Odd dimensional gauge theories and current algebra

ANNALS

OF PHYSICS

204, 281-314

Odd Dimensional GERALD Center

for

Theoretical Massachusetts

(1990)

Gauge

Theories

and Current

Algebra*

V. DUNNE+ AND CARLO A. TRUGENBERGER~ Ph~~sies. Institute

Laboratory for of Technology,

Received

Nuclear Science and Department qf Physics, Cambridge. Massachusetts 02139

April

27, 1990

We quantize gauge field theories in odd dimensional space-time with actions including both a ChernSimons term and a Yang-Mills term. Such theories will be referred to as CSYM theories. We show that there are deep connections between these theories and chiral anomalies and current algebra in even dimensional space-time. The classical canonical structure of the CSYM theories is intimately related to the algebraic properties of the consistent and covariant chiral anomalies. The quantization of the CSYM theories involves a one-cocycle which is the Wess-Zumino functional and, depending on the dimension of spacetime and the gauge group, the consistent realization of gauge invariance at the quantum level imposes a quantization condition on the Chern-Simons coupling parameter. The associated cocycle behavior of physical states may be understood in terms of an Abehan functional curvature on the space of all spatial gauge fields. By considering the CSYM theories on a space-time with a spatial boundary we show that the algebra of Gauss law generators acquires a boundary-valued anomaly which is cohomologous to the Faddeev-Shatashvili proposal for the anomaly in the equal-time commutator of Gauss law operators in the theory of massless chiral fermions interacting with a gauge field in even dimensional space-time. i‘ 1990 Academic Press. Inc.

I. INTRODUCTION Odd dimensional gauge field ‘theories are of particular interest because of the possibility of including a Chern-Simons term in the action. In this paper we consider gauge theories defined in odd dimensional space-time with action consisting of both a Chern-Simons term and a conventional Yang-Mills term. We shall refer to such theories as Chern-Simons-Yang-Mills (CSYM) theories. The three dimensional CSYM theory was first studied in Ref. [l], in which it was shown that the appearance of the Chern-Simons term in the action has several remarkable consequences,notably a quantization condition for parameters in the action and the (gauge-invariant) generation of a mass for the gauge field. This theory has become known as “topologically massive gauge theory.” The three

* This work is supported in part by funds provided by the U.S. Department Contract DE-AC02-76ER03069. + Research supported by a George Murray Post-Doctoral Fellowship. t Research supported by the Swiss National Science Foundation.

of Energy

(D.O.E.)

under

281 0003-4916190 $7.50 Copyright PI 1990 by Academic Press. Inc. All rights of reproductmn m any form reserved.

282

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dimensional gauge theory with the Chern-Simons term as the entire action was first considered in Ref. [2]. Interest in such theories, now known as pure ChernSimons (CS) theories, has recently been revived mainly through the seminal work of Witten [S] who discovered important mathematical applications (to knot theory and to the study of three dimensional manifolds), as well as a deep connection between these theories and two dimensional conformal field theory. Following Witten’s work there has been extensive study of pure CS theories in three dimensions, mainly using path integral [4], canonical [S-7], and perturbative [S] approaches. In this paper, we consider the CSYM gauge theories defined in (2n - 1) dimensional (n z 2) space-time. We shall show that the CSYM theories in general odd dimensions possess an interesting canonical and geometrical structure which is intimately related with (2n - 2) dimensional chiral anomalies [9, lo]. A canonical analysis in the Schrodinger representation [ 111 reveals that the Gauss law constraint of (2~ - 1) dimensional CSYM theory is identical to the (2n - 2) dimensional consistent anomaly equation [12]. This leads immediately to a close link between physical states of the CSYM theory and the Euclidean vacuum functional of chiral fermions coupled to an external gauge field (in one dimension lower). We show that, due to the Chern-Simons term in the action, the space of all static gauge connections acquires an Abelian functional curvature which produces non-trivial holonomies for physical state functionals and which may also be understood as a non-vanishing electric field commutator. A general expression for this curvature may be given in terms of a functional connection related to the fermionic chiral currents in (2n - 2) dimensions. It has been known for some time that there exists some relation between (even dimensional) chiral anomalies and (odd dimensional) Chern-Simons terms. For example, the Chern-Simons form is known to play a crucial role in the differential geometric approach [9, lo] to higher dimensional chiral anomalies. The gauge properties of states in the three dimensional topologitally massive gauge theories were shown in Ref. [ 131 to be closely related to two dimensional (chiral) fermionic determinants. Five dimensional CSYM theories have arisen previously in the work of Faddeev and Shatashvili [14] and Niemi and Semenoff [15] in relation to algebraic and field theoretic (respectively) approaches to four dimensional chiral anomalies.’ It was first shown by Mickelsson [ 161 that the deep mathematical relationship between chiral anomalies and CSYM theories generalizes to higher dimensions. One aim of our paper is to unify and extend these results within a general framework. Another important connection with even dimensional chiral physics arises in the three dimensional pure CS theory which has been shown to be intimately related to the chiral current algebra of two dimensional conformal field theory [3]. This relationship may be understood in several (related) ways. First, when the pure CS theory is quantized on a space-time of the form R x C, where C is a two-dimen’ In fact, in each of these papers higher dimensions.

the authors

conjectured

that

their

results

could

be generalized

to

ODD DIMENSIONAL GAUGETHEORIES

283

sional manifold without boundary, the theory has only a finite number of physical degrees of freedom (the moduli of flat gauge connections on Z:) and the finite dimensional Hilbert space is in one-to-one correspondence with the conformal blocks of two dimensional conformal field theory on Z [3, 6, 71. Second, if C is taken to have a boundary there are additional degrees of freedom living on the boundary. In this case the physical Hilbert space is infinite dimensional and carries a representation of the chiral algebra of the two dimensional conformal field theory

c3, 7, 171. Given that the connection between odd dimensional CSYM gauge theories and even dimensional chiral anomalies is a general feature (see Section III) it is natural to ask whether the aforementioned connection between three dimensional pure CS theory and two-dimensional current algebra is a general feature which persists in higher dimensions. Floreanini, Percacci, and Rajaraman have indeed found a similar connection between live dimensional Abelian pure CS theory and an Abelian current algebra on the boundary of a four space [lS]. Unfortunately, the possibility of establishing such a relation for a non-Abelian theory in higher dimensions is made difficult by the fact that the symplectic structure of pure CS theory is degenerate in live and higher dimensions [ 191, which meansthat there is no simple direct procedure for quantization. In this paper we propose an alternative route. Namely, we consider instead the CSYM theories (which have perfectly acceptable symplectic structure) and show that it is possible to extract (2n - 2) dimensional current algebra from CSYM theories defined on space-time manifolds with spatial boundaries. Such an approach has previously been considered by Mickelsson [ 161. In fact, the commutator algebra of the Gauss law generators in such a CSYM theory has a boundary valued anomalous contribution which is cohomologous to the algebraically-motivated conjecture of Faddeev and Shatashvili [14] for the anomalous equal-time commutator algebra of the fixed-time Gauss law operators in a second quantized theory of masslesschiral fermions interacting with a gauge field. This paper is organized as follows. Section II gives the classical canonical structure of the CSYM theories. The quantization of the CSYM theories using the Schrodinger representation and the origin of the relationships with even dimensional chiral anomalies are discussedin Section III. In Section IV we discuss the appearance and implications of the Abelian functional curvature in the space of static gauge fields. In Section V we consider the Abelian CSYM theories in greater detail. Section VI describes the connection between the CSYM theories and current algebra.

