Shadows and Twisted Variables

Shadows and Twisted Variables

Laurent Baulieua ∗ and Guillaume Bossarda†

aUniversite Pierre et Marie Curie, LPTHE, CNRS4 place Jussieu, F-75252 Paris Cedex 05, France

We explain how a new type of fields called shadows and the use of twisted variables allow for a better

description of Yang–Mills supersymmetric theories.

1. Introduction

Non-linear aspects and the non-existenceof a supersymmetry-preserving regulatormake the definition of supersymmetric the-ories a subtle task. We explain in theselectures notes that the introduction of newfields, called shadows, clarify the construc-tion of Yang–Mills supersymmetric theories.

In the formalism that we develop, a super-symmetric theory is defined in terms of clas-sical fields (gauge fields and matter fields),Faddeev–Popov ghosts and shadow fields.Gauge invariance is expressed by the BRSTinvariance, with a graded differential opera-tor s . The shadows fields permit the replace-ment of the notion of the supersymmetry gen-erators by that of a differential operator Q,consistent with s . The operator Q acts asan ordinary supersymmetry transformationon the gauge invariant functions of the phys-ical fields. Moreover, there exist gauges forwhich Q annihilates both the classical actionand the s -exact gauge-fixing action.

The advantage of having both operators sand Q acting on the extended set of fields isthat two independent Slavnov–Taylor identi-ties can be associated with supersymmetryand BRST invariances. Observables can beappropriately defined for understanding theirgauge and supersymmetry covariance : theyare the cohomology of the BRST symmetry.Anomalies and renormalization can be con-

∗Theoretical Division CERN, CH-1211 Geneva 23,Switzerland†Instituut voor Theoretische Fysica, Valckenierstraat65, 1018XE Amsterdam, The Netherlands

ventionally analyzed, considering insertionsof arbitrary composite operators. This de-fines an unambiguous renormalization pro-cess of Yang–Mills supersymmetric theory,for any given choice of the regularization ofdivergences.

Shadows can be used to demonstrate non-renormalization theorems. Moreover, theproofs are greatly simplified by twisting thespinor fields in tensors. In fact, twistedvariables permit one to determine off-shellclosed sub-sectors of supersymmetry algebrathat are relevant for the non-renormalizationproperties.

Both differential operators s and Q of su-persymmetric theories satisfy extended cur-vature conditions, analogous to those of thetopological BRST operator of topologicalquantum field theory. This similarity sug-gests that some of the relevant equations forthe non-renormalization theorems have a ge-ometrical meaning .

2. Introducing the shadow fields

To fix ideas, consider the N = 4, d = 4supersymmetric action in flat space. Thephysical fields of this gauge invariant the-ory with SO(3, 1) Lorentz symmetry are thegauge field Aμ, the SU(4)-Majorana spinorλ, and the six scalar fields φi in the vectorrepresentation of SO(6) ∼ SU(4). All fieldsare in the adjoint representation of a compactgauge group that we will suppose simple. Theclassical action is uniquely determined by su-persymmetry, Spin(3, 1)×SU(4) global sym-

Nuclear Physics B (Proc. Suppl.) 171 (2007) 164–175

0920-5632/$ – see front matter © 2007 Published by Elsevier B.V.

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doi:10.1016/j.nuclphysbps.2007.06.010

metry and gauge invariance. It reads

S ≡

∫d4xTr

(−

1

4FμνFμν

−1

2DμφiDμφi +

i

2

(λ /Dλ

)

−1

2

(λ[φ, λ]

)−

1

4[φi, φj ][φi, φj ]

)(1)

with φ ≡ φiτi and the supersymmetry trans-formations δSusy

δSusyAμ = i(εγμλ

)δSusyφi = −

(ετ iλ

)δSusyλ =

(/F + i /Dφ +

1

2[φ, φ]

)ε (2)

For the sake of convenience, we can chosethe parameter ε as a commuting spinor. Inthis way, δSusy

2represents the commutator of

two supersymmetry transformations, with

δSusy2≈ δgauge(ε[φ − i /A]ε) − i(εγμε)∂μ (3)

Here ≈ stands for the equality modulo equa-tions of motion.

In view of the last equation, the quest ofa quantum field theory with supersymmetryimplies the following remarks.

The presence of equations of motion in theright-hand-side of (3) is a rather annoyingtechnical difficulty. However, it can always beturned around in quantum field theory, by us-ing the Batalin-Vilkowiski formalism. More-over, as we will shortly see, even in the casewhere no auxiliary fields exist, it can be prac-tically resolved in the proofs for the consis-tency of the quantum theory by using twistedvariables.

The existence of the field dependent gaugetransformation in the commutator of two su-persymmetry transformations (3) is a deeperproblems. It concretely implies that onecannot give sense to the notion of a δSusy -invariant gauge-fixing action. This fact ex-plicitly shows up when one uses the Faddeev-Popov procedure. Suppose that one fixesthe gauge, say in a Feynman–Landau gauge.This process is independent of supersymme-try and gives an action

Sgf = S +

∫Tr

((∂A)2

2α− Ω∂DΩ

)(4)

This lagrangian breaks gauge invariance inthe desired way, but one cannot find a defi-nition of δSusy acting on the scalar Faddeev–Popov ghosts Ω and Ω that is compatiblewith the closure relation (3). This forbids oneto define the Ward identities associated to su-persymmetry with usual techniques. There-fore, one must improve the techniques cur-rently used for ordinary global symmetriescoupled to gauge invariance. Since there arecases where an off-shell superfield formalismdoes not exist (in particular for the N = 4theory) and since no regulator exist that canmaintain both supersymmetry and gauge in-variance, such improvement must follow fromnew ideas.

