1.1, the action given in eq. (3.2.6) is invariantunder the W2 (Virasoro) algebra. The x(t) variable can be regarded as a primary fieldof weight −1/2 and λ(t) plays the role of the energy-momentum tensor. The secondterm in the r.h.s. of eq. (3.2.11b) is the Schwarzian derivative of f , coming from thecentral-extension term present in the definitory relation (eq. (1.3.3a)) of the classicalW2 algebra.

There is an alternative way to find the previous finite transformations. The finiteform of the residual transformations will now be found from the finite gauge transfor-mations obtained before imposing the partial gauge-fixing conditions. The first stepis to redefine the gauge parameters in order to show the diffeomorphism invariancebefore the gauge-fixing. Secondly, the finite form of these transformations is found.Finally, the gauge conditions are imposed. In this way the finite form of the rem-nant transformations is obtained from the finite form of the transformations before thegauge-fixing.

The above-mentioned change of parameters before the gauge-fixing is (3.1.22):

β1 = λ1 ε, β2 = σ + λ2 ε, β3 = ρ + λ3 ε. (3.2.12)

The gauge transformations (3.2.3) in terms of these new parameters read:

δλ1 = ε λ1 + ε λ1 + 2 σ λ1, (3.2.13a)

δλ2 = ε λ2 + ε λ2 + σ + ρ λ1, (3.2.13b)

δλ3 = ε λ3 + ε λ3 − 2σ λ3 + 2 ρ λ2 + ρ, (3.2.13c)

δx = ε x + σ x. (3.2.13d)

The ε transformation is just a world-line reparametrization where x transforms as ascalar and the Lagrange multipliers transform as vectors. The weights of x and λ under

3.2 W2 model and finite gauge transformations 45

reparametrization (3.2.13) are different from the ones of after the gauge-fixing (3.2.10)because, for instance, in the latter the variable x is no longer a scalar.

The finite forms of the new transformations (3.2.13) are found for each ε, σ and ρ

transformations:

• Diffeomorphisms:

λ′a(t) = f(t) λa(f(t)), a = 1, 2, 3, (3.2.14a)

x′(t) = x(f(t)). (3.2.14b)

• Local scale transformations:

λ′1 = e2σ λ1, λ′2 = λ2 + σ, λ′3 = e−2σ λ3, (3.2.15a)

x′ = eσ x. (3.2.15b)

• Local redefinition of Lagrange multipliers:

λ′1 = λ1, λ′2 = λ2 + ρ λ1, λ′3 = λ3 + 2 ρ λ2 + ρ + ρ2 λ1, (3.2.16a)

x′ = x. (3.2.16b)

The gauge-fixing condition (3.2.5) can be imposed at this stage. Notice thateq. (3.2.12) reduces to eq. (3.2.8) on the gauge-fixing surface. Any arbitrary con-figuration in the gauge orbit can be realized using a composition of the above finitetransformations with generic functions f(t), σ(t) and ρ(t). If the following compositionis considered:

¤ ρ→ ¤′ σ→ ¤′′ f→ ¤′′′ ≡∼¤, (3.2.17)

then the complete finite transformation is given by:

λ1 = f(t) e2σ(f(t)) λ1(f(t)), (3.2.18a)

λ2 = f(t) (λ2(f(t)) + ρ(f(t))λ1(f(t)) + σ(f(t))) , (3.2.18b)

λ3 = f(t) e−2σ(f(t))(λ3(f(t)) + ρ(f(t)) + 2 ρ(f(t))λ2(f(t)) + λ1(f(t)) ρ2(f(t))

),

(3.2.18c)

x = eσ(f(t)) x(f(t)), (3.2.18d)

Imposing the gauge-fixing conditions (3.2.5) on these transformations one obtains thefinite form of the conditions (3.2.9) for the finite gauge parameters:

σ(t) = −12

ln f(f−1(t)), ρ(t) = −σ(t). (3.2.19)

Using this restriction in the composition of finite gauge transformations (3.2.18) onearrives at the finite residual transformations (3.2.11). The interesting point here is thatthe infinitesimal transformations (3.2.10) have been integrated without actually doingit.


3.3 sp(4,R) models

This section is devoted to the study of the particle mechanics models arising whenM = 2, i.e. when the quadratic constraints span the sp(4,R) algebra. There are threedifferent classes of sl(2,R) embeddings in sp(4,R) (see appendix A), which will leadto three different gauge-fixings. It is worth noticing that not every element of theseequivalence classes will produce a gauge-fixed model written in terms of coordinatesand velocities. Only those that produce a non-singular matrix A after the gauge-fixingwill have this property (see eq. (3.1.13)).

The following labelling of the gauge-parameters matrix will be used in this section:

βA =

(β2 β10

β10 β5

), βB =

(β3 β9

β8 β6

), βC =

(β1 β7

β7 β4

). (3.3.1)

3.3.1 (0,1) embedding

The sl(2,R) embedding in sp(4,R) with characteristic (0, 1) is given by (A.11). Thereare four remnant fields after the gauge-fixing, namely, T1, H and G±. The gauge-fixedmatrix Λr will be chosen in this way:

Λr =6~c1

H 0 0 ~c16

0 −H ~c16 0

2G+ −T1 −H 0−T1 2G− 0 H

. (3.3.2)

Here:

T1 ≡ T1 +6~c1

H2. (3.3.3)

In this gauge the action (3.1.1) becomes (x+ ≡ x1, x− ≡ x2, F+ ≡ F1, F− ≡ F2):

S =∫

dt

[(x+− 6

~c1Hx+

)(x−+

6~c1

Hx−

)+

6~c1

(G+x2

+ − T1x+x− + G−x2−)− F+F−

].

(3.3.4)

The equations of motion for the matter variables x± from this action are: [L]x+

[L]x−

=

12~c1 G+ −( d

dt + 6~c1 H)2 − 6

~c1 T1

−( ddt − 6

~c1 H)2 − 6~c1 T1

12~c1 G−

x+

x−

= 0.

(3.3.5)

These can be regarded as the Lax equations for the W-algebra associated with thisembedding, that will be denoted as W(0,1)

sp(4,R). The corresponding Lax operator is given

3.3 sp(4,R) models 47

in a 2× 2 matrix form. Matrix Lax operators have been treated in [64] (see chapter 4).They, in general, are associated with non-local W-algebras, but W(0,1)

sp(4,R) is a local W-algebra.

There are four residual gauge transformations. The remnant parameters live inker adE+ and are β10, β5, β2 and β6−β3. The following redefinitions of these parametersare obtained after taking into account eq. (3.1.22):

ε ≡ β10, η ≡ β6 − β3 = β6 − β3 +12~c1

εH − k ε, (3.3.6a)

γ+ ≡ −β2 = −β2, γ− ≡ −β5 = −β5. (3.3.6b)

The four residual transformations are:


δT1 = ε ˙T 1 + 2 ε T1 +~c1

12...ε , (3.3.7a)

δH = ε H + ε H − ~c1

12k ε, δG± = ε G± + (2± k) ε G±, (3.3.7b)

δx± = ε x± − 12

(1± k) ε x±, δF± = ε F± +12

(1± k) ε F±. (3.3.7c)

An anti-symmetric combination of the equations of motion has been introducedin the transformation of the matter and auxiliary variables (see eq. (3.1.26)).

Here k ≡ 16(kβ − kα) is an arbitrary constant that realizes a freedom in choosing

the weight of the fields as a consequence of a mixing of the ε with the η trans-formations displayed below (see subsection 3.1.1). The generator of these diffeo-morphism transformations is the Sugawara-improved energy-momentum tensorT1 − kH. Only when k = 0 all the variables (except the energy-momentumtensor) transform as primary fields: H and G± have weight one and two, respec-tively, whereas the matter and auxiliary variables transform as weight −1/2 and1/2 fields, respectively.

• Transformations generated by H:

δT1 = η H, (3.3.8a)

δH = −~c1

12η, δG± = ±η G±, (3.3.8b)

δx± = ∓12

η x±, δF± = ±12

η F±. (3.3.8c)


• Transformations generated by G±:

δT1 = γ± G± + 2 γ±G±, (3.3.9a)

δH = ∓γ±G±, δG± = 0, (3.3.9b)

δG∓ = γ±

(12

T1 ∓ 12

H ∓ 12~c1

HT1 +24~c1

HH ∓ 144(~c1)

2 H3

)+

+ γ±

(T1 ∓ 3

2H +

24~c1

H2

)∓ 3

2γ±H +

~c1

24...γ±, (3.3.9c)

δx± = −γ±

(x∓ ± 12

~c1H x∓ + F±

)+

12

γ± x∓, δx∓ = 0, (3.3.9d)

δF± = 0, δF∓ = γ±(F± − [L]x±

)+

12

γ± F±. (3.3.9e)

The algebra of these residual transformations is:

[ δε , δε′ ] = δε′′ , ε′′ = ε′ ε − ε ε′, (3.3.10a)

[ δε , δη ] = δη′ , η′ = −ε η, (3.3.10b)

[ δε , δγ± ] = δγ±′ , γ±′ = (1± k) γ± ε− ε γ±, (3.3.10c)

[ δη , δγ± ] = δγ±′ , γ±′ = ±η γ±, (3.3.10d)

[ δη , δη′ ] = [ δη , δµ ] = [ δγ± , δγ±′ ] = [ δγ± , δµ ] = 0, (3.3.10e)

and:

[ δγ+ , δγ− ] = δε′ + δη′ + δµ′ , (3.3.10f)

with:

ε′ = µ′ = −12

(γ+γ− − γ+ γ−)− 12~c1

H γ+ γ−, (3.3.10g)

η′ =12

(γ+ γ− − (1− k) γ+ γ− − (1 + k)γ− γ+)− 12~c1

(2− k)γ+ γ−H +

+12~c1

(2 + k) γ+γ−H − 12~c1

γ+ γ−

(T1 +

36~c1

H2 − kH

), (3.3.10h)

Here the µ transformation is a trivial transformation, i.e. it is proportional to theequations of motion. It is explicitly given by:

δT1 = δH = δG± = 0, (3.3.11a)

δx± = µF∓, (3.3.11b)

δF± = −2µ

(F± ± 6

~c1H F±

)− µ F± − µ [L]x± . (3.3.11c)

The non-closure of the present algebra is due to the introduction of equations of motionin the definition of the ε transformation in terms of the original β transformations. Note


the appearance of field-dependent structure functions in the algebra of transformations,which is a characteristic feature of W-algebras.

The gauge symmetry algebra of the action (3.3.4), W(0,1)sp(4R), is of type W(2, 2, 2, 1),

according to the weights of its generators (see eq. (3.3.7)). The following OPE’s displaythe quantum version of this W-algebra [65]:

1~

T1(z) T1(w) =~c1/2

(z − w)4+

2T1(w)(z − w)2

+∂T1(w)z − w

, (3.3.12a)

1~

T1(z) H(w) =H(w)

(z − w)2+

∂H(w)z − w

,1~

T1(z)G±(w) =2G±(w)(z − w)2

+∂G±(w)z − w

,

(3.3.12b)

1~

H(z) H(w) =~κ1

(z − w)2,

1~

H(z) G±(w) = ±G±(w)z − w

,1~

G±(z) G±(w) = 0,

(3.3.12c)

1~

G+(z) G−(w) = − 3~κ1

(z − w)4− 3H(w)

(z − w)3+

+1

(z − w)2

(κ1 + 1κ1 − 1

T1(w)− 32

∂H(w)− 2κ1 − 1~κ1 (κ1 − 1)

(HH) (w))

+

+1

z − w

(κ1 + 1

2 (κ1 − 1)∂T1(w)− 1

2∂2H(w) +

κ1 + 1~κ1 (κ1 − 1)

(H T1) (w)−

− 2κ1 − 1~κ1 (κ1 − 1)

(H∂H) (w)− 1~2κ1 (κ1 − 1)

(H (HH)) (w))

.

(3.3.12d)

The central charge c1 and the κ1 constant satisfy the following relation:

c1 =12κ2

1 − 16κ1 + 21− κ1

, (3.3.13)

therefore the classical limit ~→ 0, c1 →∞ implies κ1 → − c112 .

In the following, the finite forms of previous infinitesimal transformations are dis-played. The strategy is to perform the gauge-fixing on the finite sp(2M,R) transforma-tions, which are displayed in subsection 3.1.2, as it was done in the case of sl(2,R) (seesection 3.2). Explicitly, one can find the following four sets of finite transformations:

• Diffeomorphisms: The residual finite diffeomorphisms are obtained by the follow-ing composition of finite sp(4,R) transformations (ω ≡ β3 + β6):

Xβ7−→ ¤ ω−→ ¤ diff−→ X, (3.3.14)

where X stands for any variable. After the gauge-fixing one can express the β7

and ω parameters in terms of the finite Diff parameter f(t) obtaining the following


residual transformations:

T1 → f2 T1(f) +~c1

12

(f (3)

f− 3

2f2

f2

), (3.3.15a)

H → f H(f)− ~c1

12k

f

f, G± → f2±k G±(f), (3.3.15b)

x± → f−12(1±k) x±(f), F± → f

12(1±k) F±(f). (3.3.15c)

• Transformations generated by H: The finite transformations corresponding tothe η-sector are the same before and after the gauge-fixing:

T1 → T1 + η H − ~c1

24η2, (3.3.16a)

H → H − ~c1

12η, G± → e±η G±, (3.3.16b)

x± → e∓12η x±, F± → e±

12η F±. (3.3.16c)

• Transformations generated by G±: The residual finite γ± transformations areobtained by the following composition of finite transformations:

γ+ : Xβ4,β7−→ ¤ β9−→ ¤ β2−→ X, (3.3.17a)

γ− : Xβ1,β7−→ ¤ β8−→ ¤ β5−→ X, (3.3.17b)

and are given by:

T1 → T1 + γ± G± + 2 γ±G±, (3.3.18a)

H → H ∓ γ±G±, G± → G±, (3.3.18b)

G∓ → G∓ + γ±

(12

T1 ∓ 12H ∓ 12

~c1HT1 +

24~c1

HH ∓ 144(~c1)

2 H3

)+

+ γ±

(T1 ∓ 3

2H +

24~c1

H2

)∓ 3

2γ±H +

~c1

24...γ± +

+ γ2±

(12G± ∓ 18

~c1HG± ∓ 12

~c1HG± +

6~c1

T1G± +

216(~c1)

2 H2G±)

+

+ γ±γ±

(52

G± ∓ 48~c1

HG±)

+74

γ2±G± +

32

γ±γ±G± +

+12~c1

γ3±

(G±G± ∓ 12

~c1H

(G±)2

)+

24~c1

γ2±γ±G± +

36(~c1)

2 γ4±

(G±)3

(3.3.18c)


and:

x± → x± − γ±

(x∓ ± 12

~c1H x∓ + F±

)+

12

γ± x∓ +12~c1

γ2±G± x∓, (3.3.18d)

x∓ → x∓, F± → F±, (3.3.18e)

F∓ → F∓ + γ±(F± − [L]x±

)+

12

γ± F± +6~c1

γ2±G± F±. (3.3.18f)

Notice the appearance of the Schwarzian derivative in the diffeomorphism transfor-mation of T1 . These transformations are actually finite symmetry transformations ofthe action, under which the Lagrangian changes by a total derivative term and the setof equations of motion remains invariant. The finite transformations parametrized byη and γ± are a parametrization of the specific W-transformations. One might expect,according to the algebra of the infinitesimal transformations (see eq. (3.3.10f)), that thecomposition of γ+ and γ− transformations should give a finite diffeomorphism trans-formation, but clearly this is not the case. In this sense, the finite W-transformationsdisplayed above are parametrized in a rather non-standard way.

3.3.2 (1,1) (principal) embedding

The gauge-fixing induced by the principal sl(2,R) embedding (see eq. (A.14)) in sp(4,R)is given by:

Λr = −12~c

0 0 0 −√

3 ~c12

0 0 −√

3 ~c12 T

3W4

√3

2 T 0 0√3

2 T −~c6 0 0

. (3.3.19)

The two remnant fields are T and W4. Here numerical factors are taken for convention.The action (3.1.20) is given in this case by:

Spgf =∫

dt

[x1x2√

3+

6~c

T

(x2

1

3−√

3x1x2 + F 22

)− 18~c

W4 x21 + x2

2 −√

3F1F2

].

(3.3.20)

The residual transformations are parametrized by ε and ρ, which are related to theremnant β parameters in the following way:

ε− 172

ρ− 12~c

ρ T =1

2√

3β10 − 1

4β4, ρ =

35

β2. (3.3.21)

The diffeomorphism transformations are parametrized by ε. The T generator is a quasi-primary weight-two field, and W4 is a primary weight-four field. The matter (xi) and


auxiliary (Fi) variables are not primary fields. However, a set of primary fields (xi andFi) can be obtained by mixing them:

x1 ≡ x1, x2 ≡ x2 − 3√

3~c

T x1 − 12√

3x1, (3.3.22a)

F1 = F1 − 2√

3~c

T F2, F2 = F2. (3.3.22b)

The action (3.3.20) then becomes (neglecting total derivative terms):

Spgf =∫

dt

[2~c

T ˙x21 −

112

(¨x1 +

18~c

T x1

)2

− 18~c

W4 x21 + x2

2 −√

3F1F2

]. (3.3.23)

Notice that this action is of a higher order in x1. Its equation of motion is a fourth-orderdifferential equation:

x(4)1 +

60~c

T ¨x1 +60~c

T ˙x1 +18~c

(12W4 + T +

18~c

T 2

)x1 = 0. (3.3.24)

On the other hand, the matter variable x2 decouples and disappears on-shell. There arethe same number of physical degrees of freedom as in (3.3.20) and it has the followingtwo residual symmetries (up to equations of motion):


δT = ε T + 2 ε T +~c12

...ε , δW4 = ε W4 + 4 εW4, (3.3.25a)

δx1 = ε ˙x1 − 32

ε x1, δx2 = ε ˙x2 +12

ε x2, (3.3.25b)

δF1 = ε ˙F1 +12

ε F1, δF2 = ε ˙F2 +12

ε F2. (3.3.25c)

• Transformations generated by W4:

δT = 3 ρ W4 + 4 ρ W4, (3.3.26a)

δW4 =ρ

(136

...W4 +

1432

T (5) +143~c

(TW4)˙ +

1318~c

T...T +

5936~c

T T +48

(~c)2T 2T

)+

+ ρ

(536

W4 +5

324T (4) +

283~c

TW4 +88

27~cT T +

295108~c

T 2 + 32T 3

)+

+ ρ

(14

W4 +7

162...T +

499~c

T T

)+ ...

ρ

(16

W4 +7

108T +

4927~c

T 2

)+

+35648

ρ(4) T +7

324ρ(5) T +

~c15552

ρ(7)

(3.3.26b)


and:

δx1 = ρ

(9

2~cT x1 +

413~c

T ˙x1 +518

...x1

)− ρ

(236~c

T x1 +536

¨x1

)+

+118

ρ ˙x1 − 172

...ρ x1, (3.3.26c)

δx2 = 0, δF1 = 0, δF2 = 0. (3.3.26d)

The infinitesimal transformations of T and W4 are those induced by the classicalW(2,4)algebra. This algebra is the classical limit of the quantum W-algebra given by thefollowing OPE’s [12] [66]:

1~ T (z)T (w) = ~c/2

(z−w)4+ 2 T (w)

(z−w)2+ ∂T (w)

z−w , 1~ T (z)W4(w) = 4 W4(w)

(z−w)2+ ∂W4(w)

z−w ,

(3.3.27a)

1~ W4(z) W4(w) = 1

α(c)

[ ~c/4

(z−w)8+ 2 T (w)

(z−w)6+ ∂T (w)

(z−w)5+

+ 1(z−w)4

(α(c)W4(w) + 3(c−4)

2(5c+22) ∂2T (w) + 42~(5c+22) (TT ) (w)

)+

+ 1(z−w)3

(α(c)

2 ∂W4(w) + c−253(5c+22) ∂3T (w) + 42

~(5c+22) (T∂T ) (w))

+

+ 1(z−w)2

((5c+64)α(c)

36(c+24) ∂2W4(w) + 5c3+261c2−4259c+17886(5c+22)(2c−1)(7c+68) ∂4T (w)+

+ 28α(c)3~(c+24) (TW4) (w) + 176c2+3573c−1904

~(5c+22)(2c−1)(7c+68)

(T∂2T

)(w)+

+ 295c2+9504c+32962~(5c+22)(2c−1)(7c+68) (∂T∂T ) (w)+

+ 24(72c+13)~2(5c+22)(2c−1)(7c+68)

(T (TT )) (w))

+

+ 1z−w

((c−4)α(c)36(c+24) ∂3W4(w) + 5c3+1152c2−10818c+3436

40(5c+22)(2c−1)(7c+68) ∂5T (w)+

+ 14α(c)3~(c+24) ∂ (TW4) (w) + 3(13c2+301c−264)

~(5c+22)(2c−1)(7c+68)

(T∂3T

)(w) +

+ 3(59c2+3456c+940)2~(5c+22)(2c−1)(7c+68)

(∂T∂2T

)(w)+

+ 36(72c+13)~2(5c+22)(2c−1)(7c+68)

(T (T∂T )) (w))]

,

(3.3.27b)

where α(c) ≡ 54(c+24)(c2−172c+196)(5c+22)(2c−1)(7c+68) .

Thus, the action (3.3.23) provides a particle model in which the W(2,4) symmetryis implemented. Consequently, eq. (3.3.24) is the Lax equation corresponding to theW(2, 4) algebra [28].

The matrix of gauge fields (3.3.19) can be transformed into the form correspondingto the (0,1) embedding (3.3.2) with H = 0 and G− = constant. This is achieved by


performing finite sp(4,R) transformations (see subsection 3.1.2). Firstly, a Bβ trans-formation with

eBβ =

1√6

0

2√

6~c T

√2

(3.3.28)

is made in order to set the A submatrix in the form A′ =

(0 11 0

). The next step is

to perform a Cβ transformation with

Cβ =

(−12~c T 0

0 0

). (3.3.29)

After these gauge transformations the form of the matrix of gauge fields becomes:

Λ′′ =6~c

0 0 0 ~c6

0 0 ~c6 0

−36~c W + 96

~c T 2 + 2 T −5T 0 0−5T ~c

6 0 0

, (3.3.30)

which is formally equivalent to eq. (3.3.2) when H = 0 and G− = constant. In otherwords, the W(0,1)

sp(4,R) algebra contains the W(2, 4) algebra and the action of eq. (3.3.4)can be further reduced yielding the model of eq. (3.3.23). This is an example of whatis called a secondary reduction, which is part of the subject of the next chapter. Afterthis secondary reduction the weight 4 field G+ is no longer primary but is given interms of the weight 4 primary field W4 and the weight 2 quasi-primary field T as shownin (3.3.30).

