Pro eedings of the Fall Workshop

on Di�erential Geometry

and its Appli ations

Bar elona '93

X. Gr�a ia, M.C. Mu~noz and N. Rom�an, editors

Departament de Matem�ati a Apli ada i Telem�ati a

Universitat Polit�e ni a de Catalunya

editors:

Xavier Gr

�

a ia Sabat

�

e

Miguel C. Mu

~

noz Le anda

Nar iso Rom

�

an Roy

Departament de Matem�ati a Apli ada i Telem�ati a

Universitat Polit�e ni a de Catalunya

Campus Nord, edi� i C3

08071 Bar elona

Catalonia, Spain

e-mail:

xgra ia�mat.up .es

matm ml�mat.up .es

matnrr�mat.up .es

Pro eedings of the Fall Workshop on Di�erential Geometry and its

Appli ations, Bar elona '93

ISBN 84{605{1284{3

Printed in Bar elona by CPET, September 1994

Dip�osit legal B{5477{94 �

Contents

Prefa e v

S hedule vii

List of parti ipants ix

1 Symple ti topology 1

A. Ibort

2 Tangent supermanifold 9

J. Monterde and O. A. S�an hez Valenzuela

3 Lagrangian BRST ohomology 15

J. Mar��n Solano

4 Reparametrization algebras 21

X. Gr�a ia, J.M. Pons and J. Ro a

5 W symmetry and parti le models 29

J. Gomis, J. Herrero, K. Kamimura and J. Ro a

6 Quantization of onstrained systems 37

J.F. Cari~nena

7 Material bodies and elasti ity 47

M. de Le�on and M. Epstein

8 Strati�ed symple ti spa es 55

M. Saralegi

9 String theory and enumerative geometry 63

S. Xamb�o-Des amps

10 Classi al and quantum anomalies 75

M. Asorey

iv DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Prefa e

In O tober 1992 a workshop organised by the Universidad Compluten-

se de Madrid and the Consejo Superior de Investiga iones Cient��� as was

held in Madrid. The aim of the meeting was to stimulate the ollaboration

between di�erent groups working in various �elds of di�erential geometry

and its appli ations to several areas of physi s, as well as the parti ipation

of young resear hers. Some re ent resear h works on these subje ts were

presented.

With the same purpose, a se ond \Fall Workshop" was held in Bar elona

on September 20 and 21, 1993. It was organised by the resear h group on

di�erential geometry and its appli ations of the Department of Applied

Mathemati s and Telemati s of the Universitat Polit�e ni a de Catalunya.

The program in luded ten one-hour talks on the above mentioned subje ts.

We regret that some other interesting ontributions ould not be presented

due to the limited available time.

This volume ontains the summaries of the seminars given by the speak-

ers. We are grateful to them for their agreement to prepare a manus ript

for these pro eedings. Their willingly attitude would have made this book

mu h larger, if limits on their extension had not been pla ed.

As a te hni al remark, the authors sent their ontributions via e-mail.

They were written in di�erent T

E

X diale ts, and we have tried to adapt

them to a uniform style. Under this pro ess, some minor hanges may have

o urred; we hope that they will not be signi� ant. (Well, at times an

obvious misprint has also been orre ted.)

The organisation of the workshop was �nan ially supported by the fol-

lowing institutions:

� Dire i�o General de Re er a of the Generalitat de Catalunya.

� Vi e-re torat de re er a of the Universitat Polit�e ni a de Catalunya.

� Centre de Re er a Matem�ati a (Institut d'Estudis Catalans).

vi DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

To all of them we must express our a knowledgement.

We are also indebted to the S hool of Tele ommuni ations Engineering

of Bar elona, whi h allowed us to use an appropriate room, and to the

Department of Applied Mathemati s and Telemati s of our university, whi h

en ouraged and helped the realisation of the meeting.

At the time of writing these lines, another workshop is being prepared,

now at the Universidad de Granada. This is good news that we hope to be

repeated in forth oming years.

X. Gr�a ia, M.C. Mu~noz and N. Rom�an

Organising Committee

Bar elona, June 1994

S hedule

Monday 20

9h30 Re eption

10h L. Alberto Ibort, Universidad Complutense de Madrid

Topolog��a simpl�e ti a: problemas a tuales y perspe tivas

[Symple ti topology: present problems and perspe tives℄

11h Co�ee break

11h30 Juan Monterde, Universitat de Val�en ia

Problemas de integrabilidad en supervariedades

[Integrability problems on supermanifolds℄

12h30 Jes�us Mar��n, Universitat de Bar elona

Cohomolog��a BRST para sistemas lagrangianos

[BRST ohomology for lagrangian systems℄

14h Lun h

16h Josep M. Pons, Universitat de Bar elona

Transforma iones de gauge �nitas: el �algebra de reparametriza-

iones

[Finite gauge transformations: the reparametrisation algebra℄

17h Jaume Ro a, Universitat de Bar elona

Simetr��as W y modelos de part�� ulas

[W symmetries and parti le models℄

18h Co�ee break

18h30 Jos�e F. Cari~nena, Universidad de Zaragoza

Cuantiza i�on de sistemas on ligaduras

[Quantisation of onstrained systems℄

viii DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Tuesday 21

9h Manuel de Le�on, Consejo Superior de Investiga iones Cient��� as

Cuerpos materiales: elasti idad y geometr��a diferen ial

[Material bodies: elasti ity and di�erential geometry℄

10h Martin Saralegi, Consejo Superior de Investiga iones Cient��� as

Variedades estrati� adas simpl�e ti as

[Symple ti strati�ed manifolds℄

11h Co�ee break

11h30 Sebasti�a Xamb�o, Universitat Polit�e ni a de Catalunya

Sobre variedades de Calabi-Yau

[On Calabi-Yau manifolds℄

12h30 Manuel Asorey, Universidad de Zaragoza

Anomal��as l�asi as y u�anti as en el efe to Hall u�anti o

[Classi al and quantum anomalies in quantum Hall e�e t℄

List of parti ipants

Manuel ASOREY, Universidad de Zaragoza

Jos�e Vi ente BELTRAN, Universitat Jaume I

Mar��a Angustias CA

~

NADAS-PINEDO, Universidad de Granada

Jos�e F. CARI

~

NENA, Universidad de Zaragoza

Carles CURR

�

AS, Universitat de Bar elona

Arturo ECHEVERR

�

IA, Universitat Polit�e ni a de Catalunya

Jos�e FERN

�

ANDEZ N

�

U

~

NEZ, Universidad de Oviedo

Eduardo GALLEGO, Universitat Aut�onoma de Bar elona

Ja ek GANCARZEWICZ, Uniwersytet Jagiello�nski

Joan GIRBAU, Universitat Aut�onoma de Bar elona

Joaquim GOMIS, Universitat de Bar elona

Xavier GR

�

ACIA, Universitat Polit�e ni a de Catalunya

Gregori GUASP, Universitat Aut�onoma de Bar elona

Blas HERRERA, Universitat Aut�onoma de Bar elona

Luis Alberto IBORT, Universidad Complutense de Madrid

Manuel de LE

�

ON, Consejo Superior de Investiga iones Cient��� as

Jes�us MAR

�

IN, Universitat de Bar elona

David MART

�

IN de DIEGO, Consejo Superior de Investiga iones Cient��� as

Eduardo MART

�

INEZ, Universidad de Zaragoza

Eugenio MERINO, Universidade da Coru~na

Juan MONTERDE, Universitat de Val�en ia

Juan Jos�e MORALES, Universitat Polit�e ni a de Catalunya

Miguel C. MU

~

NOZ, Universitat Polit�e ni a de Catalunya

Jordi PAR

�

IS, Universitat de Bar elona

x DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Josep M. PARRA, Universitat de Bar elona

I~naki PELAYO, Universitat Polit�e ni a de Catalunya

Josep M. PONS, Universitat de Bar elona

Rafael RAM

�

IREZ, Universitat Polit�e ni a de Catalunya

Manuel F. RA

~

NADA, Universidad de Zaragoza

Agust�� REVENT

�

OS, Universitat Aut�onoma de Bar elona

Jaume ROCA, Universitat de Bar elona

Nar iso ROM

�

AN ROY, Universitat Polit�e ni a de Catalunya

Alfonso ROMERO, Universidad de Granada

Ceferino RUIZ, Universidad de Granada

Mart��n SARALEGI, Consejo Superior de Investiga iones Cient��� as

Antonio VALD

�

ES, Consejo Superior de Investiga iones Cient��� as

Carles VICT

�

ORIA, Universitat Polit�e ni a de Catalunya

Sebasti�a XAMB

�

O, Universitat Polit�e ni a de Catalunya

Frederi ZAMORA, Universitat de Bar elona

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 1{8 1

Symple ti topology: some problems and

perspe tives

Alberto Ibort

Departamento de F��si a Te�ori a,

Universidad Complutense,

Madrid

1 Introdu tion

In this le ture we will dis uss some of the fas inating aspe ts emerging

in the interplay of Geometry, Physi s and hard Analysis in the arena of

Symple ti Topology. Even if there is no a formal de�nition yet, Symple ti

Topology, or at least some of the problems that most ertainly will de�ne

it, have their origin in Poin ar�e's geometri al theorem.

Theorem 1 [Po12℄[Bi13℄ Any area preserving map of the annulus on itself

su h that it moves its boundary ir les on opposite dire tions, has at least

two �xed points.

It is easy to noti e that the area preserving ondition is ru ial be ause

it is simple to show a di�eomorphism of the annulus moving the boundary

ir les on opposite dire tions having no �xed points. It is unne essary to

re all that �xed point theorems play a entral role in many areas of Math-

emati s and its appli ations. In parti ular if we were able to obtain a proof

for the existen e of a ertain number of �xed points for all di�eomorphisms

of a di�erentiable manifold, we would have obtained a (smooth) topologi al

invariant. In this sense it is very interesting to ompare Poin ar�e's geomet-

ri al theorem with Lefs hetz �xed point theorem for a ompa t manifold

whi h is obtained from interse tion theory.

e-mail: fite207�sis.u m.es

2 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Theorem 2 Any (di�erentiable) map on an orientable smooth manifold

su h that its Lefs hetz number is nonzero, has at least one �xed point. More-

over, if the map is Lefs hetz and homotopi to the identity, the number of

�xed points is bounded below by the absolute value of the Euler-Poin ar�e

hara teristi of the manifold.

2 Arnold's onje ture

Let P be a ompa t manifold and a nondegenerate losed 2-form,

this is, a symple ti stru ture on P . A di�eomorphism � of P will be

alled symple ti if �

�

= . The set of all symple ti di�eomorphisms

form a group, denoted by Sp(P;), that is a lo ally ontra tible topologi al

group. The onne ted omponent of the identity, Sp(P;)

0

, is made of all

symple ti di�eomorphisms � su h that there is an isotopy �

t

onne ting it

with the identity and su h that it is made of symple ti di�eomorphisms.

The time-dependent ve tor �eld X

t

on P de�ned by the formula,

d

dt

�

t

(m) = X

t

(�

t

(m)); 8m 2 P

is lo ally hamiltonian. The isotopy �

t

will be alled hamiltonian if X

t

is

a hamiltonian ve tor �eld; in su h ase there will exist a time-dependent

hamiltonian h:P � [0; 1℄ ! R su h that i(X

t

) = dh

t

, where h

t

(m) =

h(m; t) for every t 2 [0; 1℄. A symple ti di�eomorphism � will be said to

be exa t if there exists a hamiltonian isotopy �

t

su h that � = �

1

. The

set of all exa t symple ti di�eomorphisms is the ommutator of Sp(P;)

0

[Ba78℄.

A �xed point p 2 P of � is nondegenerate if 1 is not an eigenvalue of

its tangent linear map at p, T

p

�, this is, p is nondegenerate if the graph

of � in P � P uts transversally the diagonal set � at p. A symple ti

di�eomorphism will be said to be nondegenerate if all its �xed points are

nondegenerate. Arnold's onje ture an now be stated as follows:

Arnold's onje ture. [Ar72℄[Ar76℄ If � is a nondegenerate exa t sym-

ple ti di�eomorphism on the ompa t symple ti manifold P , then � has

at least as many �xed points as the Morse number of P .

The Morse number of P , denoted by M(P ), is de�ned as the sum of all

Betti numbers B

k

(P ) of P and represents the minimum number of riti al

points that generi ally a smooth fun tion on P has.

1. SYMPLECTIC TOPOLOGY 3

2.1 Lagrangian interse tion theory

Arnold's onje ture an be equivalently stated as an interse tion prob-

lem of Lagrangian submanifolds. A submanifold i:L ! P of a symple ti

manifold (P;) is Lagragian if i

�

= 0 and has maximal dimension [We71℄.

The di�eomorphism � is symple ti if its graph, graph� = f(p; �(p))jp 2 Pg,

is a Lagrangian submanifold of P �P with respe t to the symple ti stru -

ture �

�

1

� �

�

2

where �

i

denote the anoni al proje tions on ea h fa tor.

The �xed points of � are in one-to-one orresponden e with the interse tion

of graph� and the diagonal � = f(p; p)jp 2 Pg that is a Lagrangian sub-

manifold itself. Weinstein's stru ture theorem for Lagrangian submanifolds

[We73℄ shows that there is a (essentially unique) simple ti identi� ation

of a tubular neighborhood on P of the Lagrangian submanifold L and a

tubular neighborhood of the zero se tion Z

L

of the otangent bundle T

�

L

with its anoni al symple ti stru ture.

It the symple ti di�eomorphism � is lose enough (C

1

-small) to the

identity, its graph will be ontained in a tubular neighborhood of the diag-

onal Lagrangian submanifold �, that ould be identi�ed in this way with

a Lagrangian submanifold of T

�

P . From the exa teness of � we obtain

immediately that there exists a smooth fun tion S de�ned in P su h that

its riti al points are pre isely the interse ting points of graph� and �, and

Arnold's onje ture is proved for this family of di�eomorphisms. Noti e

that the nondegenera y ondition for � is equivalent to the fun tion S be-

ing a Morse fun tion and then the number of riti al points of S is bounded

below by the Morse number of P [Bo82℄.

2.2 Status of Arnold's onje ture

Arnold's onje ture has not been proved yet. Weinstein [We84℄ was able

to prove the onje ture for symple ti di�eomorphisms C

0

-small extending

the ideas used to prove the onje ture for di�eomorphisms C

1

-small. The

�rst proof for all exa t symple ti di�eomorphisms ame in [Co83℄ for the

torus T

2n

using a ompletely new set of ideas for the analysis of the a tion

fun tional. Again using very di�erent ideas the onje ture (together with

the degenerate situation) was proved for the omplex proje tive spa e C P

n

in [Fo85℄. Later on, in an extraordinary tour de for e A. Floer was able

to prove the onje ture for symple ti manifolds su h that their se ond

homotopy group vanishes [Fl87℄ and for monotone symple ti manifolds

4 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

[Fl89℄. Floer's ideas are based in a fresh use of Morse theory su h as it

was rephrased by E. Witten [Wi82℄ and that we will sket h in the next

se tion. Re ently, some new results have been obtained for weakly monotone

symple ti manifolds [On94℄ and tori symple ti manifolds [Gi93℄.

2.3 The Arnold's manifesto

After some pioneering work [We77℄, V. Arnold, in the paper [Ar86℄, dis-

ussed a series of problems and established a family of onje tures and re-

sults that had emerged from the study of symple ti and onta t manifolds.

Su h paper was really a manifesto pro laiming that Symple ti Topolgy was

born. Simultaneously, some ru ial results for the development of the foun-

dations of Symple ti Topology were ommuni ated by Gromov in [Gr86℄.

The following paragraphs are verbatim opies of small fragments of the ited

papers.

\By symple ti topology I mean the dis ipline having the same relation

to ordinary topology as the theory of Hamiltonian dynami al systems has to

the general theory of dynami al systems. The orresponden e here is similar

to that between real and omplex geometry..." V. I. Arnold in \First steps

in symple ti topology", (op. it.)

\It seems diÆ ult (if possible at all) to assign a pre ise metamathemati-

al meaning to the notions of relative softness and hardness of an argument

or of a theory...`hard' refers to a strong and rigid stru ture...while `soft'

suggests some weak general property of a vast lass of obje ts...inequalities

and estimates are softer than identities...real analysis is softer than omplex

analysis..." M. Gromov in \Soft and hard symple ti geometry", (op. it.)

3 Morse-Smale-Witten-Floer theory

Morse theory [Bo82℄ was reformulated by Witten in the seminal paper

[Wi82℄ by means of the onstru tion of a omplex whose ohomology o-

in ides with the de Rham ohomology of the manifold. If P is a ompa t

smooth manifold and f is a Morse fun tion on P , we will denote by C(f)

the set of (nondegenerate) riti al points of f and we will use an arbitrary

Riemannian metri g to onstru t the gradient ve tor �eld rf . The index

of a riti al point p is de�ned as the dimension of the negative eigenspa e

of the Hessian quadrati form of f at the riti al point p and it will be

denoted by ind(p). We will also denote by M(p; q) the set of solutions (t)

1. SYMPLECTIC TOPOLOGY 5

of the equation, _ = �rf( ) going from p to q, i.e., (t)! p when t! �1,

and (t) ! q when t ! +1. Su h solutions are alled \instantons" in the

jargon of Quantum Me hani s. There is a natural a tion of R on the set

M(p; q) by traslations. Finally, we will denote by I(p; p) the set:

I(p; q) =

�

0 if ind(p)� ind(q) 6= 1

M(p; q)=R if jind(p)� ind(q)j = 1:

Denoting by C

�

the linear spa e over the �eld Z

2

generated by the points

on C(f) and the homomorphism of linear spa es Æ:C

�

! C

�

de�ned as

Æ(p) =

X

q2C

�

hp; qiq

where hp; qi =

P

2I(p;q)

�( ) and �( ) is �1 depending on the instanton

preserving or not the orientation, we have:

Theorem 3 [Wi82℄ Given an arbitrary ompa t orientable di�erentiable

manifold and a Morse fun tion, there exists a �nite dimensional omplex

(C

�

; Æ), onstru ted using the riti al points of the fun tion and the instan-

tons onne ting them, whose ohomology is pre isely the de Rham ohomol-

ogy of the manifold.

Witten's insight provided a quantum me hani al interpretation of the

ohomology operator Æ as well as a heuristi omputation tool by using path

integral te hniques to express it.

A. Floer [Fl87℄ extended Witten's ideas to the a tion fun tional de�ned

on a symple ti manifold with �

2

(P ) = 0, orresponding to an arbitrary

hamiltonian. If L(P ) denotes the free loop spa e on P and h

t

is a time-

dependent hamiltonian, we will de�ne the a tion fun tional on L(P ) as

A( ) =

Z

�

�

Z

h

t

dt

where �� = . The riti al points of A orresponds to periodi orbits of the

hamiltonian ve tor �eld de�ned by h

t

and onsequently, to the �xed points

of �

1

the exa t symple ti di�eomorphism obtained by the ow �

t

of the

previously de�ned hamiltonian ve tor �eld.