One method for handling the problemscaused by the gauge transformations inthe closing relations for the supersymmetrytransformations of classical fields is by intro-ducing an additional anticommuting scalarfield c valued in the Lie algebra of the gaugegroup. On can define in this way a differen-tial operator Q out of δSusy , which is nilpotentmodulo a translation [1]

Q2 ≈ −i(εγμε)∂μ (5)

The way to do so is to define the action of Qon all the physical fields ϕ and c as follows

Qϕ = δSusy (ε)ϕ − δgauge(c)ϕ

with

Qc = (ε[φ − i /A]ε) − c2 (6)

The field c will be called the shadow field,and its presence will allow one to solve atonce all questions discussed above, with theconclusion that the notion of the operatorδSusy must be replaced by that of the differ-ential Q at the quantum level, in a way thatis analogous to the enhancement of gauge in-variance into BRST symmetry.

We see that the action of Q on the clas-sical fields is linear in the global parametersε and on the field c. Since, for the classi-cal fields, Q is the sum of a supersymmetrytransformation and a gauge transformation,

L. Baulieu, G. Bossard / Nuclear Physics B (Proc. Suppl.) 171 (2007) 164–175 165

δSusy invariance is the same as Q invariancefor gauge invariant quantities.

The action of Q on c is quadratic both in cand ε, and Qε = 0. We have the existence ofa grading equal to the shadow number, whichis zero for the classical fields, and one for cand ε.

In practice, one must do computationswith a BRST invariant gauge-fixed the-ory, where interacting Faddeev–Popov ghostspropagate. In fact, renormalization gener-ally mixes gauge invariant operators withnon gauge-invariant BRST-exact operators.Thus, observables must be defined throughthe cohomology of the BRST operator s forordinary gauge symmetry. To control thecovariance under supersymmetry of observ-ables, the BRST Ward identity and the su-persymmetry Ward identities must be disen-tangled. It follows that Q and s must be in-dependent and consistent operators (i.e., Qand s must anticommute). Therefore thescalar field c cannot be identified with theFaddeev–Popov ghost Ω.

The idea of shadows [1] is thus to introducenew fields, in the form of BRST doublets, inorder not to affect physical quantities, and toredefine the supersymmetry transformationsof classical fields by addition of a compensat-ing gauge transformations with a parameterequal to the shadow field c. Moreover, Eq. (5)must be satisfied for all fields.

The action of the BRST operator s on allphysical fields is nothing but a gauge trans-formation of parameter Ω with

s ϕ = −δgauge(Ω)ϕ s Ω = −Ω2 (7)

and since the shadow c must not affect thephysical sector of the theory we introduce thecommuting scalar μ such that (c, μ) builds atrivial BRST doublet

s c = μ s μ = 0 (8)

We want to impose Eq. (5) on all fields, aswell as

s 2 = s Q + Q s = 0 (9)

In fact, by a direct computation, we find that

the algebra (5) and (9) is satisfied with

QΩ = −μ − [c, Ω] (10)

Qμ = −[(εφε), Ω] + i(εγμε)DμΩ − [c, μ]

We will shortly write a curvature equationthat explains these transformation laws, andin particular the property

s c + QΩ + [c, Ω] = 0 (11)

In order to define the Ward identities asso-ciated to supersymmetry, we need a BRST-exact gauge-fixing that is Q-invariant. Suchgauge-fixing will be said to be supersymmet-ric. To define it, we introduce the trivialquartet μ, c, Ω, b, with

s μ = c

Qμ = Ω

s c = 0

Qc = −b

s Ω = b

QΩ = −i(εγμε)∂μμ

s b = 0

Qb = i(εγμε)∂μc

(12)

The quantum field theory has an internalbigrading, the ordinary ghost number and thenew shadow number. The Q transformationof fields depend on the constant commutingsupersymmetry parameter. The latter is un-derstood as an ordinary gauge parameter forthe quantum field theory, but observables willnot depend on them, owing to BRST invari-ance.

3. Supersymmetric shadow dependent

lagrangians

In order to control supersymmetry andrenormalize the theory, we start from a renor-malizable s and Q invariant gauge-fixed ac-tion, which determines the Feynman rules. Aclass of such actions is of the form:

Sgf [ϕ, Ω, Ω, b, c, c, μ, μ]

= S[ϕ] − s Q∫

Tr μ(∂A +

α

2b)(13)

L. Baulieu, G. Bossard / Nuclear Physics B (Proc. Suppl.) 171 (2007) 164–175166

One has indeed

− s Q∫

Tr μ(∂A +

α

2b)

= − s∫

Tr(Ω

(∂A +

α

2b)

+ μQ(∂A +

α

2b))

=

∫Tr

(−

α

2b2 − b∂A − Ω∂DΩ + . . .