3.3.3 (12,0) embedding

The gauge-fixing induced by this embedding (A.12) has six remnant fields, namely T2,C±, D± and D0, and is given by

Λr = − 3~c2

0 0 −~c23 0C− −D0 0 2D+

4T2 −C+ 0 −C−

−C+ 2D− 0 D0

, (3.3.31)

where

T2 ≡ T2 +3

2~c2

((D0

)2 + 4D+D−)

. (3.3.32)

Constants (such as c2) are introduced for future convenience.


The action (3.1.1) becomes:

Spgf =12

∫dt

[x2

1 −~c2

61

D+x2

2 −C−

D+x1x2 +

D0

D+x2x2 +

3~c2

(2C+ +

C−D0

D+

)x1x2−

− 3~c2

(2T2 +

(C−)2

2D+

)x2

1 −3~c2

(2D− +

(D0

)2

2D+

)x2

2 − F 21 +

6~c2

D+F 22

].

(3.3.33)

A characteristic feature of this embedding is that the primary field D+ appears indenominators. The equations of motion for x1 and x2 are written in matrix form:

([L]x1

[L]x2

)= B

(x1

x2

), (3.3.34a)

B≡

− d2

dt2− 3

2~c2

(2T2 + (C−)2

2D+

)− C−

2D+ddt + 3

~c2(2C+ + C−D0

D+

)

C−2D+

ddt +

(C−2D+

).+ 3~c2

(C+ + C−D0

2D+

)~c26D+

(d2

dt2− D+

D+ddt

)−

(D0

2D+

).−

− 3~c2

(2D− + (D0)2

2D+

)

.

(3.3.34b)

The action (3.3.33) is invariant under six infinitesimal gauge transformations, whichare:

• Diffemorphisms (ε = β2):

δT2 = ε ˙T2 + 2 ε T2 +~c2

12...ε , (3.3.35a)

δD0 = ε D0 + ε D0 − ~c2

3k ε, δD± = ε D± + (1∓ 2k) ε D±, (3.3.35b)

δC± = ε C± +(

32± k

)ε C±, (3.3.35c)

δx1 = ε x1 − 12

ε x1, δx2 = ε x2 − k ε x2, (3.3.35d)

δF1 = ε F1 +12

ε F1, δF2 = ε F2 + k ε F2. (3.3.35e)

The twisting in the weights of the generators is due to the arbitrariness in choosingthe ε parameter out of the βi’s (see subsection 3.1.1: k ≡ 1

6(kβ − kα)). Alterna-tively it can be regarded as a Sugawara-like redefinition of the energy-momentumtensor T2. The generator of the above diffeomorphism transformations is thenT2 − kD0.


• Transformations generated by D0 (λ0 = −β6):

δT2 = λ0 D0, (3.3.36a)

δD0 = −~c2

3λ0, δD± = ∓2λ0 D±, (3.3.36b)

δC± = ±λ0 C±, (3.3.36c)

δx1 = 0, δx2 = −λ0 x2, δF1 = 0, δF2 = λ0 F2. (3.3.36d)

• Transformations generated by D+ (λ+ = −β4):

δT2 = λ+ D+, (3.3.37a)

δD0 = 2 λ+ D+, δD+ = 0, δD− = −λ+ D0 − ~c2

6λ+, (3.3.37b)

δC+ = −λ+ C−, δC− = 0, (3.3.37c)

δx1 = 0, δx2 = 0, δF1 = 0, δF2 = 0. (3.3.37d)

• Transformations generated by D− (λ− = β5):

δT2 = λ−D−, (3.3.38a)

δD0 = −2λ−D−, δD+ = λ−D0 − ~c2

6λ−, δD− = 0, (3.3.38b)

δC+ = 0, δC− = −λ−C+, (3.3.38c)

δx1 = 0, δx2 = λ−

(1

D+

(−~c

6x2 +

12

D0 x2 − 12

C− x1

)+ F2

),

(3.3.38d)

δF1 = 0, δF2 =~c2

6λ−D+

(F2 − 3

~c2D0 F2 − [L]x2

)+~c2

6λ−D+

F2.

(3.3.38e)

• Transformations generated by C+ (ρ+ = −β10):

δT2 =12

ρ+ C+ +32

ρ+ C+, (3.3.39a)

δD0 = −ρ+ C+, δD+ = ρ+ C−, δD− = 0, (3.3.39b)

δC+ = 2 ρ+ D− + 4 ρ+ D−, (3.3.39c)

δC− = −ρ+

(2 T2 +

6~c2

((D0

)2 + 4 D+D−)− D0

)+ 2 ρ+ D0 − ~c2

3ρ+,

(3.3.39d)

δx1 = ρ+

(1

D+

(~c2

6x2 − 1

2D0 x2 +

12

C− x1

)− F2

), (3.3.39e)

δx2 = −ρ+

(x1 +

3~c2

D0 x1 + F1

)+ ρ+ x1 (3.3.39f)


and:

δF1 = ρ+

(F2 +

3~c2

D0 F2 − [L]x2

), (3.3.39g)

δF2 =ρ+

D+

(−~c2

6F1 +

12

D0 F1 − 12

C− F2 +~c2

6[L]x1

)− ~c2

6ρ+

D+F1.

(3.3.39h)

• Transformations generated by C− (ρ− = β9):

δT2 =12

ρ− C− +32

ρ−C−, (3.3.40a)

δD0 = ρ−C−, δD+ = 0, δD− = ρ−C+, (3.3.40b)

δC+ = ρ−

(2T2 +

6~c2

((D0

)2 + 4 D+D−)

+ D0

)+ 2 ρ−D0 +

~c2

12ρ−,

(3.3.40c)

δC− = −2 ρ− D+ − 4 ρ−D+, (3.3.40d)

δx1 = ρ− x2, δx2 =6~c2

ρ−D+ x1, (3.3.40e)

δF1 = − 6~c2

ρ−D+ F2, δF2 = −ρ− F1. (3.3.40f)

The action (3.3.33) is invariant under a classical W-algebra that will be denoted as

W( 12,0)

sp(4,R). The OPE’s for the corresponding quantum W-algebra read:

1~

T2(z) T2(w) =~c2/2

(z − w)4+

2T2(w)(z − w)2

+∂T2(w)z − w

, (3.3.41a)

1~

T2(z) Da(w) =Da(w)

(z − w)2+

∂Da(w)z − w

,1~

T2(z) C±(w) =3/2 C±(w)(z − w)2

+∂C±(w)z − w

,

(3.3.41b)

1~

Da(z) Db(w) =~κ2γ

ab

(z − w)2+

fabcD

c(w)z − w

, (3.3.41c)

1~

D0(z) C±(w) = ±C±(w)

z − w,

1~

D±(z) C±(w) = −C∓(w)

z − w,

1~

D±(z) C∓(w) = 0,

(3.3.41d)

1~

C±(z) C±(w) = ±4D∓(w)

(z − w)2± 2 ∂D∓(w)

z − w, (3.3.41e)

1~

C+(z) C−(w) =

4~κ2

(z − w)3+

2 D0(w)(z − w)2

+

+1

z − w

(−4κ2 + 10

2κ2 + 3T2(w) +

4~ (2κ2 + 3)

γab(DaDb)(w) + ∂D0(w))

.

(3.3.41f)


The Da(z) fields span a sl(2,R) Kac–Moody subalgebra. The conventions for thesl(2,R) generators are:

[t0, t±] = ±t±, [t+, t−] = 2 t0, (3.3.42)

Indices are raised and lowered with γab = 12h∨ gab (f c

ab , gab and h∨ = 2 are the structureconstants, the Cartan–Killing metric, and the sl(2,R) dual Coxeter number, respec-tively). The central charge c2 is related to the sl(2,R) Kac–Moody level κ2 throughthe following equation:

c2 = −6κ2 (2κ2 + 1)2κ2 + 5

. (3.3.43)

The classical limit ~→ 0, c2 →∞ implies κ2 → − c26 .

3.4 sl(3,R) models

The W(2, 4, . . . , 2M) gauge transformations can be obtained by considering the prin-cipal sl(2,R) embedding in a general sp(2M,R) algebra. It is also possible to constructparticle-like models having symmetries related to other W algebras. In order to obtainmodels related to the AN−1 series (for instance, theWN algebras) it is necessary to lookfor sl(N,R) subalgebras in the symplectic algebras. There is a canonical embedding,namely,

sl(N,R)⊕ u(1) ⊂ sp(2N,R). (3.4.1)

Explicitly, the set of matrices of the form(B 00 −B>

)(3.4.2)

are a subalgebra of sp(2N,R) which is isomorphic to sl(N,R)⊕u(1). A particle mechan-ics model can be constructed by taking this specific form of the gauge fields matrix,but the procedure outlined in sect. 3.1 can not be implemented because A = 0 soeq. (3.1.13) is no longer valid to eliminate the p variables through their equations ofmotion. Indeed, by putting all the momenta on-shell, a null Lagrangian is obtained.

However, there are other embeddings of sl(N,R) in sp(2N,R) which will allow theconstruction of particle mechanics models. For example, one can consider a canonicalaction (3.1.1) with M = 3 taking Λ as:

Λ =

λ7 0 0 0 λ1 λ3

0 λ7 − λ8 λ5 λ1 0 00 λ2 λ8 λ3 0 00 λ4 λ6 −λ7 0 0λ4 0 0 0 λ8 − λ7 −λ2

λ6 0 0 0 −λ5 −λ8

. (3.4.3)

3.4 sl(3,R) models 59

Only a part of the φAij constraints is then considered (see appendix A for details).They still close under Poisson bracket giving a realization of the sl(3,R) algebra—itis a sl(3,R) subalgebra of sp(6,R). The gauge transformations still are those given ineq. (3.1.9a) and (3.1.9b) with the following β matrix:

β =

β7 0 0 0 β1 β3

0 β7 − β8 β5 β1 0 00 β2 β8 β3 0 00 β4 β6 −β7 0 0β4 0 0 0 −β7 + β8 −β2

β6 0 0 0 −β5 −β8

= β + εΛ. (3.4.4)

3.4.1 W3 model

The principal sl(2,R) embedding in sl(3,R) (see eq. (A.6)) induces the following gauge-fixing:

λ1 = λ2 = 1, λ3 = λ7 = λ8 = 0, (3.4.5a)

λ4 = λ5 = −12~c

T, λ6 = −24~c

√−5

2W3. (3.4.5b)

Again one can not use eq. (3.1.13) because detA = 0 but nevertheless a (higher order)particle-like Lagrangian can be obtained after eliminating the momenta variables:

Spgf =∫

dt

[12

(x1 x3 − x1 x3) +12~c

T (x1 x3 − x1 x3) +24~c

√−5

2W3 x1 x3

]. (3.4.6)

The equations of motion for the xi variables are:

[L]x1 =...x3 +

24~c

T x3 +24~c

(√−5

2W3 +

12

T

)x3, (3.4.7a)

[L]x3 = −...x1 − 24

~cT x1 − 24

~c

(−

√−5

2W3 +

12

T

)x1. (3.4.7b)

They are two copies of the Lax equation for W3. This Lax operator has been encoun-tered when dealing with the generalized KdV differential equations (see eq. (2.3.25)).

This action is invariant under the following two gauge transformations, being theremnant parameters ε and ρ ≡ β3:


δT = ε T + 2 ε T +~c12

...ε , (3.4.8a)

δW3 = ε W3 + 3 εW3, (3.4.8b)

δx1 = ε x1 − ε x1, δx3 = ε x3 − ε x3. (3.4.8c)


• Transformations generated by W3:

δT = 2 ρ W3 + 3 ρ W3, (3.4.9a)

δW3 = ρ

(115

...T +

325~c

T T

)+ ρ

(310

T +325~c

T 2

)+

12

ρ T +13

...ρ T +

~c360

ρ(5),

(3.4.9b)

δx1 = −√−2

5

(ρ

(x1 +

16~c

T x1

)− 1

2ρ x1 +

16

ρ x1

), (3.4.9c)

δx3 =

√−2

5

(ρ

(x3 +

16~c

T x3

)− 1

2ρ x3 +

16

ρ x3

). (3.4.9d)

These transformations are generated by T and W3 through the Poisson-bracket algebragiven in eq. (1.3.3), which is the classical limit of the quantum W3 algebra.

3.4.2 W23 model

It can also be constructed a model exhibiting the symmetry associated with the W-algebra generated through the only non-principal sl(2,R) embedding in sl(3,R), namelyW2

3 . The corresponding gauge-fixing, which is given by (see eq. (A.7)):

λ1 = λ2 = 0, λ3 = −1, λ4 = − 6~c

√32

B+, (3.4.10a)

λ5 =6~c

√32

B−, λ6 =6~c

(T +

184~c

A2

), λ7 = −λ8 =

3~c

A, (3.4.10b)

yields the following action:

Spgf =

√23

∫dt

1B−

[12

Ax1x2 − ~c6 x1x2 +

(12

A +~c6

B−

B−

)x1x2+

+

(A−A

B−

B− +3~c

A2

)x1x2 +

(T +

3~c

A2 − 12

AB−

B−

)x1x2 −

− 6~c

(AT +

12

AA +6~c

A3 − 12

A2 B−

B− +32B+B−

)x1x2

].

(3.4.11)

The equations of motion for x1 and x2 read:

...x1 − B−

B− x1 +6~c

(T − 3

2A + A

B−

B−

)x1 +

+6~c

(9~c

B+B− +

(ddt− B−

B− +6~c

A

)(T − 1

2A +

6~c

A2

))x1 = 0 (3.4.12a)

3.4 sl(3,R) models 61

and:

...x2 − 2

B−

B− x2 +6~c

(T − 3

2A + A

B−

B− −~c6

(ddt− B−

B−

)B−

B−

)x2 −

− 6~c

(9~c

B+B− +

(ddt− B−

B−

)(ddt− B−

B− +6~c

A

)A+

+6~c

A

(T − 1

2A +

6~c

A2

))x2 = 0. (3.4.12b)

The previous action is invariant under the following four gauge transformations:


δT = ε˙T + 2 ε T +

~c12

...ε , (3.4.13a)

δA = ε A + ε A− ~c3

k ε, δB± = ε B± +(

32± 3k

)ε B±, (3.4.13b)

δx1 = ε x1 −(

12

+ k

)ε x1, δx2 = ε x2 − 2 k ε x2. (3.4.13c)

• Transformations generated by A (α ≡ 32 β5):

δT = α A, δA = −~c9

α, δB± = ±α B±, (3.4.14a)

δx1 = −13

α x1, δx2 = −23

α x2. (3.4.14b)

• Transformations generated by B+ (β+ ≡√

32 β2):

δT =12

β+B+ +32

β+B+, (3.4.15a)

δA = −β+B+, (3.4.15b)

δB+ = 0, δB− = β+

(23T +

12~c

A2 − A

)− 2 β+A +

~c9

β+, (3.4.15c)

δx1 = β+

(1

B−

(23

T − 13

A +4~c

A2

)x1 − 2

3A

B− x1 +

√23

x1

), (3.4.15d)

δx2 =β+

B−

((23

A +8~c

A2 − 23

AB−

B−

)x2 −

(23

A− ~c9

B−

B−

)x2 − ~c9 x2

)+

+β+

B−

(~c9

x2 − 23

Ax2

). (3.4.15e)


• Transformations generated by B− (β− ≡ −√

32 β6):

δT =12

β−B− +32

β−B−, (3.4.16a)

δA = β−B−, (3.4.16b)

δB+ = −β−(

23

T +12~c

A2 + A

)− 2 β−A− ~c

9β−, δB− = 0, (3.4.16c)

δx1 = 0, δx2 = 0. (3.4.16d)

The conclusion is that the action (3.4.11) is invariant under the classical W23 algebra.

The corresponding quantum OPE’s are the following [32] [46]:

1~

T (z) T (w) =~c/2

(z − w)4+

2 T (w)(z − w)2

+∂T (w)z − w

, (3.4.17a)

1~

T (z) A(w) =A(w)

(z − w)2+

∂A(w)z − w

,1~

T (z) B±(w) =3/2B±(w)(z − w)2

+∂B±(w)z − w

,

(3.4.17b)

1~

A(z) A(w) =~κ

(z − w)2,

1~

A(z) B±(w) = ±B±(w)z − w

,1~

B±(z) B±(w) = 0,

(3.4.17c)

1~

B+(z) B−(w) =− 2 ~κ

(z − w)3− 2A(w)

(z − w)2+

+1

z − w

(2 (κ + 1)3κ− 1

T (w)− 4~ (3κ− 1)

(AA) (w)− ∂A(w))

.

(3.4.17d)

The constant κ is related to the W23 central charge c through the following expression:

c =κ (7− 9κ)

1 + κ. (3.4.18)

The differential equations (3.4.12) are invariant under the W23 algebra. They can be

regarded as the Lax equations for W23 because they play the same role of eq.(3.4.7)

with respect to the W3-invariant action (3.4.6).

Chapter 4

Relations between W-algebras

This chapter is devoted to the investigation of some new features that appeared in theliterature, mainly concerning the existence of new W-structures or relations betweenknown W-algebras. Specifically, three apparently disconnected algebraic aspects ofW-algebras will turn out to be related:

• The inclusion of a W algebra into a bigger one, both coming from the sameKac–Moody algebra via DS reduction. This inclusion can also be regarded as a(further) DS reduction of the bigger W-algebra leading to the smaller one. Thisprocedure is called secondary (DS) reduction [67].

• The enlargement of a (non-linear) W-algebra with one or more extra generatorsso that the resulting algebra becomes linear after a suitable change of basis [68].It will be shown in this chapter that this process of linearization of W-algebrashappens to be, at least in some cases, closely related to the secondary reductionscheme.

• The existence of non-local W-type algebras [64]. It will be proved that some ofthese new W-algebras, which are associated with matrix differential equations,can be understood as intermediate states in secondary reduction processes too.

The equations of motion of the particle mechanics models presented in chapter 3 willplay a central role in the discovery of these relations. Furthermore, by working with dif-ferential equations instead of the corresponding W-algebras, the explicit computationswill become much easier.

The procedure of linearizing a W-algebra will be exemplified in sect. 4.1 by con-sidering the case of W3. Examples of non-local W-type algebras are given in sect. 4.2.The secondary reduction scheme and its links with linear and non-local algebras areshown in the remaining of the chapter for the W-algebras which are related to sl(3,R)and sp(4,R). The main original results displayed in this chapter have been publishedin ref. [69].

63

64 Relations between W-algebras

4.1 Linearizing W3

The principal difficulty in handling the usual examples of W-algebras is their non-linearity. A possible way for overcoming this difficulty was presented in ref. [68]. Theidea is to treat a given non-linear W-algebra as a subalgebra of a bigger W-algebrathat exhibits the remarkable property of being linear. As a consequence, for instance,every representation of the linear W-algebra will provide a representation of the formernon-linear W-algebra. The method was originally introduced in the linearization of theW3 andW2

3 algebras [68] and, then, applied to other cases [70] [71] [72] (see also ref. [73]for further developments and ref. [74] for applications of the linearization method to(W)-string theory and to the study of relations between different string theories). Atthe moment, there is no systematic procedure for obtaining the linearizing algebra ofany given non-linear W-algebra. Nevertheless, as it will be illustrated in sect. 4.3 [69],the DS reduction scheme encodes the way of constructing, at the classical level, thelinearizing algebra of a great number ofW-algebras (see also ref. [75], [71], [76] and [72]).In this section, following ref. [68], the linearizing algebra for W3 is constructed as anexample.

A new weight-one (non-primary) field J(z) is added to the W3-algebra (see the W3-algebra OPE’s in eq. (1.2.1)). Conformal invariance dictates the form of the singularparts in the three new OPE’s involving J(z) up to fourteen arbitrary parameters. Fi-nally, commutativity and associativity of the radial product determine these parametersyielding the following expressions:

1~

T (z) J(w) =~ (1 + 3c0)(z − w)3

+J(w)

(z − w)2+

∂J(w)z − w

, (4.1.1a)

1~

J(z) J(w) =~ (1 + c0)(z − w)2

, (4.1.1b)

1~

W3(z) J(w) = α(c0)

[−~ (1 + 3c0)

2

2 (z − w)4− (1 + 3c0) J(w)

(z − w)3+

+(

2 (1 + c0) T (w) +1 + 3c0

2∂J(w)− 2

~(JJ) (w)

)1

(z − w)2−

−(

1α(c0)

W3(w)− 3 + c0

2∂T (w) +

1− 3c0

3 (1 + c0)∂2J(w)− 2

~(TJ) (w)+

+2 (1− c0)~ (1 + c0)

(∂J J) (w) +4

3~2 (1 + c0)(J (JJ)) (w)

)1

z − w

]. (4.1.1c)

Here:

α(c0) ≡√

23− 22c0 − 45c2

0

, (4.1.2)

4.1 Linearizing W3 65

and c0 is related to the central charge c appearing in the W3 OPE’s through thefollowing equation:

c = 2− 4 (3c0 + 1)2

c0 + 1. (4.1.3)

As a consequence, two different weight-one extensions ofW3 can be defined correspond-ing to the two solutions for c0 of eq. (4.1.3). In the following it will be adopted thevalue

c0 = − 172

(c + 22 +

√(c− 2) (c− 98)

). (4.1.4)

This extension of the W3-algebra is a new algebra that, when expressed in the basisT (z), W3(z), J(z), will be referred to as W1+3. The next step is to define an invertiblechange of basis in the extended algebra from the set T (z),W3(z), J(z) to the setT (z), W+(z), J(z), where:

W+(z) ≡ W3(z)− 2~

α(c0)(

(JT ) (z)− 23~ (1 + c0)

(J (JJ)) (z) +1 + 3c0

1 + c0(J ∂J) (z)−

−14~ (1 + 3c0)

(∂T (z) +

2 (1 + 3c0)3 (1 + c0)

∂2J(z)))

.