II. CLASSICAL CANONICAL

STRUCTURE OF CSYM THEORIES

In this paper we consider gauge field theories defined in odd dimensional ( 2 3) space-time with action consisting of a Chern-Simons term and a Yang-Mills term &svh4 = SC, + &vl

(2.1 1

284

DUNNE

AND

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For ease of presentation of formulae, we shall consider only flat (2n - 1) dimensional space-times (n 2 2) with Minkowskian metric diag( 1, - 1, - 1, .... - 1). However, most of the analysis may be straightforwardly generalized to curved space-times or even to space-times of the form Rx C, where the (2n -2) dimensional manifold Z may have non-trivial topology. The zeroth direction is to be identified with time in a canonical treatment, the remaining directions being identified with space. This means that the fixed-time surfaces are (2n-2) dimensional Euclidean spaces with negative definite metric diag( - 1, - 1, .... - 1). The Yang-Mills part of the action in (2.1) is 1 s YM -_ - 2e2

s

dv

(2.2)

~ 1 W,,F~‘“),

2n

where FGy are the components2 of the curvature gauge field one-form A = A,&?‘,

two-form

F corresponding

to the

FE ;Fp,dxp dx”

= &,A, - a,~, + [A,, A,]) dx’ dx”.

(2.3)

The non-Abelian gauge field is Lie algebra valued, A, = A; T”, and we work gauge group generators T” which are normalized according to [T”, Tb]

=fobT,

tr( T”T’)

= - $Yb.

(2.4)

The coupling parameter e2 in (2.2) has mass dimension 5 - 2n. The Chern-Simons part of the action in (2.1) is most concisely expressed the differential form notation of (2.3).

&s=k. s (2n - 1)-form Q,,-

using

(2.5)

Q2n-,,

where the Chern-Simons

with

r has the explicit form [lo]

.?I a

2n-1=(n-

1)1(2n)“p1

F< = 5 dA + t2A2,

’ We use Greek letters to denote spatial indices

p(. v, .._, to denote space-time (i = 1, 2, ___,2n - 2).

Io

’ d5 tr(A(F&+‘), (2.6)

Fs=, =F,

indices

(p = 0, 1, ___,2n - 2) and Latin

letters

i, ,j, .. ..

ODD

DIMENSIONAL

and K is a dimensionless coupling Chern-Simons three-form is3 n,=-&tr

=

-

( 1 47c

-Ek'yr

GAUGE

parameter.

AdAftA’

tr

i

285

THEORIES

For

example,

when

n = 2 the

>

‘4,,Ql,,+$4,,A,J,,

nv,,

(2.7)

>

and the CSYM action in (2.1) concides with that of the topologically massive three dimensional gauge theories of Ref. [ I]. When n = 3, the Chern-Simons five-form is

and the CSYM action in (2.1) coincides with an action considered previously in Refs. 114, 151. The ChernSimons form Q;22,2P, is related to the 2n dimensional Chern class Ch,,,(Fh Ch2,,(J’) =

i” tr(F”), n! (27~)~~ ’

=d.Qz,-,, where we have chosen4 to normalize s

ChJ

(2.9)

the integrated Chern class as

F) = 27cm,

nz E z.

(2.10)

An important mathematical property of the ChernSimons form Q2,,_ 1 is that is invariant under homotopically trivial gauge transformations although f LIZ,, - , (i.e., those continuously connected to the identity), it is nor necessarily invariant “large” gauge transformations. However, the under homotopically non-trivial ’ Our convention for the totally antisymmetric c-tensor in (2n - 1) dimensions is co’ IZn-11, 1; and it is numerically related to the totally antisymmetric E-tensor in (2n-2) dimensions by E” ‘2nmZ’= W-L?,, 1 P 4A more conventional normalization in the mathematics liternture is ChZ,(F) = (i”/n!(2rr)“) tr(F”), in which case the integrated Chern class is an integer. However. for our purposes (2.9b(2.10) is more convenient.

286

DUNNE

AND

TRUGENBERGER

variation of i Sz,,, _, under such a gauge transformation, the following simple form: i”(-l)“-’

(n-l)! s

=(-l)“-‘27rm,

tr((g-’

A + g - ‘Ag + g - ’ dg, has

dg)‘“-‘)

m E Z.

(2.11)

The possible values taken by the integer m depend on both the dimension of spacetime and the gauge group. This fact has far-reaching consequences in the quantizarion of gauge theories involving a Chein-Simons term-see Section III. However, for classical CSYM theories we see that the action (2.1) yields a sensible gauge theory because the equations of motion obtained by varying ScsvM with respect to the gauge field are gauge covariant

; (D, FP’“)O f

inx (n-l)!

(47~)“~’

E’~““~~~-~ tr(T”E;,,,.--F,,“~,,,“-,)=O.

(2.12)

Here D, is the usual (gauge) covariant derivative operator D, = a, + [A,, 1. The key observation in developing a canonical formalism for the CSYM theories is that the Chern-Simons part of the Lagrangian may be written in the following form,5 linear in both A, and ki ( =&,Ai):

L,, = iti

dV,, _ 2(A;f J’“(A) sspace

+ &P(A)).

(2.13)

To understand this decomposition we first recall from the definition (2.6) that the Chern-Simons form Q,,- , is linear in A, and k, because when (2.6) is written out in terms of components all space-time indices are contracted with the totally antisymmetric &-symbol. The most remarkable feature of the expansion (2.13) is that the multipliers 2 and Xi (of A, and ki, respectively) are special expressions in the (2n - 2) spatial components Ai of the gauge field which play fundamental roles in the theories of chiral fermions interacting with a gauge field Aj in (2n - 2) dimensional Euclidean space-time. To be more precise, explicit expressions for gU and Xi” are (2.14)

(2.15) ’ Note that in writing (2.13) we have assumed that we can ignore boundary terms arising from spatial integration by parts. These boundary terms are shown explicitly in the Appendix and will indeed play an important role in the discussion of current algebra in Section VI.

ODD

DIMENSIONAL

GAUGE

287

THEORIES

From these expressions, we recognize .s?~ as the (2n - 2) dimensional Euclidean couariarzt anomaly [9, 123 for one left-handed chiral fermion,’ $L = &( 1 - y’” ’ ) $, interacting with the gauge field A i having field strength F(, = ajAi - i3,Ai + [A ;, A,]. Furthermore, x” given by (2.15) is the expression (local in the components Aj of the Euclidean (2~ - 2) dimensional gauge field) whose covariant divergence relates the cooarinnt anomaly .$’ to the consistent anomaly .c4” [9, 10, 121: .a” = .d” + (D;xy. The consistent anomaly .d” is the anomalous left-handed’ non-singlet chiral current j:, .d"=

covariant

(2.16) divergence of the consistent

(D, j;)“.

(2.17)

The consistent current jl, is defined by functional variation of the effective action for one massless Dirac fermion interacting with the gauge field A, via a left-handed chiral coupling: (2.18)

(2.19a)

=

(2.19b)

e”j”1.41

Note that we may alternatively think of W,[A] as the effective action for one lefthanded Weyl fermion interacting with the gauge field Ai via a vector coupling, in which case the vacuum functional Z,[A] may be interpreted as the “square-root” of the determinant of the Dirac operator (8 - L ). For simplicity of terminology we shall refer to W,[A] as the left-handed chiral effective action. The epithet “consistent” refers to the fact that the anomaly & satisfies the Wess-Zumino consistency conditions [20,9] which may be deduced from the fact that the current is obtained by functional differentiation of the vacuum functional ZJA]. The general proof of the claims (2.13 t(2.15) is in the Appendix. By way of illustration, we present here the specific examples of n = 2 and n = 3, as these correspond to the familiar cases of the chiral anomalies in 2- and 4-dimensional

b Our conventions for the (2n -2) dimensional Euclidean (negative i;,‘,;“)= -26” and y’“-‘= -j”+$I . ..$-z. ’ For right-handed fermions the signs of .o/. .g and A” are reversed.

signature)

y-matrices

are

288

DUNNE

(respectively) Euclidean the form (2.13) as L,,=

AND

space-time.