)(14)

Here, the dots stand for terms that imply apropagation of the pairs of shadows μ, μ andc, c. They are given by an easy computa-tion. They imply ε-dependent propagatorsand vertices. However, observables are de-fined by the cohomology of the BRST oper-ator s , so that their expectation values areindependent on the values of ε, since the lateroccur through an s -exact term.

In the absence of anomaly, one can enforceboth Ward identities for the s and Q invari-ances. This means that one can concretelyimpose renormalization conditions which en-force these identities at any given finite orderof perturbation theory, within the frameworkof any type of regularization for divergences.

The prize one has to pay for having shad-ows is that they generate a perturbative the-ory with more Feynman diagrams. If weconsider physical composite operators thatmix through renormalization with BRST-exact operators, the latter can depend onall possible fields that propagate, and wehave in principle to consider a dependenceon the whole set of fields in order to computethe supersymmetry-restoring non-invariantcounterterms. For certain “simple” Greenfunctions, which cannot mix with BRST-exact composite operators, there exist gaugesin which some of the additional fields can beintegrated out, in a way that justifies, a pos-teriori, the work of Stockinger et al. for theN = 1 theories [2]. By doing this elimination,one loses the algebraic meaning, but one maygain in computational simplicity.

The shadow dependent methodology issuitable for non-ambiguously computing thenon-invariant counterterms that maintain su-persymmetry, BRST invariance and the R-symmetry. It applies to the renormalizationof all supersymmetric theories.

4. Renormalization

4.1. Ward identities for the theory

By introducing sources associated to thenon-linear s , Q and s Q transformations offields, we get the following ε-dependent ac-tion, which initiates a BRST-invariant super-symmetric perturbation theory3

Σ ≡1

g2S

−

∫d4xTr

(b∂μAμ +

α

2b2

− c∂μ(Dμc + i(εγμλ)

)−

iα

2(εγμε)c∂μc + Ω∂μDμΩ

−μ∂μ(Dμμ + [DμΩ, c] − i(εγμ[Ω, λ])

))

+

∫d4xTr

(A(s)

μ DμΩ+λ(s)

[Ω, λ]−φ(s)

i [Ω, φi]

+ A(Q)

μ QAμ − λ(Q)

Qλ + φ(Q)

i Qφi

+ A(Qs)μ s QAμ − λ

(Qs)s Qλ + φ(Qs)

i s Qφi

+ Ω(s)Ω2 − Ω(Q)QΩ − Ω(Qs) s QΩ

− c(Q)Qc + μ(Q)Qμ

+g2

2(λ

(Q)

−[λ(Qs)

, Ω])M(λ(Q)−[λ(Qs), Ω])

)

(15)

Because of the s and Q invariances, theaction is invariant under the both Slavnov–Taylor identities defined in [1], which are as-sociated respectively to gauge and supersym-metry invariance, S(s)(Σ) = S(Q)(Σ) = 0. Forthe sake of illustration, let us present the su-persymmetry Slavnov–Taylor operator of the

3M is the 32 × 32 matrix M ≡1

2(εγμε)γμ +

1

2(ετiε)τ

i − εε. It occurs because Q2 is a purederivative only modulo equations of motion. Thedimension of Aμ, λ, φi, Ω, Ω, b, μ, μ, c and c

are respectively 1, 3

2, 1, 0, 2, 2, 1

2, 3

2, 1

2and 3

2.

Their ghost and shadow numbers are respectively(0, 0), (0, 0), (0, 0), (1, 0), (−1, 0), (0, 0), (1, 1),(−1,−1), (0, 1) and (0,−1).


N = 4 theory4

S(Q)(F ) ≡

∫d4xTr

(δRF

δAμ

δLF

δA(Q)μ

+δRF

δλ

δLF

δλ(Q)

+δRF

δφi

δLF

δφ(Q)

i

+δRF

δc

δLF

δc(Q)

+δRF

δμ

δLF

δμ(Q)+

δRF

δΩ

δLF

δΩ(Q)−A(s)

μ

δLF

δA(Qs)μ

+ λ(s) δLF

δλ(Qs) − φ(s)

i

δLF

δφ(Qs)

i

+ Ω(s) δLF

δΩ(Qs)

− bδLF

δc+ Ω

δLF

δμ

−i(εγμε)(−∂μA(Qs)

ν

δLF

δA(s)ν

+∂μλ(Qs) δLF

δλ(s)

−∂μφ(Qs)

i

δLF

δφ(s)

i

+∂μΩ(Qs) δLF

δΩ(s)−∂μc

δLF

δb

+ ∂μμδLF

δΩ+ A(Q)

ν ∂μAν + λ(Q)

∂μλ

+ φ(Q)

i ∂μφi

+ Ω(Q)∂μΩ + c(Q)∂μc + μ(Q)∂μμ))

(16)

If no anomaly occurs, the Slavnov–Tayloridentities S(s)(Γ) = S(Q)(Γ) = 0 completelydetermines all ambiguities of the supersym-metric effective action Γ, order by order inperturbation theory.

4.2. Anomalies

In [1,3], we showed the absence of anomalyfor the N = 2, 4 and the stability of the N =1, 2, 4 action Σ under renormalization. Thus,all Green functions of the complete theoryinvolving shadows and ghosts can be renor-malized, in any given regularization scheme,so that supersymmetry and gauge invarianceare preserved at any given finite order.

Let us sketch the proof that no supersym-metry anomaly can exist for N = 2, 4, andthat for N = 1 the only possible anomaly isthe Adler–Bardeen anomaly.