(4.1.5)

The singular parts in all the OPE’s of this algebra are now the following:

1~

T (z) T (w) =~c/2

(z − w)4+

2T (w)(z − w)2

+∂T (w)z − w

, (4.1.6a)

1~

T (z) W+(w) =3W+(w)(z − w)2

+∂W+(w)

z − w, (4.1.6b)

1~

T (z) J(w) =γ1(c)

(z − w)3+

J(w)(z − w)2

+∂J(w)z − w

, (4.1.6c)

1~

W+(z) J(w) = −W+(w)z − w

,1~

W+(z) W+(w) = 0,

1~

J(z) J(w) =γ2(c)

(z − w)2,

(4.1.6d)

where:

γ1(c) = ~ (1 + 3c0) , γ2(c) = ~ (1 + c0) . (4.1.7)

The remarkable feature is that all non-linearities have disappeared from the r.h.s. ofthe OPE’s. The algebra defined by eq. (4.1.6) is essentially the same algebra as W1+3

because they are related through a mere invertible change of basis. However a dif-ferent name (W lin

3 ) will be used to denote this algebra when it is spanned by theT (z), W+(z), J(z) basis as a way of stressing its apparent linear character. Note


that W lin3 , despite its linear character, is a W-algebra in the sense of the definition

given in chapter 1.A twist of the stress-energy tensor T (z) can be performed in order to get a basis

where both W+(z) and J(z) become primary fields:

T0(z) ≡ T (z) +1 + 3c0

2 (1 + c0)∂J(z). (4.1.8)

The new OPE’s are:1~

T 0(z) T 0(w) =γ0(c)

(z − w)4+

2T0(w)(z − w)2

+∂T0(w)z − w

, (4.1.9a)

1~

T 0(z)W+(w) =λ(c)W+(w)

(z − w)2+

∂W+(w)z − w

,1~

T 0(z)J(w) =J(w)

(z − w)2+

∂J(w)z − w

,

(4.1.9b)

where:

γ0(c) =~

(1− 4c0 − 9c2

0

)

2 (1 + c0)and λ(c) =

32

+1

1 + c0. (4.1.10)

In this new basis the primary field W+(z) acquires a (in general) non-rational weightλ(c) depending on the value of c0.

So far the linear algebra has been defined at the quantum level. All the consider-ations can also be made after a classical limit. The behaviour of c0 when c → ∞ isextracted from eq. (4.1.4):

c →∞ ⇒ c0 → − c

36, (4.1.11)

and the classical limit

c →∞, ~→ 0, ~c = constant, (4.1.12)

of eq. (4.1.1) reads:T (z), J(w)

= −~c

24∂2

wδ(z − w) + J(w) ∂wδ(z − w) + ∂J(w) δ(z − w), (4.1.13a)

J(z), J(w)

= −~c

36∂wδ(z − w), (4.1.13b)

W3(z), J(w)

= −

√−2

5

[~c144

∂3wδ(z − w) +

12

J(w) ∂2wδ(z − w)+

+(

23

T (w) +12

∂J(w) +24~c

J(w)2)

∂wδ(z − w)+

+(√

−52

W3(w) +16

∂T (w)−

−24~c

T (w) J(w)− 24~c

J(w) ∂J(w)−(

24~c

)2

J(w)3)

δ(z − w)

]. (4.1.13c)

4.2 Non-local V-algebras 67

The change of basis (4.1.5) and the twist (4.1.8) become:

W+(z) = W3(z)− 24~c

√−2

5

(J(z) T (z) +

24~c

J(z)3 + 3 J(z) ∂J(z)+

+~c48

(∂T (z) + 2 ∂2J(z)

)), (4.1.14a)

T0(z) = T (z) +32∂J(z), (4.1.14b)

and the Poisson brackets defining the classical version of W lin3 are directly read from

the OPE’s in eq. (4.1.6) and eq. (4.1.9) with

γ0(c) =~c8

, γ1(c) = −~c12

, γ2(c) = −~c36

, (4.1.15a)

and λ(c) =32. (4.1.15b)

Note that in this classical version the weight of the W+(z) field does not depend on c.

4.2 Non-local V-algebras

Symmetry algebras associated with matrix generalizations of the Lax operator displayedin eq. (2.3.16) were studied in ref. [64]. (Further results on this topic can be found inref. [77] and ref. [78].) These symmetry algebras share the non-linearity and conformalproperties of ordinary W-algebras, but they are, in general, non-local. These algebraswill be referred to as V-algebras. In this section, several examples of V-algebras arepresented for future convenience. See ref. [64] for further details.

A Vn,m-type algebra is the symmetry algebra associated with a Lax operator of theform:

L0 = I ∂m +m−2∑

k=0

Uk ∂k, (4.2.1)

where Uk are n×n matrices. The Poisson brackets defining a Vn,m algebra are obtainedby generalizing the Adler mapping (see sect. 2.3) to the matrix case. The simplest non-local algebra appears when m = 2 and n = 2. The symmetry algebra associated withthe operator

L0 = I ∂2 + U0 =

(1 00 1

)∂2 +

6~c

(V2 + V3 2V +

2V − V2 − V3

)(4.2.2)


will be named V(0)2,2 . It is given by the following Poisson brackets:

V2(z), V2(w)

=~c12

∂3wδ(z − w) + 2V2(w) ∂wδ(z − w) + ∂V2(w) δ(z − w), (4.2.3a)

V2(z), V ±(w)

= 2V ±(w) ∂wδ(z − w) + ∂V ±(w) δ(z − w), (4.2.3b)

V2(z), V3(w)

= 2V3(w) ∂wδ(z − w) + ∂V3(w) δ(z − w), (4.2.3c)V ±(z), V ±(w)

=

12~c

ε(z − w) V ±(z) V ±(w), (4.2.3d)

V ±(z), V ∓(w)

= −12

~cε(z − w)

(V ±(z) V ∓(w) +

12V3(z) V3(w)

)+

+~c24

∂3wδ(z − w) + V2(w) ∂wδ(z − w) + ∂V2(w) δ(z − w), (4.2.3e)

V3(z), V ±(w)

=

12~c

ε(z − w) V ±(z) V3(w), (4.2.3f)

V3(z), V3(w)

=

24~c

ε(z − w)(

V +(z) V −(w) + V −(z) V +(w))

+

+~c12

∂3wδ(z − w) + 2V2(w) ∂wδ(z − w) + ∂V2(w) δ(z − w). (4.2.3g)

A Virasoro subalgebra, which is generated by V2(z), is included in V(0)2,2 . All fields have

weight two with respect to V2(z). The non-locality is encoded in the ε(z−w) function,which is the inverse of ∂wδ(z − w), i.e.,

∂zε(z − w) = δ(z − w). (4.2.4)

A simpler V2,2-type algebra arises as the symmetry algebra associated with the followingoperator:

L1 =

(1 00 1

)∂2 +

6~c

(V2 2V +

2V − V2

). (4.2.5)

This algebra, which will be named V(1)2,2 , can be regarded as the reduction of V(0)

2,2 whenthe second-class constraint

V3 = 0 (4.2.6)

is introduced in the latter algebra. The V(1)2,2 algebra is defined by:

V2(z), V2(w)

=~c12

∂3wδ(z − w) + 2V2(w) ∂wδ(z − w) + ∂V2(w) δ(z − w), (4.2.7a)

V2(z), V ±(w)

= 2V ±(w) ∂wδ(z − w) + ∂V ±(w) δ(z − w), (4.2.7b)

V ±(z), V ±(w)

=

12~c

ε(z − w) V ±(z) V ±(w), (4.2.7c)

V ±(z), V ∓(w)

= −12

~cε(z − w) V ±(z) V ∓(w) +

+~c24

∂3wδ(z − w) + V2(w) ∂wδ(z − w) + ∂V2(w) δ(z − w). (4.2.7d)

4.3 Secondary reductions of W23 69

4.3 Secondary reductions of W23

The W23 algebra contains, besides a stress-energy tensor T (z), two weight-3/2 fields

(B+(z) and B−(z)), and a weight-one current, A(z). Its OPE’s are displayed ineq. (3.4.17). After a classical limit ~ → 0, c → ∞ (κ → − c

9), ~c = constant, thisalgebra becomes:

T (z), T (w)

=~c12

∂3wδ(z − w) + 2T (w) ∂wδ(z − w) + ∂T (w) δ(z − w), (4.3.1a)

T (z), A(w)

= A(w) ∂wδ(z − w) + ∂A(w) δ(z − w), (4.3.1b)

T (z), B±(w)

=

32B±(w) ∂wδ(z − w) + ∂B±(w) δ(z − w), (4.3.1c)

A(z), A(w)

= −~c

9∂wδ(z − w),

A(z), B±(w)

= ±B±(w) δ(z − w), (4.3.1d)

B±(z), B±(w)

= 0, (4.3.1e)

B+(z), B−(w)

=~c9

∂2wδ(z − w)− 2A(w) ∂wδ(z − w) +

+(

23T (w) +

12~c

A(w)2 − ∂A(w))

δ(z − w). (4.3.1f)

The W23 algebra is almost a linear algebra because the only nonlinearity is a A(w)2

term appearing in (4.3.1f). Therefore, it is easy to check that the algebra spanned byT (z), B+(z) and A(z) is a linear subalgebra of W2

3 . Moreover this algebra is preciselyW lin

3 provided the following identifications are taken into account:

A(z) ↔ J(z), T (z) ↔ T0(z), B+(z) ↔ W+(z), ~c ↔ ~c4

. (4.3.2)

Consequently one has the following chain of inclusions:

W23 ⊃ W lin

3 = W1+3 ⊃ W3. (4.3.3)

It is worth stressing that the inclusion of W lin3 into W2

3 is valid only at the classicallevel.

4.3.1 From W23 to W3

An interesting by-product of this process is the interpretation of the W3 algebra asa subalgebra of W2

3 . This fact can be nicely visualized from the particle mechanicsmodels formalism developed on chapter 3 as follows.

The action displayed in eq. (3.4.11) is invariant under the W23 algebra. The corre-


sponding equations of motion for the xi(t) variables are:

...x1 − B−

B− x1 +6~c

(T − 3

2A +

B−

B− A

)x1 +

+6~c

(9~c

B+B− +

(ddt− B−

B− +6~c

A

)(T − 1

2A +

6~c

A2

))x1 = 0, (4.3.4a)

...x2 − 2

B−

B− x2 +6~c

(T − 3

2A +

B−

B− A− ~c6

(ddt− B−

B−

)B−

B−

)x2 −

− 6~c

(9~c

B+B− +

(ddt− B−

B−

)(ddt− B−

B− +6~c

A

)A+

+6~c

A

(T − 1

2A +

6~c

A2

))x2 = 0. (4.3.4b)

After performing a change of variables, namely,

x1 ≡(B−) 1

3+γ

y+, x2 ≡(B−) 2

3+γ

y−, γ = constant, (4.3.5)

the equations of motion are written:

...y± + 3γ

B−

B− y± +

(24~c

T + 3γ

(ddt

+ γB−

B−

)B−

B−

)y± +

+

24~c

(±

√−5

2W3 +

12T + γ

B−

B− T

)+ γ

(ddt

+ γB−

B−

)2B−

B−

y± = 0, (4.3.6)

or:

(ddt

+ γB−

B−

)3

+24~c

T

(ddt

+ γB−

B−

)+

24~c

(±

√−5

2W3 +

12T

) y± = 0, (4.3.7)

where

T =~c4~c

T +

92~c

A2 − 32

ddt

(A− ~c

9B−

B−

)− 9

2~c

(A− ~c

9B−

B−

)2 , (4.3.8a)

√−5

2W3 =

~c4~c

(9~c

B+B− +

(12

ddt

+6~c

(A− ~c

9B−

B−

))(T +

92~c

A2

)+

+14

(ddt

+6~c

(A− ~c

9B−

B−

))2 (A− ~c

9B−

B−

)). (4.3.8b)


If the constant γ is chosen to be zero then equations (4.3.7) are actually invariantunder W3 transformations —they look like the equations of motion for the W3 particlemechanics model, see eq. (3.4.7):

...y± +

24~c

T y± +24~c

(±

√−5

2W3 +

12T

)y± = 0. (4.3.9)

It can be explicitly proved that T and W3, as defined in eq. (4.3.8), span a subalgebra ofW2

3 that is precisely W3. Therefore, by manipulating the equations of motion comingfrom particle mechanics models one can guess a relation between two W-algebras;moreover, the explicit expressions linking these two W-algebras can be obtained in thesame way.

The transformation of eq. (4.3.4) into eq. (4.3.9), which exhibit a smaller numberof symmetries, suggests that one can ‘gauge away’ some degrees of freedom in the W2

3

particle mechanics model, or, in other words, that the passage from the W23 algebra to

the W3 algebra can be alternatively regarded as a DS reduction, namely, a secondaryreduction. Accordingly, it can be easily checked from the gauge transformation lawsgiven in subsection 3.4.2 that the new variables y± transform under the action of onlytwo of the four W2

3 generators. The basic idea of the secondary reduction of a W-algebra is a generalization of the ordinary DS reduction for obtaining a W-algebraout of a Kac–Moody algebra, which was explained in sect. 2.1. Now the startingpoint, instead of a Kac–Moody algebra, is another W-algebra. After introducing aset of first-class constraints in this W-algebra, the corresponding gauge freedom isfixed by means of another set of constraints such that the whole set of constraintsbecomes second-class. The corresponding Dirac brackets define another W-algebra.It is clear that the starting W-algebra is a sort of intermediate state in the directreduction of the original Kac–Moody algebra to the second W-algebra. In this wayone can establish a number of links between the W-algebras that come from the sameKac–Moody algebra. The (classical) secondary reduction procedure was introduced inref. [67]. A detailed study of quantum secondary reductions has been performed in [75].The relationship existing between secondary reductions and the linearization methodwas observed simultaneously in ref. [75], [71] and [69] (see also ref. [76] and [72]).

In the case at hand, the set of second-class constraints to be introduced in W23 for

obtaining W3 is:

B−(z) = constant ≡ 4 (~c)2

9~c

√−5

2, (4.3.10a)

A(z) = 0. (4.3.10b)

An argument for choosing these constraints, besides its second-class character, is thatwhen they are implemented, eq. (4.3.4) are reduced to a form which is equivalent to


eq. (4.3.9). It can be explicitly tested that the corresponding (Dirac-bracket) reducedalgebra is precisely W3.

4.3.2 Non-local V(2)2,2 algebra

A different reduced algebras arises when only the second-class constraint (4.3.10b) isconsidered. The resulting Dirac brackets are:

T (z), T (w)

? =~c12

∂3wδ(z − w) + 2T (w) ∂wδ(z − w) + ∂T (w) δ(z − w), (4.3.11a)

T (z), B±(w)

? =32B±(w) ∂wδ(z − w) + ∂B±(w) δ(z − w), (4.3.11b)

B±(z), B±(w)

? =9~c

B±(z) B±(w) ε(z − w), (4.3.11c)

B+(z), B−(w)

? =~c9

∂2wδ(z − w) +

23T (w) δ(z − w)− 9

~cB+(z) B−(w) ε(z − w).

(4.3.11d)

The presence of the ε(z − w) function implies that this algebra is non-local. The non-locality is an indication of the fact that the introduction of the constraint (4.3.10b)does not produce an acceptable DS reduction in the sense that it is not the gauge-fixingof any gauge symmetry generated by a previous first-class constraint in the space offields. The existence of this reduction can also be viewed from the W2

3 -model equationsof motion. As it has been shown before (see eq. (4.3.6)), they can be written in a wayin which only three independent generators can be identified: the A(z) field does notappear. Furthermore, eq. (4.3.6) can be interpreted as a matrix differential equation.By setting γ = 1

3 and multipliying by B−, eq. (4.3.6) is written, schematically, in thefollowing way:

B− ...y + B− y + P1 y + P0 y = 0. (4.3.12)

Here y stands for y+ (or y−) and P0 and P1 are shorthands for special combinationsof the W2

3 fields that can be read off from eq. (4.3.6) and (4.3.8). Equation (4.3.12) isequivalent to a system of differential equations:

y − 1B− z = 0, (4.3.13a)

˙z + P1 y + P0 y = 0. (4.3.13b)

A first derivative on eq. (4.3.13b) and the change of variables

z(t) ≡ z(t) +12

(P1(t) +

∫ t

P0(t′) dt′)

y(t), (4.3.14)


transform the previous system in L2

(y(t)z(t)

)= 0, where:

L2 =

(1 0

0 1

)d2

dt2+

12B−

(P1 +

∫P0

) − 1B−

12

(P0 − P1

)+ 1

4B−

((∫P0

)2 − P 21

)1

2B−(P1 −

∫P0

)

.

(4.3.15)

The operator L2 is of the same order of L0 in eq. (4.2.2). As a consequence, the non-local algebra displayed in eq. (4.3.11) is a new V2,2-type algebra that will be namedV(2)

2,2 . 1 Again, simple manipulations of the equations of motion for a particle mechanicsmodel provide the matching between complicated non-linear algebras. In this case, bycomparing eq. (4.3.15) with eq. (4.2.2), one gets the expression of the V(2)

2,2 -algebra

generators T , B+, B− in terms of the V(0)2,2 -algebra basis V2, V

+, V −, V3:

T =2~c~c

(V2 +

~c6

ddt

(V +

V +

)), B− = −~c

121

V +, (4.3.16a)

B+ =83

(~c~c

)2[

13V2 V + + V2 V + + V3 V + − V3 V ++

+~c9

ddt

(V + d

dt

(V +

V +

))+~c81

(V +

)3

(V +)2

]. (4.3.16b)

The L2 operator is obtained from the L0 operator in eq. (4.2.2) when the four fieldsV2, V3, V + and V − are restricted through this second-class constraint:

det U0 +6~c

V + ∂2

(V2 − V3

V +

)= 0. (4.3.17)

The remaining constraint (4.3.10a) now becomes second-class and, therefore, thisalgebra can be further reduced. The non-local terms disappear and, of course, one getsthe W3 algebra.

4.3.3 The change of variables connecting W1+3 and W lin3

On the other hand, after introducing only the first-class constraint (4.3.10a), equations(4.3.8) are written:

T =~c4~c

(T − 3

2A

), (4.3.18a)

W3 = B+ +3~c

2 (~c)2

√−2

5

(AT +

32AA +

6~c

A3 +~c12

(˙T +

12A

)). (4.3.18b)

1This non-local algebra was not explicitly considered in ref. [64] and it has recently recognized as

the symmetry algebra for a system of parafermions interacting with an abelian Toda model in ref. [78],

where it has been denoted as V(1,1)3 .


These are precisely equations (4.1.14) defining the change of basis from W lin3 to W1+3

under the identifications given in eq. (4.3.2). This change of basis was originally found—at the quantum level— by considering all the possible pieces in the algebra of fieldshaving weight three with arbitrary coefficients [68]. These coefficients are determined byadjusting the corresponding OPE’s to those of W1+3. This procedure, although purelytechnical, is very cumbersome. An alternative —and simplest— way of obtaining thischange of basis is the method explained above: the form of the W1+3 generators canbe directly read from the embedding of Lax-type differential equations or, equivalently,equations of motion of the corresponding particle mechanics models.

Summing up, there are three ways of going from the W23 algebra to the W3 algebra

(see Figure 4.1). The direct path is a secondary reduction that discards two of theW2

3 generators. In the other two ways the same process is done in two steps. Byfirstly discarding the B− generator, the linearizing algebra of W3 is obtained as anintermediate state, whereas the elimination of the A field in the first step provides anon-local algebra.

W23

T , A, B+, B−

V(2)2,2

T , B+, B−W lin

3

T , A, B+= W1+3

T, J,W3

W3

T, W3

?

©©©©©©©©¼

©©©©©©©©

ZZ

ZZ

ZZ

ZZ~

ZZ

ZZ

ZZ

Z

@@

@@

@@R@@

©©©©

©©©©

©©©©¼©©

A=0

B−=const.

A=0B−=const.

Figure 4.1: The various ways of going from W23 to the W3 algebra. Dashed-line arrows

stand for inclusion relations between algebras. Solid-line arrows mean (secondary)reductions.

4.4 Secondary reductions of W(0,1)sp(4,R) 75

4.4 Secondary reductions of W (0,1)sp(4,R)

The equations of motion of the corresponding particle mechanics model (see eq. (3.3.5))can be written as:

A(

x+

x−

)= 0, (4.4.1a)

where

A ≡(

1 00 1

)d2

dt2+

12~c1

(−H 00 H

)ddt

+

+6~c1

(T1 − H + 6

~c1 H2 −2G−

−2G+ T1 + H + 6~c1 H2

). (4.4.1b)

One can express x− in terms of x+ by allowing division by G−:

x− =1

G−

(~c1

12x+ −H x+ +

12

(T1 − H +

6~c1

H2

)x+

). (4.4.2)

Then eq. (4.4.1a) becomes a fourth order differential equation which is precisely theequation of motion for the W(2, 4) model (see eq. (3.3.24)):

(d4

dt4+

60~c

Td2

dt2+

60~c

Tddt

+18~c

(12W4 + T +

18~c

T 2

))(G−)− 1

2 x+ = 0, (4.4.3)

with:

T =~c

5~c1

(T1 − 2 ˙H − 6

~c1H2

), (4.4.4a)

W4 =~c

90~c1

(¨T1 +

12

...H − 60

~c1G+G− +

485~c1

T 21 +

575~c1

˙H 2 +18~c1

H ¨H +1085~c1

T1˙H+

+30~c1

˙T1H +1224

5(~c1)2T1H

2 +1152

5(~c1)2H2 ˙H +

17285(~c1)3

H4

), (4.4.4b)

H ≡ H − ~c1

12G−

G− . (4.4.4c)

This shows that W(2, 4) is included in the W(0,1)sp(4,R) algebra. Furthermore, W(2, 4)

can be also interpreted as the resulting algebra after a secondary reduction of theW(0,1)

sp(4,R) algebra. Indeed, T and W4 have zero Poisson brackets with both H and G−.This implies that T and W4 are combinations which are invariant under the η and γ−transformations displayed in eq. (3.3.8) and (3.3.9) and, therefore, one can gauge fix


H and G− independently of T and W4. In other words, by setting the second-classconstraints

G− = constant, H = 0, (4.4.5)

the corresponding Dirac-bracket algebra is preciselyW(2, 4) in the classical form comingfrom the OPE’s in eq. (3.3.27).

4.4.1 Linearizing W(2, 4)

Exactly in the same way as in the W3 case one easily recognizes linear subalgebras ofthe W(0,1)

sp(4,R) algebra. In particular, the generators T1, H and G+ span one of theselinear subalgebras (see the OPE’s in eq. (3.3.12)). The relations of eq. (4.4.4) becomean invertible change of variables between T1, G

+,H and T,W4,H when the G−

field is frozen to a constant value, which is the first of the constraints in eq. (4.4.5).Furthermore, the W(2, 4)-algebra structure remains unchanged due to the fact that G−

has zero Poisson brackets with T and W4. Therefore, the algebra spanned by T,W4,His a W(2, 4, 1)-type algebra which includes the W(2, 4) algebra and is equivalent to thelinear algebra generated by T1, G

+,H. For this reason the latter algebra linearizesW(2, 4) and will be denoted as W(2, 4)lin. 2

Equations (4.4.4) when G− = constant (i.e. H = H), which are directly obtainedfrom the analysis of the equations of motion of the W(0,1)

sp(4,R) model, establish the linkbetween W(2, 4) and its linearizing algebra W(2, 4)lin. These relations were found inref. [70] (at the quantum level) by directly tuning all the coefficients of the change ofgenerators (eq. (4.4.4)) in order to get the W(2, 4) algebra.

