Available at: http://www.ictp.trieste.it/~pub-off IC/2000/8IU-MSTP/40

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ANOMALY CANCELLATION CONDITIONIN LATTICE GAUGE THEORY

Hiroshi SuzukiDepartment of Mathematical Sciences, Ibaraki University,

Mito 310-8512, Japan*and

The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

We show that, to all orders of powers of the gauge potential, a gauge anomaly Adefined on 4-dimensional infinite lattice can always be removed by a local counterterm,provided that A depends smoothly and locally on the gauge potential and that A re-produces the gauge anomaly in the continuum theory in the classical continuum limit:The unique exception is proportional to the anomaly in the continuum theory. Thisfollows from an analysis of nontrivial local solutions to the Wess-Zumino consistencycondition in lattice gauge theory. Our result is applicable to the lattice chiral gaugetheory based on the Ginsparg-Wilson Dirac operator, when the gauge field is sufficientlyweak ||C/(n, /i) — 1|| < e', where U(n, fj,) is the link variable and e' a certain small positiveconstant.

MIRAMARE - TRIESTE

February 2000

• E-mail: hsuzuki@ictp.trieste.itf On leave of absence from.

1. Introduction

If one puts Weyl fermions on lattice while respecting desired physical properties, one hasto sacrifice the 75-symmetry [1,2]. This implies that the gauge symmetry is inevitably brokenon lattice when Weyl fermions are coupled to the gauge field. This is rather expected, becausewe know in the continuum theory there exists the gauge anomaly [3-8]. However, even ifthe anomaly in the continuum theory cancels tr#_£ Ta{Tb,Tc} = 0 [5-8], the fermionicdeterminant is not gauge invariant in general when the lattice spacing is finite a ^ 0. Thenthe gauge degrees of freedom do not decouple and it becomes quite unclear whether propertiesof the continuum theory (such as the unitarity) are reproduced in the continuum limit, aftertaking effect of dynamical gauge fields into account. Basically this is the origin of difficultiesof chiral gauge theories on lattice [9]. It is thus quite important to understand the structureof breakings of the gauge symmetry on lattice, which will be denoted by A, while keepingthe lattice spacing finite.

What is the possible structure of A for a ^ 0? This question appears meaningless unlessone imposes certain conditions on A. After all, uniqueness of the gauge anomaly in thecontinuum theory [5-8,10-17] is lost for a finite ultraviolet cutoff and the explicit form of thebreaking A is expected to quite depend on details of the lattice formulation. But what kindof conditions can strongly constrain the structure of A? And, under such conditions, is itpossible to relate A and the anomaly in the continuum theory? It seemed almost impossibleto answer these questions. (This statement is not completely true: If one restricts operatorswith the mass dimension < 5 (we assign one mass dimension to the ghost field), the completeclassification of possible breakings has been known in the context of the Rome approach [18].)

The atmosphere has been changed after Liischer's theorem on the 75-anomaly in theabelian lattice gauge theory G = U(l) appeared [19]. Assuming smoothness, locality andthe topological nature of the anomaly, he showed the theorem for 4-dimensional infinitelattice, that is equivalent to

A = SBlnDetM'[A]

W [a + / V ^ W + l^^PaF^{n)Fpa{n + ju + v) + A*

where DetM' is a fermionic determinant and SB is the BRS transformation [10] correspond-ing to the gauge transformation in the abelian lattice gauge theory, SBA^H) = AMc(n)and 5BC(7I) = 0; c(n) stands for the abelian Faddeev-Popov ghost field defined on lattice.In eq. (1.1), a, (3^ and 7 are unknown constants and k^n) in the last term is a local andgauge invariant current. Note that eq. (1.1) holds for finite lattice spacing and the structureis quite independent of details of the formulation. In this sense, this theorem provides a

The meaning of the locality is of course different from that of the continuum theory. We will explainthis terminology in detail in the next section.For our notation, see appendix A.

