Superspace Gauge Fixing of Topological ang-Mills Theories

26 Pages·2003·0.59 MB·English

CBPF-NF-036/03
hep-th/0310184
UFES-DF-OP2003/4
Superspace Gauge Fixing of
Topological Yang-Mills Theories
a1 a;1 b;1
Clisthenis P. Constantinidis , Olivier Piguet and Wesley Spalenza
a
Universidade Federal do Esp(cid:19)(cid:16)rito Santo (UFES)
b
Centro Brasileiro de Pesquisas F(cid:19)(cid:16)sicas (CBPF)
Abstract
We revisit the construction of topological Yang-Mills theories of the Witten type
witharbitraryspace-timedimensionandnumberof \shiftsupersymmetry"generators,
using a superspace formalism. The super-BF structure of these theories is exploited
in order to determine their actions uniquely, up to the ambiguities due to the (cid:12)xing of
the Yang-Mills and BF gauge invariance. UV (cid:12)niteness to all orders of perturbation
theory is proved in a gauge of the Landau type
Key-words: Quantum (cid:12)eld; Topological theory; Gauge theory.
1
SupportedinpartbytheConselhoNacionaldeDesenvolvimentoCient(cid:19)(cid:16)(cid:12)coeTecnol(cid:19)ogicoCNPq{Brazil.
CBPF-NF-036/03 1
1 Introduction
Observables in topological theories possess a global character, such as the knot invariants of
Chern-Simons theory, the Wilson loops, etc. The problem of (cid:12)nding alltheses invariants is a
problem of equivariant cohomology, as proposed by Witten in 1988 [1] for Yang-Mills topo-
logical theory in four-dimensional space-time. Equivariant cohomology is the cohomology
of a BRST-like operator { the \shift supersymmetry operator", associated to a local shift
transformation of the connection (cid:12)eld { in a space of gauge invariant (cid:12)eld polynomials. A
superspace formulation of Witten’s model was proposed by Horne [2] and developed later
on, in particular by Blau and Thompson [3, 4], who extended it to the cases of more than
one supersymmetry generator and in di(cid:11)erent space-time dimensions. In various cases these
topological theories are seen to arise from super-Yang-Mills theories through some twist of
group representations [1, 3, 4], possibly accompanied by dimensional reduction. The reader
maysee[5]forthesystematicconstructionoftopologicaltheoriesfromsuper-Yang-Millsones
using this technique. Our proposal is to systematize the superspace construction of actions
in the most general setting involving an arbitrary number NT of topological supersymmetry
generators in any space-time dimension D. Our construction will be direct, not passing
through the twist procedure. The question of the existence, in each case, of a corresponding
super-Yang-Mills theory will not be touched.
The theory intends to describe gauge (cid:12)eld con(cid:12)gurations with null curvature { or also
selfdual curvature, in the 4-dimensional case. The null or selfdual curvature condition is
implemented through a Lagrange multiplier (cid:12)eld B which has the same hierarchy of zero-
modes as the B (cid:12)eld of a BF type theory [6]. The point of view usually adopted in the
literature [1, 2, 3, 4] is that of considering the supersymmetry generator(s) as BRST oper-
ator(s) associated to the local shift invariance, and (cid:12)xing the latter with suitable Lagrange
multiplier (cid:12)elds. In the present paper, in order to avoid certain ambiguities which may arise
in the usual scheme, we shall consider the theory as a rigid supersymmetric theory with two
gauge invariances, namely the usual Yang-Mills gauge invariance and the gauge invariance
of the B (cid:12)eld, like the one encountered in BF-theories. Both invariances are supergauge
invariances, their parameters being superspace functions. We shall see that this is enough to
de(cid:12)ne the theory in an unambiguous way, apart from the freedom in the choice of a gauge
(cid:12)xing procedure. Moreover, in the case of a gauge (cid:12)xing of the Landau type, we shall show,
using supergraph techniques, that perturbative radiative corrections are completely absent.
The theory thus turns out to be obviously ultraviolet (cid:12)nite.
A very important point is the systematic characterization of all observables of a topolog-
ical theory. This has been fully done in [7] for the NT = 1 theories. Partial results exist in
the literature. In particular, a set of observables has been given in [8] for the case of NT = 2
in a 4-dimensionalKa(cid:127)hler manifold. In the present paper we shallonly show a rather general
set of observables, however without determining if it represents the most general one.