4.4.2 Non-local V(1)2,2 algebra

After introducing the second-class constraint H = 0 in the W(0,1)sp(4,R) model, the first-

derivative term of the A operator (see eq. (4.4.1b)) disappears. Therefore, this Aoperator becomes a Lax operator of the family presented in sect. 4.2 and the corre-sponding Dirac-bracket algebra will be a V2,2-type algebra. This algebra is preciselyV(1)

2,2 (see eq. (4.2.5) and (4.2.7)) with the following identifications:

T1 ↔ V2, G± ↔ −V ∓, ~c1 ↔ ~c. (4.4.6)

As a consequence, the further reduction of the V(1)2,2 algebra with the (now second-class)

constraint V + = constant produces the W(2, 4) algebra.

2Note that the Poisson-bracket structure of W(2, 4)lin is the same as that of W(0,1)

sp(4,R), The relation-

ship between them is of inclusion, not of reduction, as long as G− = constant is a first-class constraint.

4.5 From W( 12,0)

sp(4,R) to W(0,1)sp(4,R) 77

4.5 From W ( 12 ,0)

sp(4,R) to W (0,1)sp(4,R)

The equations of motion for the W( 12,0)

sp(4,R) particle mechanics model (see eq. (3.3.34))can be written in a more convenient form as follows:

−[L]x1(

6~c2 D+

) 12 [L]x2

= B

x1(

6~c2 D+

)− 12x2

= 0, (4.5.1a)

where B is a matrix differential operator:

B ≡(

1 00 1

)d2

dt2+

(0 S−

S− 0

)ddt

+

+

(6~c2 T2 + (S−)2 − 3

~c2 (2S+ + DS−)3~c2 (2S+ + DS−) + S− − 3

~c2(D + 3

~c2(D2 + 4D+D−))

), (4.5.1b)

and

S± ≡(

32~c2

) 12

C± (D+

)± 12 , D ≡ D0 − ~c2

6D+

D+. (4.5.1c)

The six generators of the W( 12,0)

sp(4,R) algebra are arranged in the equations of motion sothat only four independent combinations of them explicitly appear. Therefore two of thesix original gauge transformations can be regarded as being simply ‘internal’ symmetriesthat leave invariant the definition of these four generators, and do not affect the mattervariables. The symmetry algebra of eq. (4.5.1a) is then a four-dimensional algebra,

which is a subset of W( 12,0)

sp(4,R). This four-dimensional algebra is precisely W(0,1)sp(4,R), as it

can be verified when rewriting the above equations of motion as

A(

z+

z−

)= 0, (4.5.2a)

after a mere redefinition of the xi variables:

z+ ≡ x1 +(

6~c2

D+

)− 12

x2, z− ≡ x1 −(

6~c2

D+

)− 12

x2. (4.5.2b)

Here A is the matrix differential operator appearing in eq.(4.4.1b), and, again, this pro-cedure directly provides the explicit expressions of the embedding ofW(0,1)

sp(4,R) (generated


by T1, H and G±) in W( 12,0)

sp(4,R):

T1 =~c1

2~c2

(T2 +

32~c2

((D0

)2 −D2)− 1

2D

), (4.5.3a)

H = −~c1

12S−, (4.5.3b)

G± =~c1

4~c2

(±2S+ − T2 − 1

2D ± S−D +

~c2

6

(±S− − (

S−)2

)−

− 32~c2

((D0

)2 + D2 + 8 D+D−))

. (4.5.3c)

Alternatively, the W(0,1)sp(4,R) algebra can be regarded as the Dirac-bracket algebra arising

when constraining the W( 12,0)

sp(4,R) algebra with the following second-class constraints:

D+ = constant, D0 = 0. (4.5.4)

A summary of the relations found between the W-algebras that are related withsp(4,R) is displayed in Figure 4.2.

4.5 From W( 12,0)

sp(4,R) to W(0,1)sp(4,R) 79

W( 12,0)

sp(4,R)

T2, D0, D+, D−, C+, C−

W(0,1)sp(4,R)

T1,H, G+, G−

V(1)2,2

V2, V+, V −

W(2, 4)lin

T1, H, G+= W(2, 4, 1)

T,H, W4

W(2, 4)

T, W4

?

?

©©©©©©©©¼

©©©©©©©©

ZZ

ZZ

ZZ

ZZ~

ZZ

ZZ

ZZ

Z

@@

@@

@@R@@

©©©©

©©©©

©©©©¼©©

D0=0D+= const.

H=0

V +=const.

H=0G−=const.

Figure 4.2: Links between theW-algebras related with sp(4,R). The meaning of arrowsis the same as in Fig. 4.1.

Chapter 5

W3 Geometry

The conformal algebra is deeply associated with geometrical concepts: it can be realizedas the group of coordinate transformations (or reparametrizations) on a manifold that,when considering only the holomorphic part of the conformal algebra, can be consideredas one-dimensional —see sect. 1.1. Fields transforming under the conformal algebracan be translated into geometrical language by identifying them as sections in a certainfiber bundle. As W-algebras are extensions of the conformal algebra, the search forgeometrical structures behind W-algebras is then a natural problem.

The setting of aW-geometry is, however, a difficult task. The presence of nonlineari-ties and higher-order derivatives in the definition ofW-algebras is the main obstruction.Moreover, the existence of an infinite number of W-algebras, and the lack of a completesystematization of them, imply the existence of, a priori, several W-geometries. Thenatural starting point is the minimal extension of the conformal algebra, i.e., to try todefine a W3 geometry.

However, even when dealing only with the W3 algebra, one should talk about W-geometries due to the presence in the literature of several proposals, hardly ever equiv-alent, for solving the problem. Here is a schematic and simplified list of the mainapproaches:

• The formulation of a geometry based on tensor densities of rank n ≥ 2 generalizingthe role played by the ordinary metric gµν [79]. These tensor densities wouldbe the building blocks for ‘covariantizing’ the light-cone gauge W-string actionsintroduced in ref. [20] (see sect. 1.4).

• The interpretation of W-transformations as deformations of flags defined on jetbundles [80] [81]. This approach is a sequel of the (light-cone) W-gravity devel-opments [22] [23]. It starts from a geometrical reading of the Ward identities forthese induced W-gravity theories (that can be written as zero-curvature condi-tions [82]) by identfying generalized Beltrami differentials. (See also ref. [83].)

81

82 W3 Geometry

• The construction of the moduli space for W-gravity theories [84], also as a con-sequence of the study of induced W-gravities. Progress in this direction has beenmade in ref. [85] by considering a class of generalized Teichmuller spaces. Otherworks that relate W-symmetries with the geometry of higher-genus Riemann sur-faces are [86].

• The W-particle construction of ref. [87], that links W-geometry with the extrinsiccurvature of some curves.

• The interpretation of W-geometry (in the light-cone gauge) as the extrinsic ge-ometry of a certain class of affine curves and surfaces embedded in a higher-dimensional space [88].

• The study of embeddings of surfaces in a (higher-dimensional) Kahler mani-fold [89]. This approach is somehow related to the previous one but it startsfrom the formulation of W-gravity in the conformal gauge and, therefore, Todamodels are an essential ingredient in this interpretation.

The main results of the proposal for a W3 geometry that will be immediately developedwere published in [90] (see also [91]). The basic idea is very simple: defining a geo-metrical framework for the W3 algebra is equivalent to finding a base-space realizationof W3 transformations. This natural generalization of the geometrical character of theconformal algebra is achieved by extending the base space to an auxiliary space, witha precise relation with the original one-dimensional base space, such that in this new‘W-space’ all the W-transformations —not only the ordinary reparametrizations sub-set of them— have a natural implementation as transformations defined in the space.Then, the next step will be to endow all the W-objects with some clear geometricalcharacter inherited from the basic structure defined in that extended base space. Thisapproach, therefore, should be included in the last points of the list presented aboveand shares to some extent the philosophy of ref. [92].

5.1 Extension of the base space

The aim of this section is to motivate theW-geometry proposal. The strategy is twofold.Firstly, it is natural to try to generalize the W2 framework to the W3 case. Secondly,the familiar idea of the introduction of a superspace in the context of supersymmetrictheories will be a fruitful guideline in the process. Thus, in the first part of this sectionthe W2 case is reviewed. Then, the straightforward generalization of its basic featuresto the W3 setting is presented, showing how a two-dimensional extended base spacenaturally arises. Finally, this idea of an extension of the base space is exemplified inthe context of a particle mechanics model [61] exhibiting a supersymmetry as well as

5.1 Extension of the base space 83

invariance under reparametrizations, in order to motivate the developments of the nextsection.

5.1.1 W2

The equation of motion for the lagrangian (3.2.6) is: 1

(d2

dt2+ T (t)

)x(t) = 0. (5.1.1)

The symmetry of the lagrangian implies that this differential equation is invariantunder reparametrization (W2) transformations:

δx = ε x− 12

ε x, (5.1.2a)

δT = ε T + 2 ε T +12

...ε , (5.1.2b)

or, in finite form (f(t) ' t + ε(t)):

x′(t) = f−1/2 x(f(t)), (5.1.3a)

T ′(t) = f2 T (f(t)) +12

( ...f

f− 3

2f2

f2

). (5.1.3b)

Invariant under W2 here means that

1. if x(t) is a solution of (5.1.1) then x′(t) is also a solution of the same differentialequation with T ′(t) instead of T (t), and

2. two successive transformations (5.1.3) labelled with f and g can be grouped as onetransformation with g f , i.e. the composition law is that of reparametrizations.

Every second order differential equation is equivalent to a first order matricial differ-ential equation:

x + a1(t)x + a0(t)x(t) = 0 ⇔ ~x = M ~x, (5.1.4)

where ~x =

(x

y

), M =

(0 1−a0 −a1

).

The general solution of a differential equation like (5.1.4) is expressed in terms of twoarbitrary functions s0(t) and s1(t):

~x(t) = U ~x(0), (5.1.5)

with U =

(s0 s1

s0 s1

), ~x(0) =

(a

b

)= constant.

1The link with sect. 3.2 is established with the identification T ≡ 6~c

λ and a rescaling of the Poisson-

bracket structure: ·, ·(new) ≡ ~c6·, ·(old). For practical purposes this is equivalent to setting ~c ≡ 6.

84 W3 Geometry

The M matrix is M = UU−1 and then:

a0 =s1s0 − s0s1

Wr(s0, s1), a1 = − d

dtln Wr(s0, s1), (5.1.6)

where Wr(s0, s1) = s0s1 − s1s0 = detU is the wronskian of s0 and s1. Therefore, inorder to reproduce the particular differential equation (5.1.1), the wronskian Wr(s0, s1)must be a constant —that will be fixed to be the unity—, and T (t) has to be expressedas a function of s0 and s1:

T (t) = s1s0 − s0s1. (5.1.7)

The condition Wr(s0, s1) = detU = 1 reduces to one the number of arbitraryfunctions, in accordance with the fact that (5.1.1) —or, more precisely, the lagrangianthat has (5.1.1) as its equation of motion— exhibits only one gauge invariance and,therefore, the general solution of that differential equation must depend on only onearbitrary function, k(t). Taking this fact into account one can choose, without loss ofgenerality, the following parametrization for the si functions:

s0(t) = 1 ·Wr(1, k)−1/2 = k−1/2, (5.1.8a)

s1(t) = k(t) ·Wr(1, k)−1/2 = k−1/2k, (5.1.8b)

giving:

x(t) = k−1/2 (a + b k(t)) and T (t) =12

( ...k

k− 3

2k2

k2

). (5.1.9)

The gauge transformations (5.1.3) are easily recovered after prescribing the followingsimple transformation law for the arbitrary function k(t):

k′ = k f (5.1.10)

All these transformation laws can be interpreted as being induced by base-space trans-formations t → t′ parametrized by the f diffeomorphism in this way:

t = f(t′) (5.1.11)

Indeed, the object k behaves as a scalar function under base-space transformations, x

is a scalar density of weight −1/2 and T is the well-known object transforming as aweight-2 scalar density with the Schwarzian derivative extension. In this sense one saysthat the W2 transformations have a clear geometrical meaning.

There is a particular set of reparametrization (W2) transformations which will bereferred to as global transformations. They are:

f(t) =a t + b

c t + d, a, b, c, d = constants, ad− bc = 1. (5.1.12)

These transformations


1. cancel the Schwarzian derivative:

S(t) ≡...f

f− 3

2f2

f2= −2f1/2 d2

dt2f−1/2, (5.1.13)

or, in other words, preserve the condition T (t) = 0 (see eq. (5.1.3b)); and

2. are a realization of the SL(2,R) group on the projective space RP 1 ∼= S1:

t ≡ y1

y0, with

(y′0y′1

)=

(d c

b a

)(y0

y1

). (5.1.14)

Infinitesimally:

δt = −ε(gl)(t) with ε(gl)(t) ≡ ε1 + ε2 t + ε3 t2. (5.1.15)

The name ‘global transformations’ precisely reflects the fact that these are the onlyglobally defined transformations in the entire projective RP 1 space.

5.1.2 W3

The generalization of the equation (5.1.1) to W3 is the following differential equation(see eq. (3.4.7)): 2

(d3

dt3+ T (t)

ddt

+ W (t) +12T

)x(t) = 0, (5.1.16)

which is invariant under W3 transformations. The reparametrization sector is givenby:

δx = ε x− ε x, (5.1.17a)

δT = ε T + 2 ε T + 2...ε , (5.1.17b)

δW = ε W + 3 εW. (5.1.17c)

These transformations can be integrated:

x′(t) = f−1 x(f(t)), (5.1.18a)

T ′(t) = f2 T (f(t)) + 2

( ...f

f− 3

2f2

f2

), (5.1.18b)

W ′(t) = f3 W (f(t)). (5.1.18c)

2Here, for simplicity, W ≡ ±q− 5

2W3 and ~c ≡ 24 (see the footnote on page 83).

86 W3 Geometry

The specific W3 transformations are, infinitesimally:

δx = ρ(x + 2

3 T x)− 1

2 ρ x + 16 ρ x, (5.1.19a)

δT = 2 ρ W + 3 ρW, (5.1.19b)

δW = −ρ(

16

...T + 2

3 T T)− ρ

(34 T + 2

3 T 2)− 5

4 ρ T − 56

...ρ T − 1

6 ρ(5). (5.1.19c)

The equation (5.1.16) is equivalent to:

~x = M ~x, ~x =

x

y1

y2

, M =

0 1 00 0 1

−(W + 1

2 T)

−T 0

. (5.1.20)

The solution is:

~x(t) = U ~x(0), U =

s0 s1 s2

s0 s1 s2

s0 s1 s2

, ~x(0) =

a

b

c

= constant. (5.1.21)

As in the sl(2,R) case, the absence of the term with a next-to-leading order derivativeof x implies:

Wr(s0, s1, s2) = constant ≡ 1, (5.1.22)

and the si functions are expressed in terms of two arbitrary functions, k1(t) and k2(t),carrying the two gauge invariances of (5.1.16):

s0(t) = 1 ·Wr(1, k1, k2)−1/3 =(k1k2 − k2k1

)−1/3≡ K(t)−1/3, (5.1.23a)

s1(t) = k1(t) ·Wr(1, k1, k2)−1/3 = K−1/3 k1, (5.1.23b)

s2(t) = k2(t) ·Wr(1, k1, k2)−1/3 = K−1/3 k2. (5.1.23c)

The general solution and the coefficients of the differential equation (5.1.16) are thenparametrized in the following way:

x(t) = K(t)−1/3 (a + b k1(t) + c k2(t)) , (5.1.24a)

T (t) =K

K− 4

3K2

K2+

1K

(k1

...k2 − k2

...k1

), (5.1.24b)

W (t) =56

KK

K2−

...K

6K+

56

K

K2

(k1

...k2 − k2

...k1

)− 20

27K3

K3+

12K

(....k 1k2 −

....k 2k1

).

(5.1.24c)

The object k(t) ≡ K(t)1/3x(t) satisfies the following differential equation:(

d3

dt3+ L(t)

d2

dt2+ M(t)

ddt

)k(t) = 0. (5.1.25)


The arbitrary functions k1(t) and k2(t), together with an arbitrary constant, can beregarded as the three solutions of this differential equation (see eq. (5.1.24a)):

k(t) = a + b k1(t) + c k2(t). (5.1.26)

The arbitrary fields L(t) and M(t) can also be expressed in terms of the ki(t) functions:

L(t) = −K

K, M(t) =

1K

(k1

...k2 − k2

...k1

), (5.1.27)

and there is a non-invertible relation with T (t) and W (t):

T (t) = M − 13 L2 − L, (5.1.28a)

W (t) = −12 M + 1

3 L L + 16 L + 2

27 L3 − 13 LM. (5.1.28b)

As in the W2 case one can get the reparametrization transformations of the x,T and W objects (5.1.18) from the following simple transformation laws for the ki

functions:

δki = ε ki, ⇔ k′i = ki f, i = 1, 2. (5.1.29)

The specificW3 transformations are more cumbersome. One can check from eq. (5.1.19)that the following infinitesimal transformation laws are the realization of the specificW3 transformations on k1 and k2:

δki = ρ

(ki − 2

3K

Kki

)− 1

2ρ ki, i = 1, 2. (5.1.30)

These transformation laws can also be obtained by computing the gauge symmetriesof the differential equation (5.1.25):

δk = ε k + ρ

(k +

23

L k

)− 1

2ρ k, (5.1.31a)

δL = ε L + ε L− 3 ε + ρ

(2M − L− 2

3LL

)+ ρ

(2M − 3

2L− 1

3L2

)− 1

2ρ L +

12...ρ ,

(5.1.31b)

δM = ε M + 2 εM − ε L− ...ε + ρ

(M − 2

3LM +

23

LM − 23

LL− 23

...L

)+

+ ρ

(52

M +13

LM − 43

LL− 2L

)+ ρ

(2M − 2

3L2 − 2L

)− 1

6...ρ L +

12

....ρ .

(5.1.31c)

The reparametrization transformations rely on the geometry of the base space.Every change of coordinate defined on the base space (t → t′, parametrized by (5.1.11))

88 W3 Geometry

induces a transformation on the fundamental objects x, T and W via (5.1.29). From thispoint of view, the specific W3 transformations (5.1.30) are essentialy different: at firstsight they are not induced from any transformation or change of coordinates defined onthe base space. Intuitively the reason is clear: the reparametrization transformations(5.1.11) exhaust all possible transformations on the base space and therefore there is noroom for treating all the W3 transformations on the same footing. A natural way outfor this situation is to extend the dimensionality of the base space. The forthcominggeneralization of theW2 global transformations (5.1.12) to theW3 case provides furthersupport to this idea.

The sl(3,R) algebra is related to W3 as the sl(2,R) algebra is connected with W2.Moreover, although the complete finite form of a Schwarzian-type derivative is unknownfor theW3 case, one can check from theW3 transformation rules (see equations (5.1.18)and (5.1.19)) that the preservation of the conditions T = 0 and W = 0 involves aneight-parameter group.

Therefore, when trying to define a kind of global transformations for the W3 situ-ation one is led to consider SL(3,R) as the group of transformations. Consequently, anatural generalization of the W2 case points to the two-dimensional projective spaceRP 2 as the base space where SL(3,R) acts.

Let y0, y1, y2 (y0 6= 0) be a set of (local) coordinates for R3 and let ξ1, ξ2 be theassociated coordinates for RP 2 with:

ξ1 =y1

y0, ξ2 =

y2

y0. (5.1.32)

The defining action of SL(3,R) on R3

y′0y′1y′2

= A

y0

y1

y2

, detA = 1, (5.1.33)

induces the SL(3,R) projective transformations on RP 2. Infinitesimally:

δy0

δy1

δy2

=

ε0 − 13 ρ0 2 ε+ − ρ+1 −4 ρ+2

−ε− + 12 ρ−1

23 ρ0 2 ε+ + ρ+1

−ρ−2 −ε− − 12 ρ−1 −ε0 − 1

3 ρ0

y0

y1

y2

, (5.1.34)

yielding:

δξi ≡ δεξi + δρξi,

δεξ1 = −ε− − ε0 ξ1 + 2 ε+(ξ2 − ξ2

1

), (5.1.35a)

δρξ1 = 12 ρ−1 + ρ0 ξ1 + ρ+1

(ξ2 + ξ2

1

)+ 4 ρ+2 ξ1ξ2, (5.1.35b)

δεξ2 = −ε− ξ1 − 2 ε0 ξ2 − 2 ε+ξ1ξ2, (5.1.35c)

δρξ2 = −ρ−2 − 12 ρ−1 ξ1 + ρ+1 ξ1ξ2 + 4 ρ+2 ξ2

2 . (5.1.35d)


The parametrization of the sl(3,R) matrix in (5.1.34) has been chosen in such a way thatthe ε+, ε0 and ε− constants live in the principal sl(2,R) embedding of sl(3,R). Likewise,the subindex of the ρi constants reflects the gradation level of the corresponding sl(3,R)element with respect to the central element of that sl(2,R) embedding (see eq. (2.2.7)and appendix A for sl(3,R) conventions). Intuitively, those εi constants should span thesl(2,R) global transformations (in RP 2) belonging to the reparametrization subgroupof the W3 transformations because of the special role played by the sl(2,R) embeddingin generating the W algebra, as stated on sect. 2.2. Consequently, the ρi constantsshould define the global specific W3 transformations when acting on RP 2.

Within this two-dimensional image both the (global) reparametrizations and the(global) specific W3 transformations are implemented at the level of the base space.Of course one should be able to find a way of going back to the usual one-dimensionaldescription —a kind of truncation F (ξ1, ξ2) = 0. After performing this truncation onlythe reparametrization transformations (δε) retain a clear base-space interpretation and,therefore, only for these reparametrization transformations it is reasonable to imposethe stability of the truncation: δεF (ξ1, ξ2)|F=0 = 0. This gives:

(ξ2 − 1

2ξ21

)∣∣∣∣F=0

= 0, (5.1.36)

so that, after defining

t ≡ ξ1, (5.1.37a)

z ≡ ξ2 − 12

ξ21 , (5.1.37b)

the truncation can be taken to be:

z = 0. (5.1.38)

Thus, the global W3 transformations when acting on the original one-dimensional basespace are:

δt|z=0= −ε(gl)(t) +12

ρ(gl)(t), (5.1.39)

where:

ε(gl)(t) ≡ ε− + ε0 t + ε+ t2, (5.1.40a)

ρ(gl)(t) ≡ ρ−2 + ρ−1 t + ρ0 t2 + ρ+1 t3 + ρ+2 t4. (5.1.40b)

Also one can explicitly show that the specific W3 transformations do not preserve thetruncation:

δz|z=0= −ρ(gl)(t). (5.1.41)

90 W3 Geometry

On the other hand, when T = 0 and W = 0, a trivial solution of eq. (5.1.16) isgiven by:

k1(t) = t, k2(t) =12t2. (5.1.42)

The ε and ρ infinitesimal parameters appearing in eq. (5.1.40) define the transformationspreserving the values T = 0 and W = 0. These global transformations, when appliedto the values of ki given above, yield (see eq. (5.1.29) and (5.1.30)):

δk1 = ε(gl) − 12ρ(gl), δk2 =

(ε(gl) − 1

2ρ(gl)

)t + ρ(gl). (5.1.43)

This is precisely the behaviour (up to a sign) of the RP 2 coordinates ξ1 and ξ2, re-spectively, under the truncated global W3 transformations (see eq. (5.1.37), (5.1.39)and (5.1.41)). Therefore, the ki(t) functions are somehow fundamental objects deeplyconnected with the geometrical structure of RP 2. This key idea will be important inthe forthcoming developments. As a by-product, the analogy of eq. (5.1.43) with theξi global (and truncated) transformations leads to a characterization of the set of ki(t)functions that preserve the T = 0 and W = 0 conditions:

ki(t) =ai0 + ai1 t + ai2

t2

2

a00 + a01 t + a02t2

2

, i = 1, 2 (5.1.44)

where amn = const., m, n = 0, 1, 2 and det(amn) = 1.