universal characterization of the gauge anomaly in the abelian lattice gauge theory. More-over, the theorem asserts that the anomaly cancellation in the abelian lattice gauge theoryis (almost) equivalent to that of the continuum theory: The last term of the breaking (1.1)can be removed by adding the local counterterm B = ^2n A^njk^n) to the effective actionlnDetM' ->• lnDetM' + tf, because SBB = ^ n A ^ c ( n ) ^ ( n ) = - E n c ( n ) A £ M n ) - I f t h e

anomaly is pseudoscalar quantity, the first two constants vanish a = (3^ = 0. The term pro-portional to 7 vanishes if ^R e\ — ^L e\ = 0, here e# stands for the U(l) charge, becausewe have absorbed the U(l) charge in c and in F^. This argument shows that the effectiveaction with finite lattice spacing can be made gauge invariant if (and only if) the fermionmultiplet is anomaly-free! This remarkable observation was fully utilized in the existenceproof of an exactly gauge invariant lattice formulation of anomaly-free chiral abelian gaugetheories

In this paper, we attempt to generalize the above theorem (1.1) for general (compact)gauge groups. Our scheme is somewhat different from that of refs. [19,21]. In ref. [21],this problem in nonabelian theories was shown to be equivalent to a classification of gaugeinvariant topological fields in (4+2)-dimensional space, here 4-dimensions are discrete and2-dimensions are continuous. In this paper, instead, we analyze general nontrivial localsolutions to the Wess-Zumino consistency condition [22] in lattice gauge theory. For a genericgauge group, the BRS transformation is defined by:

5BU(n,ii) = U(n,ii)c(n + 'j2)-c(n)U(n,ii), SBc(n) = -c(n)2. (1.2)

Since this BRS transformation is nilpotent 8B = 0, the breaking A = SBlnDet M' mustsatisfy the Wess-Zumino consistency condition

SBA = 0, (1.3)

as in the continuum theory [22,10]. In the continuum theory, consistency and uniquenessof anomaly-free chiral gauge theories in perturbative level follow from detailed analyses ofthe consistency condition [10-17] (for a more complete list of references, see ref. [17]). Wewill see below that the consistency condition (1.3), combined with the locality in the senseof ref. [19], strongly constrains the possible structure of A, as does in the continuum theory.Our basic strategy is to imitate as much as possible the procedure in the continuum theory,especially that of ref. [16]. Of course, there are many crucial differences between continuumand lattice theories and how to handle these differences becomes key of our "algebraic"approach.

The organization of this paper is as follows. Our main theorems which generalize eq. (1.1)are stated in section 3. Our theorems are applicable only if the gauge anomaly A depends

This transformation is obtained by parameterizing the gauge transformation parameter in U(n,fj,)g(n)~1U(n, jj)g{n + //) by g = exp(Ac) where A stands for an infinitesimal Grassmann parameter.

locally on the gauge field. The only framework known to present which possesses this prop-erty is the formulation of refs. [20,21] based on the Ginsparg-Wilson Dirac operator [23-25],or equivalently the overlap formulation [26,27]. Therefore, in section 2, we summarize basicproperties of the gauge anomaly in the formulation of ref. [21]. At the same time, we in-troduce notions of admissibility and of locality. We also introduce the gauge potential anddefine the "perturbative configuration." The sections from section 3 to section 6 are entirelydevoted to a task to determine general nontrivial local solutions to the consistency conditionin the abelian theory G = U(1)N. In section 3, we give some preliminaries. In section 4,we prove several lemmas concerning De Rham and BRS cohomologies on infinite lattice.Here the technique of noncommutative differential calculus [28-31] turns to be a powerfultool [32,33]. Utilizing these lemmas, in section 6, we first determine a complete list of non-trivial local solutions to the consistency condition in the abelian theory. Here the ghostnumber of the solution is arbitrary. Then we restrict the ghost number of the solution unity.After imposing several conditions, we obtain the content of the theorem for the abelian the-ory. Section 7 is devoted to the nonabelian extension. In section 7.1, we derive a basic lemmawhich guarantees the adjoint invariance of nontrivial solutions. In section 7.2, under severalassumptions, we show uniqueness of the nontrivial local anomaly to all orders of powers of thegauge potential. This establishes the content of our theorem for nonabelian theories, whichwill be stated in section 3. In section 7.3, we explicitly write down such a nontrivial localanomaly by utilizing the interpolation technique of lattice fields [34,35]. The last section isdevoted to concluding remarks. Our notation is summarized in appendix A. In appendix B,we explain the calculation of the Wilson line which appears in the integrability conditionof ref. [21].