The plan of the paper is the following. After reminding the principal features of the
originalWitten-Donaldson’s topologicalYang-Millstheory [1] in section 2 and of superspace
CBPF-NF-036/03 2
formalism for topological theories in section 3, we shall show the construction of the action
as a super-BF one, with the appropriate gauge (cid:12)xing, in section 4. Examples of observables
are given in section 5, and the ultraviolert problem is dealt with in section 6. A discussion
of the results is done in the concluding section. Some of our conventions and notations are
given in two appendices.
2 Shift Supersymmetry
We are going to review here, for illustrative purpose, how \shift supersymmetry" may de-
scribe the gauge (cid:12)xing of gauge (cid:12)eld con(cid:12)gurations with null curvature, or alternativelywith
selfdualcurvature. We shallconcentrate on the originalDonaldson-Wittenmodel[1, 9], with
one supersymmetry generator in four dimensional space-time.
2.1 Transformation rules and invariant actions
We recall that this model implies, beyond the gauge connection a(cid:22) associated to some gauge
group G, a fermion 1-form (cid:22) and a 0-form (cid:30), with \shift" supersymmetry de(cid:12)ned by the
in(cid:12)nitesimal transformations
Q~a = ; Q~ = D(a)(cid:30) (d(cid:30)+[a;(cid:30)]) ; Q~(cid:30) = 0 : (2.1)
(cid:0) (cid:17) (cid:0)
All (cid:12)elds here and in the rest of the paper are valued in the Lie algebra of the gauge group
{ assumed to be a compact Lie group. Details on the notation are given in appendix ??.
The usualYang-Millsgaugetransformationsread, writtenasBRST transformationswith
ghost c:
2
Sa = D(a)c ; S = [c; ] ; S(cid:30) = [c;(cid:30)] ; Sc = c : (2.2)
(cid:0) (cid:0) (cid:0) (cid:0)
Whereas the BRST operator S is nilpotent, the fermionic generator Q~ is nilpotent modulo
a (cid:30)-dependent gauge transformation:
Q~2a = D(a)(cid:30) ; Q~2 = [ ;(cid:30)] ; Q~2(cid:30) = 0 : (2.3)
(cid:0) (cid:0)
This means that Q~ is nilpotent when restricted to gauge invariant quantities. Following
Witten, we may thus interpret the shiftsupersymmetry invarianceas a BRST-like invariance
in the space of gauge invariant (cid:12)eld functionals. The 1-form represents the ghost of local
shiftinvarianceand (cid:30) isitsghost of ghost. Then the followingcounting ofdegrees offreedom
holds: counting 4 degrees of freedom for a, 4 for the ghost and 1 for the ghost of ghost (cid:30),
(cid:0)
we arrive to a total of 1 degree of freedom, which corresponds to the scalar mode of the (cid:12)eld
a { which in turn is eliminated thanks to the usual Yang-Mills gauge invariance. The (cid:12)nal
number zero of local degrees of freedom is of course characteristic of a topological theory.
In view of the absence of local degrees of freedom, the theory may be de(cid:12)ned through an
actionwhichwillbepurelyofagauge(cid:12)xingtype. This(cid:12)xingofthelocalshiftsupersymmetry
CBPF-NF-036/03 3
0 1 0 (cid:0)1
may be done introducing Lagrange multipliers (cid:12)elds b2, (cid:21)1, (cid:21)0 and (cid:17)0, together with
(cid:0)1(cid:22) 0(cid:22) (cid:0)1(cid:22) (cid:0)2(cid:22)
the corresponding \antighost" (cid:12)elds b2, 1, 0 and (cid:30)0, where the lower right index
denotes the form degree and the upper left one the degree of supersymmetry or \SUSY-
number". The latter number corresponds to a ghost number in the interpretation of shift
supersymmetry as a BRST transformation. Each \antighost" transforms under Q~ into its
corresponding Lagrange multiplier:
Q~ (cid:0)1(cid:22)b2 = 0b2 ; Q~ p(cid:0)1 (cid:22)p = p(cid:21)p (p = 1;2) ; Q~ (cid:0)2(cid:30)(cid:22)0 = (cid:0)1(cid:17)0 ;
(2.4)
~ 0 (cid:0)1(cid:22) ~ p p(cid:0)1(cid:22) ~ (cid:0)1 (cid:0)2(cid:22)
Q b2 = [ b2;(cid:30)] ; Q (cid:21)p = [ p;(cid:30)] ; Q (cid:17)0 = [ (cid:30)0;(cid:30)] ;
(cid:0) (cid:0) (cid:0)
the transformation rules of the Lagrange multipliers assuring the nilpotency of Q~ modulo
a (cid:30)-dependent gauge transformation. If one intends to study the instanton gauge (cid:12)eld
con(cid:12)gurations, i.e. those withselfdualcurvature F = P+F, where P+ isde(cid:12)ned by (A.6), the
(cid:0)1(cid:22)
associated \antighost" andLagrangemutiplierhave tobe chosen as anti-selfdual: P+ 0 =
0
0, P+ (cid:21)0 = 0.