Up to now several heuristic arguments have been presented for justifying the pos-sibility of constructing a two-dimensional auxiliary ‘W3-space’, namely RP 2, where ageometry for the W3 transformations may be defined. Before going on, however, itmay be instructive to examine a more familiar situation where the base space is alsoextended in order to exhibit a geometrical description.

5.1.3 Super-Virasoro

A supersymmetric generalization of the particle mechanical models presented in chap-ter 3 has been considered in [61]. The simplest of these models is obtained after intro-ducing a set of quadratic constraints —spanning the osp(1|2) algebra— defined in thephase space of one even (x(t)) and one odd (ψ(t)) variables. The resulting equationsof motion, namely,

d2x

dt2− TB(t) x(t) + TF (t) ψ(t) = 0, (5.1.45a)

dψ

dt− TF (t)x(t) = 0, (5.1.45b)

are invariant under the (N = 1) super-Virasoro algebra:


• Reparametrization transformations:

δεx = ε x− 12 ε x, (5.1.46a)

δεψ = ε ψ, (5.1.46b)

δεTB = ε TB + 2 ε TB − 12

...ε , (5.1.46c)

δεTF = ε TF + 32 ε TF , (5.1.46d)

• Supersymmetric transformations:

δωx = ω ψ, (5.1.47a)

δωψ = ω x− ω x, (5.1.47b)

δωTB = ω TF + 3 ω TF , (5.1.47c)

δωTF = ω TB − ω. (5.1.47d)

The general solution of the equations (5.1.45) is expressed in terms of two arbitraryfunctions, namely, k(t) (even) and λ(t) (odd), and two even (a, b) and one odd (γ)constants:

x(t) =(k + λλ

)−1/2(a + b k(t) + γ λ(t)) , (5.1.48a)

ψ(t) =λ

k(a + b k(t) + γ λ(t))− b λ(t) + γ, (5.1.48b)

TB(t) = −12

( ...k

k− 3

2k2

k2

)+ λλ

...k

2k2− λ

...λ

12k− 3

2λλ

k2

k3− 3

2λλ

1k

+32λλ

k

k2, (5.1.48c)

TF (t) = k−1/2

(λ− λk

k+

λλλ

2k

). (5.1.48d)

This situation is somehow similar to the W3 case because the finite reparametrizationtransformations (5.1.46) are easily realized at the level of the k and λ functions:

k′ = k f, λ′ = λ f, (5.1.49)

but the supersymmetric ones are not:

δωk = −ω λ k1/2, δωλ = −ω(k + λλ

)1/2. (5.1.50)

Furthermore the relevant (super)group, now OSP(1|2), also acts projectively on a two-dimensional (super)space. Choosing a local set of coordinates ξ ≡ y1

y0and θ ≡ χ

y0(even

and odd, respectively), the natural action of osp(1|2), namely,

δy0

δy1

δχ

=

12ε0 ε+ α2

−ε− −12ε0 −α1

−α1 −α2 0

y0

y1

χ

, (5.1.51)

92 W3 Geometry

defines these global (infinitesimal) transformations:

δξ = −ε− − ε0 ξ − ε+ ξ2 − α1 θ − α2 ξ θ, (5.1.52a)

δθ = −12ε0 θ − ε+ ξ θ − α1 − α2 ξ. (5.1.52b)

Similarly to the sl(3,R) case the εi parameters label the transformations generatedby the sl(2,R) embedding in osp(1|2). The only truncation that is stable under theseglobal reparametrization transformations is:

θ = 0, (5.1.53)

so that the one-dimensional coordinate t can be identified with ξ. After the truncationthe infinitesimal global transformations can be written:

δt|θ=0= −ε(gl)(t), δθ|θ=0= −ω(gl)(t), (5.1.54)

where ε(gl)(t) ≡ ε− + ε0 t + ε+ t2, and ω(gl)(t) ≡ α1 + α2 t.

The next step is to characterize general (or local) transformations in the superspacesuch that their projection to the original one-dimensional space provides the expres-sions (5.1.46) and (5.1.47), taking advantage of the fact that now this two-dimensionalsuperspace is ‘wide enough’ to allow a base-space realization of both the reparametriza-tion and the supersymmetry transformations. However, as will be shown later, itis ‘too much wide’ in the sense that only a subset of the two-dimensional super-diffeomorphisms will suffice for describing all those one-dimensional transformations.A general change of coordinates in the superspace, (t, θ) → (t′, θ′), can be parametrizedby:

t = f(t′, θ′), θ = ϕ(t′, θ′), (5.1.55)

or, infinitesimally,

δt = −ε(t, θ), δθ = −ω(t, θ). (5.1.56)

A straightforward generalization of the relation (5.1.54) to general transformations is:

ε(t, θ)|θ=0= ε(t), ω(t, θ)|θ=0= ω(t), (5.1.57)

where ε(t) and ω(t) are the arbitrary infinitesimal parameters appearing in eq. (5.1.46)and (5.1.47).

Similarly, one can define the objects x(t, θ), ψ(t, θ), TB(t, θ) and TF (t, θ) havingan obvious relationship with their counterparts living in the original one-dimensionalspace after the truncation (5.1.53). They are assumed to be solutions of a system ofdifferential equations of the type (5.1.45):

∂2t x(t, θ)− TB(t, θ) x(t, θ) + TF (t, θ) ψ(t, θ) = 0, (5.1.58a)

∂tψ(t, θ)− TF (t, θ) x(t, θ) = 0. (5.1.58b)


Therefore they can be written in terms of two functions k(t, θ) and λ(t, θ) —which arethe superspace generalization, in the same sense as before, of k(t) and λ(t)— througha set of relations which are the same as (5.1.48) but for the tilded objects. Inspiredby the natural realization of the reparametrization transformations on k(t) and λ(t)(see eq. (5.1.49)) one is tempted to treat in the same way also the supersymmetrytransformations in the superspace by imposing: 3

δk = ε(t, θ) ∂tk(t, θ) + ω(t, θ) ∂θk(t, θ), δλ = ε(t, θ) ∂tλ(t, θ) + ω(t, θ) ∂θλ(t, θ).(5.1.59)

Then expressions (5.1.50) dictate:

∂θk(t, θ) = −λ k1/2, ∂θλ(t, θ) = −(k + λλ

)1/2, (5.1.60)

so that k(t, θ) and λ(t, θ) are completely determined in terms of k(t) and λ(t). Theseequations will be denoted as extension equations because they prescribe the way ofextending the original objects to the two-dimensional space. From relations (5.1.48)one can easily compute the extension equations for the x(t), ψ(t), TB(t) and TF (t)objects:

∂θx(t, θ) = ψ(t), ∂θTF (t, θ) = TB(t), (5.1.61a)

∂θψ(t, θ) = x(t), ∂θTB(t, θ) = TF (t), (5.1.61b)

and check that they are compatible with the system (5.1.58). The previous expressionscan be rewritten as:

x(t, θ) = x(t) + θ ψ(t), TF (t, θ) = TF (t) + θ TB(t), (5.1.62a)

ψ = Dx, TB = DTF , (5.1.62b)

where D ≡ ∂θ + θ∂t is the superderivative. Then both systems (5.1.45) and (5.1.58)with the extension equations can be encoded in one equation involving the superfieldsx and T :

(D3 − TF (t, θ)

)x(t, θ) = 0. (5.1.63)

Therefore this differential equation is invariant (see eq. (5.1.46) and (5.1.47)) under thefollowing infinitesimal transformations:

δx = ε(t, θ) ∂tx + ω(t, θ) ∂θx− (∂tε− ∂θω) x(t, θ) (5.1.64a)

δTF = ε(t, θ) ∂tTF + ω(t, θ) ∂θTF + 3 (∂tε− ∂θω) TF (t, θ)− ∂2t ω(t, θ), (5.1.64b)

where ε(t, θ) = ε(t) + θ ω(t), and ω(t, θ) = ω(t) +12

θ ε(t). (5.1.64c)

3All derivatives are understood to be left derivatives.

94 W3 Geometry

An important issue to stress at this point is the fact that not every diffeomorphismof the type (5.1.56) is allowed. The infinitesimal quantities ε(t, θ) and ω(t, θ) haveto satisfy the restriction (5.1.64c), i.e., whereas a general super-diffeomorphism willdepend on four arbitrary functions of t, the special type of diffeomorphisms that makethe differential equation (5.1.63) invariant depend on only two functions of t. It isworth mentioning here that the restriction (5.1.64c) can also be obtained by imposinginvariance of equations (5.1.60) under the infinitesimal transformations (5.1.59) andtaking into acount relations (5.1.57). The finite form of these special diffeomorphismsis:

f(t, θ) = f(t) + θ ϕ(t) f1/2, ϕ(t, θ) = ϕ(t) + θ(f + ϕϕ

)1/2, (5.1.65)

and they satisfy:

J(t, θ) = Dϕ(t, θ), Df(t, θ) = ϕ(t, θ) Dϕ(t, θ), (5.1.66)

where J is the superdeterminant of the Jacobian matrix corresponding to the base-spacetransformation (t, θ) → (t′, θ′) defined by the superdiffeomorphism in (5.1.55):

J(t′, θ′) ≡ sdet

(∂t′ f(t′, θ′) −∂θ′ f(t′, θ′)∂t′ϕ(t′, θ′) ∂θ′ϕ(t′, θ′)

). (5.1.67)

Those special super-diffeomorphisms constitute the so-called superconformal transfor-mations. It is easily shown that they are a closed subgroup of the general super-diffeomorphisms and can be characterized by the property that under a superconformaltransformation the superderivative D transforms homogeneously (see eq. (5.1.66)):

D′ = J(t′, θ′)D. (5.1.68)

The infinitesimal transformations (5.1.64) evidence that in the superspace it is possibleto assign a clear geometrical status to every object. Indeed, x(t, θ) behaves as a scalardensity of weight minus one under base-space diffeomorphisms of the type (5.1.56)whereas TF (t, θ) is a weight-three scalar density but with an inhomogeneous term —the super-Schwarzian derivative S(t, θ). In finite form the transformations read:

x′(t, θ) = J(t, θ)−1 x(f(t, θ), ϕ(t, θ)), (5.1.69a)

T ′F (t, θ) = J(t, θ)3 TF (f(t, θ), ϕ(t, θ))− S(t, θ), (5.1.69b)

By demanding invariance of (5.1.63) under (5.1.69) one gets the well-known expressionfor the super-Schwarzian derivative [93]:

S(t, θ) =D4ϕ

Dϕ− 2

D3ϕ

Dϕ

D2ϕ

Dϕ, (5.1.70)


or, in a more compact form:

S(t, θ) = −J(t, θ)D3J(t, θ)−1. (5.1.71)

After the truncation (5.1.53) one essentially gets the ordinary Schwarzian derivative.

The global transformations (5.1.52) correspond to the assignations:

f(t) ≡ a t + b

c t + d, ϕ(t) ≡ α2 t + α1

c t + d, ad− bc = 1, (5.1.72)

that satisfy:

J(t, θ) =1 + 1

2α1α2

c t + d+ θ

α2 d− α1 c

(c t + d)2. (5.1.73)

A straightforward computation shows that indeed these transformations cancel thesuper-Schwarzian derivative. Conversely, from the knowledge of the superdeterminantof the Jacobian corresponding to the global transformations (5.1.73) one could guessthe expression of the super-Schwarzian derivative. For doing this one realizes that thepolynomial expression of lowest degree in t and θ that can be constructed out of J(t, θ)is J(t, θ)−1:

J(t, θ)−1 =(

1− 12α1α2

)(c t + d) + θ (α1 c− α2 d) , (5.1.74)

From this fact, and taking into account that the super-Schwarzian is cancelled only bythe global transformations, one could directly state that:

S(t, θ) ∼ O J(t, θ)−1, (5.1.75)

where O is a second-order differential operator. The simplest guess for it is O = D3.

Note that this direct method for obtaining the form of the Schwarzian derivativeout of the global transformations [94] also works for the sl(2,R) situation. In that casethe Jacobian of the global transformations (5.1.12) is:

J(t) =df

dt=

1(c t + d)2

, (5.1.76)

resulting that the polynomial expression of lowest degree connected with J(t) is:

J(t)−1/2 = c t + d. (5.1.77)

This can be cancelled with a second-order differential equation. Therefore:

S(t) ∼ d2

dt2f−1/2, (5.1.78)

96 W3 Geometry

in agreement with equation (5.1.13).

The proposal for a W-geometry that will be immediately developed shares a highnumber of features with the superspace formulation of supersymmetries. This is thereason why the super-Virasoro case has been treated in so much detail. In both casesthe geometrical description arises after extending the base space. However, whereasthis is an odd extension in the case of the superspace, the W-space will be constructedby adding new even dimensions to the original base space. This does not mean thatthe new coordinates should be considered as indistinguishable from the original one. Infact, the role of the parity (odd/even) in the superspace will now be played by a kind ofgradation between the coordinates, gradation that ultimately will be inherited from thegradation of the associated vacuum preserving algebra induced by the correspondingsl(2,R) embedding. In this sense the geometrical description for the W-symmetriesthat will be developed in the next section nicely relies on the algebraic construction ofthe W-algebras.

5.2 W3 diffeomorphisms in the W3 space

Having in mind the arguments presented in the previous section —especially the su-perspace construction—, the starting point shall be the definition of a set of RP 2

objects x(t, z), T (t, z), W (t, z) and ki(t, z) (i = 1, 2), that are subject to the followingconditions:

1. they are equal to the corresponding untilded objects after the truncation (5.1.38),

2. the objects x, T and W are connected through a differential equation that gen-eralizes (5.1.16) to RP 2:

(∂3

t + T (t, z) ∂t + W (t, z) +12

∂tT (t, z))

x(t, z) = 0, (5.2.1)

3. the relation of x, T and W with ki(t, z) is established by means of a set ofequations formally equivalent to (5.1.24) but for the tilded objects.

The last sentence implies that eq. (5.2.1) is associated with a generalization of (5.1.25):(∂3

t + L(t, z) ∂2t + M(t, z) ∂t

)k(t, z) = 0, (5.2.2)

where:

k(t, z) ≡ K(t, z)1/3 x(t, z) = a + b k1(t, z) + c k2(t, z). (5.2.3)

In order to define the actual dependence on z of all the objects it suffices to imposea particular z-dependence for ki(t, z). As in the supersymmetric case the larger number

5.2 W3 diffeomorphisms in the W3 space 97

of dimensions of the space will be exploited in order to prescribe a scalar-type behaviourfor the ki(t, z) quantities, i.e., one imposes invariance of the differential equation (5.2.2)under the transformation law:

δki(t, z) = ε(t, z) ∂tki + ρ(t, z) ∂zki, i = 1, 2. (5.2.4)

This simple behaviour of the ki functions under a change of the base-space coordinatesis coherent with the conclusions stated after the analysis of the global RP 2 transforma-tions (see page 90). The matching with the one-dimensional transformation laws willrestrict the dependence on z of the ki(t, z) functions so that ∂zki will depend on ∂tki.After performing the truncation (5.1.38) one gets:

δki(t) = ε(t, z)|z=0 ki + ρ(t, z)|z=0 ∂zki

∣∣∣z=0

. (5.2.5)

The comparison of this equation with the infinitesimal transformations of ki(t) (seeeq. (5.1.29) and (5.1.30)) yields:

∂zki

∣∣∣z=0

= ki(t) + qK(t)K(t)

ki(t), (5.2.6a)

ρ(t, z)|z=0 = ρ(t), (5.2.6b)

ε(t, z)|z=0 = ε(t)− 12

ρ(t)− (q +23) ρ(t)

K(t)K(t)

, (5.2.6c)

where q is some constant. The following extension equations for ki(t, z), namely,

∂zki(t, z) = ∂2t ki + q

∂tK

K∂tki, (5.2.7)

are the simplest ones, fulfilling eq. (5.2.6a), that can be built with the ingredientsat hand. These extension equations are somehow the analogue of eq. (5.1.60) in thesuper-Virasoro case. The relations between ki(t, z) and the objects K(t, z), L(t, z) andM(t, z), i.e.,

K(t, z) ≡ ∂tk1 ∂2t k2 − ∂tk2 ∂2

t k1 = ∂tk1 ∂zk2 − ∂tk2 ∂zk1, (5.2.8a)

L(t, z) = − 1K

∂tK, M(t, z) =1K

(∂2

t k1 ∂3t k2 − ∂2

t k2 ∂3t k1

), (5.2.8b)

dictate the following extension equations:

∂zK(t, z) = ∂2t K − 2 KM − 3q K ∂tL− q L ∂tK, (5.2.9a)

∂zL(t, z) = (3q + 1) ∂2t L− 2(q + 1) L ∂tL + 2 ∂tM, (5.2.9b)

∂zM(t, z) = ∂2t M − 2(q + 1) M ∂tL + q L ∂2

t L− q L ∂tM + q ∂3t L, (5.2.9c)

98 W3 Geometry

and the transformation laws:

δK(t, z) = ε ∂tK + ρ ∂zK + (∂tε + ∂zρ) K, (5.2.10a)

δL(t, z) = ε ∂tL + ρ ∂zL + ∂tε L− 3 ∂2t ε +

+ ∂tρ(2 M + 3(2q + 1) ∂tL− (q + 1) L2

)+ 3 (q +

23) ∂2

t ρ L− ∂3t ρ,

(5.2.10b)

δM(t, z) = ε ∂tM + ρ ∂zM + 2 ∂tε M − ∂2t ε L− ∂3

t ε +

+ ∂tρ(3 ∂tM + 3q ∂2

t L− (2q + 1) L M + 2q L ∂tL)

+

+ ∂2t ρ

(3 M + 3q ∂tL + q L2

)+ q ∂3

t ρ L. (5.2.10c)

Similarly, the relation (5.2.3) connecting x(t, z) with ki(t, z) yields:

∂zx(t, z) = ∂2t x +

23

T x− (q +23)

(L ∂tx− ∂tL x

), (5.2.11)

and

δx(t, z) = ε ∂tx + ρ ∂zx− 13

(∂tε + ∂zρ) x, (5.2.12)

where the definition of T (t, z) in terms of L(t, z) and M(t, z) has been taken intoaccount, definition which is inherited from eq. (5.1.28):

T (t, z) = M − 13

L2 − ∂tL, (5.2.13a)

W (t, z) = −12

∂tM +13

L ∂tL +16

∂2t L +

227

L3 − 13

L M . (5.2.13b)

According to the extension equations (5.2.7) and (5.2.11), the second-order deriva-tives are interpreted as first-order ones. At first sight, the main difficulty in establishinga geometrical interpretation of the W-transformations is ultimately related to the pres-ence of higher-order derivatives in the expressions (see, for instance, eq. (5.1.19)). Thispeculiarity has led to a number of authors to renounce to an ordinary tangent-spacedescription in favour of a jet-space formulation of the problem [80] [81]. Neverthe-less, the ‘reduction’ via eq. (5.2.7) and (5.2.11) of the order of the derivatives thatare present in the theory opens a possibility for a tangent-space interpretation of thegeometry of W-transformations. It will be immediately shown that this is the case.This approach not only manages to get rid of the higher-order derivatives, but also theannoying nonlinearities —another feature of the W-algebras— unexpectedly disappearfrom the transformation laws of the objects —see, for instance, eq. (5.2.25) below.

Only the subset of two-dimensional diffeomorphisms of the form (5.2.4) preservingthe extension equation (5.2.7) will produce W3-transformations after the truncation


z = 0. In other words, transformation (5.2.4) should be a symmetry transformationof both eq. (5.2.2) and eq. (5.2.7). Therefore ε(t, z) and ρ(t, z) can not be arbitraryinfinitesimal quantities but must satisfy the following conditions:

∂zρ(t, z) = ∂2t ρ + 2 ∂tε− 3(q +

23) ∂tρ L, (5.2.14a)

∂z ε(t, z) = (1 + 3q) ∂2t ε− 2(q + 1) ∂tρ T + q ∂3

t ρ−

− (q +23)

((q + 1) ∂tρ L2 + 6(q +

12) ∂tρ ∂tL + 3q ∂2

t ρ L

). (5.2.14b)

The subset of the RP 2 diffeomorphisms defined by the above conditions will be denotedas W3-diffeomorphisms. The parallelism with the supersymmetric setting is evidenthere too: it is tempting to regard W3-diffeomorphisms, defined through eq. (5.2.14),as the W counterpart of superconformal transformations and of restriction (5.1.64c).Furthermore, it can be checked that W3-diffeomorphisms form a closed subset of RP 2

diffeomorphisms. However, W3-diffeomorphisms exhibit a peculiar feature, namely thepresence of the fields in its definitory relations; this is an indication that they forma quasigroup (see page 104). Anyway, one can consistently define a subspace of RP 2

which is conserved under the action of W3-diffeomorphisms, thanks to the closeness oftheir algebra. This subspace will be referred to as the W3-space.

Equations (5.2.9) ensure the compatibility of the extension equation (5.2.7) withthe differential equation (5.2.2). It is worth noticing that this compatibility is fulfilledfor any value of q. From now on it will be adopted the value:

q = −23. (5.2.15)

With this particular choice:

• The extension equation for the x(t, z) object (5.2.11) does not depend on theL(t, z) field:

∂zx(t, z) = ∂2t x(t, z) +

23

T (t, z) x(t, z). (5.2.16)

As a consequence, the L(t, z) and M(t, z) fields are also absent in the extensionequations for T (t, z) and W (t, z) (see eq. (5.2.20) below) and all the furtherdevelopments can be consistently made with the ‘fundamental’ set of variablesx, T , W only.

• The relationship of the RP 2 infinitesimal parameters with the one-dimensionalones (see eq. (5.2.6)),

ρ(t, z)|z=0 = ρ(t), ε(t, z)|z=0 = ε(t)− 12

ρ(t), (5.2.17)

is consistent with the global transformations (5.1.39) and (5.1.41).