TRUGENBERGER

When n = 2, we recall (2.7) and write

--f-/dV,a”“Ptr

A,L’,A,+?A,A,A,

=iKldV*{A;(

-&l?F;)+ky(

When n = 3, we recall (2.8) and write L,,

L,,

in

> -kE”A/U)].

(2.20)

in the form (2.13) as

A,a,A,a,A,+~A,A,,Apa,A,+~A,,A,ApA,Aj, =iKjdVI{Ag(

-&EQ’tr(T“FVFk,))

+A;

a,AkA,+A,r?,Ak+~AjAkA,

.

(2.21 )

From (2.21) we recognize Ja=

-

1 32x2

-

EYki

tr(

as the familiar covariant anomaly in four Furthermore, the expression Xi” given by -,p, converts

1 - -sUk’tr 24n2

Ja to its consistent

(2.22 1

TaFvFk.)

Euclidean

space-time

dimensions.

i3jAkA,+A,a,Ak+iA,AkA,

(2.23)

form

&pya= &?a - (D;Xi)n Aja,A,+iAjAkA,

I> .

(2.24)

We now return to the task of formulating a canonical analysis of the CSYM gauge theories. Given the result (2.13) it is a simple matter to see how the combination of Chern-Simons plus Yang-Mills actions differs from the familiar purely Yang-Mills theories. Since the Chern-Simons Lagrangian (2.13) is first order in time derivatives it does not contribute to the total CSYM Hamiltonian,’ which therefore retains the usual Yang-Mills form ‘Another space-time momentum

way to see this is that, since the Chern-Simons part of the action metric (as Q2.- L is a (2n - l)-form), it produces no contribution of the system.

is independent to the total

of the energy-

ODD

H

DIMENSIONAL

GAUGE

289

THEORIES

1 0 CSYM= 42e’ space

(2.25)

However, we also see from (2.13) that the effect of the Chern-Simons part of the total CSYM Lagrangian is to modify the canonical structure arising from the Yang-Mills part in two ways. First, due to the term linear in k, appearing in L,,, the canonical momenta are modified to

When we express the total Hamiltonian (2.25) in terms of the canonical momenta we obtain 1 H CSYM = e2(Z7y + itiX~)(H~ + itiXy) + & FG.F“” . (2.27) 2 Note that by setting ti =0 in (2.26) and (2.27) we regain the familiar Yang-Mills relations. Second. the term in L,, which is linear in the non-dynamical field A, means that the conventional Gauss law constraints of Yang-Mills theory (associated with the passage to the Weyl (A, = 0) gauge) are also modified to 1 G” E - 1 (D,E’)” e = 0.

- ix,g” (2.28 )

The Gauss law constraints G” may be re-expressed in terms momenta using (2.26), G” = (D,n’)“+ iti(Dixi)u - iti,&?‘” = (Diz7’)”

- ihd”.

In passing to the final form in (2.29b) we have used the covariant anomaly 2“ and the consistent anomaly dO. setting ti = 0 in (2.28) and (2.29) we regain the familiar A succinct way to summarize the canonical structure is to express the CSYM Lagrangian L,,,, = L,, + L,,

of the canonical (2.29a) (2.29b)

relation (2.16) between the Notice once again that by Yang-Mills relations. described by (2.25)-(2.29) in first order form

(2.30)

290

DUNNEAND

III. QUANTIZATION

TRUGENBERGER OF CSYM THEORIES

In this section we study the quantization of the classical CSYM theories described in the previous section. One of the interesting features of the quantized CSYM theories is the fact that in certain situations (to be described below) the CS coupling parameter ti must take quantized values in order to maintain gauge invariance of the quantum theory. To define a sensible classical gauge theory the action S must lead to gauge covariant equations of motion (see, e.g., Eq. (2.12)). But for a quantum gauge theory we must also require that the phase exponential of the action exp(iS) be gauge invariant. For example, this requirement leads to Dirac’s quantization condition relating the electric and magnetic charges of the magnetic monopole [21]. For the quantized CSYM theories we note that the Yang-Mills part (2.2) of the action is gauge invariant, but the Chern-Simons part (2.5) is not always gauge invariant (the qualification “not always” will be explained below). Indeed, from (2.11) we see that under the gauge transformation A +g-‘Ag +g-’ rig the CSYM action transforms as (n-l)! (21r;n-, (2n- l)! I tr(W’ ‘II

S CsVM’&SYM+(-l)n-‘~=ScsvM+(-1)“P’27rm~,

m = integer.

W*“-7 (3.1)

The fact that the integral .?I Z(g)

=

(2n)-l

~

= 27rm,

(n - 1)! (2n- *)! s tr(W’

&)2”p’)

m = integer

(3.2)

always produces 27~times an integer is a deep mathematical result from the theory of characteristic classes[22]. The particular possible values taken by the integer m depend both on the dimension (2n - 1) of space-time’ and the gauge group 3. The determination of these possible values involves an interplay between the cohomology classesof both S’” - ’ and 3 and the homotopy classesof maps g : S*” - ’ + 9. For example, when the gauge group is Abelian Z(g) is identically zero (and so the action SCSYMis gauge invariant) for all space-time dimensions with n 2 2. This is related to the fact that the homotopy groups r~~~-,(U(l)) are trivial for n 3 2. When 9 is a (compact and simple) non-Abelian gauge group the situation is more complicated. In three space-time dimensions (n=2) the integer m in (3.2) ranges over all the integers, this integer being the “winding number” of g which characterizes the homotopy classesof maps g : S3 -+ ?!?and also the integral cohomology classesof S3. The relevant homotopy group n,(3) is isomorphic to Z, the infinite 9 With suitable boundary conditions on the gauge fields and gauge transformations dimensional space-time may be regarded as S*” ‘, the (2n - 1) dimensional sphere.

the flat (2n - 1)

ODD

DIMENSIONAL

GAUGE

THEORIES

291

cyclic group of the integers, for any compact non-Abelian gauge group 3. In higher space-time dimensions (n 3 3) we shall restrict our attention to the special unitary groups 9 = SU(N) (the situation is more involved for the orthogonal, symplectic, and exceptional groups). For g taking values in SU(N) it is a fundamental result 1231 that 0 N< n (3.3a) 4g)= 271,n (3.3b) Nan, i where nz in (3.3b) runs over all integer values. The SU(N) principal bundles over S”‘+ ’ with N >, n are called “stable” and have associated homotopy groups x2,,-,(SU(N))=Z [241. We can now apply these mathematical results to the physical task of defining a consistent quantum CSYM theory. In three space-time dimensions and with a (compact) non-Abelian gauge group we see that under a gauge transformation s CSYM changes by 27~ times an integer. This is also the case in (2n - 1) space-time dimensions and with gauge group SU(N) with N&n. In these cases the CS coupling parameter ti must take quantized values KEZ

(3.4)

to ensure that exp( is,,,, ) is gauge invariant. We stressthat this argument does not lead to any quantization condition on ti in Abelian CSYM theories]’ or in (2n - 1) dimensional CSYM theories with gauge group SU(N), N < n. In five dimensions (n = 3) the situation is especially simple because both the integral I(g) and the Chern-Simons term itself are proportional to the totally symmetric traces 8” = tr( T”{ Th, r’> ). These quantities dub” vanish identically for all representations of all simple Lie algebras except for SU(N) with N 2 3. Therefore Z(g) vanishes for SU(2) in live dimensions-note that this is a trivial realization of (3.3a). Furthermore, we see that in live dimensions the coupling parameter ti is quantized as in (3.4) for all simple non-Abelian gauge groups for which the ChernSimons form is non-vanishing. We now proceed to the explicit quantization of the CSYM theories using the Schrodinger representation [ 111. We shall make extensive use of the canonical formulation of the classical CSYM theories described in the previous section. In the Schrodinger representation we adopt the Weyl (A,=O) gauge and represent the states by wavefunctionals Y[A] of the gauge field components Ai. We postulate canonical (equal-time) commutation relations between the “coordinates” A and the canonical momenta IZ: [~‘~‘(x),A’“(y)]=f6”6”“s’2”~2’(X-y).