A gauge invariant and Q~- invariant action may be taken as (following [4])
~ (cid:0)1(cid:22) (cid:0)2(cid:22) 0(cid:22) (cid:0)1(cid:22) (cid:0)1(cid:22) 0 (cid:22)
Q Tr b2 F(a)+ (cid:30)0 D(a) + 1 D(a) b2 + 0 D(a) 1
Z (cid:16) (cid:3) (cid:3) (cid:3) (cid:17)
0 (cid:0)1 1 (cid:0)1(cid:22) 0 0 (cid:22)
= Tr b2F(a)+ (cid:17)0 D(a) + (cid:21)1 D(a) b2 + (cid:21)0 D(a) 1
Z (cid:16) (cid:3) (cid:3) (cid:3) (2.5)
(cid:0)1(cid:22) (cid:0)2(cid:22) 0(cid:22) 0 (cid:0)1(cid:22) 1
+ b2 D(a) + (cid:30)0 D(a) D(a)(cid:30)+ 1 D(a) b2 + 0 D(a) (cid:21)1
(cid:3) (cid:3) (cid:3)
(cid:0)2(cid:22) 0(cid:22) (cid:0)1(cid:22) (cid:0)1 (cid:22) 0(cid:22)
+ (cid:30)0[ ; ] 1[ ; b2] 0[ ; 1] ;
(cid:3) (cid:0) (cid:3) (cid:0) (cid:3)
(cid:17)
2
where F(a) = da+ a is the curvature of the Yang-Mills connection a and is the Hodge
0(cid:3)
duality operator (see appendix ??). One sees that the Lagrange multiplier b2 implements
the zero-curvature condition F(a) = 0 { or the selfduality condition, as in the original
(cid:0)1 1
Witten’s paper. (cid:17)0 is the Lagrange multiplier (cid:12)xing the zero-mode of . Morover, (cid:21)1
(cid:0)1(cid:22) 0 0(cid:22) 0(cid:22) 0
(cid:12)xes the zero-mode of b2, (cid:21)0 that of 1. Finally, 1 (cid:12)xes the zero-mode of b2 and
(cid:0)1(cid:22) 1
0 that of (cid:21)1.
This action corresponds to a generalized \Landau gauge" (cid:12)xing. However, it is still
possible to add one invariant term quadratic in the Lagrange multipliers without spoiling
gauge invariance, supersymmetry and SUSY-number conservation. It reads
Q~ Tr (cid:24) (cid:0)2 (cid:22)2 0b2 = Tr (cid:24) 0b2 0b2 + (cid:0)2 (cid:22)2 [ (cid:0)2 (cid:22)2;(cid:30)] : (2.6)
Z 2 (cid:3) Z 2 (cid:3) (cid:3)
(cid:0) (cid:1)
In this case, which corresponds to a generalized Feynman gauge, the Lagrange mmultiplier
0
b2 becomes an auxiliary (cid:12)elds, whose elimination through its equation of motion gives rise,
for (cid:24) = 1, to the original action of Witten [1], which describes selfdual con(cid:12)gurations when
0
choosing the Lagrange multiplier b2 and its corresponding \antighost" as anti-selfdual
2-forms.
CBPF-NF-036/03 4
2.2 Observables
According to Witten, the algebra of observables of the theory is generated by sets of gauge
(n)
invariant forms wp (0 p 4, n integer) obeying "descent equations"
(cid:20) (cid:20)
Q~wp(n) +dwp(n(cid:0))1 = 0 (4 p 1) ; Q~w0(n) = 0 ; (2.7)
(cid:21) (cid:21)
and are uniquely (cid:12)xed up to total derivatives by
(n) (n)
w0 = C ((cid:30)) ;
(n) (n)
where C ((cid:30)) is an invariant corresponding to a Casimr operator of the gauge group.
n C
Each p-form wp being then integrated on some p-dimensional submanifold Mp, represents
an equivariant cohomology class and de(cid:12)nes a basis element of the algebra of observables.