2. Gauge anomaly in the Ginsparg-Wilson approach

2.1. ADMISSIBILITY, LOCALITY AND THE GAUGE POTENTIAL

The "admissible" gauge field is defined by [21]

\\P(n,/J,,U) — 1\\ < e, for all n, //, v, (2.1)

where P(n, //, v) is the plaquette variable in the representation to which the Weyl fermionis belonging and e a certain small positive constant. In this expression, \\O\\ is the operatornorm that is defined by [36]

i ! ^ , (2.2)

where the norm in the right hand side is defined by the standard norm for vectors. Thereason for this restriction of field space is two-fold.

Consider a finite lattice. Let us suppose that the Dirac operator satisfies a kind of indextheorem. Namely, a difference of numbers of normalizable zero modes of the Dirac operator

with opposite chirality is equal to the topological charge of the gauge field configuration. Theindex is an integer and thus inevitably jumps even if the gauge field configuration changessmoothly. This argument suggests that such a Dirac operator cannot be a smooth functionof the gauge field. Smoothness of the Dirac operator and in turn that of the gauge anomalyare thus expected to hold only within a restricted field space. In fact, a detailed analysis [37]of Neuberger's overlap Dirac operator [25], which satisfies the index theorem [38,39], showsthat the Dirac operator depends smoothly and locally on the gauge field when e < 1/30in eq. (2.1). Our proof is valid only when the gauge anomaly depends on the gauge fieldsmoothly and locally.

Closely relating to the above point, any configuration of the lattice gauge field cansmoothly be deformed into the trivial one U(n, /i) = 1 and thus topology of the gauge fieldspace is trivial if no restriction is imposed. On the other hand, it has been known [40] that,under the condition (2.1), one can define a nontrivial principal bundle over periodic latticeand the field space is divided into topological sectors. For example, for the fundamentalrepresentation of SU(2), t < 0.015 is enough for the construction of ref. [40] to work. Laterwe utilize the interpolation method of ref. [34] which is based on the section of the principalbundle of ref. [40].

Note that eq. (2.1) is a gauge invariant condition. The gauge equivalences of an ad-missible configuration are all admissible. However, structure of the space of admissibleconfigurations is quite complicated and no simple parameterization in terms of the gaugepotential has been known except for abelian cases [19]. This is the reason that our theoremfor nonabelian theories is in practice applicable only for the "perturbative configurations"which will be explained below.

As noted in the introduction, our basic strategy is to imitate the argument in the con-tinuum theory. The first important difference from the continuum theory is the notion oflocality. The anomaly is a local quantity when the ultraviolet cutoff is sent to infinity. Butof course this is not the case for a ^ 0 and we need an appropriate notion which works withlattice. Here we follow the definition of ref. [19] (see also ref. [41]). Suppose that 4>(n) isa field on lattice that depends on link variables U. The field 4>{n) may depend on the linkvariable U(m, (i) at a distant link from the site n. We say that 4>{n) locally depends on thelink variable, if this dependence on U(m,(i) becomes exponentially weak as \n — m\ —>• oo.To be more precise, consider the following decomposition:

) , (2.3)k=l

where 4>kin) depends only on link variables U inside a block of the size k centered at thesite n (such a field </>fc(n) is called ultra-local). If all these fields ^{n) a n ( i their deriva-tives (f>k(n; mi> A*ij •" • j mJV5 A*JV) with respect to the link variables U(mi, /ii), . . . , U(rriN, MJV)

are bounded as

< CNkPN exp(-9k), (2.4)

by the constants CV, PN a n d 9 all being independent of link variable configurations, thenwe say that 4>(n) locally depends on the link variable. In what follows, we introduce also thegauge potential and the ghost field. The same terminology will be used by simply replacing"link variable" by the name of each field. When no confusion arises, we say simply that4>{n) is local. Also when a functional is given by a sum of such local fields, <fr = Y]n (j){n),we simply say that <fr is local. If 4>{n) is a local field, the effective range of dependences isa finite number in the lattice spacing. Therefore, physically, this locality can be regardedas equivalent to the ultra-locality. The technical reason for this definition of locality isthat the Dirac operator which satisfies the Ginsparg-Wilson relation cannot be ultra-localin general [42,43] and, on the other hand, we can apply the Poincare lemma of ref. [19], ifdependences are exponentially weak.

The basic degrees of freedom in lattice gauge theory are link variables. But we stick tothe gauge potential, because its use is essential in arguments in the continuum theory. Tokeep validity of our argument as wide as possible, we consider the following two cases.