” Note. however, that there may be other reasons for quantizing non-trivial topology, depending on how one chooses to represent on physical states 1251.

(3.5) the CS parameter K if space-time has the action of gauge transformations

292

DUNNE

AND

TRUGENBERGER

The gauge fields Ajb act on the wavefunctionals momenta Z7jn act by functional differentiation

by multiplication

while

the

I y> - ‘YCAI, A’” 1Iv) t, A’T[A],

(3.6)

In the Weyl gauge, the spatial components of the CSYM equations of motion (2.12) are just Heisenberg equations of motion; however, the zeroth component of (2.12) is the Gauss law constraint (2.28))(2.29) and must be imposed as a condition on physical state functionals Y[A] . . ..A.={;(D&)Y-k.rgOj

!P[A]

= 0.

(3.7)

The Gauss law constraints G” are represented as functional differential operators which must annihilate physical states. They also have another important interpretation as the generators of static (time-independent) gauge transformations 6A,= DiA which correspond

(3.8)

to the residual gauge invariance i

dV 2np2AbGb, A;(x) [s

in the Weyl gauge

1

= (Din)’

(x)

= GA;(x).

(3.9)

The algebraic properties of the generators G” are closely related to the gauge transformation properties of the chiral anomalies &“ and &?a. For example, expressing the generators as functional differential operators as in (3.7) we see that the condition that the generators follow the gauge algebra i[G”(x),

Gb(y)]

is equivalent to the Wess-Zumino anomaly JP,

=fobcGc(x) consistency

c~‘~“-~‘(x - y) condition

-f”b’d’(X) 6@-yx -y)] = 0.

(3.10)

[20, 91 on the consistent

(3.11)

ODD

The conservation

follows

DIMENSIONAL

GAUGE

THEORIES

393

of the constraints

from the identity”

satisfied by the consistent anomaly .d”. From (3.10) and (3.12) we conclude that there is no apparent obstruction to demanding the physical state conditions (3.7) in the quantum theory. However, we shall see below that the consistent realization of the Gauss law constraints (3.7) relies crucially on the quantization condition (3.4) for the dimensionless parameter K which multiplies the ChernSimons part (2.5) of the action. Having expressed the Gauss law constraints in the suggestive form (3.7) we can deduce the gauge dependence of the physical state functionals. In fact, by comparing with the consistent anomaly equation (2.17) we see that the ti th power of the vacuum functional for one left-handed chiral fermion in (2n - 2) Euclidean dimensions satisfies the constraint (3.7)

= 0.

(3.14)

Thus a general physical state satisfying Y[A] where

p[A]

is a gauge invariant

(3.7) may be expressed as

= elh*‘LIAIp[A],

(3.15)

functional, ’ p[A]=O.

(3.16)

The factor W,[A] in the exponential in (3.15) is given by (2.19). and the exponential is the tith power of the left-handed” chiral determinant (3.17)

” We shall return to this interesting identity in Section IV. “We could have chosen to use the effective action for a right-handed W,[A]= Pilndet(PPA(l +yZn- ‘)/2). in which case a general physical state may -“~dAlp[A]. Y[.4]=e

595.203.2-4

chiral fermion be expressed as

294

DUNNE

AND

TRUGENBERGER

This shows that the Euclidean effective action for JCflavors of chiral fermions in (2n - 2) dimensions is a physical state of the CSYM theory in (2~ - 1) dimensions. An immediate and important consequence of (3.15) is that even though the physical state functionals are annihilated by the generators of static gauge transformations, they are not invariant under static gauge transformations of their arguments. Indeed, under the gauge transformation A,+Ag=g-‘Aig+g-‘a,g the physical state functional

(3.18)

(3.15) transforms Y[A”]

as

= ,iw-Z(A:d$u[A],

(3.19)

where the one-cocylce (3.20) expresses the gauge non-invariance of the chiral effective action W,[A] and is just the familiar Wess-Zumino action functional [20, lo]. The one-cocycle condition on a2n-29

cr2,~2(A;gl)+a2,~~2(Ag’;g2)-CL2n~2(A;gl is equivalent

to the integrated

g2)=O(mod2n),

version of the Wess-Zumino

(3.21)

consistency

condition

specified by the consistent

anomaly,

c20, 91. The infinitesimal

form of a2,,_ 2 is completely %-,(A;

1 +;I)=

-2ijdVzne2

trA&.

The full Wess-Zumino action cxZn_2(A; g) is, however, complicated to specify for arbitrary n (see below, Eq. (3.36)) and an explicit expression for the non-Abelian effective action W,[A] is only available in two dimensions. By means of illustration of Eqs. (3.15) and (3.19) we present here the case of n = 2 for which we can write [26] l3 W,[A]=&

Z(h)+2iJdV2rr(A+Ae)

>

,

where Z(h) = -2i +f

1 dV, tr(k’

fB

d, hk’

dV,sP”P tr(h-’

I3 Our conventions for the “light-cone” coordinates negative definite metric are: 3, E l/& (-ia, + a,), A,

a,/#-’

8-h) 8,hh-’

a,h)

in the-two-dimensional = l/JZ(

-iA,

&A,).

Euclidean

space with

ODD

DIMENSIONAL

GAUGE

Here the group element 11 (in the complexification

295

THEORIES

of the gauge group)

is defined

by A- =h-‘a_h, and B is a three-volume whose boundary is two dimensional and (3.24) we can check that the consistent currents

(3.25) space. From

(3.23) (3.26a) (3.26b)

satisfy the consistent

anomaly

equation

= -&(d+A:-&A’;) i --$a 871

A‘! ‘J’

Thus we can check that erK’vLCA1 does indeed satisfy the physical state condition (3.7). The one-cocycle c(~+~(A; g) appearing in the gauge transformation law (3.19) for physical states may be computed from (3.23) and is the two dimensional (Euclidean) Wess-Zumino action

zzn-z(A;g)= W,CA”l1 =G

W,CAl

dV, tr($

ai gg-‘A,)

+ ; s, dv, EMvptr(g~‘a,gg~‘J,gg~‘a,g)

1

.

(3.28)

Another way to understand the appearance of the cocycle factor eiK’Zn-z(A;g’ in the gauge transformation property (3.19) of physical states is to consider the variation of the Lagrangian under a static gauge transformation. A general result [27] states that if the Lagrangian changes by a total time derivative under such a gauge transformation, then this change is just the time derivative (d/dt)(mznp2) of the one-cocycle. So to determine a general expression for c(~” _ 2( A; g) we first consider the change in the CSYM Lagrangian under a general gauge transformation A -+ AS = gP ‘Ag + g ~ ’ dg. Under such a gauge transformation the Yang-Mills part of the Lagrangian is unchanged while the Chern-Simons form sZ,,- 1 changes as [lo] Q zn -,(A”)-a,,-,(A)=dp,,~,(A;g)+a,,-,(g-’dir),

(3.29)

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F;,c= dA,,, + A&,

(3.30)

At,r-tA-[dgg-‘.