N
3 T-Extended Supersymmetry
Our purpose in this section is to review and develop a superspace formalism describing
topological theories such as Wittens’s theory described in section 2 and generalizations of
it for more than one supersymmetric generators and for any space-time dimension, starting
from the formalism described in [2, 4, 7].
3.1 NT Superspace formalism
NT supersymmetry is generated by the fermionic charges QI, I = 1;:::;NT obeying the
2
Abelian superalgebra
[QI;QJ] = 0 ; (3.1)
commuting with the space-time symmetry generators and the gauge group generators. The
gauge group is some compact Lie group.
A representation of supersymmetry is provided by superspace, a supermanifold with D
3 (cid:22)
bosonic and NT fermionic dimensions . The respective coordinates are denoted by (x ,
I
(cid:22) = 0;:::;D 1), and ((cid:18) , I = 1;:::;NT). A super(cid:12)eld is by de(cid:12)nition a superspace
(cid:0)
function F(x;(cid:18)) which transforms as
@
QIF(x;(cid:18)) = @IF(x;(cid:18)) IF(x;(cid:18)) (3.2)
(cid:17) @(cid:18)
under an in(cid:12)nitesimal supersymmetry transformation.
2
The bracket is here an anti-commutator.
3
Notations and conventions on superspace are given in appendix ??.
CBPF-NF-036/03 5
I
An expansion in the coordinates (cid:18) of a generic super(cid:12)eld reads
N
1 I1 In
F(x;(cid:18)) = f(x)+ (cid:18) :::(cid:18) fI1:::In(x) (3.3)
n!
Xn=1
where the space-time (cid:12)elds fI1:::In(x) are completely antisymmetric in the indices I1:::In.
We recall that all (cid:12)elds (and super(cid:12)elds) are Lie algebra valued. We shall also deal with
superforms. A p-superform may be written as
p
^ I1 Ik
(cid:10)p = (cid:10)p(cid:0)k;I1:::Ikd(cid:18) d(cid:18) ; (3.4)
(cid:1)(cid:1)(cid:1)
Xk=0
where the coeÆcients (cid:10)k(cid:0)1;I1:::Ik are (Lie algebra valued) super(cid:12)elds which are space-time
formsof degree (p k). They arecompletelysymmetricintheirindices since, the coordinates
(cid:0) I
(cid:18) being anti-commutative, the di(cid:11)erentials d(cid:18) are commutative. The superspace exterior
derivative is de(cid:12)ned as
I (cid:22)
^
d = d+d(cid:18) @I ; d = dx @(cid:22) ; (3.5)
^2
and is nilpotent: d = 0.
The basic super(cid:12)eld of the theory is the superconnection A^, a 1-superform:
I
A^ = A+EId(cid:18) ; (3.6)
(cid:22)
with A = A(cid:22)(x;(cid:18))dx a 1-form super(cid:12)eld and EI = EI(x;(cid:18)) a 0-form super(cid:12)eld. The
superghost C(x;(cid:18)) is a 0-superform. We expand the components of the superconnection
(3.6) as
N
1 I1 In
A = a(x)+ (cid:18) :::(cid:18) aI1:::In(x) ; (3.7)
n!
Xn=1
where the 1-form a is the gauge connection, and the 1-forms aI1:::In its supersymmetric
partners. The expansions of EI and of the ghost super(cid:12)eld C read
N
1 I1 In
EI = eI(x)+ (cid:18) :::(cid:18) eI;I1:::In(x) ;
n!
Xn=1
N (3.8)
1 I1 In
C = c(x)+ (cid:18) :::(cid:18) cI1:::In(x) :
n!