Case I. When the gauge group is abelian G = U(1)N. If we take 0 < e < 1 in eq. (2.1) orequivalently (the superscript a here labels each f/(l) factor in G)

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|| Ln Pa(n, fi, is)|| < —, for all a, n, fi, v, (2.5)

there exists the relatively simple prescription [19] which allows a complete parameterizationof the space of admissible gauge fields. Under the condition (2.5), one can associate theabelian gauge potential such that

TTa(n n\ — PYTI Aa(n\ ort <T — Aa(n\ <T rvi (9 f\\U I I I . I I I — C A U ^ 1 , , { I I I . LXJ \ ^ 1 , , [III \ LXJ, I Z . L ) I

and moreover

Ln Pa(n, ii, v) = A^A^n) - AvA^(n). (2.7)

From this relation and eq. (2.5), if a configuration Aa is admissible, the rescaled one tAa

with 0 < t < 1 is also admissible. In this prescription [19] (a closely related prescription for2-dimensional periodic lattice was first given in ref. [35]), the abelian gauge potential Aa^(n)corresponding to the given link variables Ua{n,n) is not unique. Also this mapping doesnot preserve the locality. Nevertheless, as far as gauge invariant quantities are concerned,such an ambiguity disappears and also the locality becomes common for both variables.See refs. [19,44] for details.

Case II. For a general (compact) gauge group G, we define

U(n,fi) = expA^n), | |^(n)| | < TT, (2.8)

and we further impose

1||A,(n)|| < - l n ( l + e) <TT, for all fj, and n. (2.9)

4By noting | | 00 ' | | < ||O||||O'|| [36], one can see that configurations which satisfy eq. (2.9) is

in fact admissible, i.e., they satisfy eq. (2.1). But note that eq. (2.9) is a very restrictive con-dition and it contains only a portion of admissible configurations; in fact, the condition (2.9)is not gauge invariant. We call configurations which satisfy eq. (2.9) "perturbative." For per-turbative configurations, all link variables are close to unity \\U(n, fi) —1|| < e' = (1+e)1 '4 — 1.Unfortunately, our theorem for nonabelian theories is applicable only for this restricted space,when the admissibility (2.1) is required.

2.2. FERMIONIC DETERMINANT AND THE GAUGE ANOMALY

In this subsection, we study basic properties of the gauge anomaly appearing in theformulation based on the Ginsparg-Wilson Dirac operator [20,21], with a particular choiceof the integration measure. As noted sometimes [45,46], this formulation can be reinterpretedin terms of the overlap formulation [26,27]. Therefore it must be possible to repeat a similarargument also in the context of the overlap formulation.

Following refs. [20,21], we define the fermionic determinant as

D e t M ' = / d[\jj\d\$\ exp - (2.10)

where the Dirac operator D satisfies the Ginsparg-Wilson relation 75D + D75 = D75D [23].We assume that the Dirac operator D is gauge covariant and local in the sense of ref. [20] andit depends smoothly on the gauge field. Thus we assume the admissibility (2.1) for gaugefield configurations. The chirality of the fermion is defined with respect to the Ginsparg-Wilson chiral matrix 75 = 75(1 — D) [47,41,48]. Namely, the projection operator has beendefined by PJJ = (1 + e#75)/2. The chirality of the anti-fermion is on the other hand definedby the conventional 75 matrix.

The integration measure for the fermion d[tp] in eq. (2.10) thus depends on the gaugefield nontrivially due to the condition Pffip = ij). But this condition alone does not specifythe integration measure uniquely. For definiteness, we make the following choice which startswith the particular "measure term" [49]

r1

\ dsTrPH[d8PH,5r,PH], (2.11)Jo

where Tr stands for the summation over lattice points ^n of the diagonal (n, n) componentsas well as traces over the gauge and the spinor indices. In this expression, 77 stands for the

• C identically vanishes when the representation of the Weyl fermion is (pseudo-)real [49].

infinitesimal variation of link variables

6TIU(n,fi) = rifi(n)U(n,fi). (2.12)

We have to specify also the s-dependence in eq. (2.11). As a simple choice, we set

U(n,s,fJ) = exp[sAti(n)], 0 < s < 1, (2.13)

for both cases I (2.6) and II (2.8) above. Note that the line in the configurationspace U(n,s,(i) which connects 1 and U(n,(i) is contained in the admissible space (2.1)and, for the case II, in the perturbative region (2.9). The functional (2.11) smoothly andlocally depends on the gauge potential due to the assumed properties of the Dirac operator(C'JJ does not contain the inverse of the Dirac operator). Since the functional C^ is linearin rjfj,, it may be written as

»#(«) . VM = <HT«. (2.14)

This current jff depends smoothly and locally on the gauge potential.