The symmetrized

trace S tr in (3.30) means C tr(B(PC1)l . . . B(P(“)l) pxms.

S tr(B(‘), ....

(3.31)

For example, with n = 2 we have (3.32)

BAA;g)=&tr(&g-‘A), while for n = 3, we have 8&M=&

tr ( dgg-‘(AdA+dAA)+dgg-‘A3 -;dgg-lAdggplA-dgg-’

dgg-‘dgg-‘A

>

(3.33)

.

We also note that Q,,- r( g- ’ dg) in (3.29) may be written locally as Q,,- ,(g-’

dir) = a,o“k)

Therefore, from (3.29) we deduce the variation static gauge transformation to be

(3.34)

dV,n-1. in the CSYM Lagrangian

under a

where we are now regarding flzn- 2(A; g) given by (3.30) as a (2n - 2)-form depending only on the spatial components Aj of the gauge field. The second term in (3.35) may be written as an integral over a (2n - 1) dimensional space B whose boundary is our (2n -2) dimensional Euclidean space, giving a general expression for the one-cocycle Q,, _ *(A; g),

=sA,-,(A;g)+(--l)“-’

i”(n - 1 )! (2x)Hp 1 (2n _ l)! I B tr(W’

dd’“-‘1. (3.36)

ODD

DIMENSIONAL

GAUGE

297

THEORIES

Given this general expression for cxZn_ r(A; g) we can now understand the origin of the quantization condition (3.4) for the CS coupling parameter ti in the Hamiltonian formalism as follows [27]. We first recall (3.19) that the physical wavefunctionals of the CSYM theory transform under a (2n - 2) dimensional gauge transformation g with a phase

Now consider a “loop” of (2n - 2) dimensional gauge transformations g, parametrized by a homotopy parameter T, such that g,= + % = 1. From (3.37) we see that the wavefunctional Y[A J acquires the phase

ij+,)“-Ii”

(n - l)! (271)n-, (2n- l)!

1

(3.38)

where the additional coordinate in the ball B is identified with the homotopy parameter r. Requiring the physical wavefunctions to be single-valued means this phase factor must be unity. From (3.2) we see that this extra phase factor is always of the form exp(2nixm), where m is some integer. For non-Abelian CSYM theories in three space-time dimensions and for SU(N) CSYM theories in (2n - 1) spacetime dimensions with N 3 n we saw in (3.3b) that m runs over all the integers. Therefore, we arrive at the conclusion that single-valuedness of the physical wavefunctionals requires K to be quantized in integer values. For Abelian CSYM theories or for SU( N) CSYM theories in (2n - 1) dimensions with N < II the integer m is identically zero, so the physical state functionals are single-valued without imposing any quantization condition on K. It is worth noting that these conditions on K agree with those found earlier (3.4) using Dirac’s argument requiring gauge invariance of exp( iScsvM). Having computed the cocycle in the gauge transformation properties of the CSYM physical states (3.15) we should note that a complete solution of the CSYM theory would involve the additional determination of those gauge invariant functionals p[A] for which Y[A] in (3.15) is also an eigenstate of the CSYM Hamiltonian (2.27)

(3.39)

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This, however, is a very diflicult task, and even in the pure Yang-Mills case (where K = 0) this is an unsolved problem. A complete solution for all energy eigenstates is only available for the Abelian theory in three space-time dimensions (see Section V).

IV. FUNCTIONAL

CONNECTION

AND CURVATURE

In this section we discuss a geometric interpretation of the expression Xi* defined in (2.15)-(2.16). We shall see that X’“(A) may be interpreted as an Abelian functional connection on the space L’y - * of all static spatial gauge connections. The restriction to static spatial gauge connections A4 = A?(x) is natural in our canonical approach because we work at fixed time and in the A, = 0 gauge. Such a geometrical role for Xi0 has been noted previously by Bao and Nair [ 191, who showed that X corresponds to a special one-form on .ZF-’ whose curvature gives a natural presymplectic structure on Z”,” -*. It is interesting to note that their conclusions, based on self-consistency arguments within a geometric quantization approach, may be obtained from a straightforward canonical analysis of a particular (2n - 1) dimensional gauge theory, namely the CSYM theory. In the fixed-time (and Weyl gauge) Schrbdinger representation described in Section III, the physical energy eigenstate wavefunctionals of the CSYM theory are solutions Y[A] of both the Gauss law constraints (3.7) and the functional Schrodinger equation (3.39). The theory is invariant under a functional “Abelian gauge transformation” in which the wavefunctionals are multiplied by an arbitrary phase (4.la) Y’CA1 + e’““[“‘Y[A] provided Xi” is also transformed

as (4.lb)

It is easy to see that such a transformation does not affect the functional Schrbdinger equation (3.39) since the canonical momentum appears in the combination (P + ircX’“), reminiscent of the conventional gauge covariant momentum (p + eA) in quantum mechanics. The Gauss law constraint conditions also involve this “minimally coupled” combination as (recalling (2.29a)) we may write (3.7) as {Dy’(ni + ik-X’)b - ~JcJ~} Y[A]

= 0.

(4.2)

Note that the norm I/ YyII of the wavefunctional is also invariant under the phase transformation (4. la). Thus we are led to the natural interpretation of i.%?(A) as a functional connection on CT - *, the space of all (spatial) gauge connections Ai. It is an Abelian connec-

ODD

DIMENSIONAL

GAUGE

299

THEORIES

tion, since [x”, J?‘] = 0, and the Chern-Simons coefficient K is the corresponding coupling constant. As discussedin the previous section, K may or may not be quantized depending on the gauge group and the dimension of space-time. Note that iX’” is a particular function of the gauge potentials A, (see Eq. (2.15))-it is not an arbitrary, functional connection. Indeed, it is not a “pure functional gauge,” since it cannot be expressed as a functional derivative of some functional of A. The Schrodinger equation (3.39) may be thought of as a “gauged” functional Schrodinger equation with the specific form (2.15) for the functional connection ix. It is natural to consider the “functional covariant derivative” operator 9 PA”

6 = ___ - KxyX). SA”(x)

The functional curvature associated with the connection LX’ may be written

(4.4)

In two dimensions (n = 2) we recall that X”(A) curvature is

= - (i/SK) EVA;, so the associated

In four dimensions (n = 3) the connection G?(A) sponding curvature is

is given by (2.23) and the corre-

‘jk’ tr( ( T”, T”] Fk,(x))

dC4’(x - y).

(4.6)

In fact, a general expression for this curvature in (2~ - 2) dimensions is [ 191, 2i” qb(x,

Y) = (n _ 1 )! (471)nP 1

$k,.

kZ.-J

The fact that this curvature is generically non-vanishing reflects the aforementioned comment that the connection iX’ is not a pure functional gauge. The effect of this background functional curvature is to cause non-trivial holonomy of the physical state functionals (3.15). This phenomenon is a functional analogue of the

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conventional Aharanov-Bohm effect. In this light, the quantization condition (3.4) on the coupling parameter IC is analogous to the quantization of the magnetic charge of a Dirac monopole [21]. The functional curvature 9$(x, y) also manifests itself as a non-vanishing commutator of electric fields. The inclusion of the ChernSimons term in the action changes the canonical momenta from the conventional Yang-Mills form - (l/e’) E’ to the CSYM form (2.26). This means that the commutator of electric fields is proportional to the functional curvature, [E’“(x),

Ejb(y)]

= - ie4k-F$(x,

y).