Xn=1
The in(cid:12)nitesimal supergauge transformations of the superconnection are expressed as the
nilpotent BRST transformations
SA^ = d^C [C;A^] ; SC = C2 ; S2 = 0 : (3.9)
(cid:0) (cid:0) (cid:0)
In terms of component super(cid:12)elds we have
2
SA = dC [C;A] ; SEI = @IC [C;EI] ; SC = C : (3.10)
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
CBPF-NF-036/03 6
The supercurvature
^ ^^ ^2 I I J
F = dA+A = F(A)+(cid:9)I d(cid:18) +(cid:8)IJ d(cid:18) d(cid:18) (3.11)
transforms covariantly:
SF^ = [C;F^] ;
(cid:0)
as well as its components
2 1
F(A) = dA+A ; (cid:9)I = @IA+D(A)EI ; (cid:8)IJ = (@IEJ +@JEI +[EI;EJ]) ; (3.12)
2
where the covariant derivative is de(cid:12)ned by D(A)( ) = d( )+[A;( )]
(cid:1) (cid:1) (cid:1)
For further use and comparisons with the literature, let us give the explicit examples of
NT = 1; 2.
Example 1 { Case NT = 1
The superconnection (3.6) and the expansions (3.7 - 3.8) read
0
A(x;(cid:18)) = a(x)+(cid:18) (x) ; E(x;(cid:18)) = (cid:31)(x)+(cid:18)(cid:30)(x) ; C(x;(cid:18)) = c(x)+(cid:18)c(x) : (3.13)
The BRST transformations of the component (cid:12)elds are
0 0 0
Sa = D(a)c ; S = [c; ] D(a)c ; S(cid:30) = [c;(cid:30)] [(cid:31);c] ; S(cid:31) = [c;(cid:31)] c ;
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
2 0 0
Sc = c ; Sc = [c;c]
(cid:0) (cid:0)
(3.14)
As for the supersymmetry transformations de(cid:12)ned by (3.2), we have:
0 0
Qa = ; Q = 0 ; Q(cid:31) = (cid:30) ; Q(cid:30) = 0 ; Qc = c ; Qc = 0 : (3.15)
The supercurvature components (3.12) read
F(A) = F(a) (cid:18)D(a) ;
(cid:0)
(cid:9) = +D(a)(cid:31) (cid:18)(D(a)(cid:30) [ ;(cid:31)]) ; (3.16)
(cid:0) (cid:0)
2
(cid:8) = (cid:30)+(cid:31) +(cid:18)[(cid:30);(cid:31)] :
Example 2 { Case NT = 2
The superconnection (3.6) and the expansions (3.7 - 3.8) now read (with I = 1;2)
I 1 2 I 1 2
A(x;(cid:18)) = a(x)+(cid:18) I(x)+ 2(cid:18) (cid:11) ; EI(x;(cid:18)) = (cid:31)I(x)+(cid:18) (cid:30)IJ(x)+ 2(cid:18) (cid:17)I ;
(3.17)
I 1 2
C(x;(cid:18)) = c(x)+(cid:18) cI(x)+ 2(cid:18) cF :
CBPF-NF-036/03 7
The BRST transformations of the component (cid:12)elds are
IJ
Sa = D(a)c ; S I = [c; I] D(a)cI; S(cid:11) = [c;(cid:11)] D(a)cF +(cid:15) [cI; J] ;
(cid:0) (cid:0) (cid:0) (cid:0) (cid:0)
S(cid:31)I = [c;(cid:31)I] cI ; S(cid:30)IJ = [c;(cid:30)IJ] (cid:15)IJcF +[(cid:31)I;cJ] ;
(cid:0) (cid:0) (cid:0) (cid:0)
JK
S(cid:17)I = [c;(cid:17)I] [cF;(cid:31)I]+(cid:15) [cJ;(cid:30)IK] ;
(cid:0) (cid:0)
2 1 IJ
Sc = c ; ScI = [c;cI] ; ScF = [c;cF]+ 2(cid:15) [cI;cJ] :
(cid:0) (cid:0) (cid:0)
(3.18)
The supersymmetry transformations read
QIa = I ; QI J = (cid:15)IJ(cid:11) ; QI(cid:11) = 0 ;
(cid:0)
QI(cid:31)J = (cid:30)JI ; QI(cid:30)Jk = (cid:15)IK(cid:17)J ; QI(cid:17)J = 0 ; (3.19)
(cid:0)
QIc = cI ; QIcI = (cid:15)IJcF ; QIcF = 0 :
(cid:0)
The supercurvature components (3.12) read now
I 1 2 1 IJ
F(A) = F(a) (cid:18) D(a) I + 2(cid:18) (D(a)(cid:11) 2(cid:15) [ I; J]) ;
(cid:0) (cid:0)
J
(cid:9)I = I +D(a)(cid:31)I +(cid:18) ((cid:15)IJ(cid:11) D(a)(cid:30)IJ +[ J;(cid:31)I])
(cid:0)
1 2 KJ
+ 2(cid:18) (D(a)(cid:17)I (cid:15) [ K;(cid:30)IJ]+[(cid:11);(cid:31)I]) ; (3.20)
(cid:0)
1 K
(cid:8)IJ = (cid:30)IJ +(cid:30)JI +[(cid:31)I;(cid:31)J]+(cid:18) ((cid:15)IK(cid:17)J +(cid:15)JK(cid:17)I +[(cid:30)JK;(cid:31)I]+[(cid:30)IK;(cid:31)J])
2
(cid:0) 1 2 KL