Choosing a suitable metri over P , it is possible to write the equation

d =ds = �rA( ) as

� (s; t)

�s

= J( (s; t))

� (s; t)

�t

+rh

t

( (s; t));

6 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

where J is an almot omplex stru ture de�ned on P by means of and

the metri g hosen previously. We denote by C

�

the linear spa e over

the �eld Z

2

generated by the riti al points of A. The spa e of integral

urves (instantons) of the ve tor �eld rA on L(P ) onne ting the periodi

solutionn

p

with the periodi solution

q

will be denoted byM(p; q). Su h

spa e is R-invariant and we de�ne h

p

;

q

i as the number of points (mod 2)

in M(p; q)=R if dimM(p; q) = 1. The operator Æ:C

�

! C

�

is de�ned as

Æ(

p

) =

X

q

2C

�

h

p

;

q

i

q

and satis�es Æ

2

= 0. See [Ma90℄ for a more detailed des ription of these

ideas in the realm of Witten's topologi al �eld theories. Moreover we have,

Theorem 4 [Fl88℄ Let P be a ompa t symple ti manifold su h that

�

2

(P ) = 0, and � a nondegenerate exa t symple ti di�eomorphism. Let

C

�

be the linear spa e (over Z

2

) generated by the �xed points of �, then,

there exists a linear homomorphism Æ:C

�

! C

�

su h that Æ

2

= 0 and more-

over

ker Æ=ImÆ = H

�

(P;Z

2

):

4 Some re ent results

Some properties of the �xed point sets for equivariant symple ti di�eo-

morphisms have been studied re ently in [Ib93℄. In parti ular an equivariant

version of the results obtained in se tion 2.1 for di�eomorphisms C

1

-small

are des ribed, whose extension to the equivariant ase an be onsidered as

an equivariant version of Arnold's onje ture.

The problem posed by su h equivariant Arnold's onje ture is intimately

related with the relation existing between symple ti redu tion and sets of

�xed points of ertain symple tomorphisms. We have shown that Arnold's

onje ture an be lifted to a universal setting by means of an inverse sym-

ple ti redu tion. In parti ular it has been shown that for every ompa t

symple ti manifold P and every exa t symple ti di�eomorphism �, there

exists an spa e R

2n

and a time-dependent hamiltonian H

t

, su h that the

�xed points of � are in one-to-one orresponden e with ertain riti al sets

of the a tion fun tional de�ned in R

2n

by H

t

[Ib94℄.

A knowledgements. I would like to express my gratitude here to M.

Mu~noz, N. Rom�an and X. Gr�a ia, organizers of this se ond edition of the

1. SYMPLECTIC TOPOLOGY 7

Workshop de Oto~no, for their e�ort in ontinuing these meetings and for

their stubborn determination without whi h these notes would have never

been written.

Referen es

[Ar72℄ V. Arnold. Comment in On a geometri al theorem, in H. Poin ar�e,

Colleted works, vol. II, 987-989, Nauka, Mos ow (1972).

[Ar76℄ V. Arnold. Les m�ethodes math�ematiques de la M�e anique Classique.

Mir, Mos ow (1976).

[Ar86℄ V. Arnold. First steps in symple ti topology. Russ. Math. Surv.,

41, 1-21 (1986).

[Ba78℄ A. Banyaga. Sur la stru ture du groupe des di��eomorphismes que

pr�eservent une forme symple tique. Comment. Math. Helveti i 53,

174-227 (1978).

[Bo82℄ R. Bott. Le tures on Morse theory, old and new. Bull. Am. Math.

So ., 7, 331-358 (1982).

[Bi13℄ G. D. Birkho�. Proof of Poin ar�e's geometri theorem. Trans. Am.

Math. So ., 14, 14-22 (1913).

[Co83℄ Conley, Zendher. The Birkho�-Lewis �xed point theorem and a on-

je ture of V.I. Arnold. Invent. Math., 73, 33-49 (1983).

[Fl87℄ A. Floer. Fixed points of symple ti di�eomorphism. Bull. Am.

Math. So ., 16, 279-281 (1987).

[Fl88℄ A. Floer.Morse theory for Lagrangian interse tions. J. Di�. Geom.,

28, 513-547 (1988).

[Fl89℄ A. Floer. Symple ti �xed points and holomorphi spheres. Com-

mun. Math. Phys., 120, 575-611 (1989).

[Fo85℄ B. Fortune. A symple ti �xed point theorem of C P

n

. Inv. Math.,

81, 29-46 (1985).

[Gi93℄ A. B. Givental. A symple ti �xed point theorem for tori manifolds.

To appear in the volume to the memory of Andreas Floer. Preprint

1993.

[Gr86℄ M. Gromov. Soft and hard symple ti geometry. Pro . ICM, Berke-

ley, 81-98 (1986).

[Ib93℄ A. Ibort, C. Martinez-Ontalba. Fixed point sets of equivariant sym-

ple tomorphisms. Preprint. UCM (1993).

8 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

[Ib94℄ A. Ibort, C. Martinez-Ontalba. An universal setting for Arnold's

onje ture. C. R. A ad. S i. Paris. 318, S�erie II, 561-566 (1994).

[Ma90℄ J. Mateos-Guilarte. The supersymmetri sigma model, topologi al

quantum me hani s and knot invariants. J. Geom. Phys., 7, 255-

302 (1990).

[On94℄ K. Ono. On the Arnold onje ture for weakly monotone symple ti

manifolds. Preprint 1994.

[Po12℄ H. Poin ar�e. Sur un th�eor�eme de g�eom�etrie. Rend. Cir . Math.

Palermo, 33, 375-407 (1912).

[We71℄ A. Weinstein. Symple ti manifolds and their Lagrangian subman-

ifolds. Adv. Math., 6, 329-346 (1971).

[We73℄ A. Weinstein. Lagrangian submanifolds and hamiltonian systems.

Ann. of Math., 98, 377-410 (1973).

[We77℄ A. Weinstein. Le tures on symple ti manifolds. Reg. Conf. Ser.in

Math., 29, Am. Math. So ., Providen e, R.I. (1977).

[We84℄ A. Weinstein. C

0

perturbation theorems for symple ti �xed

points and lagrangian interse tions. S�eminaire Sud-Rhodanien de

G�eom�etrie III. Travaux en Cours 3, Hermann, 140-144 (1984).

[Wi82℄ E. Witten. Supersymmetry and Morse theory. J. Di�. Geom., 17,

661-692 (1982).

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 9{14 2

On the Bat helor trivialization of the tangent

supermanifold

Juan Monterde

Departament de Geometria i Topologia,

Universitat de Val�en ia,

Burjassot

O. A. S

�

an hez-Valenzuela

Centro de Investiga i�on en Matem�ati as,

Guanajuato,

M�exi o

1 Introdu tion

By Bat helor's theorem (also proved by K. Gawedzki [Ga℄), any smooth

graded manifold (M;A

M

) is isomorphi (although not anoni ally) to

(M;�E), where �E is the sheaf of se tions of the exterior algebra bun-

dle �E ! M of a smooth ve tor bundle E ! M de�ned by A

M

. (M;�E)

is then alled the Bat helor trivialization of (M;A

M

).

Our aim in this note is to obtain the Bat helor trivialization of the

tangent supermanifold ST (M;A

M

) of (M;A

M

) in terms of the initial data

M and E, given the fa t that its orresponding stru ture sheaf is Der�E .

We show that the underlying smooth manifold of ST (M;A

M

) is not TM

but TM � E

�

. This re e ts the intrinsi property that the fermioni part

of a graded manifold produ es a new (nontrivial and non-expe ted) bosoni

part in its tangent graded manifold. Furthermore, we ompletely des ribe

the Bat helor bundle as the pullba k to TM � E

�

of the Whitney sum

T

�

M � E � E ( f. Theorem 2 below). In parti ular dimST (M;A

M

) =

(2dimM + rkE; dimM + 2rkE).

e-mail: monterde�vm. i.uv.es, saval�redvax1.dgs a.unam.mx

10 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

2 Chara terization of the derivations

Let E ! M be a rank-n ve tor bundle over M , and let E = �(E)

be its sheaf of se tions. We shall also write E

�

, and �E , for the sheaves

�(E

�

), and �(�E), respe tively. Finally, we shall write X

M

for the sheaf of

se tions of the tangent bundle to M . For this part we shall follow the ideas

of [MoMo℄, [Ro1℄, and [Ro3℄ (We shall refer the reader to [Ko℄ for de�nitions

of graded manifolds and all the related topi s ex ept the on ept of tangent

supermanifold; for the latter we refer to [SV℄).

The sheaf of derivations Der�E is a lo ally free �E-module ( f. [Ko℄).

Moreover, there is a natural in lusion,

0! �E E

�

! Der�E; (1)

de�ned by letting the elements of E

�

a t on �E by ontra tion. There is

also a proje tion of �E-modules,

Der�E ! �E X

M

! 0; (2)

given on homogeneous elements as follows: Let Der

k

�E be the sheaf of

those se tions of Der�E that in rease the degree by k. Let X 2 Der

k

�E ,

and let f 2 �

0

E ' C

1

(M). Then, X f 2 �

k

E , and for any k-tuple of

se tions of E

�

, ('

1

; : : : ; '

k

), the mapping,

f 7! i('

k

) Æ � � � Æ i('

1

)(X f); (3)

de�nes a derivation of C

1

(M). Denote by

^

X('

1

; : : : ; '

k

) this derivation.

It is easy to he k that the map ('

1

; : : : ; '

k

) 7!

^

X('

1

; : : : ; '

k

) is C

1

-linear,

and alternating; it therefore de�nes a se tion of �

k

E X

M

. The maps (1),

and (2), �t together into an exa t sequen e,

0! �E E

�

! Der�E ! �E X

M

! 0: (4)

When a onne tion r in the bundle E is given, this sequen e splits and

therefore,

Der�E ' �E (X

M

� E

�

): (5)

In this des ription one manifestly reads the fa t that Der�E is a �E-module

of rank (m;n). Note that the stru ture of the super otangent sheaf an be

dedu ed from (5):

(Der�E)

�

= Hom (Der�E ;�E) ' �E (

1

M

� E); (6)

where,

1

M

denotes the sheaf of se tions of the otangent bundle to M .

2. TANGENT SUPERMANIFOLD 11

3 The tangent supermanifold

We shall now use the stru tures found in (5), and (6) to produ e

two supermanifolds|the supertangent, and super otangent manifolds to

(M;�E), respe tively|and two submersions|one from ea h of these su-

permanifolds onto (M;�E)|in su h a way that the sheaf-theoreti se tions

of Der�E , and (Der�E)

�

orrespond in a one-to-one fashion with the geo-

metri se tions of these submersions. Thus,

Der�E $ �

�

(M;�E); (STM;�A)

�

(Der�E)

�

$ �

�

(M;�E); (ST

�

M;�B)

�

:

(7)

In order to determine these superve tor bundles we shall take into a ount

the following ( f. [SV℄):

1. Superve tor bundles over (M;�E) orrespond fun torially to lo ally

free sheaves of �E-modules over M , and this fun tor ommutes with

Hom, , and �.

2. There is a universal obje t in the ategory of supermanifolds, R

1j1

,

su h that

�E $Maps

�

(M;�E);R

1j1

�

;

3. Superve tor bundles are lo ally produ ts of the base with a �ber; the

latter being isomorphi to a �xed supermanifold.

Now, the determination of the underlying smooth manifolds STM , and

ST

�

M follows from general prin iples: ea h supermanifold (M;�E) omes

equipped with a sheaf epimorphism, �E ! C

1

M

and hen e, with an exa t

sequen e,

0! N ! �E ! C

1

M

! 0; (8)

where N denotes the nilpotent ideal of �E . The sheaf E of se tions of

the Bat helor bundle E an be re overed from this sequen e by looking at

C

1

� N � N

2

� � �, and observing that E ' N=N

2

. This has the stru ture

of the odd part of the super otangent sheaf sin e N is ontained in the

maximal ideal of vanishing superfun tions ( f. [Ro2℄).

The anoni al epimorphism �E ! C

1

M

an be used to also de�ne a

fun tor from the ategory of lo ally free �E-modules, into the ategory of

lo ally free C

1

M

-modules over M ; namely, any lo ally free �E-module, M,

gives rise to the lo ally free C

1

M

-module,M=(N M). For the supertangent,

and the super otangent sheaves, this fun tor produ es,

Der�E 7! Der�E=(N Der�E) = X

M

� E

�

(Der�E)

�

7! (Der�E)

�

=

�

N (Der�E)

�

�

=

1

M

� E

(9)

12 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Lemma These isomorphisms are independent of the onne tion used to

split the sequen e (4)

In parti ular, the underlying manifolds to the supertangent and to the

super otangent spa es to (M;�E), are respe tively given by,

STM = TM �E

�

; and, ST

�

M = T

�

M �E; (10)

whi h are the ordinary Whitney sums of the given smooth ve tor bundles

over M .

To understand the stru ture of the Bat helor bundles A, and B, of

(STM;�A), and (ST

�

M;�B) we refer ourselves to Proposition 2.9 of [SV℄.

If we apply the general results there obtained to the supertangent and su-

per otangent sheaves, we obtain:

Theorem Let (M;�E) be a graded manifold of graded dimension (m;n).

The Bat helor trivializations of the tangent and otangent supermanifolds

are

(STM;�A) =

�

TM �E

�

;�e�

�

(

1

M

� E � E)

�

(ST

�

M;�B) =

�

T

�

M �E;�e�

�

(X

M

� E

�

� E)

�

:

(12)

Note: The graded dimension of the tangent supermanifold is (2m +

n; 2n+m).

Brief review of the argument Let (M;�E) be a supermanifold. Let

F

0

! M , and F

1

! M be two smooth ve tor bundles of �nite rank over

M (say, p, and q, respe tively), and let F

0

, and F

1

be their orresponding

sheaves of smooth se tions. Let M be a lo ally free sheaf of �E-modules

over M , and assume it has the following stru ture:

M' �E (F

0

�F

1

); (13)

so that M has ZZ

2

-rank (p; q). Then, there is a anoni al isomorphism,

M' Hom

�

(F

0

�F

1

)

�

;�E

�

: (14)

In parti ular, ea h se tion of the sheaf Hom

�

(F

0

� F

1

)

�

;�E

�

extends

uniquely to a se tion,

ZZ

2

-Alg

�

�(F

0

�F

1

)

�

;�E

�

; (15)

of ZZ

2

-graded algebra homomorphisms between sheaves of ZZ

2

-graded al-

gebras. The se tions of the latter, in turn, are in one-to-one orrespon-

den e with maps from the base supermanifold (M;�E), into a superman-

ifold whose stru ture sheaf is �(F

0

� F

1

)

�

. The laim is that these are

2. TANGENT SUPERMANIFOLD 13

pre isely the lo al geometri se tions of the superve tor bundle: maps from

the base into the super�ber.

In fa t, if M is to give rise to a supermanifold (F;�F), equiped with a

supermanifold epimorphism �: (F;�F) ! (M;�E), in su h a way that the

geometri se tions (i.e., maps �: (M;�E) ! (F;�F) su h that � Æ � = id)

orrespond to the sheaf theoreti se tions of M, then, there must be a

anoni al embedding �E ! �F that de�nes �. This must be so, sin e ea h

se tion � gives rise to a superalgebra epimorphism, �

�

: �F ! �E , su h

that, �

�

Æ �

�

= id

�

. In other words, �E must be a anoni al summand

|and in fa t, a subalgebra| of �F . Therefore,

�F ' �(� � � � E): (16)

This yields the global result of the assertion that the supermanifold (F;�F)

must be lo ally trivial; i.e., lo ally the produ t, (M;�E) � (V;�V), of the

base with the super�ber (V;�V). In this situation,

�

�

(M;�E); (F;�F)

�

$Maps

�

(M;�E); (V;�V)

�

$ ZZ

2

-Alg

�

�V ;�E

�

:

(17)

This result, together with (15), ompletes the pi ture given by (16); namely,

�F ' �

�

(F

0

�F

1

)

�

� E

�

: (18)

The only te hni al point is that the Whitney sum of the bundles F

�

0

, F

�

1

,

and E now o urs over the underlying total spa e of the superve tor bundle;

i.e., over F = F

0

�F

1

. This is done by taking the pullba k of su h bundles

along e�:F

0

� F

1

!M .

Corollary Let (M) = �(�T

�

M) be the Cartan algebra of di�erentiable

forms on a smooth manifold M . The tangent supermanifold of the graded

manifold (M;(M)) is

�

TM � TM;�e�

�

(

1

M

�

1

M

�

1

M

)

�

:

Referen es

[Ba℄ Bat helor, M., \The Stru ture of Supermanifolds", Trans.

Amer. Math. So . 253 (1979) 329{338.

[Ga℄ Gawedzki, K., \Supersymmetries|Mathemati s of supergeome-

try", Ann. Inst. H. Poin ar�e Se t. A 27 (1977) 335{366.

[Ko℄ Kostant, B., \Graded Manifolds, Graded Lie Theory and Pre-

quantization", Le ture Notes in Math. vol. 570, (Bleuler, K. and

14 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Reetz, A., eds.), Pro . Conf. on Di�. Geom. Methods in Math.

Phys., Bonn 1975, Springer Verlag, Berlin and New York, 1977,

pp. 177{306.

[MoMo℄ Monterde, J. and Montesinos, A., \Integral Curves of Deriva-

tions", Annals of Global Analysis and Geometry 6 no. 2 (1988)

177{189.

[Ro1℄ Rothstein, M., \Deformations of omplex supermanifolds",

Pro . Amer. Math. So . 95 no. 2 (1985) 255{260.

[Ro2℄ Rothstein, M., \The axioms of supermanifolds and a new stru -

ture arising from them", Trans. Amer. Math. So . 299 (1987) 387{

396.

[Ro3℄ Rothstein, M., \The Stru ture of Supersymple ti Supermani-

folds", Le ture Notes in Physi s vol. 375, (Bartozzi, Bruzzo and

Cian i, eds.), Pro . XIX Int. Conf. on Di�. Geom. Methods in

Math. Phys., Rapallo Italy 1990, Springer Verlag, Berlin and New

York, 1991, pp. 331{343.

[SV℄ S

�

an hez-Valenzuela, O. A., \On Supergeometri Stru tures",

Ph.D. thesis, Harvard University{Cambridge (1986);

\On Superve tor Bundles", Comuni a iones T�e ni as IIMAS-

UNAM (Serie Naranja) 457 (1986).

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 15{19 3

BRST Cohomology in Lagrangian Formalism

Jes

�

us Mar

�

�n Solano

Departament de Matem�ati a E on�omi a, Finan era i A tuarial,

Universitat de Bar elona,

Bar elona

1 Introdu tion

The aim of this paper is to des ribe dynami al systems de�ned by a

Hamiltonian fun tion in a presymple ti manifold in terms of the BRST

ohomology. A BRST formalism an be developed for these systems ex-

tending the ideas of Forger and Kellendok [2℄. This approa h works as fol-

lows. Given a manifold M and a submanifold C, together with a Lie group

a ting on M and leaving C invariant, the algebra of G-invariant fun tions

on C is des ribed in terms of ohomology when C is de�ned as the zero

level of an irredu ible set of (G- ovariant) onstraints. This onstru tion is

independent of any additional stru tures su h as, for example, a symple ti

stru ture on M , and therefore provides, with the appropiated hanges, a

natural framework for a uni�ed des ription of BRST ohomology both for

Lagrangian and Hamiltonian systems.

The s hedule of this paper is as follows. First, we will onstru t the

BRST omplex in the sense of Forger et al [2℄. And se ondly, we will make

use of these ideas to give the general lines to in lude Lagrangian systems

and, in general, presymple ti dynami al systems. This pro ess ontains

two steps: a Hamiltonian regularization and the BRST onstru tion.

e-mail: jmarin�ris d2.ub.es

16 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

2 Constru tion of the BRST omplex for a ertain

lass of dynami al systems

We will assume that the onstraint submanifold C is de�ned as the zero

level of a map ' :M ! V

�

, whereM is a manifold and V is a ve tor spa e.

That is,

C = '

�1

(0):

There exists also a Lie groupG a ting onM . Our obje tive is to des ribe

the G-invariant fun tions on C, that is, C

1

(C=G).