(4.8)

This should be compared with the vanishing of the commutator of electric fields in the pure Yang-Mills theories (where K = 0). A quantum mechanical analogue of (4.8) is the non-vanishing commutator of velocities in the field of a Dirac magnetic monopole [27]. There is, however, a crucial difference between this quantum mechanical example and the field theoretic relation (4.8). While the velocity commutator in the field of a Dirac monopole does not satisfy the Jacobi identity, signaling the presence of a three-cocycle [27], the electric field commutator (4.8) satisfies the Jacobi identity [E’“(x),

[EJb(y),

P”(z)]]

+ {cyclic permutations}

= 0.

(4.9)

At this point it is worth mentioning that FGb is the even dimensional analogue of the odd dimensional functional curvature and non-commuting electric fields found in fixed-time or equal-time investigations of chiral anomalies [28]. However, the important difference is that PGb arises canonically in the even dimensional space resulting from a fixed-time treatment of odd dimensional (space-time) CSYM gauge theory, and there is no three cocycle. In Ref. [28] the functional curvature arises in the odd dimensional space resulting from non-canonical fixed-time or equal-time treatments of the even dimensional (space-time) theory of chiral fermions interacting with a gauge field. In these cases, a non-trivial three cocycle is found to occur. We conclude this section with some comments concerning the transformation properties of the connection X’(A) when the “coordinates” A undergo a gauge transformation. This involves an interesting relationship between the covariant anomaly d” and the functional curvature Szb. From the expressions (2.14) and (4.7) it is straightforward to derive the following relation: &P(x) ~ = - rly(y) GAjb(y)

qyx,

y).

Using the identity &a(x) -= dA”(y)

-~ Mb(y) GAj”(x)

(4.11)

ODD

DIMENSIONAL

GAUGE

THEORIES

301

and the relation (2.16) between the covariant anomaly .a’ and the consistent anomaly &‘“, we can convert (4.10) into an expression for the functional derivative of d”: cw”( x) -= dAjh(y)

-ylyx)

sJ+(Y) 6n”o+~“b’X”(X)6’2n~2’(X-y).

(4.12)

This is precisely the identity (3.13) quoted previously which is needed to prove that the Gauss law constraints G” are conserved. The relation (4.12) reveals how X’(A) transforms under an infinitesimal gauge transformation of the gauge potentials A,. Indeed, rewriting (4.12) we obtain the change in X”’ as

(4.13 ) [(L),~)“x’“(y)]=1.““‘x”(x)6’““~“(x-y)~~iiiiiiiiiiiiiiiiiiiiiiiiiiiii The presence of the second term on the right-hand side of (4.13) indicates the gauge non-covariance of X’h. It is straightforward to check that (4.12) together with (2.16) implies the Wess-Zumino consistency condition (3.11) for the consistent anomaly .d.

V.

CSYM

ABELIAN

THEORIES

When the gauge group is AbelianL4 much of the analysis described in Sections II and III simplifies considerably. The Lagrange density in (2~ - 1) dimensions is

The gauge properties of the theory are less complicated than in the non-Abelian theory because now the action ScsvM = j dV,,,- , 9&M is invariant under all gauge transformations. (Mathematically speaking, this reflects the fact that the (2n - 1 )th homotopy group nzn ~ I of U( 1) is trivial.) The Chern-Simons part of the Lagrangian may be written in the form (2.15) L,, = ix

dVzn sspace

kh,~‘(A)

+ &Y’(A)),

where

and -i

J-1 zz

n(n-2)! I4 We use the convention

(27r)“-’

that the Abelian

” ~~‘Z’-‘Ai,(d,,A,,) E

... (a,LnmJA,2,,mi).

gauge field A, is real-valued.

(5.2)

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The expression (5.3) for 2 is the covariant anomaly in (2n - 2) dimensional Euclidean space-time. In the Abelian theory the distinction between “covariant” and “consistent” forms of the anomaly is simply a matter of overall normalization because (see (2.16))

1 =-- 8. n

(5.5)

A Schriidinger representation analysis may be formulated as in Section III. The Gauss law constraints associated with the passage to the Weyl gauge must be imposed on physical states as before-see Eq. (3.7):

(

fa,f&d

>

Y[AI =o.

(5.6)

Here the Abelian case is significantly simpler than the non-Abelian case because we can immediately write down an explicit expression for such a physical state !P[A]. In fact it is straightforward to check that

=exp

( J -IC

satisfies the physical written as

dV,,_,sdLA,

state condition

a. V2

>

(5.6). Thus a general physical

Y’CAI = ‘J’ot-Al ~‘c4,

(5.7) state may be (5.8)

where p[A] is gauge invariant. Comparison of (5.7t(5.8) with (3.15)(3.17) shows that !Po[A] has the same gauge transformation properties as the Kth power of the left-handed chiral Abelian Dirac determinant. For example, in two dimensions (n = 2) we have, from (5.7), yOIA]=exp(

-f-~dV2{AP~AP-A+~A+})

=exp(-k-dV2{A-$L-A+A)) xexp(gJdV,{AP>AP+A+2A+-2A+A-)).

(5.9)

We recognize the first factor in (5.9) as the rcth power of the usual expression for the left-handed fermionic determinant in the chiral Schwinger model, while the

ODD

DIMENSIONAL

GAUGE

303

THEORIES

second factor is the (ti/2)th power of the (gauge invariant) fermionic determinant in the vector Schwinger model [ 121. Given (5.7))(5.8) we can write down a simple expression for the one-cocycle x2,*~ r(A; g) appearing in the gauge transformation law (3.19) for a physical state. Since (5.10) we deduce

1 =n!(27$

’

dV

2”-2E

il --&? ‘(d,,A,,)

Just as in Section III for the non-Abelian the one-cocycle KU~,, z(A; g) by considering under a static gauge transformation

...(a,,,m,A,2nm2)i’

(5.11)

CSYM theories, we can also compute the change in the CSYM Lagrangian

A,+A;+iY,A;

i, =o.

(5.12)

From (5.1) and (5.2) we see that under such a gauge transformation L CSYM -+ L

CSYM

a’ + IK z

dV 2,-2J”-9y’ , >

(5.13)

from which we deduce the one-cocycle to be

which agrees with the expression (5.11) obtained from considering the gauge transformation properties of the physical states (5.8). As mentioned earlier, in order to give a complete solution of the CSYM theory we would need to specify all those gauge invariant functionals p[A] such that the physical state (5.8) is also an eigenstate of the CSYM Hamiltonian. This eigenstate condition is just the functional Schrodinger equation (3.39) I’dV+,{

-;(-&r&)(~-ti,i)+-$Piifilj

Y[A]=&!F[A].

(5.15)

Given that physical states must have the form (5.8) we can write a simplified Schrodinger equation for p[A] = p[A.] just in terms of the physical gauge

304

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invariant gauge field degrees of freedom, the transverse components Ai, of the gauge field. If we decompose Ai into its longitudinal and transverse parts as Ai= dip + AT, (5.16) A,?=(b,y)A,, then the Schrodinger

equation

(5.15) becomes

(5.17)

= &Y[A]. Substituting p[AT]:

in the form (5.8) for Y[A]

this reduces to a Schrodinger

equation for

(5.18)

= &p[AT], where

-i E~‘.+-~A;(~~~A;) =n(n - 2)! (27~)“~ ’

... (cY,,~-,A~,~,).

(5.19)

For arbitrary n it is still not possible to find explicitly all the solutions tional Schrodinger equation. However, when n = 2, (5.18) reduces to

of this func-

p[B]=&p[B], where e”’ = e2rc/2rr and B is the magnetic field B = - EqaiAj. Equation

(5.20) (5.20) has the

ODD DIMENSIONAL GAUGETHEORIES

30.5

form of a Schr6dinger equation for a functional harmonic oscillator, as is most clearly seen by writing the Hamiltonian in momentum space,

BEG

(5.21) / I

1 i

d2xem “‘“B(x).