+ 2(cid:18) ([(cid:31)I;(cid:17)J]+[(cid:17)I;(cid:31)J] (cid:15) [(cid:30)IK;(cid:30)JL]) :
(cid:0)
(cid:1)
Counting the number of degrees of freedom:
The numbers ofdegrees offreedom, i.e. the numbers ofcomponent (cid:12)elds {rememberingthat
a p-form has D!=[p!(D p!)] components { are shown in Table 1. If we were considering the
(cid:0)
Fields: aI1(cid:1)(cid:1)(cid:1)In(x) A(x;(cid:18)) eI;I1(cid:1)(cid:1)(cid:1)In(x) EI(x;(cid:18)) cI1(cid:1)(cid:1)(cid:1)In(x) C(x;(cid:18))
Numbers NT NT NT NT NT NT
D n D 2 NT n NT 2 n 2
of (cid:12)elds:
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Table 1: Numbers of component (cid:12)elds. D = space-time dimension, NT = number of super-
symmetry generators.
present theory asausualsupersymmetric gaugetheory, with(super)gauge invariancede(cid:12)ned
by the BRST transformations (3.10), the number of physical degrees of freedom would be
given by the total number of components of the forms A and EI minus the number of
components of the superghost C. However, considering it as a topological theory we have to
treat supersymmetry as a local invariance, too, all (cid:12)elds excepted the Yang-Mills connection
a being ghosts or ghosts of ghosts, as in the example shown in section 2. The SUSY-number
CBPF-NF-036/03 8
4
s is thus a ghost number as well as the usual ghost number g. Thus the e(cid:11)ective ghost
number is equal to s +g and, in the counting of the physical degrees of freedom, we must
s+g
therefore assign a sign ( ) to the number of degrees of freedom of a (cid:12)eld, as shown in
(cid:0)
Table 2. One sees that there is a complete cancellation of the local degrees of freedom, as it
Fields: aI1(cid:1)(cid:1)(cid:1)In A eI;I1(cid:1)(cid:1)(cid:1)In EI cI1(cid:1)(cid:1)(cid:1)In C
SUSY #: n 0 n+1 1 n 0
Ghost #: 0 0 0 0 1 1
Degrees of n NT n+1 NT n+1 NT
( 1) D n 0 ( 1) NT n 0 ( 1) n 0
freedom: (cid:0) (cid:0) (cid:0)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Table 2: Numbers of physical degrees of freedom. D = space-time dimension, NT = number
of supersymmetry generators.
should in a topological theory.
3.2 Wess-Zumino gauge
The contact with the formalism described in section 2 is made by choosing a special gauge
(cid:12)xing [4] of the Wess-Zumino type [10]. The BRST transformations of the component (cid:12)elds
can be calculated from the super(cid:12)eld expressions (3.10). They are explicitly given, for
NT = 1 and 2, by (3.14,3.18). We shall only write explicitly the linear part { or Abelian
approximation { of the transformations in the general case, which will be suÆcient for our
argument:
Sa = dc+ ; SaI1:::IN = dcI1:::IN + ;
(cid:0) (cid:1)(cid:1)(cid:1) (cid:0) (cid:1)(cid:1)(cid:1)
SeI = cI + ; SeI;I1:::IN = cII1:::IN + ; (3.21)
(cid:0) (cid:1)(cid:1)(cid:1) (cid:0) (cid:1)(cid:1)(cid:1)
Sc = ; ScI1:::IN = ;
(cid:1)(cid:1)(cid:1) (cid:1)(cid:1)(cid:1)
where the dots represent nonlinear terms. These transformaions indicate that eI(x) and the
completely antisymmetric part of the (cid:12)elds eI;I1:::In(x) are pure gauge degrees of freedom. A
possiblegauge (cid:12)xing istherefore of settingthese (cid:12)elds tozero. This de(cid:12)nes the Wess-Zumino
(WZ) gauge:
eI = 0 ; e[I;I1:::In] = 0 (1 n NT) : (3.22)
(cid:20) (cid:20)
This (cid:12)xes the gauge degreees of freedom corresponding to the ghosts cI1:::In (1 n NT).