4The linearized Slavnov–Taylor operator S(Q)|Σ [1]

verifies S(Q)|Σ2 = −i(εγμε)∂μ, which solves in prac-

tice the fact that Q2 is a pure derivative only moduloequations of motion.

An anomaly in a supersymmetry theorycan only occur if a pair of local functionalsA and B of the fields and sources can vio-late the pair of Ward identities for both sand Q invariances. For instance, when onerenormalizes the theory at the one-loop level,the result of the computation can violate theWard identities by local terms A and B, asfollows

S(s)|ΣΓ1 loop = �

∫A

S(Q)|ΣΓ1 loop = �

∫B (17)

If either A and B cannot be eliminated byadding local counterterms to Γ1 loop, whichmeans that they are not S(s)|Σ and S(Q)|Σ ex-act, one has an anomaly, and the theory can-not be renormalized while maintaining eithersupersymmetry or gauge invariance, or both.In [1,3], we proved that the solution A andB of Eq. (17), modulo S(s) |Σ and S(Q)|Σ ex-act terms, can only depend on the fields, andthus, the consistency relation for s and Qimplies:

s∫

A = 0 Q

∫B = 0

Q

∫A + s

∫B = 0 (18)

In fact, the first equation implies that A mustbe the consistent Adler-Bardeen anomaly,which descends formally from the Chern classTr FFF . But then, the Q symmetry is sodemanding that the second and third equa-tions have no solution B �= 0 for N = 2, 4.Thus there cannot be an anomaly for thesecases. For N = 1, the constraint is weaker,and the Adler-Bardeen anomaly admits asupersymmetric counterpart B. However,the Adler–Bardeen theorem holds, and ifthe one-loop coefficient of the Adler–Bardeenanomaly cancels, it will cancel to all order.

Of course, these are well known facts.However, by having introduced the shad-ows, both Ward identities for supersymme-try and gauge invariance allow a safe verifi-cation of the status of gauge and supersym-metry anomalies by the standard consistency


argument, valid to all order of perturbationtheory.

4.3. Ward identities for the observables

Observables of a super-Yang–Mills theoryare Green functions of local operators in thecohomology of the BRST linearized Slavnov–Taylor operator S(s) |Σ. From this definition,these Green functions are independent of thegauge parameters of the action, including ε.Classically, they are represented by gauge-invariant polynomials of the physical fields[1,4]. We introduce classical sources u forall these operators. We must generalize thesupersymmetry Slavnov–Taylor identity forthe extended local action that depends onthese sources. Since the supersymmetry al-gebra does not close off-shell, other sourcesv, coupled to unphysical S(s)|Σ-exact opera-tors, must also be introduced. We define thefollowing field and source combinations ϕ∗

A∗μ ≡ A(Q)

μ − ∂μc − [A(Qs)μ − ∂μμ, Ω]

φ∗i ≡ φ(Q)

i − [φ(Qs)

i , Ω]

c∗ ≡ c(Q) − [μ(Q), Ω]

λ∗ ≡ λ(Q) − [λ(Qs), Ω] (19)

They verify S(s) |Σϕ∗ = −[Ω, ϕ∗]. The col-lection of local operators coupled to the v’sis made of all possible gauge-invariant (i.e.S(s)|Σ-invariant) polynomials in the physicalfields and the ϕ∗’s. These operators haveghost number zero, and their shadow num-ber is negative, in contrast with the phys-ical gauge-invariant operators, which haveshadow number zero.

The relevant action is thus Σ[u, v] ≡ Σ +Υ[u, v], with

Υ[u, v] ≡

∫d4x

(uij

1

2Tr φiφj+uα

i Tr φiλα

+ uijk

1

3Tr φiφjφk

+ KuμijTr

(iφ[iDμφj] +

1

8λγμτ ijλ

)

+ Kuμνi Tr

(Fμνφi −

1

2λγμντ iλ

)

+ Ku5μ

1

2Tr λγ5γ

μλ

+ CuijkTr(1

3φ[iφjφk] +

1

8λτ ijkλ

)

+ CuμijTr

(iφ[iDμφj] −

1

4λγμτ ijλ

)

+ Cuμνi Tr

(Fμνφi +

1

4λγμντ iλ

)+uα

ijTr φiφjλα

+iuμ αi Tr Dμφiλα+uμν αTr Fμνλα+· · ·

+vαi Tr φiλ∗

α+vαβTr λαλ∗β +vμ

i Tr φiA∗μ

+ vijTr φiφ∗ j + ivμ αi Tr Dμφiλ∗

α

+ 0vαi Tr λαφ∗ i + ivμαβTr Dμλαλ∗

β

+ivμijTr Dμφiφ∗ j+i −1v

μαi Tr Dμλαφ∗ i+· · ·

)

(20)Here, the · · · stand for all other analogous

operators.The Slavnov–Taylor operator S(Q) can be

generalized into a new one, Sext(Q)

, by addi-tion of terms that are linear in the functionalderivatives with respect to the sources u andv, in such a way that

Sext

(Q)(Σ[u, v]) = S(Q)(Σ) + Sext

(Q)|ΣΥ

+

∫d4xTr

(δRΥ

δAμ

δLΥ

δA∗μ

+δRΥ

δλ

δLΥ

δλ∗

+δRΥ

δφi

δLΥ

δφ∗i

)= 0 (21)