This is the total Hamiltonian for an infinite system of harmonic oscillators of frequencies o(p) = Jm. The (unnormalized) ground state energy eigenstate is

(5.22) where the (infinite) zero-point energy is E,=f

jdVZ j hWp).

(5.23)

Combining (5.22) with (5.7)-(5.8) we find the physical ground state wavefunctional of the three dimensional Abelian CSYM theory to be

Yg.,.CAl = ul,c‘41 fQ.ca Kexp(

-~jdV,Bp)enp(~jdV,B~~~.B)

(5.24)

which agrees with the result of Deser-JackiwPTempleton from the study of topologically massive gauge theories (see Eq. (2.29) of Ref. [Id]).

VI. CURRENT ALGEBRA One of the reasons for the recent interest in three dimensional pure ChernPSimons theory is its close relation with two dimensional KaE-Moody algebras and the Wess-Zumino-Witten (WZW) model [29, 303. This relation arises as follows. On a space-time of the form R x C, where .X is a two dimensional manifold without boundaries, the pure Chern-Simons theory has only a finite number of physical degrees of freedom (the moduli of flat connections on 1). However, when 2 has a boundary, there are additional degrees of freedom on the boundary. These are encoded in a Lie group valued field g on the boundary, with currents jLI - tr Tag- ’ a, g (X is the local coordinate on the boundary) satisfying a

306

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KabMoody algebra with central extension IC given by the (quantized) coefficient of the pure ChernSimons action. The corresponding construction starting from the live dimensional Abelian Chern-Simons theory [18] has led to an Abelian current algebra with an operatorial extension given by the infinitesimal two-cocycle derived by dimensional descent from the six dimensional Chern class. This suggests that there is a general relation between (2n - 1) dimensional pure Chern-Simons theories and (2n - 2) dimensional current algebras. However, establishing such a relation in general is complicated by the fact that the symplectic structure of pure CS theories is degenerate in live and higher dimensions [19], which means that there is no direct procedure for quantization. In this section we show that the relation with (2n - 2) dimensional current algebra is not peculiar to pure Chern-Simons theories, but is rather a consequence of the presence of the Chern-Simons term in the action. More specifically, we show how (2n - 2) dimensional anomalous current algebras arise from (2n - 1) dimensional CSYM theories defined on space-time manifolds with spatial boundaries.” In contrast to the pure CS theories the CSYM theories, involving both a CS term and a YM term in the action, have a tractable canonical structure in live and higher dimensions. We consider space-time manifolds of the form R x D, where D is a (2n - 2) dimensional Euclidean disk with boundary B topologically equivalent to S’” ~ 3. In the case of manifolds with boundaries, some extra care is needed in formulating and quantizing the theory. First, we have to specify the boundary conditions on the fields and the gauge transformations. We leave both the values of the fields and the gauge transformations free on the boundary, although we are going to consider occasionally the special class of gauge transformations approaching unity on the boundary. A second modification arises in the first-order form of the CSYM Lagrangian, Eq. (2.30). This acquires a boundary term (6.1) where da’ denotes the standard oriented surface element on the boundary B. Correspondingly, the generators of static gauge transformations (in A, = 0 gauge) also acquire a boundary term, GB(A) = G(A) - j- do’A“fl”, B (6.2) G(A)=j

dV,,-,A”G”,

G”= (Dini)a

- itcgf”.

D

I5 The dimension count is simple: a fixed-time (2~ -2) dimensional, and the spatial boundary comes from an equal-time analysis of a (2n-2) dimensional.

analysis of the (2n - I) dimensional CSYM theory is is (2n-3) dimensional. The related current algebra dimensional chiral theory and so is also (2n-3)

ODD

By means of an integration

DIMENSIONAL

GAUGE

307

THEORIES

by parts, Gs( A ) can be rewritten

as

j dV,,~2(-(~,/i)“n’u-itinus~). D

G,(A)=

(6.3)

The additional boundary term in G,(A) compensates the boundary when integrating by parts the first term in G”. Thus we obtain

term arising

C-G,(~)> At’(x)1 = (~,~(x)Y’,

(6.4)

showing that (6.3) is indeed the correct generator of static gauge transformations. Naturally, for gauge transformations approaching unity on the boundary, (6.3) reduces to the expression valid in open space. In Section III we saw that when there are no spatial boundaries the Gauss law generators follow the gauge algebra (3.10). In terms of the generators G(A) this algebra reads

iCG(A,), G(Az)l =G(CA,, nzl).

(6.5)

We now show that when there is a spatial boundary the commutator Gauss law generators G,(A) acquires an anomalous contribution,

algebra of the

iCGB(AIL GB(~2)l=GB(C~I, ~21)+aB(~I, Ad. The anomalous a,(A,,

(6.6)

term is

AZ)=iti

D dy((D,A,(x)Y’

+s D

dx [/iI,

&Y)

- (D,A,(x))”

A,]“aP’

4(y))

ddb(Y) 6A’“o

(6.7)

In the absenceof spatial boundaries the Wess-Zumino consistency condition (3.11) implies the vanishing of this anomaly (cf. (6.5)). However, this is not the case here becausethe relation (3.11) with ,oCgiven by (2.14)-(2.16) is only true up to boundary terms. Taking this into account we find that a,(A,, A2) is non-zero and that it is purely a boundary term. For example, in two dimensions (n = 2)

(6.8) In four dimensions (n = 3),

0,(/i,, n2)=Lj48x2

n

dx&i’k’aitr([/l,,/i,](Aja,A,+a,A,A,+A,A,A,)

-(a,n,)A,/12A,+ca,n,)A,n,A,).

(6.9)

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While the anomaly a,(A,, A,) may always be written as an integral over the (2~ - 3) dimensional spatial boundary B, the explicit expression in terms of the gauge fields which is obtained from the general formula (6.7) becomes very messy in four and higher dimensions. We can in fact greatly simplify the commutator algebra by modifying the generators in the following fashion. We note first that despite the anomalous contribution appearing in the commutator algebra (6.6) the generators G,(A) still satisfy the Jacobi identity. Thus a,(A,, A,) is an infinitesimal two-cocycle. We are free to modify the form of this two-cocycle by a trivial two-cocycle, which is achieved by adding some functional of A and the gauge field to G,(A). For the purposes of later interpretation it is also important that this addition to GB(A) be a pure boundary term. We consider the following modified generators [ 161 G,(A)

= G,(A)

+ iK 1 dV,,-,

8i(AuQ’“)

D

=

where sZLa is an expression

fD

dV 2n~ 2( - (D,A )” n” + ilc cY,A%‘~),

in the gauge field components da

Note that the modified generators i[G,(A),

= aiQiu.

G,(A)

(6.10)

defined by (6.11)

still generate static gauge transformations

A:(x)]

= (DiA(x))a.

(6.12)

Furthermore, if we restrict the gauge parameters purely to the spatial boundary and consider the A, = 0 gauge (where r is the “radial” direction normal to the boundary) then the G,(A) generate gauge transformations on gauge fields restricted to the boundary. These modified generators G,(A) satisfy the anomalous algebra

~C~,(~,),~,c~,)l=e,(Cn,,n,])+ci,(/l,,/i,),

(6.13)

where 6,(/i,,

A,)=

-ilc +f

dxai([Al,A2])aSZiu

.

(6.14)

D

Since the difference G, - GB is a boundary

term, a,( A r, A,) is once again a bound-

ODD

DIMENSIONAL

GAUGE

309

THEORIES

ary-valued anomaly. Its form is much simpier than a,(,4 , , AZ). For example, in two dimensions (n = 2),

(6.15) In four dimensions

(n = 3)

j” =s[

B

dx@‘?itr({A,, dcr’@tr(

s,A,}

akA,)

[A,, a,A,> $,A,).