Now, using the Ginsparg-Wilson relation, one can show [49] that CJJ] satisfies the differ-

ential form of the integrability condition [20,21]:

'hQ = -ieH Tr PH [5nPH, SCPH]. (2.15)

Moreover, considering a one parameter family of gauge fields, Ut(n, fi) (0 < t < 1) andintroducing the transporting operator Qt by [21]

dtQt = [dtPt, Pt]Qt, Pt = PH\u^ut, Qo = 1, (2.16)

one can show (appendix B) that C'v satisfies the integrability in the integrated form [21] foran arbitrary closed loop Uo(n, fj,) = Ui(n, fj,) (here ^ ( n ) = dtUt(n, fj,)Ut(n, yu)"1)

W' = exp(i f dtC'^j = Det(l - Po + P0Qi)"e f f , (2.17)

as far as the loop Ut(n, fi) is contained within the perturbative region (2.9) for the case II.Since both the space of admissible fields (2.5) for the case I and the space of perturbativeconfigurations (2.9) for the case II are contractable, there is no global obstruction [50] whichis a lattice counterpart of the Witten's anomaly [51]. Eq. (2.17) guarantees that there

f Here we assume that r\ and ( are independent of the gauge field.

exists the integration measure dfV'MV'] which corresponds to the measure term £ [21]. Inparticular, the infinitesimal variation of the fermion determinant (2.10) is given by

' = Tr S^DPHD'1 + itHC'r (2.18)

We have completely specified the fermionic determinant (2.10) up to a physically irrel-evant proportionality constant. This fermionic determinant is however not gauge invariantin general. The resulting gauge anomaly A = 5B In Det M' is obtained simply by setting

7fc(n) = U(n, fi)c(n + £)£/(n, / i)"1 - c(n), (2.19)

in eq. (2.12). Then from eqs. (2.18) and (2.14), we have

1 } \ (2-20)

where use of the gauge covariance SBD = [D, c] for s = 1 and the Ginsparg-Wilson relationhas been made. Manifestly, this anomaly A depends smoothly and locally on the gaugepotential (and on the ghost field) from the assumed properties of the Dirac operator andfrom the properties of the current j ' a .

We need to know the classical continuum limit of A. The gauge potential in the classicalcontinuum limit A^ix) is introduced by the conventional manner

U(n, fj) = Vexp\a I duAJn + (l — u)Jia)\, (2.21)

L A J

where V stands for the path ordered product. Then the first term of eq. (2.20) produces the

covariant gauge anomaly which can be deduced from the general arguments [52,21] or from

explicit calculations using Neuberger's overlap Dirac operator [53,54] as

d4x£^patTcdf,[ AvdpAa + ^AvApAa ), (2.22)

for a single Weyl fermion. For the second term of eq. (2.20), which corresponds to a diver-gence of the so-called Bardeen-Zumino current [55] in the continuum theory, it is easier to

\ Of course, we assume that parameters in the Dirac operator has been chosen such that there is onlyone massless degree of freedom.

consider £ (2.11) instead of the divergence of the current j ' a . With the choice (2.13), we- 7 /

see in the classical continuum limit

SBPH = s[PH,c] + O(a). (2.23)

Then by using the Ginsparg-Wilson relation, we have

teH£v = -e-f- f dssdaTray5(l-D) + O(a). (2.24)1 Jo

It is possible to argue that the O(a)-term in eq. (2.23) contributes only to 0(a)-termin eq. (2.24) from the mass dimension and the pseudoscalar nature of iejj^'rj (assuming

the Lorentz invariance restores in the classical continuum limit). Then, from eq. (2.22) withthe substitution A^ —> sA^, we have

t r c dn {^KdpAa + \AvApA^ . (2.25)

Combining eqs. (2.22) and (2.25), we have the correct consistent anomaly in the continuumtheory:

r / i \

(2.26)