Indeed, if we were to compute S(Q)(Σ[u, v])without taking into account the transforma-tions of the sources u and v, the breaking ofthe Slavnov–Taylor identity would be a localfunctional linear in the set of gauge-invariantlocal polynomials in the physical fields, A∗

μ,c∗, φ∗

i and λ∗.Eq. (21) defines the transformations Sext

(Q)|Σ

of the sources u and v. Simplest examplesfor the transformation laws of the u’s are forinstance

Sext

(Q)|Σuij = −i[γμτ{iε]α∂μuα

j} + ∂μ∂μv{ij}

+2u{i|kvj}k + 2uα

{ivj}α

−i∂μ(u{i|kvμ

j}

k+ uα

{ivμ

j}α)


Sext

(Q)|Σuα

i = [ετ j ]α(uij − i∂μ( Ku

μij + Cu

μij)

)−2i[εγμ]α∂ν( Ku

μνi + Cu

μνi )

+i[γμ]βα∂μvβ

i − uij0vjα − uα

j vij

+uβi vα

β + uαβviβ

+i∂μ(uij−1vjμα − uβ

i vμαβ ) (22)

These transformations are quite complicatedin their most general expression. However,for many practical computations of non-supersymmetric local counterterms, we canconsider them at v = 0. We define Qu ≡(Sext

(Q)|Σu)|v=0

. By using δSusyΥ[u]+Υ[Qu] = 0

we can in fact conveniently compute Qu. No-tice that Q is not nilpotent on the sources,but we have the result that Υ[Q2u] is a lin-ear functional of the equation of motion ofthe fermion λ.

It is a well-defined process to compute allobservables, provided that a complete set ofsources has been introduced. This lengthyprocess cannot be avoided because there ex-ists no regulator that preserves both gaugeinvariance and supersymmetry. We mustkeep in mind that renormalization generallymixes physical observables with BRST-exactoperators, and a careful analysis must bedone [5].

5. Enforcement of supersymmetry

Once both Ward identities for the Greenfunctions of fields and of observables havebeen established, it is a straightforward (buttedious) task to adjust the counterterms thatare necessary to ensure supersymmetry andgauge symmetry at the quantum level. Thepossibility of that is warranted by the factthe theory is renormalizable by power count-ing, that no anomaly exist, and that the la-grangian is stable. The technical details aregiven in [6]. The question of not having a reg-ulator that maintains supersymmetry is irrel-evant. However, in practice, one wishes topreserve the symmetry of the bare action asmuch as it is possible, and thus, one uses di-mensional reduction regularization, as in [7].

6. Twisted variables

Using twisted variables for the spines infour dimensions allows one to extract sub-algebra of supersymmetry transformationsthat close without using equations of motion[3]. This property allows one to greatly sim-plify the proofs of finiteness in supersymmet-ric theories. Before coming to this point, letus sketch the way the twist works for theN = 4 theory, by choosing the so-called firsttwist of this theory.

6.1. N = 4 super-Yang–Mills theory in

the twisted variables

The components of spinor and scalar fieldsλα and φi can be twisted, i.e., decomposedon irreducible representations of the followingsubgroup5

SU(2)+ × diag(SU(2)− × SU(2)R

)× U(1)

⊂ SU(2)+ × SU(2)− × SL(2,H) (23)

We redefine SU(2)∼= diag(SU(2)−×SU(2)R

).

The N = 4 multiplet is decomposed as fol-lows

(Aμ, Ψμ, η, χI , Φ, Φ) (L, hI , Ψμ, η, χI) (24)

In this equation, the vector index μ is a“twisted world index”, which stands for the(12 , 1

2 ) representation of SU(2)+ × SU(2).The index I is for the adjoint representationof the diagonal SU(2). In fact, any given fieldXI can be identified as a twisted antiselfdual2-form Xμν− ,

Xμν− ∼ XI (25)

by using the flat hyperKahler structure JIμν .

All 16 components of the SL(2,H)-Majorana spinors can therefore be mappedon the following multiplets of tensors.

λ → (Ψ(1)

μ , Ψ(−1)

μ , χ(−1)

I , χ(1)

I , η(−1), η(1)) (26)

The scalars φi in the fundamental represen-tation of SO(6) decompose as follows

φi → (Φ(2), Φ(−2), L(0), h(0)

I ) (27)

5Usually, one means by twist a redefinition of theenergy momentum tensor that we do not considerhere.


where the superscript states for the U(1) rep-resentation. The 16 generators of the super-symmetry algebra and the corresponding pa-rameter ε are respectively twisted into

Q(1), Q(−1), Q(1)

μ , Q(−1)

μ , Q(1)

I , Q(−1)

I (28)

and

ε → (ω(1), �(−1), ε(1)μ, ε(−1)μ, υ(1)I , υ(−1)I)

(29)

with

δSusy = �Q+ωQ+εμQμ, +εμQμ+υIQI+υIQI

(30)

The ten-dimensional super-Yang–Mills the-ory determines by dimensional reduction theuntwisted N = 4 super-Yang–Mills theory.Analogously, the twisted eight-dimensionalN = 2 theory determines the twisted formu-lation of the N = 4 super-Yang–Mills theoryin four dimensions by dimensional reduction[8,3].