We can now restrict attention solely to the boundary and express the anomalous algebra (6.13) without the gauge functions A, and A, (and in the A, = 0 gauge). In two dimensions this gives (6.17) where -yB is the one dimensional boundary coordinate. of a KaE-Moody algebra. In four dimensions we find

Note that this has the form

where sB is the three dimensional boundary coordinate. We recognize the boundary anomalies (6.15))(6.18) as the infinitesimal twococycles derived directly by dimensional descent from the four and six dimensional Chern classes, respectively [ 121. Faddeev and Shatashvili [14] proposed these terms to represent the anomalous equal-time commutator of the fixed-time Gauss law operators in the second quantized theory of massless chiral fermions interacting with a gauge field A. We stress that (6.17))(6.18) have been derived from a gauge field theory with no fermion fields present. Our results show that the algebra (6.6) of Gauss law generators in the fixed-time (i.e., (2n - 2) dimensional) analysis of the (2~ - 1) dimensional CSYM theory has an anomaly living on the (2~ - 3) dimensional spatial boundary and that this anomalous algebra is cohomologous to the Faddeev-Shatashvili anomaly for the equal-time commutator of Gauss law operators in a (2~ - 2) dimensional theory of massless chiral fermions interacting with a gauge field. Note that explicit field theoretic analyses of the Gauss law

DUNNE AND TRUGENBERGER

310

anomaly in two and four dimensions using the BJL technique [31] have also found results cohomologous to, but not precisely isomorphic to, the Faddeev-Shatashvili form. Expression (6.14) provides a compact expression for the generalization of the Faddeev-Shatashvili cohomological form to higher dimensions. In the case of an Abelian theory we can write down a more explicit expression for the boundary-valued anomaly ci,(/i i , AZ). In the Abelian theory the generator eB given by (6.10) is &(A)=

-{

dV,,p,i?,A(n’-iK@).

(6.19)

D

Thus i[C?B(A,),

6,(/i,)]

=iK

JD

dx i, dy 8,/i,(x)

i?jA,(y) (Z-E).

(6.20)

From the general Abelian anomaly d given by (5.3)-(5.5) we deduce

Q’=- n!(27r)“z i &ii”‘.j*.~‘Aj,(d,,Aj,) ...(aj2nm4Aj2n-3).(6.21) Using this in (6.20) we obtain the anomalous extension to the commutator as

j B dai’~“-~~izn~2~,(~;~~,)(~,A,) ... (8i2n-3Ai2nmZ). (6.22) As before we may restrict our attention to the boundary and write

(6.23) where xB is the (2n - 3) dimensional boundary coordinate. We note that the n = 3 version of (6.23) agrees with the form of the results of Floreanini, Percacci, and Rajaraman [18] for pure CS theories and that the general expression (6.23) agrees with their conjectured form in higher dimensions.

VII.

CONCLUDING

REMARKS

To conclude, we summarize our results by emphasizing that the (2n - 1) dimensional CSYM gauge theories may be analyzed in a straightforward canonical formalism (in contrast to their pure CS counterparts for n 2 3). Their structure is intimately related, in two important ways, to the theories of masslesschiral fermions interacting with a gauge field in (2n - 2) Euclidean dimensions.

ODD

DIMENSIONAL

GAUGE

THEORIES

311

First, when the CSYM theory is considered on a space-time in which there is no spatial boundary, the canonical structure of the CSYM theory involves special expressions in the spatial gauge field components Ai which play key roles in the (2n - 2) dimensional Euclidean chiral theory. For example, the CSYM Gauss law constraint is precisely the Euclidean (2n - 2) dimensional anomaly equation. The closure of the Gauss law generator commutator algebra is equivalent to the Wess-Zumino consistency condition on the consistent chiral anomaly. The conservation of the Gauss law generators yields a new identity giving the functional variation of the consistent anomaly in a simple form. Second, when the CSYM theory is considered on a (2n - 1) dimensional spacetime with a (2~ - 3) dimensional spatial fixed-time boundary, the Gauss law generators satisfy an anomalous commutator algebra. The anomalous term lives on the (2vl- 3) dimensional boundary and is cohomologous to the anomalous term proposed by Faddeev and Shatashvili in the equal-time commutator of fixed-time Gauss law operators in the theory of massless chiral fermions interacting with a gauge field in (2n - 2) dimensional space-time. Starting in three dimensions (n = 2) one can consider both the puve CS theory and the CSYM theory. In each of these theories (with space-time having a spatial boundary) one finds that the Gauss law generators satisfy a KaE-Moody algebra on the boundary. In the pure CS theory one can go further and construct representations of this algebra using the physical states of the pure CS theory [3, 6, 7, 171. One motivation for the current paper is the natural question of whether or not an analogous construction works in higher dimensions. This is a difficult question since the higher dimensional pure CS theories cannot be directly quantized in any straightforward manner. However, the results of this paper suggest that it may indeed be possible to obtain representations of the higher dimensional current algebra by considering first the CSYM theories and then taking a limit e* --) 00 in which the Yang-Mills term is removed. Such a limit has been considered in three dimensions and in a quantum mechanical analogue in Ref. [32]. This proposal is currently under consideration.

APPENDIX

To prove the claims (2.13)-(2.15) for arbitrary n we first note that the Chern-Simons part of the Lagrangian Lcs dx” = Ispace Q,, _, is linear both in A, and in ki because when the definition (2.6) of the Chern-Simons form Q,, ~, is written in local coordinates all the space-time indices are contracted with the totally antisymmetric a-tensor. Therefore, it is clear that Lcs may be written in the form (2.13) if we ignore boundary terms arising from total spatial derivatives. So to determine the multipliers 2’ and XiU of Ai and kp (respectively) we simply compute 6L,-,/dAg and c?L,,/dkY. To this end, consider an arbitrary variation of the Chern-Simons form Q,, ~ i,

312

DUNNE

AND

TRUGENBERGER

n-2

+ 1 tr(A(E;;)‘6~~(,~)“-2-‘) r=O

= (n- l);(zx).-l

(tr(GA(F)“-

1 ‘) + d tr(GA Y)},

(A.1) (‘4.2)

where the (2n - 3)-form Y is given by Y=‘i2 j’d{ r=O 0

If we now consider the variation

t(Ft)‘A(FJ-2-‘.

of the Chern-Simons

(~~%s)dx~=j

(A.3) Lagrangian

dQ2n--1,

space

(A.4)

then we see that we can ignore the second term in (A.2) (in absence of spatial boundaries) except when the exterior derivative operator d involves a time derivative. This means that the space-time index of 6A can be 0 only in the first term-never in the second term. Thus

(A.5) from which we deduce the expression (2.14) for J”. To compute 6L,,/6ki from (A.2) we need only consider the case when the exterior derivative in the second term in (A.2) involves a time derivative. Then from the expression (A.3) for Y we derive

Fh+lk,+l ..,~.‘~-2k~-2 1, x tr( TaPkl5 . ..F$%A 5 ’ 5 5

(A.61

from which we deduce the expression (2.15) for F’. This is precisely the X’ satisfying (2.16) found by Bardeen and Zumino (see Eq. (3.56) of Ref. [9]).

313

ODDDIMENSIONALGAUGETHEORIES ACKNOWLEDGMENTS We bringing

would

like

to thank

references

[16]

Note added in proo$ the

role

of the

CS action

R. Jackiw and

[19]

and

I. Singer

to our

attention.

While completing as an induced

this action

for

helpful

discussions.

Thanks

manuscript we received a preprint in odd dimensional fermion-gauge

also

to R. Jackiw

[33] which systems.

for

discusses

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