We will onsider only dynami al systems su h that the set of onstraints

formed by the '

i

is irredu ible, or, in other words, 0 2 V

�

is a regular value

of '. The idea onsists in onstru ting two omplexes:

1. The se ond omplex will be simply (�g

�

; d

2

), where d

2

de�nes the

Chevalley ohomology of a Lie algebra:

0! R

d

2

! g

�

d

2

! � � �

d

2

! �

i

g

�

d

2

! �

i+1

g

�

d

2

! � � �

2. The �rst omplex (K; d

1

) will be a ertain Koszul omplex (a y li ),

that is, su h that H

k

d

1

(K) = 0 for all k 6= 0.

Now the BRST omplex (S;D) is onstru ted as the tensor produ t of

the two omplexes,

S = K �g

�

; D = d

1

+ d

2

:

With an appropiate de�nition of the a tion of d

1

and d

2

on the two

fa tors, and under ertain additional onditions, D will de�ne a ohomology

(D

2

= 0), and we will be able to des ribe the fun tions in the quotient spa e

C=G = '

�1

(0)=G in terms of the ohomology of the BRST omplex (S;D).

In our parti ular ase, the two omplexes are the following ones:

1. We have a onne ted (real) Lie group a ting onM , with Lie algebra g.

Therefore, (�

�

g; d

2

) with the Chevalley ohomology is the �rst om-

plex. Moreover, the a tion of G on M indu es an homomorphism of

Lie algebras

g �! X(M)

� 7! ��

M

via the fundamental ve tor �elds given by �

M

(m) =

d

dt

(g(t) �m j

t=0

,

where g(t) is a urve in G with g(0) = 1 and _g(0) = �.

3. LAGRANGIAN BRST COHOMOLOGY 17

2. The �rst omplex is (C

1

(M)�

�i

V; d

1

), where we take the inversed

graduation (i � 0). Therefore,

K

i

= C

1

(M) �

�i

V

and

K = �

i�0

C

1

(M) �

�i

V:

Given the mapping ' : M ! V

�

, the operator d

1

: V ! C

1

(M) is

de�ned as d

1

(v)(m) = '(m)(v), for all m 2M and v 2 V .

Then, the BRST omplex is de�ned as (S;D), where the BRST omplex

S = �

k2Z

S

k

, with S

k

= �

i+j=k

K

i

�

j

g

�

, and the BRST operator D =

d

1

+ d

2

. We de�ne that d

1

a ts trivially on the se ond fa tor, that is,

d

1

(k �) = d

1

(k) �, for any k 2 K, � 2 �g

�

. By other hand, the a tion

of d

2

on K is de�ned by

(�1)

jkj

(d

2

k)(�) = �

M

(�)k;

with � 2 g.

We have to solve two questions now:

� D

2

= 0. Be ause D

2

= d

2

1

+ d

1

� d

2

d

2

� d

1

+ d

2

2

= d

1

� d

2

+ d

2

� d

1

, then

D

2

6= 0 in general. But if ' :M ! V

�

is equivariant, or, equivalently,

�

M

(�)(d

1

(k)) = d

1

(�

M

(�)k), then D

2

= 0 [2℄.

� The a y li ity of K. It an be shown that the regularity of the zero

value is equivalent to the a y li ity ondition on K [2℄.

Therefore, we an ompute the ohomology of S using that H

k

D

(S) =

H

k

d

2

(H

0

d

1

(S)), be ause (K; d

1

) is a Koszul omplex. In parti ular, be ause

d

1

a ts trivially on the se ond fa tor �g

�

,

H

0

D

(S) = H

0

d

2

(H

0

d

1

(S)) = (H

0

d

1

(K))

g

:

But H

0

d

1

(K) = C

1

(M)=C

1

(M) � d

1

(V ) = C

1

(C). Then we have shown

that

H

0

D

(C

1

(M) �V �g

�

) = C

1

(C)

g

:

This method an be applied to des ribe degenerate Lagrangian systems

whose degenera y is entirely due to some kind of gauge invarian e, but not

to the general ase of an arbitrary distribution. In the following se tion we

will give the basi lines to over this point.

18 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

3 BRST ohomology for singular Lagrangians

Let L

0

be an ordinary degenerate Lagrangian fun tion on TQ. The Car-

tan 2-form !

L

0

is degenerate and we will assume that its kernelK = ker!

L

0

de�nes a smooth distribution, in su h a form that K de�nes an integrable

distribution whose se tions are the gauge algebra of the theory. Assum-

ing that there exists a globally de�ned (but non unique) dynami al ve tor

�eld � (i

�

!

L

0

= dE

L

), the true physi al spa e of the theory is the quotient

spa e TQ=K [1℄ [5℄. In this ase, the presymple ti system de�ned by a

presymple ti form (!

L

0

in the Lagrangian formalism) and a Hamiltonian

fun tion (the energy fun tion de�ned by L

0

in our ase) an be regular-

ized [7℄, that is, embedded in a Hamiltonian system using the oisotropi

embedding theorem [3℄.

The oisotropi embedding theorem shows the existen e of a symple ti

manifold (P; !) and an embedding i:TQ! P su h that i

�

! = !

L

0

and TQ

is a oisotropi submanifold (�rst lass) of P . The symple ti manifold P

is expli itely onstru ted as a tubular neighborhood of the zeroth se tion

of the ve tor bundle K

�

! TQ. After the regularization pro edure, we an

obtain a Hamiltonian system (P; !;H).

Now, we an take the map J : P ! K

�

, de�ned as the natural em-

bedding of P in K

�

. It is lear that TQ = J

�1

(0), and if we denote by

K = ker!

L

, then C

1

(TQ)

K

= C

1

(J

�1

(0)=K) = C

1

(TQ=K). Unfortu-

nately, K will not be in general a trivial bundle, so it will not be possible in

those ases to apply the formalism de�ned in the previous se tion. But if

we onsider the Chevalley-Gelfand-Fuks ohomology related to the in�nite-

dimensional Lie algebra de�ned by K, the results before an be extended

in an appropiated form to in lude arbitrary Lagrangian systems (and in

general presymple ti systems) with primary onstraints [7℄.

Finally, we would like to remark that, in some ases, it will be possi-

ble to understand the BRST omplex on a tangent supermanifold [8℄, the

BRST operator as a Hamiltonian superve tor �eld, and to �nd a regular

superlagrangian in this tangent supermanifold that des ribes the dynami s

on TQ=K (see [7℄ for the details).

3. LAGRANGIAN BRST COHOMOLOGY 19

Referen es

[1℄ F. Cantrijn, J.F. Cari~nena, M. Crampin, L.A. Ibort. J. Geom. Phys.,

3, 353-400 (1986).

[2℄ M. Forger, J. Kellendok. Commun. Math. Phys., 143, 235-255 (1992).

[3℄ M. Gotay. Pro . Am. Math. So . 84, 111-114 (1984).

[4℄ M. Henneaux, C. Teitelboim. Commun. Math. Phys., 115, 213-230

(1988).

[5℄ L.A. Ibort, J. Mar��n-Solano. J. Phys. A: Math. Gen. 25, 3353-3367

(1992).

[6℄ L.A. Ibort, J. Mar��n-Solano. Int. J. Mod. Phys. A, Vol. 8, 20, 3565-

3576 (1993).

[7℄ L.A. Ibort, J. Mar��n-Solano. Anales de F��si a, Monograf��as. Vols. 1

and 2. M.O., M.S. and J.M.J. (Eds). CIEMAT/RSEF, Madrid (1993).

Vol. 2, 229-232.

[8℄ L.A. Ibort, J. Mar��n-Solano. Rep. Math. Phys. 32 385-409 (1993).

[9℄ B. Kostant. Le ture Notes in Mathemati s, 570, 177 (1977).

[10℄ G. Morandi, C. Ferrari, G. Lo Ve hio, G. Marmo, C. Rubano. Phys.

Rep., 188, 147 (1990).

20 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 21{27 4

Finite gauge transformations: the

reparametrization algebras

Xavier Gr

�

a ia

Departament de Matem�ati a Apli ada i Telem�ati a,

Universitat Polit�e ni a de Catalunya,

Bar elona

Josep M. Pons

Departament d'Estru tura i Constituents de la Mat�eria,

Universitat de Bar elona,

Bar elona

Jaume Ro a

Department of Physi s,

Queen Mary and West�eld College,

London

Abstra t

Some generalizations of the one-dimensional reparametrization al-

gebra are studied by applying ertain developments on formal inte-

gration of in�nitesimal gauge transformations. The losure of these

algebras is related with the form of the �nite form of the reparamet-

rizations.

1 Introdu tion

Gauge transformations are ontinuous transformations that depend on

arbitrary fun tions of time |or spa e-time variables in �eld theory. The

presen e of these arbitrary fun tions introdu es spe i� features that are

absent in the ordinary symmetries. For instan e, let us onsider the rela-

tion between the in�nitesimal generator and the �nite transformation: both

e-mail: xgra ia�mat.up .es, pons�ebube m1.bitnet, j.ro a�qmw.a .uk

22 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

are onne ted by the exponential map, as usual, but we an ask ourselves

about the relation between the arbitrary fun tions appearing both in the

in�nitesimal generator and in the �nite transformation.

Here we will address this question in the ase of one-dimensional re-

parametrization invarian e. Besides the usual in�nitesimal transformations

asso iated to s alar, ve tor, tensor, et . obje ts, we will study a more general

family of transformations that ontain, as a parti ular ase, the generators

asso iated, in a two-dimensional onformal �eld theory, to the in�nitesimal

transformation of the analyti omponent of the energy-momentum tensor

of the theory [1℄.

All these in�nitesimal transformations will be integrated to obtain their

�nite form. In this pro ess we shall single out the spe ial {and few{ ases

where the �nite form depends on the arbitrary fun tion of the generator only

through the ow of reparametrizations. We will also relate this property

with the losure of the gauge algebra.

More details on these questions, as well as an extension to the super-

symmetri ase, an be found in [2℄.

2 Integration of reparametrization transformations

Let us �rst onsider the s alar ase. The in�nitesimal transformation is

Æq = � _q;

where �(t) is an in�nitesimal arbitrary fun tion whi h is related to an

ordinary fun tion '(t) through an in�nitesimal onstant parameter �:

�(t) = '(t)�.

Let us integrate the in�nitesimal transformation. We have to solve

�

��

q(�; t) = '(t)

�

�t

q (�; t); (1)

with the initial ondition q(0; t) = q(t). Sin e it is a linear equation, it is

trivially integrated as

q(�; t) = e

�'�

t

q(t):

It is easily he ked that this an be written as

q(�; t) = q(f

�

(t)); (2)

where the fun tion f

�

(t) is the ow asso iated to the in�nitesimal generator

'(t)�

t

:

f

�

(t) = e

�'�

t

t:

4. REPARAMETRIZATION ALGEBRAS 23

Now let us onsider the integration of

Æq = � _q + h _�q;

where the real parameter h denotes the tensor hara ter of q under t-

reparametrizations. The orresponding di�erential equation is

�

��

q(�; t) = ('�

t

+ h _')q(�; t); (3)

and it is formally integrated as q(�; t) = e

�('�

t

+h _')

q(t).

Let us insert a unit operator in front of q(t) in the form 1 = e

��'�

t

e

�'�

t

.

The e

�'�

t

part a ts on q(t) produ ing, as in the s alar ase, the result

q(f

�

(t)). On the other hand, the operator

e

�('�

t

+h _')

e

��'�

t

an be shown to a t only multipli atively, and so we an assume it to a t

on the unity. The �nite transformation then reads

q(�; t) =

�

e

�('�

t

+h _')

1

�

q((f

�

(t)):

Now using the equation satis�ed by the hth power of

_

f

�

= �

t

e

�'�

t

t it is

easily he ked that

_

f

h

�

(t) = e

�('�

t

+ _')

1:

Thus we an rewrite the transformation in terms of f

�

only:

q(�; t) = (

_

f

�

(t))

h

q(f

�

(t)): (4)

3 Integration of extended reparametrizations

Next we would like to integrate more general reparametrization trans-

formations. We shall onsider transformations of the type

Æq = � _q + h _�q + ��

(n)

; (5)

where we have introdu ed a dependen e on the nth derivative of the in-

�nitesimal gauge fun tion �. The oeÆ ient � is an arbitrary real parameter,

whi h an be taken to be 1.

Now we shall integrate these generalized transformations. The asso i-

ated di�erential equation is

�

�

q(�; t) = ('�

t

+ h _')q(�; t) + '

(n)

(t); (6)

with the initial ondition q(0; t) = q(t).

We apply the pro edure of variation of onstants: given a linear di�er-

ential equation �q=�� = A(�)q + b(�) with initial ondition q(0) = q

0

, and

24 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

a fundamental matrix V (�) with V (0) = Id |for instan e, V (�) = exp(�A)

when A does not depend on �|, the solution of the non-homogeneous equa-

tion is

q(�) = V (�)q

0

+ S(�);

where S(�) =

R

�

0

d� V (�)b(�� �).

In our problem the linear operator V is V (�)q = (

_

f

�

)

h

(q Æ f

�

), so as

S(�; t) =

Z

�

0

d� (

_

f

�

(t))

h

'

(n)

(f

�

(t)): (7)

The �nite reparametrization is therefore given by (4) plus (7).

Now we an rewrite the integral as

S(�; t) =

Z

�

0

d� (

_

f)

h

�

1

_

f

�

t

�

n

(�

�

f); (8)

where we have used the hain rule together with the basi relation

' Æ f =

�f

��

:

Noti e that the solutions q

�

(4) of the homogeneous problem (3) are

written as fun tions depending only on the ow f

�

of '�

t

. Now we address

the same question for the solutions of the generalized reparametrization

equations. This will be a hieved if and only if the integrand I in (8) is a

total �-derivative.

Leaving the spe ial ase n = 0 aside for a while, we have arrived to an

expression

I(�

t

f; : : : ; �

n

t

f ; �

�

(�

t

f); : : : ; �

�

(�

n

t

f)) = (

_

f)

h

�

1

_

f

�

t

�

n

(�

�

f);

onsidering I as a �rst-order lagrangian in the variables (�

t

f; : : : ; �

n

t

f), then

I is a total �-derivative if and only if its Euler-Lagrange equations vanish

identi ally.

The hange

e

B

= 1=

_

f ;

yields

I = �e

�hB

�

e

B

�

t

�

n�1

�

�

B; (9)

whi h is to be onsidered as a �rst-order lagrangian in the variables

(B; : : : ; �

n�1

t

B).

We begin with the Euler-Lagrange equation of the last variable, whi h

is easily found to be

[I ℄

�

n�1

t

B

= (n� 1� h)e

(n�1�h)B

�

�

B

4. REPARAMETRIZATION ALGEBRAS 25

when n � 2; this equation vanishes if and only if n� h = 1.

Next we onsider the variable �

n�2

t

B:

[I ℄

�

n�2

t

B

= e

(n�1�h)B

�

n(n� 3)

2

�

�

_

B +

(n� 1)(n� 2)

2

(n� 1� h)

_

B�

�

B

�

when n � 3; now this equation vanishes if and only if n� h = 1 and n = 3.

We on lude that if I is a total �-derivative then n � 3, and that for

n = 2 and n = 3 we have h = n � 1; these two parti ular ases an be

examined, and none of their Euler-Lagrange equations introdu e further

restri tions.

Now we onsider the ase n = 1, for whi h I = �e

�hB

�

�

B. The Euler-

Lagrange equation for B vanishes identi ally:

[I ℄

B

= he

�hB

�

�

B � �

�

(�e

�hB

) = 0;

so any value of h is allowable.

Finally we onsider the spe ial ase n = 0, for whi h

I = (

_

f)

h

(�

�

f)

is to be onsidered as a �rst-order lagrangian in two variables (f;

_

f). Now

we obtain

[I ℄

_

f

= h(

_

f)

h�1

�

�

f [I ℄

f

= �h(

_

f)

h�1

�

�

_

f;

and these equations vanish if and only if h = 0.

In on lusion, the solution q

�

(t) depends on t through the reparametriz-

ation fun tion f

�

(t) if and only if (h; n) takes one the following values:

(0; 0); (h; 1); (1; 2); (2; 3): (10)

Let us write the expli it results for S

(h;n)

(�; t) in these ases:

S

(0;0)

(�; t) = f

�

(t)� t � S

0

� t:

S

(0;1)

(�; t) = log

_

f � S

1

:

S

(h;1)

(�; t) =

1

h

(

_

f

h

� 1):

S

(1;2)

(�; t) = �

_

B

�

+

_

B

0

=

�

f

_

f

� S

2

:

S

(2;3)

(�; t) = �

�

B

�

+

1

2

_

B

2

�

=

:::

f

_

f �

3

2

�

f

2

_

f

2

� S

3

;

noti e that S

3

is the s hwarzian derivative of f .

26 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

The ases listed above are the only ones for whi h the �nite transfor-

mation is a lo al fun tional of the ow f

�

.

4 Closure of the algebra

In order to have �nite gauge transformations whi h satisfy the repara-

metrization omposition law we should he k the losure of the in�nitesimal

gauge algebra. It turns out that losure is obtained only for de�nite values of

h and n, in fa t, the same values obtained in the pre eding se tion. Indeed,

the omputation of the ommutator of two transformations (5) gives

[Æ

�

1

; Æ

�

2

℄ q = Æ

�

q +A

(h;n)

(�

1

; �

2

); (11)

where � = _�

1

�

2

� �

1

_�

2

and the fun tion A

(h;n)

is given by

A

(h;n)

(�

1

; �

2

) = �

(n+1)

1

�

2

� �

1

�

(n+1)

2

+ h

�

�

(n)

1

_�

2

� _�

1

�

(n)

2

�

�( _�

1

�

2

� �

1

_�

2

)

(n)

:

The algebra is losed when A

(h;n)

= 0. This ondition is satis�ed for the

following values of (h; n):

(0; 0); (h; 1); (1; 2); (2; 3):

There is still a question to solve: what is the relation between the

losure of the reparametrization algebra and the ow-dependen e of the

�nite reparametrizations? A formal argument follows.

We onsider the in�nite-dimensional Lie algebra X(I) of ve tor �elds on

R vanishing outside a ompa t interval I . Identifying a ve tor �eld '�=�t

with the fun tion ', the Lie bra ket is ['; ℄ = '

_

� _'.

Let Di�eo(I) be the orresponding Lie group of di�eomorphisms. Both

sets are related through the exponential map exp:X(I) ! Di�eo(I), whi h

maps '�=�t 7! exp('�=�t) = f

1

, where f

�

is the ow of '�=�t.

The generalized reparametrizations a t on the in�nite-dimensional ve -

tor spa e C

1

(I;R) of paths in R. The set aff(C

1

(I;R)) of aÆne maps

of this spa e is also an in�nite-dimensional Lie algebra, whose Lie group is

the set A�(C

1

(I;R)) of invertible aÆne maps. Both stru tures are also

related through the exponential map.

Constru ting a reparametrization from a fun tion ' is just an inje tive

linear map

X(I) ! aff(C

1

(I;R))

4. REPARAMETRIZATION ALGEBRAS 27

' 7! Y

'

; Y

'

� q = ' _q + h _'q + '

(n)

:

We have already omputed the exponential of Y

'

: it is the transformation

T

'

= exp(Y

'

) de�ned by (4) plus (7) at � = 1.

We would like to obtain a ommutative diagram with a map between

Di�eo(I) and A�(C

1

(I;R)), or at least between the images of the expo-

nential maps:

Di�eo(I)

X(I)

A�(C

1

(I;R))

aff(C

1

(I;R))

� � �>

-

Y

?

exp

?

exp

f

1

= exp('�

t

)

'

T

'

= exp(Y

'

)

Y

'

� � �>

-

? ?

The losure of the reparametrization algebra means that the image of

the linear map Y is a Lie subalgebra, and indeed then the map Y is a Lie

algebra anti-homomorphism; this indu es a Lie group anti-homomorphism

T between the �nite transformations, and this map loses our diagram.