The twisted N = 2, d = 8 symmetry con-tains a maximal supersymmetry subalgebrathat closes without the equations of motion.It depends on nine twisted supersymmetryparameters, which are one scalar � and oneeight-dimensional vector εM .

By dimensional reduction (�, εM ) decom-poses into (�, εμ, ω, υI) and the off-shell rep-resentation of supersymmetry remains. Thedimensionally reduced four-dimensional su-persymmetry with 9 parameters is

δSusy = �Q + ωQ + εμQμ + υIQI (31)

It closes independently of equations of mo-tions to

δSusy2

= δgauge(Φ(φ)+�εμAμ)+�εμ∂μ (32)

with

Φ(φ) ≡ �2Φ + ω�L + �υIhI

+(ω2 + εμεμ + υIυI)Φ (33)

Moreover, using the extended nilpotent dif-ferential d+ s +Q−�iε, the action of Q and

s on all fields is simply given by the definitionof the following extended curvature

F ≡ (d + s + Q − �iε)(A + Ω + c

)+

(A + Ω + c

)2

= F + Ψ(λ) + Φ(φ) (34)

and the Bianchi relation that it satisfies

(d + s + Q − �iε)F

+ [A + Ω + c , F ] = 0 (35)

Here the linear function of the gluini Ψ(λ) is6

Ψ(λ) ≡ �Ψ+ωΨ+υIJI(Ψ)+g(ε)η+iεχ (36)

Eqs. (34) and (35) determine respectively theaction of Q and s on A, c, Ω and on thefields on the right-hand-side of Eq. (34), byexpansion in form degree.

Few degenerate component equations oc-cur when solving Eqs. (34) and (35). Theyare solved by introducing the fields χI andη, the auxiliary fields HI , Tμ and the shadowfield μ. Notice that the auxiliary fields HI

and Tμ, carry a total of 7 = 3 + 4 de-grees of freedom. The latter compensate thedeficit between the number of off-shell gauge-invariant degrees of freedom of fermions andbosons in the theory.

Eqs. (34) and (35) determine δSusy as

δSusyA = �Ψ + ωΨ + g(ε)η + g(JIε)χI

+υIJI(Ψ)

δSusyΨ = −�dAΦ − ω(dAL + T

)+ iεF

+g(JIε)HI

+g(ε)[Φ, Φ] − υI

(dAhI + JI(T )

)δSusyΦ = −ωη + iεΨ − υI χ

I

δSusy Φ = �η

δSusyη = �[Φ, Φ] − ω[Φ, L]

+LεΦ − υI [Φ, hI ]

δSusyχI = �HI + ω[Φ, hI ]

+LJIεΦ − υI [Φ, L]

+εIJKυJ [Φ, hK ]

6Given a vector field V , one defines the 1-formg(V ) ≡ gμνV μdxν , and the vector (JI(V ))μ ≡

JIμ

νV ν . Ψ(λ)μ can be written �Ψμ + ωΨμ +υμν− Ψν − εμη + ενχμν−


δSusyHI = �[Φ, χI ]

+ω([L, χI ] − [η, hI ] − [Φ, χI ]

)−LJIεη − [Φ, iJIεΨ] + Lεχ

I

+υJ [hJ , χI ] + υI([η, L] + [Φ, η]

)−εI

JKυJ([η, hK ] + [Φ, χK ]

)δSusyL = �η − ωη + iεΨ − υIχ

I

δSusy η = �[Φ, L] + ω[Φ, Φ] + LεL

+iεT + υI

(HI + [hI , L]

)δSusy Ψ = �T − ωdAΦ − g(ε)[Φ, L]

+g(JIε)[Φ, hI ] + υIJI(dAΦ)

δSusyT = �[Φ, Ψ] + g(JIε)([η, hI ]

)+ω

(−dAη − [Φ, Ψ] + [L, Ψ]

)−g(ε)

([η, L] + [Φ, η]

)+g(JIε)

([η, hI ] + [Φ, χI ]

)+ LεΨ

+υI

([hI , Ψ] + JI(dAη + [Φ, Ψ])

)δSusyhI = �χI + ωχI − iJIεΨ − υIη

−εIJKυJχK

δSusy χI = �[Φ, hI ] + ω([L, hI ] − HI

)+Lεh

I − iJIεT + υI [Φ, Φ]

+υJ [hJ , hI ] + εIJKυJHK (37)

One can verify that, for Tμ = HI = 0,the transformation laws of δSusy in Eq. (37)are the on-shell transformation laws of thetwisted N = 4 supersymmetry. It is quiteremarkable that the supersymmetry trans-formations are the solution of the curvatureequation (34) and its Bianchi identity (35).As we will shortly sketch, these equationsplay a key role in non-renormalization the-orems.

6.2. Protected operators

Superconformal invariance implies that theso-called BPS local operators are protectedfrom renormalization and their anomalousdimensions vanish to all orders in pertur-bation theory [9]. In the N = 4 theory,these operators play an important role for theAdS/CFT correspondence, since their non-renormalization properties allows to test theconjecture.