100 W3 Geometry

In the approach developed in [89] the fundamental quantities have been taken to bethe equivalent of the ki(t, z) functions. The extension equations for them are definedfrom the very beginning to be (5.2.7), but with q = 0.

For future reference, the above-mentioned relations inherited from (5.1.24) connect-ing ki with x, T and W are displayed here:

x(t, z) = K(t, z)−1/3(a + b k1(t, z) + c k2(t, z)

), (5.2.18a)

T (t, z) =12

∂2t K

K− 1

2∂zK

K− 2

3

(∂tK

K

)2

, (5.2.18b)

W (t, z) =112

∂3t K

K+

14

∂t∂zK

K− 5

12∂tK ∂zK

K2− 1

4∂tK ∂2

t K

K2+

427

(∂tK

K

)3

.

(5.2.18c)

Note that the last two relations are expressed in terms of K(t, z) only. Furthermore,they can be rewritten in a more compact form:

T (t, z) =32

K(t, z)1/3(∂z − ∂2

t

)K(t, z)−1/3, (5.2.19a)

W (t, z) =12

K(t, z)1/3(∂3

t − 3 ∂t∂z

)K(t, z)−1/3 +

12

∂tT (t, z). (5.2.19b)

The extension equations for the T and W objects can be obtained from the com-patibility condition between eq. (5.2.1) and (5.2.16). The same result arises whenconsidering eq. (5.2.13) and (5.2.9):

∂zT (t, z) = 2 ∂tW (t, z), (5.2.20a)

∂zW (t, z) = −23

T (t, z) ∂tT (t, z)− 16

∂3t T (t, z). (5.2.20b)

After eliminating W from equations (5.2.20) one arrives at the so-called Boussinesqequation for T :

∂2z T = −1

3∂4

t T − 43∂t

(T ∂tT

), (5.2.21)

which is the simplest non-trivial equation belonging to the third KdV-type hierarchyof differential equations. The extended coordinate z plays the role of the time variableof the Boussinesq equation (see eq. (2.3.27)). This is not an accident, of course. Thefundamental differential equation (5.2.1) is nothing but the Lax equation of third orderand all the compatible time flows that can be introduced are well-known thanks to thetechnique of the pseudodifferential operators sketched in sect. 2.3. The generalizationof this construction to the WN case will rely in this fact (see sect. 5.3).


Finally, one can compute the infinitesimal transformation laws for the T (t, z) andW (t, z) objects:

δT (t, z) = ε ∂tT + ρ ∂zT + ∂tρ

(3 W +

12

∂tT

)+ ∂zρ T − 2 ∂t∂z ε + ∂2

z ρ, (5.2.22a)

δW (t, z) = ε ∂tW + ρ ∂zW − 16

∂tε ∂tT + ∂z ε T +12

∂tρ

(∂tW − 1

6∂2

t T

)+

+12

∂zρ

(3 W +

16

∂tT

)+

12

∂t∂zρ T +12

∂t∂2z ρ, (5.2.22b)

where it has been taken into account the W3-diffeomorphism relations (5.2.14), thatnow read:

∂zρ(t, z) = 2 ∂tε + ∂2t ρ, (5.2.23a)

∂z ε(t, z) = −∂2t ε− 2

3∂3

t ρ− 23

∂tρ T . (5.2.23b)

Equations (5.2.22), together with eq. (5.2.12), are the symmetry transformations of thesystem of differential equations composed by eq. (5.2.1) and eq. (5.2.16) when ε(t, z)and ρ(t, z) satisfy the relations (5.2.23). It is convenient to define:

V (t, z) ≡ W +16

∂tT . (5.2.24)

In terms of T (t, z) and V (t, z) the infinitesimal transformation laws (5.2.22) are writ-ten:

δT (t, z) = ε ∂tT + ρ ∂zT + 3 ∂tρ V + ∂zρ T − 2 ∂t∂z ε + ∂2z ρ, (5.2.25a)

δV (t, z) = ε ∂tV + ρ ∂zV + ∂z ε T + 2 ∂zρ V − ∂tε V − ∂2z ε. (5.2.25b)

Notice that the cumbersome one-dimensional W3 transformations (5.1.19) turn intothe simple expressions displayed in eq. (5.2.12) and (5.2.25) when passing to the W3-space. The geometrical status of the x(t, z), T (t, z) and V (t, z) objects appears to beclear now. On one hand x(t, z) behaves as a scalar density of weight −1/3 under W3-diffeomorphisms (see eq. (5.2.12)), whereas the infinitesimal transformations of T (t, z)and V (t, z) (eq. (5.2.25)) resemble the transformation laws for Christoffel symbols.

5.2.1 Geometrical status of the W3 fields

Consider a base-space infinitesimal transformation law:

δtµ = −εµ. (5.2.26)

102 W3 Geometry

The induced infinitesimal transformation laws for a weight-h scalar density φ and forthe Christoffel symbols Γα

βλ read:

δφ = εµ∂µφ + h∂µεµ φ, (5.2.27a)

δΓαβλ = εµ∂µ Γα

βλ + ∂βεµ Γαµλ + ∂λεµ Γα

βµ − ∂µεα Γµβλ + ∂β∂λεα. (5.2.27b)

Greek indices run over the set 1, . . . , d, where d is the dimension of the (extended)base space, and (t1, . . . , td) are its coordinates. In this case d = 2 and:

t1 ≡ t, t2 ≡ z, ε1 ≡ ε, ε2 ≡ ρ. (5.2.28)

The inhomogeneous term in the infinitesimal transformation for the V (t, z) object(5.2.25b) suggests the following identification:

V (t, z) ≡ −Γ122(t, z). (5.2.29)

This implies:

T (t, z) ≡ Γ222(t, z)− Γ1

12(t, z)− Γ121(t, z). (5.2.30)

It is easily checked that the infinitesimal transformation law (5.2.25a) is in agreementwith this relation provided:

Γ111(t, z)− Γ2

21(t, z)− Γ212(t, z) = 0. (5.2.31)

Stability of this relation under infinitesimal W3-diffeomorphisms is ensured by takinginto account eq. (5.2.23b) and imposing:

Γ211(t, z) = 1. (5.2.32)

This last relation is also preserved by infinitesimal W3-diffeomorphisms, according toeq. (5.2.23a).

5.2.2 Finite transformations and the W3-Schwarzian derivative

Conversely, one can take conditions (5.2.31) and (5.2.32) as the starting point and getthe definitory relations for W3-diffeomorphisms (5.2.23) as a consequence. Moreover,this computation can be performed by considering not only infinitesimal transforma-tions but also finite ones. In this way, the finite form of the W3-diffeomorphism condi-tions (5.2.23) can be obtained. Under general coordinate transformations, (t1, . . . , td)→(t′1, . . . , t′d), the weight-h scalar density φ and the Christoffel symbols Γα

βλ transform:

φ′(t′, z′) = Jh φ(t, z), (5.2.33a)

Γαβλ′(t′, z′) =

(∂tµ

∂t′β∂tν

∂t′λΓρ

µν(t, z) +∂2tρ

∂t′β∂t′λ

)∂t′α

∂tρ, (5.2.33b)


where J is the Jacobian determinant of the transformation:

J = det(

∂tµ

∂t′ν

). (5.2.34)

When d = 2 the coordinate transformation can be parametrized in this way:

t = f(t′, z′), z = g(t′, z′), (5.2.35)

and the Jacobian is:

J = ∂t′ f ∂z′ g − ∂t′ g ∂z′ f . (5.2.36)

From the requirement that these general coordinate transformations keep the condi-tions (5.2.31) and (5.2.32) invariant one finds a set of relations to be satisfied by thetransformation functions f(t′, z′) and g(t′, z′):

∂z′ f = ∂2t′ f −

23

∂t′J

J∂t′ f − 2

3∂t′ f ∂t′ g T (f , g)− (∂t′ g)2 V (f , g), (5.2.37a)

∂z′ g = ∂2t′ g −

23

∂t′J

J∂t′ g +

(∂t′ f

)2 +13

(∂t′ g)2 T (f , g). (5.2.37b)

These relations can also be regarded as the extension equations for f and g. A W3-diffeomorphism is a general coordinate transformation (t = f(t′, z′), z = g(t′, z′))satisfying equations (5.2.37) for given T (t, z) and V (t, z). Note again that, in con-trast to the superconformal case (see, for instance, eq. (5.1.66)), the definition of W3-diffeomorphisms involves the T (t, z) and V (t, z) objects.

The finite transformation of x(t, z) is given by eq. (5.2.33a) with h = −13 and the

Jacobian (5.2.36), which can be rewritten —after taking eq. (5.2.37) into account— as:

J = ∂t′ f ∂2t′ g − ∂t′ g ∂2

t′ f +(∂t′ f

)3 + ∂t′ f (∂t′ g)2 T (f , g) + (∂t′ g)3 V (f , g). (5.2.38)

The transformation of the Christoffel symbols (5.2.33b) induces the transformations ofthe extended objects, T (t, z) and V (t, z) under finite W3-diffeomorphisms:

T ′(t′, z′) =1J

(∂t′ f ∂2

z′ g − ∂2z′ f ∂t′ g − 2 ∂t′∂z′ f ∂z′ g + 2 ∂t′∂z′ g ∂z′ f + 3 ∂t′ f

(∂z′ f

)2+

+(∂t′ f (∂z′ g)2 + 2 ∂z′ f ∂t′ g ∂z′ g

)T (t, z) + 3 ∂t′ g (∂z′ g)2 V (t, z)

),

(5.2.39a)

V ′(t′, z′) =1J

(∂z′ f ∂2

z′ g − ∂z′ g ∂2z′ f +

(∂z′ f

)3 + ∂z′ f (∂z′ g)2 T (t, z) + (∂z′ g)3 V (t, z))

.

(5.2.39b)

Notice that T ′ and V ′ have an effective nonlinear dependence on T and V . However,these nonlinearities are hidden in the definition of the W3-diffeomorphism transforma-tions (5.2.37). It can be explicitly checked that:

104 W3 Geometry

• The composition of two W3-diffeomorphisms yields another W3-diffeomorphism.Specifically, if

(t, z) −→ (t′, z′) −→ (t′′, z′′)(T , W ) −→ (T ′, W ′) −→ (T ′′, W ′′)

(5.2.40)

are two W3-diffeomorphisms defined by (f1, g1) and (f2, g2) respectively, then

(t, z) −→ (t′′, z′′)(T , W ) −→ (T ′′, W ′′)

(5.2.41)

is a W3-diffeomorphism defined by:

f(t′′, z′′) ≡ f1(f2(t′′, z′′), g2(t′′, z′′)), g(t′′, z′′) ≡ g1(f2(t′′, z′′), g2(t′′, z′′)).(5.2.42)

• If (t, z) → (t′, z′), (T , W ) → (T ′, W ′) is a W3-diffeomorphism defined by f(t′, z′)and g(t′, z′), then the inverse transformation (t′, z′) → (t, z), (T ′, W ′) → (T , W )defined by f−1(t, z) and g−1(t, z), where

t = f(f−1(t, z), g−1(t, z)), z = g(f−1(t, z), g−1(t, z)), (5.2.43)

is a W3-diffeomorphism too.

• The identity transformation f(t′, z′) = t′, g(t′, z′) = z′ is a W3-diffeomorphism.

Therefore the set of W3-diffeomorphisms is a group under the composition law. Fromthe point of view of the transformation laws for the fields T and W ,W3-diffeomorphismsact as a quasigroup [95] because the composition law depends on the fields themselves.

When T = 0 and V = 0 the W3-diffeomorphism definitory equations (5.2.37) arevery similar to the extension equations for the ki(t, z) functions (eq. (5.2.7) with q =−2

3). Indeed, given a solution (k1, k2) of these extension equations, one can obtain aW3-diffeomorphism (f , g) defined through the following relations:

f ≡ k1, g ≡ k2 − 12

(k1

)2. (5.2.44)

The Jacobian of this particular W3-diffeomorphism is:

J = K, (5.2.45)

and the corresponding finite transformation laws (5.2.39) are expressed in terms of thisJacobian through relations that resemble eq. (5.2.19):

T ′(t′, z′) =32

J1/3(∂z′ − ∂2

t′)J−1/3, (5.2.46a)

V ′(t′, z′) =12

J1/3(∂3

t′ − 3 ∂t′∂z′)J−1/3 +

23

∂t′ T′(t′, z′). (5.2.46b)


In other words, given the objects T and W parametrized in terms of the ki functions asin eq. (5.2.19), there always exists a W3-diffeomorphism that connects with a systemof coordinates such that T ′ = 0 and W ′ = 0.

The ki functions themselves define a transformation in RP 2,

(t0, z0) → (t, z) t0 = k1(t, z), z0 = k2(t, z). (5.2.47)

linking a coordinate system (t0, z0) where Γαβλ = 0 with a coordinate system (t, z)

such that T and W are written as in eq. (5.2.19). It is clear that this transformationis not a W3-diffeomorphism because, for instance, it does not preserve the condition(5.2.32). The existence of that (t0, z0) system of coordinates in RP 2 is ensured by thefact that the curvature tensor Rα

βγλ associated with the Christoffel symbols Γαβλ defined

in eq. (5.2.29)–(5.2.32) vanishes.Global W3-diffeomorphisms are those preserving the conditions T = 0 and V = 0.

According to eq. (5.2.46) the Jacobian of a global W3-diffeomorphism is:

J(t′, z′) =1(

q t′ + r(z′ + 1

2 t′2)

+ s)3 , q, r, s = constants. (5.2.48)

Of course, they are the projective SL(3,R) transformations:

t =a t′ + b

(z′ + 1

2 t′2)

+ c

q t′ + r(z′ + 1

2 t′2)

+ s, z +

12

t2 =mt′ + n

(z′ + 1

2 t′2)

+ p

q t′ + r(z′ + 1

2 t′2)

+ s, (5.2.49)

where

q r s

a b c

m n p

= constant ∈ SL(3,R).

The analogy with the method for obtaining the super-Schwarzian derivative out of theglobal transformations (see page 95) is clear. Therefore, the expressions in eq. (5.2.46)can be regarded as the components of a W3-Schwarzian derivative.

5.2.3 A covariant system of differential equations

The fundamental system of partial differential equations, namely, (5.2.1) and (5.2.16),can be written as a system of second order differential equations:

(∂2

t − ∂z +23

T

)x(t, z) = 0, (5.2.50a)

(∂z∂t +

13

T ∂t + V − 13

∂tT

)x(t, z) = 0. (5.2.50b)

The identification of T (t, z) and V (t, z) as Christoffel symbols naturally leads to thepossibility of rewriting the system (5.2.50) by handling the covariant derivative of the

106 W3 Geometry

scalar density x(t, z). Specifically the purpose is to encode that system in the equation:

∇α∇β x(t, z) = 0. (5.2.51)

The covariant derivative of a weight-h scalar density φ is:

∇ρφ = ∂ρφ− hΓσσρ φ, (5.2.52)

and the covariant derivative of a weight-h 1-covariant density ωµ is:

∇ρωµ = ∂ρωµ − Γσµρ ωσ − hΓσ

σρ ωµ. (5.2.53)

Therefore:

∇α∇β φ =

=(∂α∂β − Γσ

βα ∂σ − h Γσσβ ∂α − hΓσ

σα ∂β − h∂αΓσσβ + hΓρ

βα Γσσρ + h2 Γρ

ρα Γσσβ

)φ.

(5.2.54)

Comparison of the ∂ρx terms in the (α = 1, β = 1) component of eq. (5.2.51) withthose in eq. (5.2.50a) yields:

Γ211 = 1, (5.2.55a)

(2h + 1) Γ111 + 2hΓ2

21 = 0. (5.2.55b)

By doing the same with the (α = 1, β = 2) component of eq. (5.2.51) and eq. (5.2.50b)one gets:

hΓ111 + hΓ2

21 + Γ212 = 0, (5.2.56a)

hΓ222 + (h + 1) Γ1

12 = −13T . (5.2.56b)

Equivalence of the (α = 1, β = 2) and the (α = 2, β = 1) components of eq. (5.2.51)implies:

Γ112 = Γ1

21, Γ212 = Γ2

21, (5.2.57a)

∂2

(Γ1

11 + Γ221

)= ∂1

(Γ1

12 + Γ222

). (5.2.57b)

The ∂ρx terms in the (α = 2, β = 2) component of eq. (5.2.51) will be compared withthe ∂ρx pieces in the compatibility condition between eq. (5.2.50a) and eq. (5.2.50b),namely:

(∂2

z + V ∂t − 13T ∂z + ∂tV − 2

3∂zT − 2

9T 2 − 1

3∂2

t T

)x(t, z) = 0. (5.2.58)

5.3 Generalization to WN 107

This comparison provides the following identifications:

(2h + 1) Γ222 + 2hΓ1

12 =13T , (5.2.59a)

Γ122 = −V . (5.2.59b)

The solution of equations (5.2.55), (5.2.56) and (5.2.59) when h 6= −13 , namely,

Γ111 = Γ2

12 = 0, Γ222 = −Γ1

12 =13T , (5.2.60)

reproduces the system (5.2.50) only when T = 0 and V = 0. In concordance witheq. (5.2.12), the non-trivial solution is obtained when the scalar density x(t, z) hasweight h = −1

3 . Moreover, when h = −13 the identifications displayed in equations

(5.2.29)–(5.2.32) are recovered:

Γ211 = 1, Γ1

11 = 2Γ212, (5.2.61a)

T (t, z) = Γ222 − 2Γ1

12, V (t, z) = −Γ122. (5.2.61b)

Comparison of the terms without derivatives of x(t, z) in eq. (5.2.50) and in eq. (5.2.51)yields:

Γ112 = Γ2

22 −(∂1 − Γ2

12

)Γ2

12, (5.2.62a)

Γ122 =

(Γ2

12

)3 + Γ212Γ

222 −

13Γ2

12∂1Γ212 −

13∂1Γ2

22 −13∂2

1Γ212. (5.2.62b)

Equations (5.2.61a) and (5.2.62) imply that only two of the six (torsionless) Christoffelsymbols, say Γ2

22 and Γ212, are independent. These two Christoffel symbols are implicitly

related to the objects T (t, z) and V (t, z). Their extension equations are derived fromeq. (5.2.57b):

∂2Γ212 =

13

(2 ∂1Γ2

22 − ∂21Γ2

12 + 2 Γ212 ∂1Γ2

12

), (5.2.63a)

and from a further compatibility condition of the fundamental equations (5.2.50b)and (5.2.58):

∂2Γ222 =

13

(10

(Γ2

12

)2∂1Γ2

12 + 6 Γ222 ∂1Γ2

12 − 2(∂1Γ2

12

)2 − 2Γ212 ∂1Γ2

22+

+2Γ212 ∂2

1Γ212 + ∂2

1Γ222 − 2 ∂3

1Γ212

). (5.2.63b)

5.3 Generalization to WN

The formalism developed in the previous section can be generalized to the WN algebra.The Lax equation for the WN algebra is:

Lx(t) =

(∂N

t +N∑

k=2

uk(t) ∂N−kt

)x(t) = 0. (5.3.1)

108 W3 Geometry

The extended base-space is (N − 1)-dimensional and the coordinates will be denotedt1, t2, . . . , tN−1 with t1 ≡ t. Tilded objects will be the extended objects with thenatural relationship with the original ones. For instance:

x(t, 0, . . . , 0) = x(t). (5.3.2)

Eq. (5.3.1) is then generalized to the extended base-space as:

L x(t1, . . . tN−1) =

(∂N

1 +N∑

k=2

uk(t1, . . . , tN−1) ∂N−k1

)x(t1, . . . , tN−1) = 0. (5.3.3)

The extension equations for x must be compatible between themselves. The resultsof sect. 2.3 (specially eq.(2.3.29)) and 5.2 (see eq. (5.2.16)) point to the direction ofidentifying the extended coordinates with the different time flows of the Nth generalizedKdV hierarchy of differential equations associated with the Lax operator in eq. (5.3.3).Therefore, the extension equations —or multi-time equations— are written:

∂kx(t1, . . . , tN−1) = Lk/N+ x(t1, . . . , tN−1), k = 2, . . . , N − 1. (5.3.4)

As a consequence, the x object plays the role of a ‘truncated’ Baker–Akhiezer functionof the corresponding KP hierarchy of differential equations (see subsect. 2.3.1).

The W-transformations in this extended ‘W-space’ are those leaving eq. (5.3.3) andeq. (5.3.4) invariant. As a generalization of subsect. 5.2.3, this fundamental system ofdifferential equations will arise from a covariant description:

∇α∇β x(t1, . . . , tN−1) = 0. (5.3.5)

These equations are supposed to have N independent solutions for x, which is assumedto behave as a scalar density of weight h = − 1

N under general coordinate transforma-tions in the (N − 1)-dimensional space:

∇αx = ∂αx− hΓσσα x. (5.3.6)

A first compatibility condition gives:

∇[α∇β] x = Sσαβ ∇σx− hRσ

σαβ x = 0. (5.3.7)

Here Sγαβ and Rλ

αβγ are the torsion and curvature tensors, respectively:

Sγαβ ≡ Γγ

αβ − Γγβα, (5.3.8a)

Rλαβγ ≡ ∂βΓλ

αγ − ∂γΓλαβ + Γλ

σβ Γσαγ − Γλ

σγ Γσαβ. (5.3.8b)

As a consequence of eq. (5.3.7), the Christoffel symbols must be torsionless:

Γγαβ = Γγ

βα. (5.3.9)

5.3 Generalization to WN 109

A further compatibility condition, namely:

∇[α∇β]∇γ x = Sσαβ ∇σ∇γ x−Rσ

γαβ ∇σx− h Rσσαβ ∇γ x = 0, (5.3.10)

yields, together with eq. (5.3.7):

Rλαβγ = 0. (5.3.11)

The vanishing of the curvature tensor implies the existence of a set of flat coordinates,kµ (µ = 1, . . . , N − 1), such that the Christoffel symbols vanish. In this coordinatesystem, eq. (5.3.5) can be trivially solved:

x(k1, . . . , kN−1) =N−1∑

µ=1

cµ kµ + cN , ca = const., a = 1, . . . , N. (5.3.12)

For a generic coordinate system tµ such that

kµ = kµ(t1, . . . , tN−1), µ = 1, . . . , N − 1, (5.3.13)

the Christoffel symbols and the scalar density x are written:

x(t1, . . . , tN−1) = Jh

N−1∑

µ=1

cµ kµ(t) + cN

, (5.3.14a)

Γρµν(t

1, . . . , tN−1) = (J−1)ρσ

∂Jσν

∂tµ, (5.3.14b)

where:

Jαβ ≡

∂kα(t)∂tβ

, J ≡ det(Jαβ). (5.3.15)

The Christoffel symbols are compatible with a metric of the form:

gµν(t) = aσρ∂kσ(t)

∂tµ∂kρ(t)∂tν

, aσρ = const. (5.3.16)

The fundamental system of differential equations that provides the multi-time de-scription of WN transformations (eq. (5.3.3) and (5.3.4)) is obtained from the covariantsystem (5.3.5) once the (torsionless) Christoffel symbols are gauge-fixed to a particularform. Here is a conjecture for the form of the gauge-fixing for any N :

Γρα1 = δρ

α+1, α + 1 ≤ ρ ≤ N − 1, α = 1, . . . , N − 2, (5.3.17a)N−1+σ−β∑

α=1

Γα+β−σαβ + h

N−1∑

α=1

Γαασ = 0, σ + 1 ≤ β ≤ N − 1, σ = 1, . . . , N − 2.