Referen es

[1℄ A.A. Belavin, A.M. Polyakov and A.B. Zamolod hikov, \In-

�nite onformal symmetry in two-dimensional quantum �eld theory",

Nu l. Phys. B 241 (1984) 333{380.

[2℄ X. Gr

�

a ia, J.M. Pons and J. Ro a, \Closure of reparametriza-

tion algebras and ow-dependen e of �nite reparametrizations", Int. J.

Mod. Phys. A 9 (1994), to appear.

28 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 29{35 5

W symmetry and parti le models

Joaquim Gomis Josep Herrero

Departament d'Estru tura i Constituents de la Mat�eria,

Universitat de Bar elona,

Bar elona

Institut de F��si a d'Altes Energies,

Catalonia

Kiyoshi Kamimura

Department of Physi s,

Toho University,

Japan

Jaume Ro a

Department of Physi s,

Queen Mary and West�eld College,

London

Abstra t

It is des ribed howW transformations an be obtained from ordi-

nary Lie algebras through redu tions of at 2d gauge onne tions in-

du ed by sl(2) embeddings. A parti le me hani s model with Sp(2M)

gauge invarian e is introdu ed whi h en odes the zero- urvature on-

dition as the gauge transformation of Lagrange multipliers. The

sl(2)-indu ed redu tions, reinterpreted as partial gauge-�xings, lead

to parti le models with W symmetry. The issue of integration of

in�nitesimal W transformations is also dis ussed.

1 Introdu tion

Extended hiral symmetries play an important role in 2d onformal �eld

theory, 2d gravity models and integrable hierar hies of non-linear di�erential

e-mail: quim�ebube m1.bitnet, herrero�ebube m1.bitnet, j.ro a�qmw.a .uk

30 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

equations. The study of extensions of the Virasoro algebra in luding bosoni

primary �elds of higher spin was �rst developed by Zamolod hikov [1℄. Su h

extensions turn out to be non-linear algebras, known as W algebras |for a

re ent review on W algebras see [2, 3℄.

Classi al W algebras are obtained by a ontra tion of W algebras

through a !1, �h ! 0 limit, keeping �h onstant; where is the entral

extension of the Virasoro algebra. Two methods have enjoyed mu h su ess

to onstru t these algebras: the Drinfeld-Sokolov (DS) Hamiltonian redu -

tion for Ka -Moody urrent algebras [4℄-[7℄ and the approa h based on the

zero- urvature ondition (ZCC) [8℄-[11℄. The latter gives a pres ription to

determining in�nitesimal W transformations from 2d gauge theories pro-

vided one identi�es a derivative along one of the spa e-time dire tions as

the gauge variation. A major short oming of the DS redu tion and the zero-

urvature method is that they give only \OPEs" of the redu ed algebras,

or equivalently, determine only the in�nitesimal transformations.

2 W transformations from the ZCC

In the zero- urvature approa h to onstru ting W transformations [9℄-

[11℄ one onsiders a maximally non- ompa t real Lie algebra G and a G-

valued gauge onne tion A(z; �z) living in a two-dimensional spa e:

A(z; �z) = A

z

(z; �z)dz +A

�z

(z; �z)d�z = A

a

(z; �z)T

a

;

where T

a

span G. It is assumed to satisfy the ZCC:

F

z�z

= �

�z

A

z

� �

z

A

�z

+ [A

z

; A

�z

℄ = 0; (1)

whi h an be regarded as the ompatibility equation of a linear system of

partial di�erential equations [9℄

�

�

� �A

�z

�

= 0; (� �A

z

) = 0: (2)

After some formal identi� ations:

A

z

! �(t); A

�z

! �(t);

�

z

! d=dt; �

�z

! Æ;

the ZCC has a natural reinterpretation as a one-dimensional transformation

law:

Æ� = _�� [�; �℄; (3)

where � is a 1d gauge onne tion and � denotes the in�nitesimal gauge

parameters.

5. W SYMMETRY AND PARTICLE MODELS 31

Following Polyakov's `soldering' pro edure to obtain lassi al hiral W

transformations [8℄ we make a restri tion on the form of the gauge onne -

tion � (partial gauge-�xing):

� =M +W; (4)

where M is a non-zero onstant element of G and W = W

b

T

0

b

. The Lie

algebra elements T

0

b

span G

W

, a subspa e of G and W

b

are the remnant

omponent �elds of �.

We are looking for the residual gauge transformations (3) preserving the

partial gauge-�xing (4). The possible gauge-�xings, i.e. the hoi es of M

and W , should satisfy the following two requirements:

� We want the gauge-�xing onservation ondition to determine the

non-remnant gauge parameters in terms of the residual ones and the

remnant �elds W

b

in a purely algebrai way.

� The residual transformations should in lude a di�eomorphism (Vira-

soro) se tor in su h a way that we ould identify a weight-two quasi-

primary �eld among the remnant �elds.

Both requirements are satis�ed if the partial gauge-�xing (4) is indu ed

by an sl(2; R) embedding [6, 12℄, S, in the original Lie algebra G. In this

ase we hoose:

M = E

+

; W 2 G

W

= ker adE

�

; (5)

where E

+

, E

�

and h are the de�ning elements of the sl(2; R) embedding:

[h;E

�

℄ = �E

�

; [E

+

; E

�

℄ = h:

Remnant parameters live in ker adE

+

.

The existen e of a di�eomorphism se tor an be shown by performing

a de omposition of parameters: �! (�; �). Consider the following hange:

� = � + ��+ _�H; (6)

whereH =

P

�

k

�

H

�

is a general onstant element of the Cartan subalgebra

H of G and � ontains the other (dim G � 1) parameters so that (6) is an

invertible hange.

The result of the partial gauge-�xing is summarized as follows:

� The �eld T living in the subspa e generated by E

�

transforms as a

quasi-primary �eld of weight two:

ÆT = �

_

T + 2_�T +

1

2

:::

�

:

32 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

� Fields W

i

living in the subspa e H \ G

W

transform as quasiprimary

weight-one �elds with a �� extension term:

ÆW

i

= �

_

W

i

+ _�W

i

+ k

i

��:

� The rest of remnant �elds, W

A

, transform as true primary �elds:

ÆW

A

= �

_

W

A

+ h

(A)

_�W

A

:

These are standard transformations of �elds under in�nitesimal repara-

metrizations t ! t + �. There is no expli it general formula for the trans-

formations generated by the other remnant parameters. They are pre isely

genuine hiralW transformations. Together with the � transformation they

form a set of in�nitesimal transformations with a losed algebra and with

�eld-dependent stru ture fun tions. They ontain a di�eomorphism se tor

with a weight-two quasi-primary �eld T .

3 The parti le model

Let us introdu e a gauge-invariant parti le me hani s model with gauge

group Sp(2M). The parti le model en odes the two-dimensional zero-

urvature ondition leading toW transformations, as the gauge transforma-

tion of a set of Lagrange multipliers.

The model is de�ned in terms of anoni al variables

�

x = (x

�

i

); p = (p

�

j

)

�

,

where �; � denote spa e-time indi es and i; j = 1; : : : ;M label the parti le

multiplet. They are subje t to the onstraints:

�

1

ij

=

1

2

p

i

p

j

; �

2

ij

= p

i

x

j

; �

3

ij

=

1

2

x

i

x

j

:

These 2M

2

+M onstraints lose under Poisson bra ket giving a realiza-

tion of the sp(2M) algebra. The simplest anoni al a tion enfor ing su h

onstraints is

S

C

=

Z

dt

�

p

i

_x

i

� �

A

ij

�

A

ij

�

; A = 1; 2; 3;

where �

A

ij

are the orresponding Lagrange multipliers.

The momenta p an be eliminated using their own equations of motion:

p = �

�1

1

( _x� �

2

x) := K:

The purely lagrangian a tion of the model is

S =

Z

dt

1

2

�

K

>

�

1

K � x

>

�

3

x� F

>

�

1

F

�

; (7)

5. W SYMMETRY AND PARTICLE MODELS 33

where the auxiliary variables F

�

i

are introdu ed to keep losed the sp(2M)

algebra of gauge transformations.

The gauge transformations an be ondensed in a matrix notation:

ÆR = �R; Æ� = _�� [�; �℄; (8)

where

R =

�

x

F +K

�

; � =

�

�

2

�

1

��

3

��

>

2

�

2 sp(2M):

Noti e that the transformation of � is identi al to (3). Hen e the model

in orporates the ZCC in a natural way, the matter transformations and

equations of motion playing the role of the ompatibility equations (2).

Gauge-�xings of this model indu ed by sl(2) embeddings lead to redu ed

models invariant under W transformations [13, 14℄.

The model is, in addition, useful for the onstru tion of �nite W trans-

formations. To �ndW transformations in their �nite form may be a step to

the understanding of the geometry underlying su h symmetry. The sp(2M)

gauge transformations (8) are easily integrated to their �nite form. Then,

�nite W transformations an be obtained by onsidering a general �nite

sp(2M) transformation and imposing in it the gauge-�xing onservation.

4 Redu ed models

The simplest ase is the sp(2) � sl(2) model.

�

We an rewrite the

matrix of Lagrange multipliers � as

� = �

1

E

+

+ 2�

2

h� 2�

3

E

�

;

whi h de�nes the embedding of sl(2) in sp(2). In this simple ase the spa e

of remnant �elds is generated by E

�

alone: G

W

= ker adE

�

= hE

�

i, the

remnant parameter belongs to ker adE

+

and the gauge-�xing is given by

�

1

= 1; �

2

= 0; �

3

:= T: (9)

The asso iated partially gauge-�xed a tion is

S

W

2

=

Z

dt

�

_x

2

2

� T

x

2

2

�

:

There is only one remnant gauge parameter, �, the one asso iated with

di�eomorphisms. The remnant symmetry transformations are

Æx = � _x�

1

2

_�x;

�

Supersymmetri extensions have been studied in [15℄.

34 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

ÆT = �

_

T + 2_�T +

1

2

:::

�

: (10)

These in�nitesimal transformations an be integrated dire tly to give their

standard �nite form under t! f(t) reparametrizations [16℄

x

0

(t) = (

_

f)

�1=2

x(f(t));

T

0

(t) = (

_

f)

2

T (f(t)) +

1

2

_

f

:::

f �

3

2

�

f

2

_

f

2

: (11)

As mentioned before, su h �nite transformations an also be obtained

from the �nite symmetry transformations of the sp(2) model as the rem-

nant transformations onserving the gauge sli e. Indeed, the �nite symme-

try transformations of the sp(2) model an be written as the omposition

of reparametrizations, lo al dilatations and lo al rede�nitions of Lagrange

multipliers, parametrized respe tively by f(t), �(t) and �(t). The omplete

�nite transformation is given by

~x = e

�(f(t))

x(f(t));

~

�

1

=

_

f(t)e

2�(f(t))

�

1

(f(t));

~

�

2

=

_

f(t) [�

2

(f(t)) + �(f(t))�

1

(f(t)) + _�(f(t))℄ ;

~

�

3

=

_

f(t)e

�2�(f(t))

[�

3

(f(t)) + _�(f(t))

+2�(f(t))�

2

(f(t)) + �

1

(f(t))�

2

(f(t))

�

: (12)

Imposing the gauge-�xing onditions (9) on these transformations we

obtain the form of the ompensating � and � transformations in terms of

the di�eomorphism f :

�(t) = �

1

2

ln

_

f(f

�1

(t));

�(t) = � _�(t):

Finally, the insertion of this restri tion in the omposition of �nite trans-

formations (12) gives the �nite residual transformations (11). The inter-

esting point here is that we have been able to integrate the in�nitesimal

transformations (10) without a tually doing it. This may be parti ularly

useful in more involved examples, where the non-linearities inherent to W

transformations make the dire t integration a diÆ ult task.

The sp(4) model has three inequivalent sl(2) embeddings and provides

redu ed models where genuineW transformations are present. They an be

obtained in their �nite form [14℄. However, it is believed that the omposi-

tion of twoW transformations should lose in a di�eomorphism transforma-

tion (plus other transformations su h as dilatations, et .), whi h hanges

5. W SYMMETRY AND PARTICLE MODELS 35

the argument of �elds from t to f(t). Finite W transformations as obtained

here do not share this feature, so, in some sense, they are not parametrized

in a suitable way. In order to obtain �nite W transformations satisfying

this property one should introdu e non-linear hanges of gauge parameters

before the gauge-�xing. This is a subje t under urrent study.

Referen es

[1℄ A. B. Zamolod hikov, Theor. Math. Phys. 63 (1985) 1205.

[2℄ L. Feh�er, L. O'Raifeartaigh, P. Ruelle, I. Tsutsui and A. Wipf, Phys.

Rep. 222 (1992) 1.

[3℄ P. Bouwknegt and K. S houtens, Phys. Rep. 223 (1993) 183.

[4℄ V. Drinfeld and V. Sokolov, J. Sov. Math. 30 (1984) 1975.

[5℄ J. Balog, L. Feh�er, P. Forg�a s L. O'Raifeartaigh and A. Wipf, Ann.

Phys. 203 (1990) 76.

[6℄ F.A. Bais, T. Tjin and P. van Driel, Nu l. Phys. B357 (1991) 632.

[7℄ J.M. Figueroa-O'Farrill, Nu l. Phys. B343 (1990) 450.

[8℄ A.M. Polyakov, Int. J. Mod. Phys. A5 (1990) 833.

[9℄ A. Bilal, V.V. Fo k and I.I. Kogan, Nu l. Phys. B359 (1991) 635.

[10℄ A. Das, W.J. Huang and S. Roy, Int. J. Mod. Phys. A7 (1992) 3447.

[11℄ J. de Boer and J. Goeree, Nu l. Phys. B401 (1993) 369.

[12℄ L. Feh�er, L. O'Raifeartaigh, P. Ruelle and I. Tsutsui, \On the Com-

pleteness of the set of Classi al W algebras obtained from DS Redu -

tions", preprint Bonn-HE-93-14 (1993).

[13℄ J. Gomis, J. Herrero, K. Kamimura and J. Ro a, Progr. Theor. Phys.

91 (1994) 413.

[14℄ J. Gomis, J. Herrero, K. Kamimura and J. Ro a, \Di�eomorphisms,

non-linear W symmetries and parti le models", preprint UTTG-23-93

(1993).

[15℄ K.M. Apfeldorf and J. Gomis, Nu l. Phys. B 411 (1994) 745.

[16℄ X. Gr�a ia, J.M. Pons and J. Ro a, \Closure of reparametrization al-

gebras and ow-dependen e of �nite reparametrizations", Int. J. Mod.

Phys. A 9 (1994), to appear.

36 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 37{45 6

Quantization of onstrained systems

Jos

�

e F. Cari

~

nena

Departamento de F��si a Te�ori a,

Universidad de Zaragoza,

Zaragoza

1 Introdu tion

The mathemati al models for Classi al and Quantum Theories seem

to be di�erent: so, in Classi al Me hani s, the states are points of a phase

spa e, observables are given by fun tions in phase spa e and dynami al evo-

lution is des ribed by Hamilton's equations, while in Quantum Me hani s

the states are rays (rather than ve tors) of a Hilbert spa e H, observables

are selfadjoint operators in H and dynami s is given by S hr�odinger equa-

tion. The mathemati al framework unifying both approa hes is the theory

of Hamiltonian dynami al systems. In the traditional Hamiltonian approa h

the phase spa e is to be geometri ally interpreted as the otangent bundle

T

�

Q of the on�guration spa e Q. It an be endowed with a symple ti

stru ture in a anoni al way. A more general approa h would be to on-

sider as fundamental obje t a symple ti manifold, whi h also allows for a

des ription of the Lagrangian approa h and even more general situations.

We re all that (M;!) is a symple ti manifold if M is a �nite-

dimensional di�erentiable manifold and ! is a losed nondegenerate 2-form,

! 2 Z

2

(M). The nondegenera y of ! is equivalent, by de�nition, to the

property that the maps !̂

u

:T

u

M ! T

�

u

M , given by h!̂

u

(v); v

0

i := !

u

(v; v

0

)

have maximal rank. Noti e that !̂:TM ! T

�

M is a base-preserving �bered

map and indu es a mapping between their se tions. If H 2 C

1

(M), the

Hamiltonian ve tor �eld X

H

asso iated with the Hamiltonian H is the

e-mail: jf � .unizar.es

38 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

unique ve tor �eld satisfying !̂(X

H

) = dH:

A separable omplex Hilbert spa e H is anoni ally endowed with an

exa t symple ti stru ture. A tually, H is a real linear spa e and T

�

H is

to be identi�ed with H, for any � 2 H, by means of the Abelian traslation

group.

For instan e, if H = C

N

, with omplex oordinates z

i

=

1

p

2

(q

i

+ ip

i

),

then (q

0

; p

0

) 2 C

N

orresponds to the ve tor q

0

�

�q

j(q;p)

+ p

0

�

�q

j(q;p)

.

Let � be the 1-form � 2

V

1

(H) de�ned by �

�

( ) = � Imh�; i: Then,

the 2-form = �d� is a symple ti form given by

�

(D

; D

�

) = 2 Imh ; �i:

In parti ular, when H = C

N

, the oordinate expressions of � and are

� =

P

N

i=1

1

2

(p

i

dq

i

� q

i

dp

i

) and =

P

N

i=1

dq

i

^ dp

i

: Therefore (q

i

; p

i

) are

Darboux ordinates.

A ontinuous ve tor �eld is just a ontinuous map A:H ! H; therefore

a linear operator A on H is a spe ial kind of ve tor �eld. In parti ular, let

A be a selfadjoint operator and onsider the real fun tion a(�) = h�;A�i:

Then, the Hamiltonian ve tor �eld orresponding to the fun tion a de�ned

by the selfadjoint operator A is X

a

(�) = �i A�, and therefore, when A

is the quantum Hamiltonian H of a system, we see that the S hr�odinger

evolution equation is nothing but the \Hamilton equations" for the lassi al

Hamiltonian dynami al system (H;; h).

The map ':A(H)! C

1

(H) is inje tive, be ause the relation h�;A�i =

h�;B�i, 8� 2 H, A;B 2 A(H), implies that A = B. If we re-

all that X

f'(A);'(B)g

= �[X

'(A)

; X

'(B)

℄; we see that X

f'(A);'(B)g

=

�[�iA;�iB℄ = �iX

'([A;B℄)

; and therefore f'(A); '(B)g = �i '([A;B℄):

In parti ular, when B = H we reobtain the so alled Ehrenfest prin iple:

X

h

'(A) = f'(A); '(H)g = �i'([A;H ℄):

In general the problem of quantization of a system is the onverse,

namely, given a lassi al Hamiltonian system (M;!;H), the aim is to

�nd a Hilbert spa e H and a map � : C

1

(M) ! A(H) su h that

�i�(ff; gg) = [�(f); �(g)℄. The idea of Geometri Pre-quantization is to

onstru t a U(1)-bundle overM and a onne tion in it su h that its urva-

ture oin ides with the pull-ba k of !. Then H is given by a linear spa e of

square-integrable se tions (with respe t to the measure !

^n

) in the bundle

and �(f) for f 2 C

1

(M) is the horizontal lift of the Hamiltonian ve tor

�eld X

f

2 X(M).

6. QUANTIZATION OF CONSTRAINED SYSTEMS 39

2 Quantization methods for onstrained systems

The aim is to onstru t a quantum system orresponding to a on-

strained lassi al system: a presymple ti system (M;!;H). Very often M

is a submanifold j :M ! N of a symple ti manifold (N;) and ! = j

�

.

For instan e, if the lassi al system is des ribed by a singular Lagrangian,

then either (TQ; !

L

; E

L

) or (FL(TQ); j

�

!