One wishes to prove that, without the as-sumption of the superconformal symmetry,

N = 4 supersymmetry implies that all 1/2BPS primary operators, and thus all theirdescendants, have zero anomalous dimension.We will sketch the proof of this statement us-ing only Ward identities associated to gaugeand supersymmetry invariance. The 1/2 BPSprimary operators are the gauge-invariantpolynomials in the scalar fields of the the-ory in traceless symmetric representations ofthe SO(5, 1) R-symmetry group.

In the gauge εμ = 0 the operator Q isnilpotent.7 The linear function of the scalarfields Φ(φ) that characterizes the field de-pendent gauge transformations that appearin the commutators of two supersymmetries,depends in this case on five parameters,

Φ(φ) = �2Φ+�ωL+�υIhI +(ω2+υIυ

I)Φ

(38)

The decomposition under the independentfunctions of the supersymmetric parametersof the invariant polynomial P in Φ(φ) givesall the gauge invariant polynomials in thescalar fields that belongs to traceless symmet-ric representations of SO(5, 1) [3]. Since Q isnilpotent with the restricted set of parame-ters, the shadow number 2 component of thecurvature equation (34) is also a curvatureequation

Qc + c2 = Φ(φ) (39)

By comparison with the Baulieu–Singer cur-vature equation in TQFT’s, one interpretsc as the component of the connexion of thespace of gauge orbits along the fundamentalvector field generating supersymmetry andΦ(φ) as the component of its curvature alongthe same fundamental vector field.8 The

7Remember that the supersymmetry parameters ap-pearing in the differential Q can be understood inquantum field theory as gauge parameters of the Q-invariant gauge-fixing action.8By this we mean the following. Given ω as the con-nection of the fiber bundle defined as the direct sumof the space of irreducible connexions and the spaceof matter fields of the theory, on which the groupof pointed gauge transformations acts freely. DefineΦ as the corresponding curvature s ω + ω2. The su-persymmetry transformations can be seen as gener-


Chern–Simons formula then implies that anygiven invariant polynomial P(Φ) can be writ-ten as a Q-exact term

P(Φ(φ)

)= Q Δ

(c, Φ(φ)

)(40)

where the Chern-Simons form Δ is given by

Δ(c, ω(ϕ)

)≡

∫ 1

0

dtP(c | tω(ϕ) + (t2 − t)c2

)(41)

Any given polynomial in the scalar fieldsbelonging to a traceless symmetric represen-tation of SO(5, 1) has a canonical dimensionwhich is strictly lower than that of all otheroperators in the same representation, madeout of other fields. Thus, by power counting,the polynomials in the scalar fields can onlymix between themselves under renormaliza-tion. Thus, if C is the Callan–Simanzik op-erator, for any homogeneous polynomial PA

of degree n in the traceless symmetric repre-sentation, renormalization can only produceanomalous dimensions that satisfy

C[PA

(Φ(φ)

)·Γ

]=

∑B

γAB

[PB

(Φ(φ)

)·Γ

]

(42)

In this notation, given a local operator O,[O · Γ

]means its insertion in the generating

functional of one-particle irreducible Greenfunctions Γ. Then, the Slavnov–Taylor iden-tities imply

C[ΔA

(c, Φ(φ)

)· Γ

]=∑

B

γAB

[ΔB

(c, Φ(φ)

)· Γ

]

+ · · · (43)

where the dots stand for possible S(Q)|Γ-invariant corrections. However, in the

ated by an anticommuting fundamental vector fieldv, such that Q = Lv ≡ [Iv, s]. With the reducedset of parameters, the vector field v commutes withitself. Then one has

LvIvω+(Ivω)2 =1

2Iv

2`s ω + ω2

´+

1

2[Lv, Iv]ω =

1

2Iv

2Φ

shadow-Landau gauge (i.e., the gauge (13)

with α = 0), ΔA(c, Φ(φ)) cannot appear inthe right-hand-side because such term wouldbreak the so-called ghost Ward identities [3].One thus gets the result that γA

B = 0

C[PA

(Φ(φ)

)· Γ

]= 0 (44)

Upon decomposition of this equation in func-tion of the five independent supersymme-try parameters, one then gets the finitenessproof for each invariant polynomial P(φ) ≡P(φi, φj , φk, · · · ) in the traceless symmet-ric representation of the R-symmetry group,namely

C[P(φ) · Γ

]= 0 (45)

Having proved that all 1/2 BPS primaryoperators have zero anomalous dimension,the Q-symmetry implies that all the opera-tors generated from them, by applying N = 4super-Poincare generators, have also vanish-ing anomalous dimensions. It follows that allthe operators of the 1/2 BPS multiplets areprotected operators.

It is worth considering as an example thesimplest case of Tr Φ(φ)2. One has

QTr(Φ(φ)c −

1

3c3

)= Tr Φ(φ)2

s QTr(Φ(φ)c −

1

3c3

)= 0

s Tr(Φ(φ)c −

1

3c3

)=

Tr

(μ(Φ(φ) − c2

)− [Ω, Φ(φ)]c

)(46)

These constraints imply that Δ(0,3)

[ 32 ]is pro-

portional to Tr(Φ(φ)c − 1

3c3). Thus the

three insertions that we have introduced canonly be multiplicatively renormalized, withthe same anomalous dimension. Moreover,the ghost Ward identities forbid the introduc-tion of any invariant counterterm dependingon the shadow field c, if it is not trough aderivative term dc or particular combinationsof c and the other fields that do not appearin the insertion Tr

(Φ(φ)c − 1

3c3). This gives

the result that

C[Tr Φ(φ)2 · Γ

]= 0 (47)


Finally, by decomposition of the gauge-invariant operators upon independent combi-nations of the parameters, we obtain that allthe 20 operators that constitute the traceless-symmetric tensor representation of rank twoin SO(5, 1) are protected operators

Tr(Φ2

), Tr

(ΦL

), Tr

(ΦΦ +

1

2L2

),

Tr(ΦL

), Tr

(Φ2

), Tr

(ΦhI

), (48)

Tr(LhI

), Tr

(ΦhI

), Tr

(δIJΦΦ +

1

2hIhJ

)

This constitutes the simplest application ofEq. (45), for P(φ) ≡ Tr

(φiφj −

16δi

jφkφk).