(5.3.17b)

110 W3 Geometry

These (N − 1)(N − 2) gauge-fixing conditions are valid, at least, for N = 3 (they giveeq. (5.2.61a)), N = 4 and N = 5 [91]. The zero-curvature condition (5.3.11) providesnew conditions on the Christoffel symbols (such as eq. (5.2.62) for N = 3) and, also, de-fines the extension equations for the remaining ones (eq. (5.2.63) for N = 3). The N−1uk(t) fields appearing in eq. (5.3.3) are expressed in terms of the remaining Christof-fel symbols and, therefore, they depend on the kµ(t) functions through eq. (5.3.14b).These expressions are regarded as the components of a WN -Schwarzian derivative. Fi-nally, WN transformations are the subset of general coordinate transformations on theextended space that preserve the form of the gauge-fixing (5.3.17).

Conclusions

A particle mechanics model in phase space has been introduced which can be recastas a one-dimensional sp(2M,R) gauge theory. Different partial gauge-fixings of thismodel induced by sl(2,R) embeddings in sp(2M,R) yield reduced theories in which theremnant Lagrange multiplier variables correspond to generators of classical W-algebrasassociated with the Cn simple Lie algebras. It has also been shown how to obtainmodels invariant under W-algebras related to other series, such as the An.

The equations of motion for the matter variables of these particle mechanics modelshave been identified, in the W2, W3 and W(2, 4) cases, with the Lax operators associ-ated with these algebras, which are then regarded as symmetry algebras of differentialequations. This image has been easily generalized for other (non-principal) classicalW-algebras by computing the equations of motion —generally in a matrix form— ofthe corresponding particle mechanics models.

A number of relations between classical W-algebras coming from the same Kac–Moody algebra have been discovered by simple manipulations of these Lax equations:secondary reductions of W-algebras are nicely understood as algebraic transformationsof differential equations.

The same method has been proved to be useful in the contextualization of non-localand linear W-algebras as intermediate states in DS (secondary) reductions of ordinaryW-algebras.

As a by-product, a straightforward method for obtaining the change of variablesthat linearizes a given W-algebra has been discovered.

In relation with the issue of finite W-transformations, the simplest particle me-chanics model, with W2 symmetry, has only one remnant Lagrange multiplier whichtransforms as a weight-two quasi-primary field. In this case, the finite form of itssymmetry transformations is easily obtained by using the finite transformations of thesp(2,R) model and restricting them to those satisfying the gauge-fixing condition.

Application of this procedure to the model associated with the (0, 1) sl(2,R) em-bedding in sp(4,R) yields finite symmetry transformations of its action. These finitetransformations are perfectly acceptable as a parametrization of the gauge freedom ofthe system and they are actually useful for building the general solution of the model.

111

112 Conclusions

However, they cannot be regarded as standard finite W-diffeomorphism transforma-tions because their composition does not give ordinary diffeomorphisms. In order toobtain the expected form of finite W-diffeomorphism transformations one might intro-duce a non-linear change of infinitesimal gauge parameters before the gauge-fixing bymodifying the Yang-Mills transformations in a similar way as it was done to extractthe ordinary diffeomorphism.

The treatment of specific W-transformations on the same footing of ordinary dif-feomorphism transformations is, however, hard to imagine unless the original one-dimensional base space is extended to a two-dimensional W-space. It has been shownhow to define this W-space in the case of W3 by exploiting the analogy with the well-known superspace.

The W3-diffeomorphisms have been characterized and the link with the theory ofintegrable systems of differential equations has been established.

A clear geometrical status for the W3-algebra generators in the W3 space has beensettled: they are interpreted as the components of a partially gauge-fixed connection.

The finite form of W3-transformations are then obtained in this framework. Simpleexpressions for a W3 generalization of the Schwarzian derivative are given.

Finally, the way of extending these results to the WN algebras has been pointedout.

To end up, it is worth stressing that the study of Lax equations, coming fromequations of motion of particle mechanics models, has been a fruitful guideline in allprocesses (from algebraic to geometrical issues).

Outlook

A number of open questions arise at the end of this work. The most obvious ones referto the generalization of the ideas contained in it. As an example, the construction ofW-geometries associated with other W-algebras —different from WN— would be aninteresting issue to investigate.

Another natural line of research would be the implementation of the W-space tech-nology developed here in the study of (W) string theories. The key idea in definingthe W-geometry has been the introduction of new (base-space) coordinates that canbe interpreted as new time-like coordinates. This idea of considering space-times withmore than one time dimension is also implicit in chapter 3 because the particle me-chanics models presented there are only consistent if the target space-time has morethan one time dimension (see appendix B for details). In reference with this it shouldbe mentioned that recent Theoretical Physics developments consider the possibility oflooking at dynamics in space-times with more than one time dimension as useful toolsin the search for a unified string theory. See, for instance, ref. [96] where one of theparticle mechanics models presented in this work is studied under this point of view.

Conclusions 113

Certainly, W-string theories should enter in that unified description and it is worthinvestigating whether the multi-time language provides some hints in this direction.

Appendix A

sl(2,R) embeddings

A sl(2,R) embedding in a given Lie algebra G is a subalgebra S ⊂ G which is isomorphicto sl(2,R):

S = E0, E+, E−, [E0, E±] = ±E±, [E+, E−] = E0. (A.1)

Two embeddings S1 and S2 are said to be equivalent if there exists an automorphismof G mapping S1 onto S2. There will be as many admissible gauge-fixings (2.2.4) asclasses of equivalent sl(2,R) embeddings.

Given a canonical decomposition of G (i.e. given a Cartan subalgebra of G, H, aset of positive roots, ∆+, and a set of simple roots, Π), 1

G =∑

α∈∆+

+ G−α + H +∑

α∈∆+

+ Gα, (A.2)

and a sl(2,R) embedding in G, S, one can always choose a member of the same classof equivalence of S such that:

E0 ∈ H, i.e. E0 = Hδ where δ =∑

β∈Π

cβ β, (A.3a)

E± =∑

γ∈Γδ

e±γ , eγ ∈ Gγ , Γδ = γ ∈ ∆+ | (γ, δ) = 1 . (A.3b)

The quantity δ is called the defining vector of such an embedding. The characteristicof this sl(2,R) embedding is constructed by writing down the number (β, δ) under thedot of the Dynkin diagram of G which represents the root β, for each β ∈ Π. Twoimportant results follow [97] [98]:

• Two sl(2,R) embeddings are equivalent if and only if their characteristics coincide.1In fact, the formalism developed here leads to the characterization of the sl(2,C) embeddings in a

complex Lie algebra. This is tantamount to finding the sl(2,R) embeddings in a maximally non-compact

real Lie algebra (see the footnote on page 22).

115

116 sl(2,R) embeddings

• The characteristic of any sl(2,R) embedding will exhibit only numbers of the set0, 1

2 , 1.

It can be shown that the potential characteristic which exhibits a 1 under every dotalways gives rise to a sl(2,R) embedding. This particular embedding is known as theprincipal sl(2,R) embedding.

The examples of the sl(2,R) embeddings in sl(3,R) and in sp(4,R) are presentedin the following.

G = sl(3,R)

The Dynkin diagram and other conventions for sl(3,R) are displayed here:

© ©α β

(A.4a)

(α, α) = (β, β) =13

(α, β) = −16, ∆+ = α, β, α + β . (A.4b)

The characteristics for the two different classes of sl(2,R) embeddings in sl(3,R) are:

© ©α β

12

12

© ©α β

1 1(A.5)

The following sets can be taken as representatives for these classes of sl(2,R) embed-dings in sl(3,R). The corresponding branchings are also presented (see eq. (2.2.7),G1 ≡ S):

• Principal (1, 1) embedding:

E0 = 6Hα + 6Hβ E± =√

6E±α +√

6E±β (A.6a)

G = G1 + G2 (A.6b)

G2 = 〈Eα+β, Eβ − Eα,Hα −Hβ, E−α −E−β, E−(α+β)〉 (A.6c)

• Non-principal (12 , 1

2) embedding:

E0 = 3Hα + 3Hβ E± =√

3E±(α+β) (A.7a)

G = G1 + G(1)12

+ G(2)12

+ G0 (A.7b)

G(1)12

= 〈Eα, E−β〉 G(2)12

= 〈Eβ, E−α〉 G0 = 〈Hα −Hβ〉. (A.7c)

sl(2,R) embeddings 117

The matrix conventions for sl(3,R) are showed here for completeness:

Hα = 16

1 0 00 −1 00 0 0

Hβ = 1

6

0 0 00 1 00 0 −1

Eα = 1√6

0 1 00 0 00 0 0

Eβ = 1√

6

0 0 00 0 10 0 0

Eα+β = 1√

6

0 0 10 0 00 0 0

E−α = 1√6

0 0 01 0 00 0 0

E−β = 1√

6

0 0 00 0 00 1 0

E−(α+β) = 1√

6

0 0 00 0 01 0 0

The sl(3,R) subalgebra of sp(6,R) that has been considered in sect. 3.4 is realized bytaking the following subset of the φAij quadratic constraints in the M = 3 case:

Eα =1√6p1p2, Eβ =

1√6p3x2, Eα+β = − 1√

6p1p3, (A.8a)

E−α = − 1√6x1x2, E−β =

1√6p2x3, E−(α+β) =

1√6x1x3, (A.8b)

Hα =16(p2x2 + p1x1), Hβ =

16(p3x3 − p2x2). (A.8c)

G = sp(4,R)

The Dynkin diagram, normalizations and positive roots set for sp(4,R) are:

© ©〈α β

(A.9a)

(α, α) =16

(β, β) =13

(α, β) = −16, ∆+ = α, β, α + β, 2α + β .

(A.9b)

There are only three classes of non-equivalent sl(2,R) embeddings. Their characteristicsare:

© ©〈α β

12

0© ©〈α β

0 1© ©〈α β

1 1(A.10)

They are referred to as the (12 , 0), (0, 1) and principal embeddings, respectively.

118 sl(2,R) embeddings

The matrix conventions used in section 3.3 for the generators of sp(4,R) are:

Hα = 112

1 0 0 00 −1 0 00 0 −1 00 0 0 1

Hβ = 1

6

0 0 0 00 1 0 00 0 0 00 0 0 −1

Eα = 1√12

0 1 0 00 0 0 00 0 0 00 0 −1 0

E−α = 1√

12

0 0 0 01 0 0 00 0 0 −10 0 0 0

Eβ = 1√6

0 0 0 00 0 0 10 0 0 00 0 0 0

E−β = 1√

6

0 0 0 00 0 0 00 0 0 00 1 0 0

Eα+β = 1√12

0 0 0 10 0 1 00 0 0 00 0 0 0

E−(α+β) = 1√

12

0 0 0 00 0 0 00 1 0 01 0 0 0

E2α+β = 1√6

0 0 1 00 0 0 00 0 0 00 0 0 0

E−(2α+β) = 1√

6

0 0 0 00 0 0 01 0 0 00 0 0 0

A representative of every class of equivalent sl(2,R) embeddings in sp(4,R) and thecorresponding branchings are displayed here (G1 ≡ S):

• (0, 1) embedding:

E0 = 6Hα + 6Hβ E± =√

6E±(α+β) (A.11a)

G = G1 + G(2)1 + G(3)

1 + G0 (A.11b)

G(2)1 = 〈E2α+β, Eα, E−β〉 G(3)

1 = 〈Eβ, E−α, E−(2α+β)〉 (A.11c)

G0 = 〈Hα〉. (A.11d)

• (12 , 0) embedding:

E0 = 6Hα + 3Hβ E± =√

3E±(2α+β) (A.12a)

G = G1 + G(1)12

+ G(2)12

+ G(1)0 + G(2)

0 + G(3)0 (A.12b)

G(1)12

= 〈Eα+β, E−α〉 G(2)12

= 〈Eα, E−(α+β)〉 (A.12c)

G(1)0 = 〈Eβ〉 G(2)

0 = 〈E−β〉 G(3)0 = 〈H−β〉. (A.12d)

sl(2,R) embeddings 119

• (1, 1) (principal) embedding:

E0 = 18Hα + 12Hβ E± =√

18E±α +√

12E±β (A.13a)

G = G1 + G3 (A.13b)

G3 = 〈E2α+β, Eα+β,√

3Eβ −√

2Eα,Hα −Hβ,√

3E−β −√

2E−α,

E−(α+β), E−(2α+β)〉. (A.13c)

This embedding produces a gauge fixing such that there is no gauge-fixed particle-mechanics Lagrangian in terms of coordinates and velocities. Therefore, anotherelement of the same conjugacy class has been considered in subsect. 3.3.2, namely:

E0 = 18Hα + 6Hβ E± =√

18E±(α+β) +√

12E∓β (A.14a)

G = G1 + G3 (A.14b)

G3 = 〈E2α+β, Eα,√

3E−β −√

2Eα+β,Hα + 2Hβ,√

3Eβ −√

2E−(α+β),

E−α, E−(2α+β)〉. (A.14c)

Appendix B

Signature of the space-time for

the particle models

The canonical action for the particle mechanics models presented in chapter 3 is:

S =∫

dt(pi,µ xµ

i − λAijφAij

), µ = 1, . . . , d, i, j = 1, . . . ,M, A = 1, 2, 3.

(B.1)

The explicit form of φAij is:

φ1ij =12

ηµνpi,µ pj,ν , φ2ij = pi,µ xµj and φ3ij =

12

ηµν xµi xν

j . (B.2)

Here ηµν is a flat metric invariant under SO(d−, d+), where d− + d+ = d, being d thedimension of the target space-time where the particles live. The number of degrees offreedom for the M particles is 2Md. The number of first-class constraints φAij = 0 is2M2 + M . Every first-class constraint eliminates two degrees of freedom so that a firstrestriction for d is:

2Md > 2(2M2 + M

) ⇒ d > 2M + 1. (B.3)

The constraints coming from eq. (B.2), even if eq. (B.3) is fulfilled, may trivialize themodel. For example, if d− = 0 then φ1ij = 0 implies pi = 0. Therefore, ηµν should bea Minkowskian metric. In order to investigate the geometrical restrictions that theseconstraints impose to the equations of motion it is convenient to write:

xµi = (~x 0

i , ~xi), pµi = (~p 0

i , ~pi), (B.4)

where ~x 0i and ~xi are d− and d+-dimensional vectors, respectively, so that:

ηµνxµi xν

i = − (~x 0

i

)2 + (~xi)2 . (B.5)

121

122 Signature of the space-time for the particle models

The constraints φ1ii = 0 and φ3ii = 0 imply:

xi = |xi|(x0i , xi) and pi = |pi|(p0

i , pi). (B.6)

The hatted vectors are unitary and |xi|2 =(~x 0

i

)2 = (~xi)2. The remaining constraints

(φ1ij = 0, φ3ij = 0 (i 6= j) and φ2ij = 0) imply:

cos(x0

i x0j

)= cos (xixj) , cos

(p0

i p0j

)= cos (pipj) , cos

(x0

i p0j

)= cos (xipj) .

(B.7)

Here x0i p

0j is an angle in a d−-dimensional space whereas xipj is defined in a d+-

dimensional space. These constraints restrict the signature of the space-time as it willbe illustrated in the following for the lower values of M .

M = 1

The space-time dimension for the sl(2,R) model should be, at least, d = 4. The onlypossibilities for the ηµν metric invariance group are SO(1,3) and SO(2,2).

SO(1,3)

The SO(3) (rigid) symmetry of the space-like sector can be used to set:

x = (1, 0, 0) and p = (cosα, sinα, 0). (B.8)

On the other hand, cos(x0p0

)= ±1 because d− = 1. Therefore, eq. (B.7) implies:

α = 0 or π, (B.9)

which is an extra, geometrical restriction that trivializes too much the equations ofmotion: there is no room for introducing the three arbitrary functions correspondingto the three sl(2,R) gauge symmetries because there are only two undetermined pa-rameters (|x| and |p|). In other words, the original 8 degrees of freedom are reducedto 6 due to φ1 = 0 and φ3 = 0. These 6 degrees of freedom are |x|, |p|, x and p. Thelast constraint xp = ±1 eliminates two degrees of freedom, instead of only one.

SO(2,2)

In this case one can take, by using the two SO(2) rigid symmetries:

x = (1, 0), p = (cosα, sinα), (B.10)

x0 = (1, 0), p0 = (cosβ, sinβ). (B.11)

Signature of the space-time for the particle models 123

Then, eq. (B.7) simply eliminates one degree of freedom: α = ±β.

By the same reasoning, one can easily see that if d > 4 then still d− 6= 1 is necessary.Therefore, the M = 1 particle model should be defined in a target space-time of, atleast, two times.

M = 2

The space-time dimension should be, at least, d = 6. The SO(1,5) case presents thesame problems of the SO(1,3) case above. The SO(2,4) case also induces too muchrestrictions:

The six rotations of the spatial SO(4) symmetry can be used to set:

x1 = (1, 0, 0, 0), (B.12)

p1 = (cosα, sinα, 0, 0), (B.13)

x2 = (cosβ, sinβ cos γ, sinβ sin γ, 0), (B.14)

p2 = (cosλ, sinλ cosφ, sinλ sinφ cosψ, sinλ sinφ sinψ). (B.15)

This, together with eq. (B.7), yields:

x01p

01 = x1p1 = α, and x0

1x02 = x1x2 = β. (B.16)

The time sector is a two-dimensional space. Therefore, eq. (B.16) implies x02p

01 = α±β.

However, from eq. (B.7) and (B.13) and (B.14) one is led to:

cos(x0

2p01

)= cosα cosβ + sinα sinβ cos γ. (B.17)

Therefore, cos γ should be ±1, which is an extra restriction.

As a consequence, the M = 2 particle model is a consistent theory in a space-timeof, at least, three times.

The natural generalization is that the sp(2M,R) particle model, that lives in aspace-time with, at least, d = 2M + 2, needs a metric invariant under SO(M + 1,M + 1).

Resum

Introduccio

La simetria conforme en dues dimensions juga un paper molt important en els temes derecerca actual de la Fısica Teorica, particularment en l’anomenada teoria de cordes. Lestransformacions conformes son aquelles que transformen la metrica de l’espai-temps enella mateixa multiplicada per un factor. La invariancia sota transformacions conformesproporciona 1

2(d + 1)(d + 2) quantitats conservades si l’espai-temps es de dimensio d,excepte en el cas que d = 2, que en proporciona infinites. Aquest fet fa que de les teoriesamb invariancia conforme en dues dimensions se’n puguin coneixer moltes caracterıs-tiques invocant nomes arguments de simetria i es pot parlar, fins i tot, de classificartotes les possibles teories conformes en d = 2 que satisfacin una serie de requerimentsfısics. En una teoria de camps conforme en dues dimensions existeixen camps que estranformen d’una manera particularment simple: son els anomenats camps quirals pri-maris, caracteritzats pel seu pes. Un camp que sempre esta present es l’anomenat tensord’energia-impuls T (z), que es el generador de la simetria conforme i te pes igual a dos.La quantitzacio de la simetria conforme es realitza mitjancant l’habitual substitucio decamps per operadors i de parentesis de Poisson per commutadors, un cop una certaprescripcio d’ordenacio normal es establerta. Aquest proces s’anomena quantitzacioradial i porta d’una manera natural —a traves de la connexio del formalisme conformeamb el calcul amb una variable complexa— a la definicio d’expansions del producte dedos operadors (EPO) en forma de series de Laurent. L’EPO de T (z) amb ell mateix,aleshores, exhibeix un terme no present en el formalisme classic que conte una quantitatanomenada carrega central. Expandint un camp en serie de potencies es calculen elsseus modes. Aquests modes verifiquen unes relacions de commutacio —heretades del’EPO corresponent— que, en el cas dels modes de T (z), defineixen una algebra de Liede dimensio infinita, anomenada algebra de Virasoro.

L’extensio de simetries es un procediment que es present molt sovint en FısicaTeorica. Un exemple d’aixo es la definicio de les anomenades supersimetries. L’extensiode la simetria conforme introduint-hi un nou operador fermionic dona lloc a la su-peralgebra de Virasoro. Un altre exemple es l’extensio de l’algebra de Virasoro —o,

125

126 Resum

equivalentment, de la simetria conforme— mitjancant la introduccio dels modes corre-sponents a camps primaris bosonics. La verificacio de les identitats de Jacobi i la con-servacio de la simetria conforme dicta gairebe unıvocament l’estructura de l’algebra es-tesa. Aquestes algebres aixı definides s’anomenen genericament algebresW quantiques.L’algebra W mes simple s’aconsegueix estenent l’algebra de Virasoro introduint-hi elsmodes d’un operador quiral primari de pes tres. Aquesta algebra s’anomena algebraW3

i fou obtinguda per A.B. Zamolodchikov l’any 1985. La caracterıstica fonamental del’algebra W3 i, en general, de totes les algebres W, es que no son lineals sino que enles relacions definitories de l’algebra de modes i en les EPO corresponents apareixentermes quadratics que fan que aquestes algebres no siguin algebres de Lie. En gene-ral, l’estructura de les algebres W es extremadament complicada, la qual cosa dificultaenormement els calculs involucrats. Hom pot realitzar lımits classics d’aquestes alge-bres tornant al llenguatge de camps i parentesis de Poisson. Un primer lımit classics’aconsegueix fent tendir la constant de Planck ~ cap a zero i el valor de la carregacentral c cap a infinit de manera que el producte ~c es mantingui finit. D’aquestaforma l’algebra corresponent, tot i ser classica, encara exhibeix termes que contenenla carrega central. Un segon lımit classic consisteix en fer tendir el producte ~c cap azero. En qualsevol cas, les algebres W classiques tambe contenen termes no lineals.