0

; H) are the presymple ti sys-

tems to be onsidered. The problem is to hoose a Hilbert spa e H and an

asso iation of selfadjoint operators in H (quantum observables) with las-

si al observables following some quantization rules. The Geometri Quan-

tization te hniques fail for many reasons: �rst, !

^n

= 0, and se ond, the

asso iation of fun tions in M with ve tor �elds in M is not one-to-one.

There are di�erent ways of solving the problem:

A: First redu tion, then quantization

It onsists on doing the lassi al Hamiltonian redu tion and then the

appli ation of GQ te hniques (or something similar else) to the redu ed

system. A disadvantage of the method is that the fa tor spa e M=Ker!

is not always a manifold, the redu tion pro ess may be diÆ ult, and even

worse, ovarian e properties are lost.

B: Dira 's quantization approa h

Dira onsidered the ase (M = FL(TQ); j

�

!

0

; H) and he suggested to

forget in a �rst step the onstraint fun tions �(q; p) de�ningM , and quantize

the system (T

�

Q;!

0

; H). On e the Hilbert spa e has been onstru ted, a

Hilbert subspa e is to be sele ted by the onditions �(q; p)jvi = 0. However,

if there exists a se ond lass onstraint fun tion , su h a Hilbert subspa e

redu es to the trivial zero ve tor. A tually, it is then possible to hoose

another onstraint � su h that f�; g = 1, and therefore, if jvi is su h

allowed ve tor and the quantization rules are preserved, [�;℄jvi = 0 =

i �hjvi: Dira solved su h diÆ ulty by eliminating the se ond lass onstraints

and introdu ing a new Poisson bra ket (now alled Dira bra ket). This

orresponds to the \Coisotropi Embedding Theorem"

Theorem (Gotay 82) Let (M;!) be a presymple ti manifold. If E is

the hara teristi bundle Ker!

L

, then there exists a symple ti stru ture

in a neighborhood of the zero se tion of the dual bundle E

�

su h that M

is embedded oisotropi ally as the zero se tion of E

�

. This embedding is

essentially unique.

Theorem (Sniaty ki 74) Let (M;!) be a pre-symple ti manifold j :

40 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

M ! P embedded in a symple ti manifold (P;) in su h a way that

! = j

�

. Then there exists a symple ti manifold (

~

P ;

~

) of P su h that M

is oisotropi ally embedded in

~

P .

As a onsequen e of this theorem we an always redu e the problem

to the ase in whi h no se ond lass onstraints appear. The se ond lass

onstraints are those determining

~

P and the original symple ti stru ture

is hanged to the symple ti stru ture in

~

P , i.e. it will in lude the se ond

lass onstraint fun tions.

In many ases both quantization pro edures oin ide but it is not always

the ase.

C: Other quantization approa hes

There exist other quantization approa hes, the most important one be-

ing the so allled BRST quantization pro edure, and group quantization

methods based on a previos hoi e of a symmetry group. We will only om-

ment the method proposed by Faddeev and Ja kiw for quantization of a

parti ular type of singular systems.

3 Faddeev-Ja kiw Quantization Metod

Faddeev and Ja kiw proposed an alternative method for quantization

of onstrained systems des ribed by �rst order in velo ities Lagrangains:

L = m

j

(q) v

j

� V (q): These Lagrangians are relevant be ause they arise

in several problems: the dynami s in light- one quantization is �rst order,

there is a des ription �rst order in time for 1+1 hiral bosons, onventional

se ond order formulations are redu ible to �rst order, the limit of a parti le

moving in a strong magneti �eld, et . The main referen e is: Faddeev L.

and Ja kiw R., Hamiltonian redu tion of un onstrained and onstrained

systems, Phys. Rev. Lett. 60, 1692 (1988).

A ording to Ja kiw, Dira 's method is umbersome and an be stream-

lined and simpli�ed. In the new method onstraints are never introdu ed

and the the distin tion of �rst lass and se ond lass onstraints is unne -

essary. Moreover, the method is generalizable in order to in lude Grassman

variables. Let us �nally remark that it is not a method of quantization but

a method of doing a lassi al Hamiltonian redu tion of dynami s. For a

re ent review, see R. Ja kiw, \(Constrained) Quantization Without Tears",

in Constraint Theory and Quantization Methods, edited by F. Colomo, L.

Lusanna and G. Marmo (World S ienti� , 1994), pp. 163{175.

6. QUANTIZATION OF CONSTRAINED SYSTEMS 41

The energy of su h Lagrangian that has no se ond order kineti term is

but V (q). The Euler-Lagrange equations de�ned by L are

�m

i

�q

k

_q

k

=

�m

k

�q

i

_q

k

�

�V

�q

i

=)

�

�m

i

�q

k

�

�m

k

�q

i

�

_q

k

= �

�V

�q

i

:

Sin e the E-L equations are �rst order one an assume that only oordinates

are relevant and if one assumes that there exists a Hamiltonian des ription,

then the evolution equation would be written _q

i

= fV; q

i

g =

�V

�q

j

fq

j

; q

i

g.

The idea of F-J is that if the matrix A

ij

=

�m

j

�q

i

�

�m

i

�q

j

is invertible, then

the Poisson bra ket matrix is to be de�ned by fq

i

; q

j

g = A

�1

ij

.

The geometri theory was developed by J.F. Cari~nena, C. L�opez and

M.F. Ra~nada, J. Math. Phys. 29, 1134 (1988). Let the tangent bundle

� : TQ ! Q be the velo ity phase spa e. The Lagrangian L 2 C

1

(TQ)

an be written as L = b�� �

�

V , with V 2 C

1

(Q), and � 2

V

1

(Q) given by

� = m

j

(q) dq

j

. The fun tion b� 2 C

1

(TQ) is de�ned by b�(q; v) = h�

�(v)

; vi.

Then the 1-form �

L

and the energy fun tion are �

L

= �

�

(�), and E

L

=

�

�

(V ):

The Hessian matrix W is null: W

ij

= �

2

L=�v

i

�v

j

� 0, and therefore

all the � -verti al ve tors are in the kernel of !

L

. The sear h for the other

elements in kernel of !

L

starts by looking for a basis fZ

a

= z

i

a

g of the

module of eigenve tors of the zero eigenvalue of A

ij

.

A) The regular ase:

The simplest ase, �rst studied by F-J, is when the matrix A is regular,

i.e., detA 6= 0; then (Q; d�) is a symple ti manifold, and Ker!

L

redu es to

the set of � -verti al ve tors. The redu ed spa e is Q itself and the redu ed

symple ti stru ture is �d�. There will be no dynami al onstraint fun tion

and the solution of the dynami al equation is X = �

i

�

�q

i

+ f

i

�

�v

i

, with the

fun tions �

i

uniquely determined by �

i

= (A

�1

)

ij

�V

�q

j

.

The Hamiltonian H is de�ned on the primary onstraint submanifold

P

1

= FL(TQ), determined by �

j

(q; p) = p

j

�m

j

(q); by the restri tion of the

fun tion

e

V . Then P

1

is the graph of the form � and with the identi� ation

of it with the base Q the pull-ba k of the anoni al 1-form !

0

in T

�

Q is

�d�. It is easy to he k that f�

j

;�

k

g = A

jk

, and therefore, when d� is

symple ti all the onstraints are of the se ond lass, and the general theory

leads again to the study of the Hamiltonian dynami al system (Q;�d�; V ),

as in the Lagrangian ase.

B) The singular ase:

42 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

In a more general ase, a basis for Ker!

L

is made from z

i

a

�

�q

i

and

�

�v

i

,

with a = 1; : : : ; n � n

0

, n

0

= rankA. The primary onstraint submanifold

M

1

is then determined by the onstraint fun tions �

a

= X

a

E

L

= z

i

a

�

i

with X

a

= z

i

a

�

�q

i

+ f

i

�

�v

i

2 Ker!

L

and �

i

=

�L

�q

i

� v

k

�

2

L

�v

i

�q

k

; that in the

ase we are onsidering are �

a

= z

i

a

�V

�q

i

, be ause �

i

= A

ik

v

k

+

�V

�q

i

: The

point is that the onstraint fun tions are basi fun tions, or in the physi s

terminology, holonomi onstraints, de�ning a submanifold Q

0

of Q. The

se ondary onstraints for se ond order equations will be given by linear

fun tions in the velo ities

d

d�

a

.

The general solution for the equation i(X)!

L

= dE

L

will be X =

�

�

i

+ �

a

z

i

a

�

�

�q

i

+f

i

�

�v

i

; with �

i

being a solution of A

ij

�

j

=

�V

�q

i

and where

�

a

and f

i

are arbitrary fun tions on TQ. Noti e however that su h a so-

lution is not the restri tion of a SODE but only in those points of TQ for

whi h A

ij

v

j

=

�V

�q

i

. The general solution of the dynami al equation is given

by

X =

�

v

i

+ �

a

z

i

a

�

�

�q

i

+ f

i

�

�v

i

;

while the SODE ondition orresponds to the hoi e �

a

= 0.

The method developed by F-J for dealing with the singular ase is based

in Darboux Theorem and therefore is only valid lo ally. The idea is that if

� is of onstant lass 2s + 1, then there exist new oordinates (�q

a

; �p

a

; z

l

),

with a = 1; : : : ; s, l = 2s + 1; : : : ; 2n, su h that the 1-form � is written

� = dz

2s+1

+ �p

a

d�q

a

. If � is of onstant lass 2s there will exist lo al

oordinates (�q

a

; �p

a

; z

l

), with a = 1; : : : ; s, l = 2s + 1; : : : ; 2n, su h that

� = dz

2s+1

+ �p

a

d�q

a

. In these new oordinates L is written L = �p

a

_

�q

a

�

�(�q

a

; �p

a

; z

l

) and the onditions

��

�z

l

= 0 an be used to eliminate some z

l

in terms of �q

a

; �p

a

. The other variables z

k

whi h annot be eliminated are

renamed �

l

and onsidered as Lagrangian multipliers and we arrive to a

Lagrangian L

0

= �p

a

_

�q

a

� �

k

�

k

(�q

a

; �p

a

), and the pro ess an be iterated.

I do not know any geometrization of this approa h. I feel it will be

based on the on ept of lass of a 1-form and a theorem like the oisotropi

embedding theorem.

Bar elos-Neto and Wotzasek (Mod. Phys. Lett. 7, 1737 (1992), Int. J.

Mod. Phys. 7, 4981 (1992)) showed by means of several examples how to

pro eed. The idea is the following:

Every su h a onstraint an be in orporated in a new Lagrangian L

de�ned in the tangent bundle of a new on�guration spa e Q of the form

6. QUANTIZATION OF CONSTRAINED SYSTEMS 43

Q = Q� R

(n�n

0

)

, by

L(q

i

; �; v

i

; �) = L(q

i

; v

i

) + �

a

d�

a

(q

i

; v

i

);

where (q

i

; �

a

; v

i

; �

a

) denote the oordinates on the tangent bundle TQ.

The expressions for !

L

and E

L

are �

L

= �

L

+ �

a

g

d�

a

=) !

L

= !

L

+

g

d�

a

^ d�

a

; and E

L

= E

L

; be ause the

d�

a

-term will not ontribute to the

energy fun tion E

L

, for �(

d�

a

)�

d�

a

= 0.

The pre eding expression shows that the rank of of !

L

may be higher

than that of !

L

and this is the starting point in the F-J approa h. When

ne essary, this pro edure should be iterated.

The Fundamental Question is whether the substitution of L by L is jus-

ti�ed. Let L 2 C

1

(TQ) be a regular Lagrangian fun tion and suppose that

the system is onstrained by �(q

i

; v

i

) = 0: Let de�ne Q = Q � R, the o-

ordinates of TQ being (q

i

; �; v

i

; �) and let L 2 C

1

(TQ) be L(q

i

; �; v

i

; �) =

L(q

i

; v

i

) + ��(q

i

; v

i

):

L is singular and it has a onstant rank i� � = ~'+ �̂; with ' 2 C

1

(Q)

and � 2

V

1

(Q). Then, !

L

= !

L

+ ~� ^ d�� �

f

d�; E

L

= E

L

� � ~'.

In the ase of holonomi onstraints, � = 0 and the Kernel of !

L

, de�ned

by !

L

is generated by Z

1

=

�

��

, and Z

2

=

�

��

, leading to the onstraint

' = 0. The same is true if there are several holonomi onstraints.

Let us now onsider the ase in whi h L is singular and all onstraint

fun tions �

a

= X

a

E

L

, a = 1; : : : ; k are holonomi . Let us onsider an

extended on�guration spa e R

k

� Q and denote � the natural proje tion

� : R

k

� Q ! Q and T� : T (R

k

� Q) ! TQ the orresponding tangent

map. If L 2 C

1

(T (R

k

�Q)) is de�ned by L

1

= (T�)

�

L+�

a

�

a

; sin e !

L

1

=

(T�)

�

!

L

; E

L

1

= (T�)

�

E

L

��

a

�

a

; it is an easy task to he k that Ker!

L

1

is generated by the set of ve tor �elds proje ting onto the ve tor �elds X

a

of Ker!

L

, plus �=��

a

and �=��

a

. The onstraint fun tions determined by

�=��

a

are then the (pull-ba k of the) original primary onstraint fun tions

�

a

. On the other hand, the other onstraints X

a

(X

b

E

L

)�

b

� 0 an be

seen as a homogeous system for the unknowns �

a

and this system has a

linear set of solutions of dimension n� n

0

� k, where k = rankX

a

X

b

(E

L

).

The di�eren e between solutions of the dynami al equations for L and L

is a ve tor �eld Z whose restri tion to the primary onstraint submanifold

satis�es i(Z)!

L

= �

a

d�

a

. The se ond order ondition leads us to onsider

the tangent bundle of the new on�guration spa e R

k

�Q

0

.

The same approa h an be done when we use

d

d�

a

instead of �

a

as

44 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

onstraint fun tions and we repla e the original Lagrangian L for L

2

=

L+ �

a

d

d�

a

. In this ase E

L

2

= (T�)

�

E

L

, and !

L

2

= (T�)

�

!

L

+ d�

a

^ d�

a

,

from whi h it is easy to see that

X = �

i

�

�q

i

+ �

a

�

��

a

+ �

i

�

�v

i

+ �

a

�

��

a

is in Ker!

L

2

i�

A� � �

a

r�

a

�W� = 0; �:r�

a

= 0; W� = 0:

The last equation is the same as in the original ase, and the se ond one says

us that we will only obtain the onstraints orresponding to ve tor �elds in

Ker!

L

tangent to the primary onstraint submanifold.

Taking into a ount that �

a

d

d�

a

=

d

d(�

a

�

a

)� �

a

�

a

, we see that L

2

may

be repla ed by L

3

= L� �

a

�

a

, whi h is quite similar to L

1

with the hange

of �

a

for its velo ity �

a

.

Example: Bar elos-Neto and Wotzasek (Mod. Phys. Lett. 7, 1737

(1992))

Let us onsider a non-relativisti point pati le moving on a sphere. The

Lagrange fun tion is L =

1

2

_q

2

+

�

2

(q

2

� 1); for whi h the Hamiltonian is

H =

p

2

2

�

�

2

(q

2

� 1); the �rst order Lagrangian des ribing the system being

L = p _q �

p

2

2

+

�

2

(q

2

� 1):

If we start with L = p _q �

p

2

2

+

�

2

(q

2

� 1); the matrix A

ij

is singular:

A =

0

�

0 �Æ

km

q

k

Æ

km

0 0

�q

k

0 0

1

A

The zero eigenve tor is (0; q

m

; 1) and therefore the onstraint is q � p, whi h

will be introdu e by means of a new multiplier � and in this step the La-

grangian is L

0

= p _q �

p

2

2

+

�

2

(q

2

� 1) +

_

� q � p and the new matrix A is

A =

0

B

B

�

0 �Æ

km

q

k

p

k

Æ

km

0 0 q

k

�q

k

0 0 0

�p

m

�q

m

0 0

1

C

C

A

whi h is a regular matrix with inverse given by

A

�1

=

0

B

B

B

�

0 Æ

km

�

q

k

q

m

q

2

�q

k

0

�Æ

km

+

q

k

q

m

q

2

q

k

p

m

�p

k

q

m

q

2

p

k

�q

k

q

m

�p

m

0 �1

0 q

m

1 0

1

C

C

C

A

6. QUANTIZATION OF CONSTRAINED SYSTEMS 45

from whi h we obtain the generalized bra kets

fq

k

; q

m

g = 0; fq

k

; p

m

g = Æ

km

�

q

k

q

m

q

2

; fp

k

; p

m

g =

q

k

p

m

� p

k

q

m

q

2

:

46 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 47{54 7

Material bodies, elasti ity and di�erential

geometry

Manuel de Le

�

on

Instituto de Matem�ati as y F��si a Fundamental,

Consejo Superior de Investiga iones Cient��� as,

Madrid

Mar elo Epstein

Department of Me hani al Engineering,

University of Calgary,

Calgary

Abstra t

The presen e of inhomogeneities is re e ted through the la k of

integrability of the asso iated G-stru ture, where G is the group of

material symmetries of the material body.

1 Introdu tion

The theory of distributions of inhomogeneities in elasti bodies was orig-

inally on eived by Kondo, Eshelby, Kroner, and others, as the result of a

limiting pro ess starting from a defe tive rystalline stru ture. It was later

obtained by Noll and Wang as a geometri theory based on the proper-

ties of the material response fun tion alone. The use of G-stru ture theory

has re�ned the formulation and fa ilitated the derivation of spe i� results.

In fa t, the presen e of inhomogeneities, su h as dislo ations and dis li-

nations, manifests itself through the la k of integrability of the asso iated

G-stru ture. In this talk we resume the results obtained in the papers

[7, 4, 5, 8℄ for materials of higher grade.

e-mail: eeml02� . si .es, mepstein�a s.u algary. a

48 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

2 Bodies of higher grade

A body B is a 3-dimensional di�erentiable manifold whi h an be overed

with just one hart. An embedding � : B �! R

3

is alled a on�guration

of B and its k-jet j

k

X;�(X)

� is alled a lo al on�guration at X . The body

is identi�ed with any one of its on�gurations, say �

0

: B �! R

3

, alled

a referen e on�guration. Given any arbitrary on�guration � : B �! R

3

,

the hange of on�gurations � = � Æ �

�1

0

is alled a deformation, and its

k-jet j

k

�

0

(X);�(X)

� is alled a lo al deformation at �

0

(X). The me hani al

behaviour of a material body is hara terized by one fun tion W whi h

depends, at ea h point of B, only on the value of the derivatives of the de-

formation evaluated at that point. The order of the highest derivative whi h

appears in the des ription is alled the grade of B. If W = W (j

k

X;�(X)

�)

then B is said to be of grade k. Materials of higher grade appears for

instan e in theories su h as so- alled polar elasti ity [18, 9℄.

B is said to be materially uniform if for two arbitrary points X;Y 2 B

there exists a lo al di�eomorphism � from a neighbourhood of Y onto a

neighbourhood of X su h that �(Y ) = X and

W (j

k

X;�(X)

�) =W (j

k

X;�(X)

� � j

k

Y;X

�) ; (1)

for all lo al deformations.

Let G(X;Y ) = fj

k

Y;X

� j (1) holds for all j

k

X;�(X)

�g. Hen e G(X) =

G(X;X) is the group of material symmetries at X . Let �

k

(B;B) � J

k

(B;B)

be the Lie groupoid of invertible k-jets with sour e and target maps � and �,

respe tively. We assume that (B) =

S

X;Y 2B

G(X;Y ) is a Lie subgroupoid

of �

k

(B;B). Noti e that the property of global uniformity is equivalent to

the transitivity of (B). The map � � � : (B) �! B � B is a surje -

tive submersion and, hen e there exist lo al se tions [12℄. A lo al se tion

P

0

(X;Y ) is alled a lo al material uniformity. In su h a ase we say that the

body B enjoys lo ally smooth uniformity. If there exists a global se tion P

0

of ��� then we say that the body B enjoys smooth uniformity [2℄ and P

0

is

alled a material uniformity. In what follows we shall assume that B admits

a global material uniformity P

0

.