6.3. Cancellation of the β function

form descent equations

To show that the coupling constant of theN = 4 theory is not rescaled by renormaliza-tion, the key point is proving that the actionS =

∫L0

4 has vanishing anomalous dimen-sion, in the sense that it cannot be renormal-ized by anything but a mixing with a BRST-exact counterterms. We will restrict here tothe proof of this lemma, that is proving theCallan–Symanzik equation

C[∫

L04 · Γ

]= S(s)|Γ

[Ψ(1) · Γ

](49)

where Ψ(1) is a functional of ghost number -1and shadow number 0. (See [3] for a completediscussion.)

To prove (49), we will use the fact thatdescent equations imply that the lagrangiandensity is uniquely linked to a combinationof protected operators (6.2), with coefficientsthat are fixed functions of the supersymmet-ric parameters.

As shown in [3], the reduced supersymme-try with the six generator Q, Q and Qμ issufficient to completely determine the classi-cal action. For simplicity, we will thus re-strict δSusy to these generators in this section(υI = 0). Because L0

4 and Ch04 = Tr(FF )

are supersymmetric invariant only moduloa boundary-term, the algebraic Poincarelemma predicts series of cocycles, which arelinked to L0

4 and Ch04 by descent equations,

as follows:

δSusyL04 + dL1

3 = 0

δSusyL13 + dL2

2 = �iεL04

δSusyL22 + dL3

1 = �iεL13

δSusyL31 + dL4

0 = �iεL22

δSusyL40 = �iεL

31 (50)

δSusyCh04 + dCh1

3 = 0

δSusyCh13 + dCh2

2 = �iεCh04

δSusyCh22 + dCh3

1 = �iεCh13

δSusyCh31 + dCh4

0 = �iεCh22

δSusyCh40 = �iεCh3

1 (51)

Using the grading properties of the shadownumber and the form degree, we convenientlydefine

L ≡ L04 + L1

3 + L22 + L3

1 + L40 (52)

Ch ≡ Ch04 + Ch1

3 + Ch22 + Ch3

1 + Ch40

The descent equations can then be written ina unified way

(d+δSusy−�iε)L = 0 (d+δSusy−�iε)Ch = 0

(53)

Note that on gauge-invariant polynomials inthe physical fields, δSusy can be identifiedto s + Q, in such way that the differential(d + δSusy − �iε) is nilpotent on them. SinceL0

4 and Ch04 are the unique solutions of the

first equation in (50), one obtains that L andCh are the only non-trivial solutions of thedescent equations, that is, the only ones thatcannot be written as (d + δSusy − �iε) Ξ fora non trivial element of the s cohomologyΞ. The expression of the cocycles Chs

4−s canbe simply obtained using the extended cur-vature (34) since the extended second Chernclass

Ch =1

2Tr

(F+�Ψ+ωΨ+g(ε)η+g(JIε)χ

I

+ �2Φ + �ωL + (ω2 + |ε|2)Φ)2

(54)

is (d + δSusy − �iε) invariant by definition.


As for determining the explicit form ofLs

4−s for s � 1, we found no other way thandoing a brute force computation. In this way,one gets [3]

L40 =

1

2Tr

((�2Φ + �ωL + ω2Φ

)2+ �2|ε|2Φ2

)

(55)

The last cocycle L40 is a linear combination

of the protected operators (6.2) and thus, itsanomalous dimension is zero. This permitsto prove that its ascendant L0

4 can only berenormalized by d-exact or S(s)|Σ-exact coun-terterms.

7. Conclusion

In the formalism that we have presented,the set of fields of a supersymmetric theoryhas been extended. With the introduction ofshadow fields, one can express supersymme-try under the form of a nilpotent differentialoperator.

This clarifies many questions that arisewhen one builds the quantum field theoryof a supersymmetric Yang–Mills theory, inparticular for defining observables and studytheir renormalization. For instance, super-symmetric observables can be defined withinthe standard point of view of the cohomol-ogy of the BRST symmetry. In this frame-work, we have been able to define unam-biguously the computation at all order inperturbation theory of all correlation func-tions, including insertions of gauge invariantlocal operators. The Slavnov–Taylor identi-ties permit one to compute the non-invariantfinite counterterms to maintain supersymme-try and gauge invariance of observables, inde-pendently of the choice of the regularizationscheme.

By twisting the spinors, one can find sub-algebra of supersymmetry with no equationsof motions in the closure relations. This per-mits to simplify the proofs of various renor-malization theorems for the N = 4 super-Yang–Mills theory.

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Shadows and Twisted Variables