Un cop introduıdes les algebres W, es plantejava la questio d’identificar models quefossin invariants sota la corresponent simetria W. La simetria conforme es present a lateoria de cordes. Per tant, el primer pas fou intentar definir una mena de teoria cordesW que, a mes d’esser simetrica sota la simetria conforme, exhibıs simetria W. Aixonomes es pot fer parcialment ja que simultaniament s’ha de resoldre el problema dela definicio d’una geometria W, problema que s’esmentara mes avall. De tota manera,hom pot obtenir resultats, com ara la construccio d’una gravetat W, teoria que s’obtemitjancant un proces totalment analeg al que se segueix per obtenir una descripciode la gravetat en dues dimensions a partir de la teoria de cordes. D’aquesta maneraes descobreixen relacions entre les algebres W i els models de Wess–Zumino–Novikov–Witten i les teories de Toda, relacions que es fan mes evidents en estudiar les reduccionshamiltonianes de Drinfel’d–Sokolov que es defineixen tot seguit.

Aquesta memoria tracta diferents aspectes de les algebres W classiques —definidessegons el primer dels lımits classics abans presentats—. El metode mes sistematicdisponible per construir algebres W classiques —molt mes simple que la construcciodirecta a partir de l’estudi de les identitats de Jacobi— es l’anomenada reduccio hamil-toniana de Drinfel’d–Sokolov. Aquest metode parteix d’una algebra de Kac–Moodyassociada a una algebra de Lie G. Hom pot definir una estructura de Lie–Poissonper l’algebra de Kac–Moody. La introduccio d’un conjunt de lligams de segona classeen aquesta estructura de Lie–Poisson —com, per exemple, la igualacio d’algun gener-ador de l’algebra de Kac–Moody a zero o a una constant— provoca la transformacio

Resum 127

de l’estructura en una algebra W, que ve definida, per tant, pels parentesis de Diraccorresponents. La seleccio dels lligams que es poden imposar a una determinada al-gebra de Kac–Moody perque produeixi una algebra W —es a dir, perque continguil’algebra de Virasoro i perque la resta de camps inclosos siguin primaris— es redueixa un problema algebraic que te la seguent solucio: cal que els lligams segueixin unarelacio de gradacio induıda per una subalgebra sl(2,R) en G. De fet, aquesta es enprincipi una condicio suficient per obtenir, al final del proces, una algebra W. De totamanera, hi ha prou indicis per pensar que tambe es una condicio necessaria o, si mesno, poques algebres W escapen a aquesta metodologia. D’aquesta manera hom potestablir una classificacio de les algebres W prou completa, classificacio que, en ultimainstancia, es recolza en la classificacio de totes les possibles inclusions no equivalentsd’una algebra isomorfa a sl(2,R) en una algebra de Lie G. Aixı, per exemple, quanG es igual a sl(2,R) obviament nomes hi ha una subalgebra sl(2,R), la qual indueixuna reduccio de Drinfel’d–Sokolov que porta a l’algebra de Virasoro. En canvi, quanG = sl(3,R), llavors sl(2,R) es pot incloure de dues maneres no equivalents a G, pro-porcionant dues algebres W diferents, una de les quals —la generada per la inclusioanomenada principal— es justament W3. L’altra inclusio dona lloc a una altra algebraW, anomenada W2

3 , que conte, a banda del tensor energia-impuls, tres camps mes, dosdels quals tenen pes igual a 3/2 i l’ultim es de pes igual a 1.

Les algebres W classiques estan sorprenentment connectades a la teoria de sistemesintegrables. Aixı, per exemple, l’equacio de Korteweg–de Vries, equacio diferencial nolineal que descriu certs fenomens hidrodinamics, es pot descriure com un sistema hamil-tonia de dimensio infinita, descripcio que porta a la seva caracteritzacio com un sistemaintegrable. L’estructura hamiltoniana corresponent a aquest sistema —o, mes ben dit,una d’elles, ja que s’en pot definir mes d’una— resulta ser l’algebra de Virasoro —ambun terme de carrega central—. Similarment, l’algebra W3 esta associada a una altraequacio diferencial, anomenada equacio de Boussinesq. Un gran nombre d’algebres Wobtingudes mitjancant la reduccio de Drinfel’d–Sokolov —sobretot aquelles que prove-nen de la inclusio principal d’una algebra sl(2,R) en una algebra de Lie— es podeninterpretar com a simetries de sistemes integrables d’equacions diferencials, anomenatsgenericament sistemes de Korteweg–de Vries generalitzats.

Resultats

Models de mecanica de partıcules amb simetries W

En aquesta memoria es presenta un metode per construir models de mecanica departıcules que exhibeixen simetries W en forma de simetries gauge. Aquest metodees recolza en la reduccio hamiltoniana de Drinfel’d–Sokolov abans descrita. El punt

128 Resum

de partida es un lagrangia expressat en termes de les coordenades i moments d’unconjunt de partıcules que conte uns multiplicadors de Lagrange, els quals implementenun conjunt de lligams en forma de combinacions quadratiques de les esmentades coor-denades i moments. Aquests lligams quadratics son de primera classe i generen, sotal’estructura simplectica habitual, l’algebra sp(2M,R), on M es el nombre de partıculesdel model. Pel fet de ser uns lligams de primera classe el model exhibeix una simetriagauge. Les transformacions dels multiplicadors de Lagrange son de tipus Yang–Millsamb sp(2M,R) com grup de gauge. Les equacions del moviment associades a aquestlagrangia permeten expressar els moments en funcio de la resta de variables. Quan aixoes fa l’algebra de les transformacions, que abans era tancada, s’obre. Una manera detornar a una algebra de gauge tancada es introduint un conjunt de camps auxiliars ques’acoblen als multiplicadors de Lagrange i no tenen terme cinetic.

Aquest model es invariant sota reparametritzacions de la coordenada temporal deque depenen les variables del sistema. Si aquesta coordenada temporal s’interpreta comuna coordenada complexa llavors aquesta invariancia sota reparametritzacions esdeveuna simetria conforme. La identificacio d’aquesta simetria sota reparametritzacions esfa en realitzar un canvi en els parametres infinitesimals de les transformacions gauge,transformacions que, com s’ha dit abans, son de tipus Yang–Mills. La relacio que lligaels nous parametres amb els antics involucra els multiplicadors de Lagrange. Despresd’aquest canvi, de les 2M2+M simetries gauge inicials del model, totes menys una con-tinuen essent de tipus Yang–Mills. Totes elles es poden integrar donant l’expressio de lestransformacions de gauge finites del model. Pel que fa a la simetria de reparametritza-cions, els multiplicadors de Lagrange es comporten —un cop es fa aquell paral·lelismeamb la simetria conforme— com camps primaris de pes 1, tot i que aquest caracter espot modificar lleument introduint un conjunt de constants arbitraries en el canvi delsparametres de gauge abans esmentat.

Les simetries tipus Yang–Mills que exhibeix el sistema es poden interpretar comsimetries induıdes per una algebra de Kac–Moody. Aquest fet possibilita l’aplicacio delmetode de reduccio hamiltoniana de Drinfel’d–Sokolov descrit en la introduccio. Aquestprocediment ara s’interpreta com una fixacio parcial del gauge en el model de partıcules,ja que alguns dels multiplicadors de Lagrange s’igualaran a zero o a una constant. Lainclusio d’una algebra isomorfa a sl(2,R) en sp(2M,R) dicta, com s’ha dit abans, dequina manera s’ha de fer aquesta fixacio parcial del gauge. Continuant amb l’analogia,el model de partıcules, un cop fixat parcialment el gauge, exhibira la simetria de gaugeassociada a una algebra W. D’aquesta manera s’obte una realitzacio lagrangiana, enun model simple, d’aquesta simetria. Seguint la pista del parametre gauge que dona lestransformacions de reparametritzacions sota el proces de fixacio parcial del gauge hompot escriure les transformacions de reparametritzacions del model reduıt, identificantels pesos dels multiplicadors de Lagrange que sobreviuen, a mes del seu caracter primari

Resum 129

—amb la salvetat descrita al paragraf anterior—.

El model d’una partıcula (M = 1), en ser sp(2,R) isomorf a sl(2,R), es invariantsota l’algebra de Virasoro despres de la fixacio del gauge. L’equacio del moviment de lapartıcula no es res mes que l’equacio diferencial generada per l’operador de Lax de segonordre —de rellevancia en la teoria d’equacions de Korteweg–de Vries generalitzades—associat a l’algebra de Virasoro. Hom pot realitzar la fixacio parcial del gauge a nivellde les transformacions de gauge finites obtenint d’aquesta manera les transformacionsresiduals de gauge del model reduıt. Aquestes transformacions finites son les que cor-responen a un camp primari de pes −1/2 —la variable de posicio de la partıcula— i aun tensor energia-impuls —el multiplicador de Lagrange que sobreviu—. La derivadaschwarziana tıpica de la transformacio d’un tensor energia-impuls apareix d’aquestamanera sense haver d’integrar les transformacions reduıdes directament.

Quan M = 2 apareixen els models corresponents a sp(4,R), les lagrangianes delsquals s’obtenen. Hi ha tres possibles inclusions de sl(2,R) en sp(4,R), cadascuna deles quals dona lloc a una algebra W diferent. L’algebra corresponent a la inclusioanomenada (0, 1) conte quatre camps. Per aquesta algebra s’inclouen les transforma-cions de gauge finites que s’obtenen seguint el proces descrit pel cas M = 1. Aquestestransformacions W finites no apareixen parametritzades d’una manera satisfactoria jaque, entre altres motius, d’acord amb l’algebra haurien de produir, sota composicio, lestransformacions de reparametritzacions, cosa que no verifiquen o, almenys, ho fan ambuna realitzacio molt poc clara d’aquestes darreres transformacions. La clau del proble-ma probablement es trobi en el fet que les transformacions de reparametritzacions jaes poden fer apareixer abans de la fixacio del gauge. Hom hauria de trobar un canvi deparametres de gauge abans de la seva fixacio parcial que fes apareixer d’alguna manerales transformacions W. Els altres dos models del cas M = 2 son el corresponent a lainclusio principal —que dona lloc a una algebra W amb un camp extra de pes quatre—i el corresponent a la inclusio (1

2 , 0). L’algebra W associada a aquest ultim model estagenerada, a mes de pel tensor d’energia-impuls, per dos camps de pes 3/2 i per trescamps de pes 1 que configuren una algebra de Kac–Moody.

Tot i que els models presentats, en ser invariants sota sp(2M,R), naturalment donenlloc a algebres W lligades a la serie Cn, tambe es poden construir altres models. Coma exemple es presenta un model que es invariant sota sl(3,R), obtingut a partir d’unainclusio d’aquesta algebra en sp(6,R). Les dues reduccions possibles donen lloc a dosmodels invariants sota W3 i W2

3 , respectivament, d’acord amb el que s’ha comentat ala introduccio.

Les equacions del moviment que es deriven de tots aquests models permeten definirels operadors de Lax correponents a les algebres W en questio. En el cas de l’algebrade Virasoro l’operador, com ja s’ha dit, es ∂2 + u, mentre que en el cas de W3 aquest

130 Resum

es ∂3 + u∂ + v. En canvi, per les algebres W no obtingudes a partir de la inclusioprincipal, com per exemple W2

3 , aquests operadors son matricials.

Relacions entre algebres WUn dels resultats exposats a la memoria es l’establiment d’unes relacions entre diferentsalgebres W. El metode de reduccio de Drinfel’d–Sokolov es pot generalitzar per tal dereduir, en comptes d’una algebra de Kac–Moody, una altra algebra W. En aquest cases parla de reduccio secundaria. Una reduccio secundaria connecta algebres W ques’obtenen a partir de la mateixa algebra de Kac–Moody.

Es mostra com aquesta reduccio secundaria es pot fer en dos passos. El primerpas dona lloc a una algebra de tipus W —perque conte camps primaris, inclou lasimetria conforme i, a mes, presenta termes no lineals— pero que no es local. Lareduccio d’aquesta algebra no local proporciona l’algebra W d’arribada. Les algebresno locals que constitueixen els passos intermitjos d’aquesta reduccio secundaria vanser estudiades per A. Bilal —que les va anomenar algebres V— i una de les sevescaracterıstiques es que el seu operador de Lax associat es matricial. Aquest fet permetfer la descripcio de la reduccio a partir de l’estudi dels models de partıcules presentatsabans, ja que, com s’ha dit, sovint proporcionen equacions de moviment matricials. Enefecte: hom pot transformar l’equacio de moviment del model invariant sota l’algebraW2

3 , per exemple, en una equacio generada per un operador de Lax dels que estanassociats a algebres no locals i que conte tres camps. Aquesta transformacio permettrobar l’expressio dels camps primaris de la nova algebra no local en termes dels deW2

3 .

D’altra banda, les equacions de moviment del model invariant sota W23 tambe es

poden transformar en l’equacio de moviment del model invariant sota W3. Aquest fetes un reflex de la reduccio secundaria abans esmentada i presenta l’avantatge que hompot escriure directament, un altre cop, la relacio que hi ha entre els generadors deles dues algebras per simple comparacio dels coeficients de les equacions de movimentrespectives.

En un altre ordre de coses, es va observar recentment que hom pot estendre l’algebraW3 amb l’addicio d’un nou camp de pes 1 de manera que, despres de fer un canvi devariables, aquesta nova algebra esdeve lineal —i, per tant, molt mes tractable—. En lamemoria es mostra que aquesta algebra lineal es una subalgebra de W2

3 . Aixo propor-ciona una tercera via per anar de W2

3 a W3 a traves, no ja de processos de reduccio,sino d’inclusio d’algebres. Tanmateix, l’avantatge de treballar amb les equacions demoviment dels models de partıcula corresponents consisteix, un cop mes, en la facilitatde relacionar els generadors de les diferents algebres. En aquest cas allo que s’obtees aquell canvi que fa lineal l’algebra W3 estesa, canvi que, per altres metodes, es

Resum 131

complicat de trobar.

Tots aquests processos de relacio entre diferents algebres W a traves de l’estudi deles equacions de moviment de models de partıcules —reduccions secundaries, algebresno locals i algebres lineals— s’exemplifiquen tambe en el cas de les algebres W lligadesa sp(4,R).

Geometria W3

L’algebra de Virasoro esta ıntimament lligada amb conceptes geometrics: es pot rea-litzar com a grup de transformacions d’una varietat de dimensio 1. Els camps quees transformen sota l’algebra de Virasoro poden traslladar-se al llenguatge geometricinterpretant-los com a seccions d’un cert espai fibrat. Donat que les algebresW son unageneralitzacio de l’algebra de Virasoro, es natural cercar una interpretacio geometricaper elles. Tanmateix, la presencia de termes no lineals i de derivades d’ordre alt en lesrelacions definitories d’aquestes algebres fa difıcil l’establiment d’una geometria W. Ames, l’existencia d’una infinitat d’algebres W diferents implica l’existencia, en principi,de geometries W en comptes d’una unica geometria W. En aquesta memoria l’estudise centra en el cas mes senzill, es a dir, W3.

La idea basica de la proposta per una geometria W3 que es presenta en aquestamemoria es bastant simple: la definicio d’una descripcio geometrica per l’algebra W3

passa per la implementacio d’aquesta algebra com a grup de transformacions en l’espaibase. Aquest espai base no pot ser de dimensio 1 ja que totes les transformacions possi-bles d’aquest espai base ja estan codificades amb la subalgebra conforme que conte W3.Aleshores, si es vol dotar les transformacions especıfiques de W3 amb una realitzacio al’espai base, esta clar que l’unica sortida es estendre aquest espai base amb mes dimen-sions. Aquesta idea no es nova a la Fısica Teorica: la construccio del super-espai tambefa us del concepte d’extensio de l’espai base. La proposta que es presenta aquı es potentendre analogament, tot i que ara la nova coordenada sera bosonica en comptes defermionica.

L’estudi de les equacions de moviment dels models de partıcula corresponents tornaa ser un guia en el proces de concrecio de l’anterior programa. Solucionant l’equacio demoviment del model de partıcula corresponent a sl(2,R) hom expressa les variables delmodel en termes d’una funcio. Si a aquesta funcio se li suposa un comportament escalarsota transformacions conformes, llavors les transformacions finites sota l’algebra de Vi-rasoro de les variables del model apareixen facilment, inclosa la derivada schwarziana dela transformacio del multiplicador de Lagrange que sobreviu. L’extensio d’aquest argu-ment al cas del model amb simetria W3 es directa. En aquest cas la solucio de l’equaciode moviment involucra dues funcions arbitraries, reflex de les dues transformacionsgauge del sistema. Igual que abans, les transformacions corresponents a la simetria

132 Resum

conforme es realitzen directament com a transformacions geometricament clares sobreaquestes dues funcions, mentre que l’altra transformacio no es realitza d’una maneraclara.

Una altra via d’estudi la constitueixen les anomenades transformacions globals. Estracta d’un subgrup de les transformacions W3 de dimensio 8 que preserven la condicioT = W = 0, on T i W son els camps generadors de W3. Aquestes transformacionsglobals son la generalitzacio de las transformacions de Mobius del cas de la simetriaconforme. Aquest ultim cas es formula convenientment com a transformacions queactuen a l’espai projectiu RP 1. Les transformacions globals deW3, aleshores, es naturalassociar-les a l’espai RP 2, que es de dimensio 2. El mecanisme de reduccio hamiltonianade Drinfel’d–Sokolov permet destriar quines tres d’aquestes vuit transformacions sonla realitzacio de la subalgebra conforme (global) de W3. L’estudi de la transformaciod’aquelles dues funcions arbitraries que apareixen a la solucio de l’equacio de movimentsota aquestes transformacions conformes apunta cap a la direccio de tractar-les com aescalars definits a RP 2.

Es per tant natural considerar l’espai RP 2 com l’espai on definir la geometria W3.Cal definir per cada funcio o variable una extensio definida en aquest espai bidimen-sional —espai W3— amb una prescripcio per tornar a l’espai unidimensional. Aquestarelacio essencialment converteix segones derivades en la variable de l’espai unidimen-sional habitual en primeres derivades en la nova variable de l’espai W3. D’aquestamanera un dels problemes de les transformacions W, com es la presencia de derivadesd’ordre alt, desapareix. Aquesta presencia havia portat a una serie d’autors a aban-donar una descripcio geometrica de fibrat tangent per les transformacions W per adop-tar el formalisme de fibrats de jets. Amb la formulacio que es proposa en la memoriaaixo no es necessari. Es mes: l’altra dificultat de les transformacions W, es a dir, lapresencia de termes no lineals, tambe desapareix. Les transformacions infinitesimalsdels objectes estesos permeten aleshores fer una interpretacio geometrica clara de lestransformacions W3: aquestes s’interpreten com a difeomorfismes d’aquest espai W3 iles variables que representen les coordenades de les partıcules (en l’espai estes) es trans-formen com densitats escalars, mentre que els generadors de la simetria es comportencom sımbols de Christoffel. Aquest comportament senzill permet trobar facilment lestransformacions finites dels objectes de la teoria i, en ultima instancia, definir unageneralitzacio de la derivada schwarziana en l’espai W3.

Si be les transformacions W3 s’entenen com a difeomorfismes en l’espai W3, notot difeomorfisme d’aquest espai sera una transformacio W3. D’altra banda, en lesrelacions definitories d’aquest espai W3 hi intervenen els camps de la teoria. Aixo faque l’estructura d’aquests difeomorfismes —anomenats difeomorfismesW3— sigui la dequasigrup. La formulacio del cas W3 en termes d’un sistema d’equacions diferencials

Resum 133

en l’espai W3 permet generalitzar la construccio a les algebres anomenades WN . Lacaracteritzacio dels difeomorfismes W s’interpreta llavors com aquell subconjunt de lestransformacions W que preserva una certa fixacio gauge d’una connexio riemannianadefinida en aquest espai.

Conclusions

S’ha introduıt un model de mecanica de partıcules que es pot formular com una teoriagauge unidimensional invariant sota sp(2M,R). Diferents fixacions parcials del gaugea aquest model, induıdes per inclusions de l’algebra sl(2,R) a sp(2M,R), donen lloc ateories reduıdes en les quals els multiplicadors de Lagrange que sobreviuen es comportencom els generadors d’algebres W associades a la serie Cn d’algebres de Lie. Tambe s’hamostrat com obtenir models invariants sota altres algebres W, com per exemple, lesassociades a la serie An.

Les equacions de moviment per a les variables que expressen la posicio de lespartıcules s’han identificat, en els casos de la inclusio principal de sl(2,R) a sp(2M,R),amb els operadors de Lax associats amb les algebres W corresponents, algebres que,d’aquesta manera, s’interpreten com algebres de simetria de certes equacions diferen-cials. Aquesta imatge s’ha generalitzat per altres algebres W tot trobant les equacionsde moviment —generalment en forma matricial— dels models de partıcula correspo-nents.

S’han descobert una serie de relacions entre algebres W classiques procedents dela mateixa algebra de Kac–Moody. Aquestes relacions s’han trobat efectuant transfor-macions senzilles de les equacions de moviment corresponents. El mateix metode haresultat eficac en la contextualitzacio d’algebres lineals i no locals en el marc de lesreduccions hamiltonianes (secundaries) de Drinfel’d–Sokolov. Com una consequenciad’aquest proces s’ha obtingut un metode directe per trobar el canvi de variables quelinearitza una algebra W.

Pel que fa a l’obtencio de transformacionsW finites, en el cas del model de partıculames senzill, amb simetria conforme, s’ha pogut obtenir la transformacio finita realitzantel proces de fixacio parcial del gauge a nivell finit. L’aplicacio d’aquest mateix proce-diment al model associat a la inclusio (0, 1) de sl(2,R) en sp(4,R) tambe proporcionales transformacions finites del model reduıt. Ara be, aquestes transformacions, tot i seruna parametritzacio acceptable de la simetria gauge residual, no es poden interpretarcom a transformacions W finites estandard ja que, en composar-les, no produeixen lestransformacions de reparametritzacio ordinaries. Per tal d’obtenir les transformacionsW estandard s’hauria de fer un canvi dels parametres infinitesimals de gauge abansde la fixacio d’aquest per tal de modificar les transformacions tipus Yang–Mills de la

134 Resum

mateixa manera a com es fa per obtenir la simetria de reparametritzacio ordinaria.

El tractament de les transformacions W especıfiques en peu d’igualtat amb lesreparametritzacions es, altrament, difıcil d’imaginar a menys que l’espai base originals’estengui a un espai W de dimensio dos. S’ha mostrat com construir aquest espaiW en el cas de W3 aprofitant l’analogia amb el super-espai. Els difeomorfismes W3

s’han calculat i un lligam amb la teoria de sistemes integrables d’equacions diferencialss’ha establert. Apareix una caracteritzacio geometrica clara pels generadors W3 enaquest espai W3: es tracta dels components d’una connexio parcialment restringida.La forma finita de les transformacions W3 s’ha obtingut en aquest formalisme. S’handonat expressions simples per la derivada W-schwarziana. Finalment, s’ha assenyalatcom estendre aquests resultats a les algebres WN .

Per acabar, cal remarcar que l’estudi de les equacions de Lax, procedents dels modelsde mecanica de partıcules, ha estat un molt profitos per obtenir tots aquests resultats.

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