Choose X

0

2 B arbitrarily and de�ne P (X;Y ) = P

0

(X;X

0

) �

P

0

(Y;X

0

)

�1

. We have P (X;Y ) 2 G(X;Y ) and P (Z; Y ) � P (Y;X) =

P (Z;X), P (X;X) = 1

X

, i.e., P (X;Y ) is a material uniformity in the sense

of Noll [15℄. Put

X

0

(B) = �

�1

(X

0

). If we de�ne P (X) = P (X;X

0

), then

P (X) is a di�erentiable global se tion of � :

X

0

(B) �! B. P (X) was

7. MATERIAL BODIES AND ELASTICITY 49

alled a uniform referen e by Noll [15℄ in the ase of material bodies of �rst

grade. We have

Proposition 1

X

0

(B) is a prin ipal �bre bundle over B with stru ture

group G(X

0

).

Let F

k

B be the frame bundle of order k of B [3, 11, 13, 20℄. If we

hoose a k-frame Z

0

2 F

k

B at X

0

then G = Z

�1

0

� G(X

0

) � Z

0

is a Lie

subgroup of G

k

(3) alled the group of material symmetries of B. A global

se tion � : B �! F

k

B is de�ned by �(X) = P (X) � Z

0

. Thus, � is a

parallelism of order k on B. If we extend �(B) by the a tion of G we obtain

a G-stru ture !

G

(B) = �(B)G of order k on B (see [11℄ as an standard

referen e for G-stru tures of higher order and [10, 16℄ for the order one).

The map !

G

(B) �!

X

0

(B), �(X)A �! P (X) � Z

0

� A is an isomorphism

of prin ipal bundles over B. The global se tions P and � are indistin tly

alled a uniform referen e on B.

A body B is said to be homogeneous if it admits a global deformation

� su h that � = �

�1

is a uniform referen e, say �(X) = j

k

0;X

(�

�1

Æ �

�(X)

),

8X 2 B, where �

�(X)

: R

3

�! R

3

denotes the translation on R

3

by the

ve tor �(X). B is said to be lo ally homogeneous if every X 2 B has a

neighbourhood whi h is homogeneous.

Theorem 1 B is lo ally homogeneous if and only if !

G

(B) is integrable.

3 Bodies of �rst grade

If B is a uniform body of �rst grade, then � : B �! FB is an ordinary

parallelism, say � = (�

1

; �

2

; �

3

), where �

1

; �

2

; �

3

are three linearly indepen-

dent ve tor �elds on B whi h de�ne a linear onne tion � by r

�

i

�

j

= 0.

� has torsion but no urvature. The torsion tensor T of � is alled the

inhomogeneity tensor [15℄.

Theorem 2 (Noll [15℄, Truesdell and Noll [18℄) If B is homogeneous, then

there exists a uniform referen e P with zero inhomogeneity. Conversely, if

B admits a uniform referen e P with zero inhomogeneity, then it is lo ally

homogeneous.

Next, we an onsider G-stru tures de�ned by extending to the whole

group G of material symmetries a given global se tion �. For a detailed

study of this ase we remit to [19, 1, 6℄.

50 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

4 Bodies of se ond grade

If B is a uniform body of se ond grade, then � : B �! F

2

B is a

parallelism of se ond order. The group G

2

(3) may be identi�ed with the

semidire t produ t Gl(3;R) / S

2

(3), where S

2

(3) is the ve tor spa e of

bilinear symmetri maps � : R

3

� R

3

�! R

3

. The group multipli ation

is given by (A;�)(A

0

; �

0

) = (AA

0

; �(A

0

; A

0

) + A�

0

) [3℄. � indu es a global

se tion p : B �! FB, and an invariant se tion q : FB �! F

2

B. Thus, p

indu es a at linear onne tion �, and q a symmetri linear onne tion �

[11℄. We all D = �� � the inhomogeneity tensor.

Theorem 3 [4, 5, 8℄ If B is homogeneous, then there exists a uniform

referen e P with zero inhomogeneity. Conversely, if B admits a uniform

referen e P with zero inhomogeneity, then it is lo ally homogeneous.

As above, we shall onsider G-stru tures de�ned by extending to the

group of material symmetries G a given global se tion �. Sin e G will be a

subgroup of G

2

(3) we �rst give a lassi� ation of su h subgroups.

4.1 The subgroups of G

2

(n)

A dire t omputation shows that (Gl(n); 0) and (1; S

2

(n)) are subgroups

of G

2

(n).

4.1.1 General subgroups

For every subgroup

�

h of G

2

(n), �g = �

2

1

(

�

h) is a subgroup of Gl(n), where

�

2

1

: G

2

(n) �! Gl(n) is the anoni al proje tion. The proje tion of

�

h by

�

2

: G

2

(n) �! S

2

(n) is denoted by �

�g

. Then the most general subgroup

of G

2

(n) is of the form

�

h = (�g;�

�g

), where �g is a subgroup of Gl(n) and �

�g

is a subset of S

2

(n). For ea h g 2 �g, we denote by �

g

the subset of S

2

(n)

de�ned by �

g

= f� 2 S

2

(n) j (g; �) 2

�

hg. It is not diÆ ult to see that the

subset (1;�

1

) is a subgroup of G

2

(n) and that �

1

is an additive subgroup

of S

2

(n). Moreover there exists a bije tion between �

1

and �

g

.

4.1.2 Toupin subgroups

Let �g be a subgroup of Gl(n), and let S be a �xed element of S

2

(n).

Then (�g; �s) � (�g; S(�g; �g)��gS) is a subgroup of G

2

(n). Indeed, it is the result

of the onjugation (�g; �s) = (1; S)(�g; 0)(1; S)

�1

. Conversely, the onjugate

of any subgroup of the form (�g; 0) by a �xed element (H;T ) is of the given

7. MATERIAL BODIES AND ELASTICITY 51

form. This kind of subgroups will be alled subgroups of Toupin, sin e he

[17℄ appears to have postulated that these are the only ones to be onsidered

as symmetries of se ond grade materials.

4.1.3 Generalized Toupin subgroups

Among the general subgroups

�

h = (�g;�

�g

) of G

2

(n), we single out those

for whi h �

1

= 0, and hen e �

g

onsists, for ea h g 2 �g, of a single el-

ement. We shall all these groups generalized subgroups of Toupin, for

obvious reasons. We an easily prove that every one-parameter subgroup

�

h = exp t(A;�) of G

2

(n) is a generalized subgroup of Toupin if A 6= 0. How-

ever, there are one-parameter subgroups whi h are not generalized Toupin

subgroups. For instan e, exp t(0; �) = (1; t�), where � 6= 0.

4.1.4 The subgroups (1;�

1

) and their onjugates

Consider a subgroup ofG

2

(n) of the form (1;�

1

). Then �

1

is an additive

subgroup of S

2

(n). Noti e that S

2

(n) is isomorphi as a ve tor spa e to R

m

,

where m =

1

2

n(n+ 1). By using a result of Morris [14℄ we dedu e that the

losed subgroups of S

2

(n)

�

=

R

m

are:

- dis rete subgroups Z

r

,

- ve tor subspa es R

r

,

- or mixed subgroups R

p

� Z

r�p

.

The onjugate subgroups of (1;�

1

) by an arbitrary element (A;S) 2

G

2

(n) is (A;S)(1;�

1

)(A;S)

�1

= (1; A�

1

(A

�1

; A

�1

)). Thus the element

S is not relevant for onjugation of subgroups of the form (1;�

1

). Hen e

we shall only onsider the onjugate subgroups obtained by onjugation of

(1;�

1

) with an element A 2 Gl(n).

4.2 Integrability of some G-stru tures of se ond order

4.2.1 Parallelisms of se ond order

The most simple

�

h-stru ture of se ond order on B is a trivial stru ture,

say a (1; 0)-stru ture. In other words, a parallelism of se ond order � :

B �! F

2

B. In su h a ase, there are no material symmetries. Then we

have

Theorem 4 A parallelism of se ond order is integrable if and only if the

inhomogeneity tensor D = �� � vanishes.

52 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

4.2.2 Toupin stru tures

We all a

�

h-stru ture on B a Toupin stru ture when the stru ture group

is a Toupin subgroup. Then we put

�

h = (�g; �s) = (�g; S(�g; �g)� �gS), where �g

is a subgroup of Gl(3) and S is a given element of S

2

(3). Sin e (�g; �s) is the

onjugate subgroup of the subgroup (�g; 0), then the (�g; �s)-stru ture

�

H

0

is

onjugate to the (�g; 0)-stru ture

�

H , i.e.,

�

H

0

=

�

H (1; S). Hen e the problem

of the integrability of

�

H

0

is redu ed to de ide about the integrability of

�

H .

Denote by

�

G the proje ted �g-stru ture of the �rst order. Then we have the

following.

Theorem 5 If

�

H is integrable then � is a lo ally at onne tion. Con-

versely, if � is a lo ally at �g- onne tion,

�

H is integrable.

4.2.3 (1;�

1

)-stru tures and their onjugates

As in the ase of Toupin stru tures, we must only he k the integrability

of

�

h = (1;�

1

)-stru tures. Consider a (1;�

1

)-stru ture

�

H on B. We an

de�ne for ea h S 2 �

1

a symmetri linear onne tion on B as follows.

Consider the global se tion �

S

= � (1; S). Hen e the global se tion �

S

determines a symmetri linear onne tion �

S

. We have

Theorem 6 If

�

H is integrable then there exists a lo al oordinate system

(x

i

) around ea h point of B su h that the Christo�el omponents of � belong

to �

1

. Conversely, if there exists an element S 2 �

1

su h that �

�S

= �

then

�

H is integrable.

4.2.4

�

h = (�g;�

�g

)-stru tures su h that (�g; 0) is a subgroup of

�

h

Let

�

h = (�g;�

�g

) a subgroup of G

2

(3) su h that (�g; 0) is a Lie subgroup

of

�

h. Of ourse, (1;�

1

) is a Lie subgroup of

�

h. Now, let

�

H a

�

h-stru ture on

B. We have the following.

Theorem 7 If

�

H is integrable then there exists a lo al oordinate system

(x

i

) around ea h point of B su h that the Christo�el omponents of � belong

to �

1

. Conversely, if � is a lo ally at �g- onne tion,

�

H is integrable.

Referen es

[1℄ F. Bloom: Modern Di�erential Geometri Te hniques in the Theory of

Continuous Distributions of Dislo ations, Le ture Notes in Math., 733,

Springer, Berlin, 1979.

7. MATERIAL BODIES AND ELASTICITY 53

[2℄ H. Cohen, M. Epstein, Remarks on uniformity in hyperelasti materi-

als, Int. J. Solids Stru tures, 20, 3 (1984), 233-243.

[3℄ L. A. Cordero, C. T. J. Dodson, M. de Le�on: Di�erential Geometry of

Frame Bundles, Mathemati s and Its Appli ations, Kluwer, Dordre ht,

1989.

[4℄ M. de Le�on, M. Epstein: On the integrability of se ond order G-

stru tures with appli ations to ontinuous theories of dislo ations, Rep.

Math. Phys. 33,3 (1993), 419-436.

[5℄ M. de Le�on, M. Epstein: Corps mat�eriels de degr�e sup�erieur, C. R.

A ad. S i. Paris I319, (1994).

[6℄ M. Elzanowski, M. Epstein, J. Sniaty ki: G-stru tures and material

homogeneity, Journal of Elasti ity, 23 (1990), 167-180.

[7℄ M. Elzanowski, M. Epstein: On the symmetry group of se ond-grade

materials, Int. J. Non-Linear Me hani s, 27, 4 (1992), 635-638.

[8℄ M. Elzanowski, S. Prishepionok: Preprint, Portland State University,

1993.

[9℄ A. C. Eringen, Ch. B. Kafadar: Polar Field Theories, Continuum

Physi s, Ed. A. Cemal Eringen, vol. IV, Part I, pp. 1-73, A ademi

Press, New York, 1976.

[10℄ A. Fujimoto: Theory of G-stru tures, Publi ations of the Study Group

of Geometry, Vol. I. Tokyo, 1972.

[11℄ S. Kobayashi: Transformation Groups in Di�erential Geometry,

Springer, Berlin, 1972.

[12℄ K. Ma kenzie, Lie groupoids and Lie algebroids in Di�erential Geome-

try, London Mathemati al So iety Le ture Note Series, 124, Cambridge

Univ. Press, Cambridge, 1987.

[13℄ P. Molino: Th�eorie des G-Stru tures: Le Probl�eme d'Equivalen e, Le -

ture Notes in Math., 588, Berlin, Springer, 1977.

[14℄ S. A. Morris: Pontryaguin Duality and the Stru ture of Lo ally Com-

pa t Abelian Groups, London Mathemati al So iety Le ture Note Se-

ries, 29, Cambridge University Press, 1977.

[15℄ W. Noll: Materially Uniform Simple Bodies with Inhomogeneities,

Ar h. Rational Me h. Anal. 27, (1967), 1-32.

[16℄ S. Sternberg: Le tures on Di�erential Geometry, 2nd edition, Chelsea,

New York, 1983.

54 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

[17℄ R. A. Toupin: Theories of Elasti ity with Couple-stress, Ar h. Rational

Me h. Anal. 17, (1964), 85-112.

[18℄ C. Truesdell, W. Noll: The Non-Linear Field Theories of Me hani s,

Handbu h der Physik, Vol. III/3, Springer, Berlin, 1965.

[19℄ C. C. Wang: On the Geometri Stru tures of Simple Bodies, a Mathe-

mati al Foundation for the Theory of Continuous Distributions of Dis-

lo ations, Ar h. Rational Me h. Anal. 27, (1967), 33-94.

[20℄ P. Ch. Yuen: Higher order frames and linear onne tions, Cahiers de

Topologie et G�eometrie Di��erentielle, XII, 3 (1971),333-371.

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 55{62 8

Strati�ed symple ti spa es

Martin Saralegi

Instituto de Matem�ati as y F��si a Fundamental,

Consejo Superior de Investiga iones Cient��� as,

Madrid

Fa ult�e Jean Perrin,

Universit�e d'Artois,

Lens

\Let M be a Hamiltonian G-spa e with momentum map J :M ! G

�

,

where G

�

is the dual of the Lie algebra G of G. Marsden and Weinstein [4℄

have shown that if 0 2 G

�

is a regular value of the momentum map and G

a ts freely on the manifold J

�1

(0), then the redu ed spa e M

0

= J

�1

(0)

has the stru ture of a symple ti manifold. If the regularity onditions are

dropped, the zero level set is no longer a manifold and there are jumps in the

orbit types of the points in J

�1

(0). However, the quotient M

0

still makes

sense topologi ally, and it an be regarded in a natural way as a subset of

the whole orbit spa eM=G. We show that the spa eM

0

de omposes into a

union of symple ti manifolds that �t together in a ` onelike' manner. We

all su h an obje t a strati�ed symple ti spa e"

In this way was presented in [5℄ the new important notion of strati�ed

symple ti spa e. In this talk we intend to give an elementary geometri

presentation of strati�ed symple ti spa es, we also present some of the

main results of the Sjamaar and Lerman's work.

We observe that a study (parallel to that of [5℄) about osymple ti

redu tion for singular values has been done in [3℄.

e-mail: saralegi� . si .es, saralegi�gat.univ-lille1.fr

56 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

1 Strati�ed spa es

A strati�ed spa e is a topologi al spa e admitting a partition in man-

ifolds, alled strata. They do not have ne essarily the same dimension.

Ea h stratum meets the other ones in a oni al way. This notion was intro-

du ed by R. Thom [6℄. This kind of spa e o urs often in the mathemati al

univers: orbit spa es of ompa t Lie groups, semialgebrai systems, omplex

algebrai varieties, simpli ial omplexes, leaf spa es of singular riemannian

foliations, Whitney strati�ed spa es . . . Before giving a pre ise de�nition of

this on ept we present some exemples.

1.1 Cone. Let M be a ompa t manifold. The one M is the produ t

M � [0; 1[ with the boundary ollapsed to a point V . This spa e is not a

manifold (if M is not a sphere!). Noti e that we have the de omposition

M = fV g

|{z}

singular stratum

[ M � [0; 1[

| {z }

regular stratum (dense)

:

Even if the one is not a so�sti ated exemple it is very important sin e it

des ribes the lo al stru ture of a strati�ed spa e (remember that the lo al

stru ture of a manifold is the one S

n

). We shall write

t

M the one of

ray t > 0, that is, the produ t M � [0; t[ with the same identi� ation.

1.2 Joint. Let M;N be two ompa t manifolds. The joint M � N is

the produ t M �N � [�1; 1℄ with the the following identi� ations on the

boundaries:

(x; y;�1) � (x

0

; y;�1) and (x; y; 1) � (x; y

0

; 1) for x; x

0

2M and y; y

0

2 N:

8. STRATIFIED SYMPLECTIC SPACES 57

M �N

M �N

q

�

N

�

p

M

t

N

t

M

T

T

T

T

T

T

T

T

T

T

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

X

X

X

X

X

X

X

X

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

�

X

X

X

X

X

X

X

X

T

T

T

T

T

T

T

T

T

T

X

X

X

X

X

X

X

X

�

�

�

�

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

�

�

�

�R

�

�

�

��

This spa e is not a manifold (ex ept when M and N are spheres): a point

p 2 M has a system of neighborhoods of the form M �

t

N and a point

q 2 N has system of neighborhoods of the form M �

t

N , t > 0. Noti e

that the transversal stru ture ofM (resp. N) is given by the one N (resp.

M). In this ase we have the de omposition:

M �N = M [N

| {z }

singular strata

[ M �N� ℄� 1; 1[

| {z }

regular stratum (dense)

:

1.3 In the above exemples, the singular part � of the strati�ed spa e is

a manifold. This is not the general pi ture, the subset � an be mu h

more ompli ated than this. In fa t, � is in general a singular spa e. For

exemple, it is enough to mix up the two above exemples; the one (M �N)

is a strati�ed spa e whose strata are: fV g; M� ℄0; 1[; N� ℄0; 1[ (singular)

and M �N� ℄� 1; 1[ � ℄0; 1[ (regular). Here, we have � = (M [N).

1.4 De�nition. A strati� ation S = fS

�

g

�2�

of a topologi al spa e Z

is a lo ally �nite olle tion of lo ally losed submanifolds verifying

1

:

(1) Z =

[

�2�

S

�

,

(2) frontier ondition: S

�

\ S

�

6= ; , S

�

� S

�

, � � �, and

(3) there exists ! 2 � with S

!

= Z.

1

� is a partially ordered set

58 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

The stratum S

!

is alled regular stratum while the others are said singu-

lar strata. A system of tubes f�

�

:T

�

! S

�

g

�2�

is a family of tubular

neighborhoods of fS

�

g

�2�

verifying

(4) the �ber of �

�

is the one of a strati�ed spa e L

�

, alled the link

of S

�

, and

(5) �

�

Æ

�

�

= �

�

on T

�

\ T

�

if � � �.

We observe that the stratum S

�

has transversally a oni al stru ture given

by the one L

�

. The last ondition indi ates that the tubular neighbor-

hoods �t together in a parti ularly ni e way. In the previous exemples, these

neighborhoods are nothing but produ t spa es.

A strati�ed spa e is a topologi al spa e Z endowed with a strati� a-

tion and a system of tubes.

2 Smooth stru ture on a strati�ed spa e

In spite of the fa t that strati�ed spa es are not manifolds, smoothness

still has a meaning in this ontext.