(cid:20) (cid:20)
The remaining gauge degree of freedom parametrized by the ghost c, which is of the usual
Yang-Mills type, can be (cid:12)xed in a usual way.
The WZ gauge condition (3.22) is not stable under supersymmetry transformations, but
one can rede(cid:12)ne the generators QI into new generators Q~I, compatible with the WZ condi-
tion, resultingfromacombinationofQI and ofa(cid:12)eld dependent supergauge transformation.
4
s and g are de(cid:12)ned by attributing s = g = 0 to the gauge connection a(x), s = 1; g = 0 for the
I
supersymmetry genratorsQI { hence s=(cid:0)1 to (cid:18) { and s=0; g =1 for the BRST generator S.
CBPF-NF-036/03 9
Thus, let us combine an in(cid:12)nitesimalsupersymmetry transformation of constant commuting
I
parameters (cid:15) with a supergauge transformation Æ(cid:3) of anticommutingparameters (fermionic
super(cid:12)eld) (cid:3)(x;(cid:18)):
I I
Q~ = (cid:15) QI +Æ(cid:3) (cid:15) Q~I : (3.23)
(cid:17)
Æ(cid:3) is in fact a BRST transformation (3.14), with C substituted by (cid:3). This will de(cid:12)ne the
modi(cid:12)ed supersymmetry generator Q~I, provided we choose (cid:3) in such a way to preserve the
WZ gauge condition (3.23). It is convenient to rewrite the WZ condition in a superspace
way:
I
(cid:18) EI(x;(cid:18)) = 0 ; (3.24)
where EI is the d(cid:18)-part of the superconnection (3.6). We shall denote by E~I the solution of
this condition, and by e~[I;I1:::In] (0 n NT) the components of its (cid:18)-expansion { which are
(cid:20) (cid:20)
therefore solutions of (3.22). The latters are tensors with mixed symmetry. Applying Q~ to
(3.24) we (cid:12)nd, after some integration by part in (cid:18):
I I J I I J I I
Q~((cid:18) EI) = (cid:18) (cid:15) @JEI @IC [C;EI] = (cid:15) EI+(cid:18) @I(cid:3)+@J((cid:15) (cid:18) EI)+[(cid:18) EI;(cid:3)]; (3.25)
(cid:0) (cid:0) (cid:0) (cid:0)
(cid:0) (cid:1)
which shows that the WZ condition (3.24) is stable if, and only if, (cid:3) obeys the equation
I I
(cid:18) @I(cid:3) = (cid:15) EI : (3.26)
The solution reads
NT
I 1 I1 In
(cid:3) = (cid:15) (cid:18) (cid:18) e~I;I1:::In(x) ; (3.27)
n!n (cid:1)(cid:1)(cid:1)
Xn=1
where the functions e~I;I1:::In(x) are the coeÆcients of the super(cid:12)eld expansion of E~I, solution
of (3.24).
One can now check that the superalgebra now closes up to (cid:12)eld dependent gauge trans-
formations Æ :
e~IJ
[Q~I;Q~J] = 2Æ :
e~IJ
(cid:0)
Physical degrees of freedom in the WZ gauge:
The numbers of component (cid:12)elds are now given in Table 3. Remember that the only re-
maining ghost is c(x), since the cI1:::In for n 1 correspond to the gauge degrees of freedom
(cid:21)
which have been (cid:12)xed. In order to count the physical degrees of freedom we must again take
Fields: aI1(cid:1)(cid:1)(cid:1)In(x) A(x;(cid:18)) eI;I1(cid:1)(cid:1)(cid:1)In(x) EI(x;(cid:18)) c(x)
Numbers NT NT NT+1 NT
D n D 2 n+1 n (NT 1)2 +1 1
of (cid:12)elds: (cid:0)
(cid:0) (cid:1) (cid:0) (cid:1)
Table 3: Numbers of component (cid:12)elds in the WZ gauge. D = space-time dimension, NT =
number of supersymmetry generators.