2.1 De�nitions. A ontinuous map f :Z ! R is said to be smooth if

the restri tion to a stratum S

�

is smooth in the ususal sense. A smooth

stru ture C

1

(Z) on Z is an algebra made up of smooth fun tions. Given

two strati�ed spa es Z and Z

0

with smooth stru tures C

1

(Z) and C

1

(Z

0

),

a ontinuous map �:Z ! Z

0

is smooth if for any f 2 C

1

(Z

0

) the ompo-

sition f

Æ

� is smooth, f

Æ

� 2 C

1

(Z).

In this ontext, a ve tor�eld on Z is just a derivation X :C

1

(Z) !

C

1

(Z). Consider a generalized urve (t) on Z (the omposition f

Æ

is a

smooth fun tion in the usual sense). For any smooth fun tion f 2 C

1

(Z)

we asso iate the smooth fun tion X(f) 2 C

1

(Z) de�ned by

X(f)( ) = (f

Æ

)

0

:

This gives a derivation X :C

1

(Z) ! C

1

(Z), that is, a ve tor�eld on Z.

The map is the ow of X .

2.2 Example. If the strati�ed spa e Z is a losed subset of a smooth

manifold, we de�ne

C

1

(Z) = ff :Z ! R j there exists F 2 C

1

(Z) with f = F j

Z

g

whi h is a smooth stru ture on Z (see [8℄).

8. STRATIFIED SYMPLECTIC SPACES 59

3 Strati�ed spa es and a tions

Let �:G�M !M be an a tion of a ompa t Lie group on a manifold

M . It is well known that the orbit spa e M=G is a strati�ed spa e (see for

example [7℄). The strata of M=G are des ribed as follows.

3.1 For a subgroup H of G denote by M

(H)

the set of all points x of

M whose stabilizer G

x

is onjugate to H . The set M

(H)

is an invariant

submanifold of M (not ne essarily onne ted). So, the quotient M

(H)

=G

is also a manifold. The strata of the orbit spa e are just the onne ted

omponents of ea h M

(H)

=G. Noti e that M is also a strati�ed spa e in

su h a way that the orbit map �:M !M=G is a strati�ed morphism. Using

an invariant metri on M (G is ompa t!) one an onstru t a system of

tubes invariant under �; this gives a system of tubes in M=G.

3.2 There is a natural smooth stru ture on this spa e M=G: the smallest

subalgebra of C

0

(M=G) ( ontinuous fun tions) making � smooth, that is,

C

1

(M=G) = ff :M=G! R j the omposition f

Æ

� is smoothg:

3.3 Example. Consider the a tion : (S

1

� S

1

� S

1

) � C

m

! C

m

given

by

(z

0

; z

1

; z

2

) � (u

0

; : : : ; u

n

1

; v

0

; : : : ; v

n

2

; w

0

; : : : ; w

n

3

) =

(z

0

u

0

; : : : ; z

0

u

n

1

; z

1

v

0

; : : : ; z

1

v

n

2

; z

2

w

0

; : : : ; z

3

w

n

3

);

where m = n

1

+ n

2

+ n

3

+ 3. Noti e that this a tion preserves the usual

metri on C

m

: So, it indu es an a tion

�: (S

1

� S

1

� S

1

)� S

2m�1

�! S

2m�1

;

in the unit sphere. Let us exhibit the strata of the orbit spa e Z =

S

2m�1

Æ

(S

1

� S

1

� S

1

): A straightforward al ulation gives

�

S

2m�1

�

f1g�S

1

�S

1

= S

2n

1

+1

;

�

S

2m�1

�

S

1

�f1g�S

1

= S

2n

2

+1

;

�

S

2m�1

�

S

1

�S

1

�f1g

= S

2n

3

+1

;

�

S

2m�1

�

S

1

�f1g�f1g

= S

2(n

2

+n

3

)+3

�

�

S

2n

2

+1

[ S

2n

3

+1

�

;

60 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

�

S

2m�1

�

f1g�S

1

�f1g

= S

2(n

1

+n

3

)+3

�

�

S

2n

1

+1

[ S

2n

3

+1

�

and

�

S

2m�1

�

f1g�f1g�S

1

= S

2(n

1

+n

2

)+3

�

�

S

2n

1

+1

[ S

2n

2

+1

�

:

So, the singular strata of Z are

C P

n

1

; C P

n

2

; C P

n

3

;

C P

n

2

� C P

n

3

�

�

C P

n

2

[ C P

n

2

�

;

C P

n

1

� C P

n

3

�

�

C P

n

1

[ C P

n

3

�

and

C P

n

1

� C P

n

2

�

�

C P

n

1

[ C P

n

2

�

:

4 Singular redu tion

We now onsider the singular redu tion of a Hamiltonian group a tion

�:G � (M;!) ! (M;!) as des ribed in the introdu tion. When 0 2 G

�

is not a regular value of J then the redu ed spa e M

0

= J

�1

(0)=G is

not a manifold but a singular manifold. Nevertheless, this spa e inherits

from (M;!) a symple ti stru ture. In order to des ribe this phenomenon

Sjamaar and Lerman introdu ed the following notion.

4.1 De�nition. A symple ti strati�ed spa e X is a strati�ed spa e

with a smooth stru ture C

1

(X) su h that:

1. Ea h stratum S is a symple ti manifold.

2. C

1

(X) is a Poisson algebra.

3. The embeddings S ,! X are Poisson.

4.2 We have already seen that M

0

is a strati�ed spa e with strata

(M

0

)

(H)

= J

�1

(0) \M

(H)

Æ

G; H subgroup of G:

By ombining x3.1 and x4.1, Arms, Cushman and Gotay [1℄ de�ned the

following smooth stru ture C

1

(M

0

). A fun tion f

0

:M

0

! R is said to be

smooth if there exists a G-invariant fun tion

~

f 2 C

1

(M) with

~

f j

Z

= �

�

f

0

.

They also showed that the algebra C

1

(M

0

) inherits a Poisson algebra

stru ture from C

1

(M): if f; g 2 C

1

(M

0

) then ff; gg

M

0

2 C

1

(M

0

) is de-

�ned by

g

ff; gg

M

0

= f

~

f; ~gg

M

: Sjamaar and Lerman proved that this Poisson

8. STRATIFIED SYMPLECTIC SPACES 61

bra ket is ompatible with the symple ti forms on the strata ofM

0

; in fa t,

they showed the following result.

4.3 Theorem. Let (M;!) be a Hamiltonian G-espa e with momentum

map J :M ! G

�

. The redu ed spa e J

�1

(0)=G is a strati�ed symple ti

spa e, with the above stru ture.

The following example is taken from [2℄.

4.4 Example. The (1,1,-1,-1)-resonan e is the ir le a tion on C

4

gener-

ated by the Hamiltonian

J(z

1

; z

2

; z

3

; z

4

) = 1=2(jz

1

j

2

+ jz

2

j

2

� jz

3

j

2

� jz

4

j

2

):

We des ribe the topology of the redu ed spa e at the zero level. The zero

level set is given by the equation jz

1

j

2

+jz

2

j

2

= jz

3

j

2

+jz

4

j

2

: Consider the unit

seven sphere S

7

in C

4

, given by the equation jz

1

j

2

+ jz

2

j

2

+ jz

3

j

2

+ jz

4

j

2

= 1:

Its interse tion with the zero level set of J is an S

1

-invariant submanifold

of C

4

, namely the produ t of two three-spheres of radius 1/2,

J

�1

(0) \ S

7

= S

3

� S

3

� C

2

� C

2

:

The S

1

-a tion on the �rst opy of S

3

is given by e

i�

� (z

1

; z

2

) = (e

i�

z

1

; e

i�

z

2

);

and on the se ond opy of S

3

is given by e

i�

� (z

3

; z

4

) = (e

�i�

z

3

; e

�i�

z

4

): The

quotient of S

3

�S

3

by S

1

is denoted by S

3

�

S

1

S

3

: It is an S

3

-bundle over the

omplex proje tive line C P

1

, asso iated to the Hopf �bration S

3

! C P

1

:

Topologi ally, the redu ed spa e (C

4

)

0

an now be written as the one

(S

3

�

S

1

S

3

):

4.5 Dynami s on the redu ed spa e. Dynami s on a strati�ed sym-

ple ti spa e are introdu ed by a Hamiltonian fun tion h 2 C

1

(M

0

). Then

the motion equations are given by:

X

h

(f) = ff; hg

M

0

for any f 2 C

1

(M

0

) ;

whi h is written in terms of the Poisson bra ket be ause M

0

is not a mani-

fold. Sjamaar and Lerman proved that, for any Hamiltonian h 2 C

1

(M

0

),

there exists a unique ve tor �eld X

h

2 X(M

0

) satisfying the motion equa-

tions. The ve tor �eld X

h

is alled the Hamiltonian ve tor �eld asso iated

62 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

to h. Its Hamiltonian ow preserves the symple ti pie es of M

0

. The re-

stri tion of to a pie e (M

0

)

(H)

equals the Hamiltonian ow of the fun tion

h j

(M

0

)

(H)

:

Referen es

[1℄ J. Arms, R. Cushman and M. Gotay: A universal redu tion pro edure

for Hamiltonian group a tions. - Group a tions, preprint 591, Univer-

sity of Utre ht, the Netherlands, 1989.

[2℄ R. Cushman and R. Sjamaar: On singular redu tion of Hamiltonian

spa es. - Pro eedings of the Colloque International �a l'honneur de Jean-

Marie Souriau, 1990.

[3℄ M. de Le�on and M. Saralegi: Cosymple ti redu tion versus symple ti

redu tion. - J. Phys. A: Math. Gen. 26(1993), 1-11.

[4℄ J. Marsden, A. Weinstein: Redu tion on Symple ti Manifolds with

Symmetry. - Rep. Math. Phys. 5 (1974), 121-130.

[5℄ R. Sjamaar and E. Lerman: Strati�ed symple ti spa es and redu tion.

- Annals of Math., 134 (1991), 375-422.

[6℄ R. Thom: Ensembles et morphismes strati��es. - Bull. Amer. Math.

So ., 75(1969), 240-284.

[7℄ A. Verona: Strati�ed mappings - Stru ture and triangulability. - Le t.

Notes in Math., Springer Verlag.

[8℄ H. Whitney: Analyti extensions of di�erentiable fun tions de�ned by

losed sets. - Trans. Amer. Math. So . 36 (1934), 63-89.

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 63{73 9

String theory and enumerative geometry

Sebasti

�

a Xamb

�

o-Des amps

Departament de Matem�ati a Apli ada II,

Universitat Polit�e ni a de Catalunya,

Bar elona

Abstra t

The goal of this talk was to all attention to one of the re ent inter-

a tions between geometry and theoreti al physi s: ideas, results and

problems pertaining to the enumerative geometry of rational urves

(mainly on a Calabi{Yau manifold) were introdu ed and their signif-

i an e dis ussed with regard to puzzling al ulations in superstring

theory.

1 Enumerative geometry

Let me begin by outlining what enumerative geometry is about

1

. Ever

sin e its formal beginning in the 1860's, the goal of this bran h of algebrai

geometry has been to solve problems that ask for the number of �gures of

a given kind that satisfy a given list of onditions (see S hubert [1879℄).

The evolution of enumerative geometry is losely tied to the evolution

of algebrai geometry: while algebrai geometry often has mustered inspi-

ration in enumerative geometry for the introdu tion of on epts or the ad-

van ement of onje tures, it is also the ase, in the opposite dire tion, that

enumerative geometry has be ome enri hed ea h time algebrai geometry

has made progress in its on eptual foundation.

S hubert's enumerative al ulus is a good example of the in uen e in

e-mail: sxd�ma2.up .es

1

For a histori al perspe tive entered on the `prin iple of onservation of number', see

Xamb�o-Des amps [1993 a℄.

64 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

the dire t sense. S hubert introdu ed a `symboli al ulus'

2

, whi h allowed

him to solve hosts of enumerative problems

3

. S hubert's symboli al ulus

great su ess ould not hide, however, that it la ked a solid mathemati al

foundation, and this was the main reason that led Hilbert to in lude the

development of su h a foundation as the problem number 15 in the list he

ompiled for the 1900 International Congress of Mathemati ians. Problem

15, in its turn, was su h a powerful stimulus for algebrai geometry that

many on epts and fundamental results introdu ed after 1900 have their

origin in it.

Conversely, the urrent foundation for enumerative geometry is the al-

gebrai geometry of the day, espe ially interse tion theory, but of ourse

often it has to develop its own methods to meet its ends.

Among the �gures studied by S hubert, rational urves o upy a distin-

guished pla e. In addition to lines and oni s, he onsidered planar singular

ubi s ( uspidal ubi s and nodal ubi s), whi h are unexpe tedly diÆ ult

to handle from an enumerative point of view, and also twisted ubi s in P

3

,

whi h by de�nition are urves C of P

3

not ontained in a plane and for

whi h there exists a surje tive map P

1

! C given by ubi polynomials. It

is not hard to see that any twisted ubi is proje tively equivalent to the

ubi given by the map

[t

0

; t

1

℄ 7! [t

3

0

; t

2

0

t

1

; t

0

t

2

1

; t

3

1

℄ :

For example, S hubert determined the hara teristi numbers of the

family of twisted ubi s. These numbers involve the following three ` har-

a teristi ' onditions: �, that the ubi meets a given line; �, that the ubi

is tangent to a given plane; and P , that the ubi goes through a given

point. These hara teristi numbers were veri�ed in Kleiman{Str�mme{

Xamb�o [1987℄. On the other hand, S hubert results and methods for uspi-

dal and nodal plane ubi s have been updated and extended in works su h

as Miret{Xamb�o [1989, 1991, 1994℄.

2

Sometimes it is alled S hubert al ulus, but today this lo ution is mostly reserved for

the appli ation of symboli al ulus to enumerative problems on erning linear varieties.

3

For example, �nding that there are: 2 lines in 3-spa e meeting 4 given lines; or 92

oni s in 3-spa e meeting 8 given lines; or 128 plane uspidal ubi s whose usp and ex

lie ea h in a given line and in addition go through 3 given points and are tangent to 2

given lines; or 640 twisted ubi s that go through 3 points, meet 4 given lines and are

tangent to 2 given planes.

9. STRING THEORY AND ENUMERATIVE GEOMETRY 65

2 The theoreti al physi s input

In the last years there has been an enormous amount of interesting work

related to string theories that is orders of magnitude beyond the ontrol of

experiment. It appears, nevertheless, that they are able to produ e highly

non-trivial and pre ise mathemati al onje tures whi h so far seem very

hard to prove rigorously (in the mathemati al sense). In this situation

it happens (see Ja�e{Quinn [1993℄) that a (mathemati al) proof of any

of those statements plays the role that in traditional physi s is played by

experiments.

Among the onje tures, here we are going to onsider only those that

predi t the number of rational urves on ertain Calabi{Yau varieties, for

example on a general quinti hypersurfa e in P

4

C

4

. String theorists al ulate

a `Yukawa oupling' series f(q) in two di�erent ways, using a prin iple alled

`mirror symmetry', and get the following two expressions:

f(q) = 5 + 2875q+ 4876875q

2

+ :::

and

f(q) = 5 +

P

k�1

n

k

k

3

q

k

1�q

k

= 5 + n

1

q + (2

3

n

2

+ n

1

)q

2

+ :::

where n

k

is the number of rational urves of degree k in the quinti threefold.

The se ond expression omes, roughly speaking, from a quantum orre tion

alled `sum over instantons' (whi h here we may take to mean rational

urves). The values gotten for the �rst four n

k

are the following:

k n

k

1 2875

2 609 250

3 317 206 375

4 242 467 530 000

In Candelas{de la Ossa{Green{Parkes [1992℄, the work where su h num-

bers were published for the �rst time, there is a table for 1 � k � 10, and

in prin iple string theorists an al ulate n

k

up to any value of k be ause it

is possible to al ulate, in prin iple, as many terms of the �rst form of the

q-expansion of f(q).

4

For a beautiful introdu tion to Calabi{Yau manifolds, and in parti ular to its inter-

se tion and Yukawa ouplings, see Morrison [1993℄; for a more elementary dis ussion of

the main ideas, see Xamb�o-Des amps [1994 a℄.

66 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

This result is very striking, even if we disregard the magi of su h al-

ulations and do not bother about what is the pre ise meaning of rational

urves whi h is used. Indeed, if we write a

k

for the k-th oeÆ ient of the

�rst form of the q-expansion, then a

2

� a

1

must be an integer divisible by

8, a

3

�a

1

must be and integer divisible by 27, and so on, whi h are unlikely

properties at the very least. In fa t there is no known a priori reason for

the a

k

to be integers and proving that they indeed are so would be very

interesting

5

3 On Clemens' onje ture

There is another reason why the results are so striking. They ta itly

say that on a general quinti threefold there are only �nitely many rational

urves for ea h degree k � 1. This was in fa t onje tured by Clemens

(see Clemens [1983, 1984℄), but its truth is far from known at present:

Katz [1986b℄ showed that the onje ture is orre t for k � 7 and re ently

the ases k = 8 and k = 9 have been settled by Johnsen and Kleiman

[1993℄, and independently by Nijsee [1993℄. For omparison let us add that

Clemens' onje ture states not only that on a general quinti threefold there

are �nitely many rational urves, but also that ea h su h urve is smooth,

that its normal bundle is O

X

(�1) � O

X

(�1), where O

X

(�1) is the tau-

tologi al line bundle on P

1

C

, and that two distin t rational urves do not

interse t . As shown by Vainsen her [1993℄, there is at least an ex eption to

the smoothness part of this onje ture: there are 17 601000 plane rational

urves of degree 5 (ea h of whi h has exa tly 6 ordinary nodes).

But matters are, as they stand, far mu h worse when it omes to al-

ulate the numbers n

k

. The good news are that n

1

= 2875 was already

5

Morrison [1993℄, p. 238, writes: \These arguments have a rather numerologi al avor.

I am reminded of the numerologi al observations made by Thompson and [by℄ Conway

and Norton about the j-fun tion and the monster group. At the time [it was 1979℄ no

onne tion between these two mathemati al obje ts was known. The q-expansion [Fourier

series℄ of the j-fun tion was known to have integer oeÆ ients, and it was observed

that these integers were integral linear ombinations of the degrees of the irredu ible

representations of the monster group. This prompted mu h spe ulation about possible

deep onne tions between the two, but at the outset all su h spe ulation had to be

hara terized as `moonshine' (Conway and Norton's term). The formal similarities to

the present work should be lear: a q-expansion of some kind is found to have integer

oeÆ ients, and these integers then appear to be linear ombinations of another set of

integers, whi h o ur elsewhere in mathemati s in a rather unexpe ted lo ation. Perhaps

it is too mu h to hope that the eventual explanation will be as pretty in this ase".

9. STRING THEORY AND ENUMERATIVE GEOMETRY 67

known by S hubert over a entury ago (it has been worked out using mod-

ern methods by many authors; see, for example, Katz [1983, 1986a℄, or

Xamb�o-Des amps [1993b℄); that n

2

had been al ulated by S. Katz [1986b,

1988℄ and on�rmed by other authors (see next referen e; see also Xamb�o-

Des amps [1994 ℄); and that the number n

3

has been al ulated, after the

string theorists dis overy, by Ellingsrud{Str�mme [1992℄

6

. The bad news

are that for k � 4 nobody seems to know a sensible line of atta k, mu h as

it happened with the ase k = 3 in the mid eighties. The prin iple behind

the �rst omputations (re ently other methods have been dis overed; we

will ome ba k to this later on) was basi ally the same, namely, to express

the rational urves on the quinti threefold of a given degree k 2 f1; 2; 3g as

the zero lo us of a se tion of a suitable ve tor bundle on the variety para-

meterizing the urves in question (the rank of the bundle oin ides with the

dimension of the variety), but of ourse the omplexity of the al ulations

in reases dramati ally from k = 1 to k = 3, sin e the varieties and bundles

involved have dimensions and ranks 6, 11 and 16, respe tively. Unfortu-

nately the methods used so far look hopeless for k � 4 be ause the variety

that parameterizes rational urves of degree 4 in P

4

is onje tured to be

singular and it is not lear how to go around this fa t (see Piene [1993℄; for

other al ulations of a similar nature, see Katz [1992, 1993℄ and Ellingsrud{

Str�mme [1993℄). In early May 1994, however, Kontsevi h [1994℄ announ es

substantial progress along these lines.

Sin e the oeÆ ients of f(q) omputed so far are indeed integers, and

the resulting values of the n

k

are also integers, and sin e for k 2 f1; 2; 3g

the values oin ide with those supplied by enumerative geometers it looks

as if something very deep is going on with su h al ulations. What is it?

4 Strings

String theory is a vast and imposing subje t (see Green{S hwartz{

Witten [1987℄, Castellani{D'Auria{Fr�e [1991℄, Kaku [1988, 1991℄, and the

large lists of referen es in them, espe ially in the �rst quoted book). Its

goal is to provide a uni�ed theory of the four known fundamental intera -

tions. It starts postulating that the basi unit of energy (whi h we know,

sin e Einstein theory of relativity, to be the same thing as matter) is a

6

For a qui k and lear introdu tion to the geometry behind these and other al ula-

tions, see Piene [1993℄.

68 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

string-like, rather than point-like, and that the di�erent parti les are just

quantum states (energy levels) of su h strings. This view is interesting in

that it gives from the start a good qualitative explanation of the known

features of parti les and intera tions, espe ially if we admit that strings an

break and re ombine.

Sin e strings have to give rise to the smallest of the known parti les

(quarks and leptons), its dimensions must be very small. It turns out that

they must be of the order of 10

�33

m (Plan k's s ale), whi h is about 15

orders of magnitude beyond the length s ale that an be probed with present

day te hniques. In any ase, the only known theoreti al guide to deal with

su h entities is quantum me hani s. Thus one way theorists deal with the

dynami s of a string is by applying a variant of the Feynmann integral,

one of the standard metods to present quantum me hani s. A ording to

this approa h, the probability for a transition from a state into another is

expressed by the integral of a `density' over the spa e of all surfa es swept

out by a possible evolution in spa e-time of the string, from the �rst state

to the se ond, the density being of su h a nature that the weight of a given

surfa e is maximum for the lassi al (Lagrangian) evolution and falls o�

qui kly to zero when the surfa e deviates more and more from the lassi al

one.

Now it turns out, when realisti onditions are imposed (heteroti string,

say), that a string theory an be onsistent only in dimension 10. Theo-

reti ians interpret this by saying that at low energy 6 of the ten dimensions

are ` ompa ti�ed'. One way to interpret this is that the 10-dimensional

manifold is of the form M � X , where M is Minkowski's spa e-time and

X is a ompa t 6-dimensional manifold whose radius is of the order of

Plan k's s ale (so it would be `visible' only after in reasing the resolution

power of the present day gear about 15 orders of magnitude). Furthermore,

in Candelas{Horowitz{Stromminger{Witten [1985℄ (see also Stromminger{

Witten [1985℄) it was shown that X must be a Calabi{Yau 3-fold, and in

prin iple any Calabi{Yau 3-fold is equally eligible. Thus the situation is that

before ompa tifying the theory of the heteroti string is basi ally unique,

but that its low energy behavior depends on a manifold X of whi h we

only know that it is a Calabi{Yau 3-fold. It is possible to impose further

onditions to this manifold, one of them being through the `number of gen-

erations' of parti les. We refer to Gepner [1987℄ for an introdu tion to this

subje t.

9. STRING THEORY AND ENUMERATIVE GEOMETRY 69

5 Mirror symmetry

Now the physi al ontent ofX turns out to be an asso iated obje t alled

a `super onformal quantum �eld theory', or SCFT for short (see Kaku

[1988, 1991℄, L�ust{Theissen [1989℄, Cuerno{Sierra{G�omez [1991℄). The rel-

evant �elds in this theory, whi h are grouped into `�elds' and `anti�elds', are

asso iated to H

1;1

= H

1;1

(X) and H

2;1

= H

2;1

(X)

7

.These �elds, however,

an be ex hanged by `supersymmetry', and so it is possible to inter hange

the physi al role of H

1;1

and H

2;1

. This observation led to the introdu -

tion of the `mirror symmetry' on ept

8

. A ording to this prin iple, there

should exist, given a Calabi{Yau 3-fold X , another Calabi{Yau 3-fold X

0

yielding the same physi al SCFT , but with reversed (hen e the `mirror'

term) ohomologies.

Thus we de�ne a Calabi{Yau 3-fold X

0

to be a mirror of a Calabi{Yau

3-fold X if the Hodge diamond of X

0

is obtained by inter hanging h

1;1

and

h

2;1

, that is, if

h

1;1

(X

0

) = h

2;1

(X) ; h

2;1

(X

0

) = h

1;1

(X) :

The `mirror symmetry onje ture' asserts that given a Calabi{Yau 3-fold X ,

there exists a mirror X

0

of X su h that the SCFT 's asso iated to X and

X

0

are the same. Thus the existen e of mirrors was predi ted on theoreti al

physi s grounds. Soon after that string theorists (and mathemati ians) have

omputed thousands of mirror pairs and there is qui k progress in this area.

Now we want to des ribe the non trivial relations gotten by the equiv-

alent physi al theories of a mirror pair. Let I be the ubi (interse tion)

form on H

1;1

and Y a normalized Yukawa oupling on H

2;1

. Consider the

`quantum interse tion form' I

Q

on H

1;1

as follows:

I

Q

(�

1

; �

2

; �

3

) = I(�

1

; �

2

; �

3

)+

P

[P

1

'

�!X℄

n(')

�3

e

�

R

'

�

w

X

R

'

�

�

1

R

'

�

�

2

R

'

�

�

3

;

where w

X

is a K�ahler form on X and where n(') is the overing degree of

'. With this terminology, the `mirror symmetry prin iple' states that

I

Q

(X) = Y (X

0

) ; Y (X) = I

Q

(X

0

) :

Sin e the evaluation of I

Q

(X) an be expressed in terms of the rational

urves on X (for the quinti 3-fold it turns out to be the se ond form of

7

See Morrison [1993℄ for notations and ba kground.

8

The term `mirror symmetry' appears for the �rst time in Greene{Plesser [1990℄; see

also Green{Plesser [1992℄.

70 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

f(q) in the introdu tion), the game be omes to al ulate a mirror X

0

of

X and its Yukawa oupling, and all this is usually done by using methods

of variation of Hodge stru ture. However it goes, su h omputations have

been arried out in many ases now, they all lead to predi tions about the

number of rational urves on a given X (and even other types of urves),

and the numbers have been on�rmed (as hinted in the introdu tion) in

relatively small (but growing) number of ases.

For more details on the ideas dis ussed in this paper, see:

Aspinwall and L�utken [1991a, b, 1992℄, Aspinwall, L�utken and Ross

[1990℄, Aspinwall and Morrison [1993℄, H�ubs h [1992℄, Manin [1993℄, Xamb�o-

Des amps [1994b, ℄, Kontsevi h and Manin [1993℄, Kontsevi h [1994℄.

Referen es

P. S. Aspinwall, C. A. L�utken [1991a℄: Geometry of mirror manifolds, Nu-

lear Physi s B353, 427-461.

P. S. Aspinwall, C. A. L�utken [1991b℄: Quantum algebrai geometry of

superstring ompa ti� ations, Nu lear Physi s B355, 482-510.

P. S. Aspinwall, C. A. L�utken [1992℄: A new geometry from superstring

theory, in Yau [1992℄, 316-341.

P. S. Aspinwall, C. A. L�utken, G. G. Ross [1990℄: Constru tion and ou-

plings of mirror manifolds, Phys. Lett. 241B, 373-380.

P. S. Aspinwall, D. Morrison [1993℄: Topologi al �eld theory and rational

urves, Commun. Math. Physi s 151, 245-262.

P. Candelas, T. Horowitz, A. Stromminger, E. Witten [1985℄: Va uum on-

�gurations for superstrings, Nu lear Physi s B258, 46-74.

Philip Candelas, Xenia C. de la Ossa, Paul S. Green, and L. Parkes [1992℄:

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in Yau [1992℄, 31-95 [reprint from Nu lear Phys. B359 (1991), 21-74; the

announ ement was made in Phys. Lett. B258 (1991), 118-126℄

L. Castellani, R. D'Auria, P. Fr�e [1991℄: Supergravity and superstrings{a

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is not �nitely generated, Publ. Math. IHES 58, 19-38.

H. Clemens [1984℄: Some results on Abel{Ja obi mappings, in \Topi s in

trans endental algebrai geometry", Prin eton University Press.

R. Cuerno, G. Sierra, C. G�omez [1991℄: Teor��a de ampos onformes, sis-

temas integrables y grupos u�anti os, Edi iones Universidad de Salaman a.

G. Ellingsrud & S. A. Str�mme [1992℄: The number of twisted ubi urves

on the general quinti threefold, in Yau [1992℄, 181-240.

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on Calabi{Yau omplete interse tions, Preprint [le tures given at Dyrkol-

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72 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

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Mathemati al Inst. Univ. of Leiden, Report W93-26.

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74 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

Pro eedings of the Fall Workshop on

Di�erential Geometry and its Appli ations

Bar elona '93

pp. 75{80 10

Classi al and quantum anomalies in the

quantum Hall e�e t

Manuel Asorey

Departamento de F��si a Te�ori a,

Universidad de Zaragoza,

Zaragoza

Abstra t

We analyse some lassi al and quantum aspe ts of a point parti le

moving on a 2-dimensional torus. The system exhibits three di�er-

ent types of anomalies in the presen e of an uniform magneti �eld.

First, a global anomaly implies the quantization of the magneti ux

a ross the torus. Se ond a lo al anomaly transforms the translation

symmetry into a Heisenberg symmetry, and �nally, a novel ommu-

tator anomaly is responsible for the quantum symmetry breaking of

translation invarian e.

1 Introdu tion

The motion of harged parti les on a plane under the a tion of a

transversal magneti �eld exhibits very pe uliar behaviours whi h give rise

to three di�erent Hall e�e ts: lassi al, integer and fra tional. The Lorentz

for e generated by the magneti �eld on the ele trons indu es a (magneti )

resistan e to the passage of the ele tri urrent along a ondu ting strip.

The in rease of resistan e along the strip is traded by an in reasing ondu -

tivity a ross the strip. This is the lassi al Hall e�e t observed by Hall in

1879 [1℄.

The quantum dynami s of this system was �rst analysed by Lev Landau

in the thirties [2℄. He remarked the quantization of energy levels (Landau

e-mail: asorey�saturno.unizar.es

76 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

levels).

More re ently, it was experimentally observed on super ondu ting

strips a dis ontinuous dependen e of the indu ed Hall ondu tivity on the

transversal magneti �eld [3℄. The jumps on the Hall ondu tivity are as-

so iated to the quantization of energy levels, ele tri urrent intensities and

magneti uxes predi ted by Landau theory (quantum Hall e�e t). The or-

re tions due to the Coulomb repulsion of the ele trons on the super ondu t-

ing strip yield a di�erent novel pattern of Hall ondu tivity dis ontinuities

(fra tional quantum Hall e�e t).

In the Hall e�e t on a strip with periodi boundary onditions the sur-

fa e e�e ts disappear and new physi al e�e ts arise due to the ompa tness

of the on�guration spa e. In this note we analyse the lassi al and quan-

tum aspe ts of three di�erent anomalies whi h appear in this ase due to

the non trivial topology of the on�guration spa e. The existen e periodi

boundary onditions means that the parti les are moving e�e tively on a

two dimensional torus T

2

.

2 Classi al Anomalies

The lassi al dynami s of a harged parti le moving on T

2

under the

a tion of an uniform magneti �eld B in governed by the Hamiltonian

H =

1

2m

2

X

i=i

p

2

i

(1)

on the otangent spa e T

�

T

2

endowed with the magneti symple ti form

w =

2

X

i=i

dp

i

^ d'

i

+Bd'

1

^ d'

2

(2)

The system is translation invariant be ause of the uniform hara ter

of the magneti �eld. The in�nitesimal generators of this U(1) � U(1)

symmetry are linear ombinations of the ommuting ve tor �elds

L

i

= ��

i

i = 1; 2

of T

�

T

2

.

L

1

and L

2

are lo ally Hamiltonian ve tor �elds with respe t to ! and

preserve the Hamiltonian H . However, the orresponding onserved mo-

menta

l

i

= p

i

+B�

ij

'

j

i = 1; 2 (3)

10. CLASSICAL AND QUANTUM ANOMALIES 77

asso iated by Noether theorem are not globally de�ned on T

�

T

2

unlike H

whi h is the global quantum momentum asso iated to time evolution.

On the other hand the Poisson bra ket of the two momenta l

1

; l

2

with

respe t to the symple ti form !

fl

1

; l

2

g = B

does not vanish. Thus, the anoni al algebra of onserved urrents de�nes

a entral extension of the Lie algebra R�R generated by L

1

and L

2

. The

extended algebra is a Heisenberg algebra with entral harge the strength

B of the magneti �eld.

These two features an be onsidered as lassi al anomalies [4℄ whi h

are in part responsible for quantum anomalies asso iated to translation

symmetry whi h will be dis ussed below.

However, from a pure lassi al point of view the fa t that the onserved

momenta are not globally de�ned or do not preserve the ommutation re-

lations of the translation group [L

1

; L

2

℄ = 0 do not imply any pathologi al

behavior of the lassi al dynami s. Indeed, translation symmetry is a real

symmetry of the lassi al dynami s, fl

i

; Hg = 0.

3 Quantization

Quantization of those systems is only possible if the symple ti form !

is an integer form, i.e. [!=2�℄ 2 H

2

(T

2

;C); k = 2�B 2 Z. In this ase the

quantum states are se tions of the line bundle E

k

(T

2

;C) with �rst Chern

lass k and the quantum Hamiltonian is given by

H = �

1

2m

�

A

(4)

where �

A

is the ovariant Lapla ian with respe t to a onne tion A of

E

k

(T

2

;C) with urvature ! = dA. The quantization of k is a quantum

onsisten y onstraint �rst pointed out by Dira for the motion of a harged

parti le in the presen e of a magneti monopole [5℄. If the quantization

ondition is not satis�ed a quantum anomaly prevents the existen e of a

selfadjoint extension of the Hamiltonian. In su h a ase the existen e of the

anomaly implies a loss of unitarity.

A similar anomaly arises in Chern-Simons theory in 2+1 dimensions [6℄

and the theory of massless Dira fermions oupled to SU(2) gauge �elds in

the fundamental representation [7℄.

78 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

The spe trum of the Hamiltonian (4) is given by the Landau levels

E

n

=

B

m

(n+

1

2

) n 2 N

as in the in�nite plane ase. However, in the present ase the degenera y

of ea h level is �nite, n + jkj, whereas in in�nite volume the degenera y

is in�nity. The degenera y of the ground state E

0

does not depend on

the metri of the torus or the form of the magneti �eld but only on the

�rst Chern lass of the line bundle E

k

(T

2

;C), where the physi al states are

de�ned.

4 Translation anomalies

The quantum generators of translation symmetries are given by

L

j

= �i(r

j

A

�

) j = 1; 2 (5)

where r

A

�

is the ovariant derivative with respe t to a onne tion A

�

with

urvature dA

�

= �!. They satisfy the Heisenberg algebra ommutation

relations

[L

1

;L

2

℄ = �iB (6)

whi h shows a new kind of quantum anomaly. The two ommuting gener-

ators of lassi al symmetries L

1

; L

2

give rise to quantum operators L

1

;L

2

whi h do not ommute.

In this ase the existen e of the anomaly implies that the lassi al Lie

algebra R�R yields upon quantization to a entral extension whose en-

tral harge is the magneti ux B. The phenomenon is possible thanks to

the proje tive nature of states and symmetries in quantum me hani s. It

is similar to the gauge anomaly whi h generates S hwinger terms in the

ommutators of hiral harges in urrent algebras [8℄ {[10℄.

5 The ommutator anomaly

Finally, there is a mu h less known anomaly in this model whi h is

essential for understanding the �nite degenera y of the energy spe trum. If

the Heisenberg algebra were a real symmetry of the quantum system the

energy levels should be in�nitely degenerated, be ause any representation

of the Heisenberg algebra (6) must be in�nite dimensional, and any energy

level supports a representation of the symmetry algebra.

10. CLASSICAL AND QUANTUM ANOMALIES 79

However on the torus the degenera y of Landau levels is �nite. There

are n + jkj states with energy E

n

= (n + 1=2)B=m. The solution of the

apparent paradox omes from the observation that although the generators

of the translation Heisenberg algebra L

1

;L

2

ommute withH on the domain

of fun tions with ompa t support on (0; 2�) � (0; 2�), the orresponding

selfadjoint extensions do not ommute.

This is possible be ause the domain of de�nition ofH is not preserved by

the a tion of L

i

. In fa t, the operators L

i

are essentially selfadjoint opera-

tors on the spa e of smooth se tions �(E

�k

) of the line bundle E

�k

(T

2

;C)

with �rst Chern-Class �k whereas H is essentially selfadjoint on �(E

k

).

Consequently, the ommutator of L

i

and H is not well de�ned and in this

sense translation invarian e is broken in the quantum system. Noti e that

all the operators H;L

1

;L

2

an densely de�ned on the same Hilbert spa e

H = L

2

(E

k

) = L

2

(E

�k

), but their domains di�er in way whi h yields to a

breaking of the lassi al symmetry in a new anomalous way.

This interpretation of the anomaly based on the anomalous behavior

of the domain of de�nition of the quantum Hamiltonian under translations

was advo ated by Esteve [11℄ (See also [12℄). In this ase the existen e

of anomalous ommutator is ru ial for the understanding of the �nite de-

genera y of energy levels in spite of the existen e of a partial translation

invarian e.

Finally, we remark the existen e of a lose relationship between lassi al

and quantum anomalies. Besides the Dira anomaly leading to ux quan-

tization whi h is a pure quantum anomaly whi h interferes the onsisten y

of the quantization pres ription, the other quantum anomalies have lassi-

al pre edents. The entral anomalous extension of translation symmetry

although it is based on the proje tive nature of quantum states and sym-

metries it an be inferred from a lassi al Poisson analysis of translation

symmetry [4℄. The ommutator anomaly also has a lassi al pre ursor. The

Hamiltonian H and the symple ti form ! are globally de�ned on T

�

T

2

,

whereas the momenta l

1

; l

2

asso iated to translation symmetry are not glob-

ally de�ned on T

�

T

2

. This is a pure lassi al property whi h is on the root

of the quantum ommutator anomaly.

It is interesting to remark that this simple Hall model exhibits on the

torus all the pathologies usually asso iated to anomalies in �eld theory.

Moreover, in the dis rete approximation the spe trum of the Hamiltonian of

this model an display a Cantor set stru ture and quantum group invarian e

80 DIFFERENTIAL GEOMETRY AND ITS APPLICATIONS

depending on the non-rational or rational hara ter of the magneti ux [13℄

The above interpretations of lassi al and quantum anomalies an be

extended to quantum �eld models although in su h a ase the analysis is

more omplex due to the need of renormalization of observables, symmetry

and states.

A knowledgements. This note is a summary of joint work with A.

Az �arraga and J.M. Izquierdo [14℄. I thank J.G. Esteve and G. Ma k for en-

lightening dis ussions. This work was partially supported by CICyT under

grant AEN93-0219.

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