This expression is for a simple gauge group. The gauge anomaly for a generic gaugegroup G = Yl Ga can be obtained by simply substituting c —>• ^2a cGa and A^ —>• Y^a ^ a

in eq. (2.26). To have the standard form of the anomaly for G = Yia^a, w e add thelocal counterterm to the effective action In Det M" = In Det M' + 5, for the measureterm ieffC" = itu£ + SVS, where

) ( a ) " 1] [U(n,p)ia) ~ 1] [U(n,v)W - 1].n

(2.27)The superscript a runs over simple groups in G and j3 denotes each U(l) factor in G. S de-pends smoothly and locally on the link variable and the modification does not affect the

For abelian cases, the relation 8BPH = S[PH,C] holds for arbitrary a.Strictly speaking, an explicit calculation of eq. (2.11) or of eq. (2.24) in the classical continuum limit,using say the Neuberger's overlap Dirac operator, has not been carried out in the literature. A corre-sponding calculation in the linearized level in the overlap formulation was given in the last referenceof ref. [27]. See also ref. [21]

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integrability, eqs. (2.15) and (2.17). The counterterm S was chosen such that its classi-

cal continuum limit becomes e# J d4[x e^vpaA^ p tr A» Apa A& /(144TT2). Then the gauge

anomaly of the modified effective action becomes

OA 2Z47T

' tr 9,

in the classical continuum limit.

We have thus observed that the anomaly on lattice A = 5B In Det M" smoothly andlocally depends on the gauge potential and on the ghost field and that A reproduces thegauge anomaly in the continuum theory in the classical continuum limit, eq. (2.28). In thefollowing sections, we show that such an anomaly A = 5B In Det M" on infinite lattice canalways be written as A = 5BB, where B smoothly and locally depends on the gauge potential,if (and only if) the anomaly in the continuum theory is canceled tr#_£ Ta{Tb,Tc} = 0 etc.This statement holds to all orders of powers of the gauge potential for nonabelian cases.This implies that, for an anomaly-free fermion multiplet, one can improve the fermionicdeterminant as

Det M"[A] ->• Det M[A] = Det M"[A] exp(-B[A]), (2.29)

so that the improved fermionic determinant DetM[A] has the exact gauge invariance.Therefore, to all orders of the gauge potential, there exists a gauge invariant lattice formu-lation of anomaly-free nonabelian chiral gauge theories, as far as the perturbative configu-rations (2.9) on infinite lattice are concerned.

In the context of the overlap formulation, our choice of the measure term (2.11) corre-sponds to a particular choice of the phase of the vacuum state. The formula correspondsto eq. (2.20) was given in the last reference of ref. [27]. The gauge anomaly and the Witten'sanomaly as local and global obstructions in the overlap formulation were studied in detailin ref. [56]. See also ref. [46].

* Note that (when r\ and ( are independent of the gauge field) (Sr,S^ — 5^8^ + ^^])<S = 0 holds for anyfunctional S of the link variable.

** Since the fermionic determinant Det Af[A] is gauge invariant, one can then regard it as a functional ofthe link variable Det M[C/] for the case I (this is trivially the case for the case II).

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3. Results

In this section, we present our main results in a summarized form. When the gaugegroup is abelian G = U(1)N, we will show the following theorem.

Theorem (Abelian theory). Let A[c, A] he the gauge anomaly A = SB lnDet M'[A] on 4-dimensional infinite hypercubic lattice, defined from a certain fermionic determinant Det M'.Suppose that A depends smoothly and locally on the abelian gauge potential A^ and onthe abelian ghost Reid ca and that it reproduces for smooth Geld configurations the gaugeanomaly in the continuum theory (2.28) in the classical continuum limit. Then A is alwaysof the form (for a single Weyl fermion)

A[c,A] = -^L J2 ^ £ w / » ^ » ^ ( « + $ + ") +SBB[A], (3.1)n abc

where the functional B depends smoothly and locally on the gauge potential Aa.

This is a natural generalization of the Liischer's result (1.1) to multi-C/(l) cases. Asa consequence of this theorem, the anomaly cancellation in the corresponding continuumtheory ^2R eReReR — ^2L e,aLeh

LecL = 0 guarantees the gauge invariance of the effective action,

after subtracting the local counterterm B.

For nonabelian gauge theories, we will show the following statement.

Theorem (Nonabelian theory). Let A[c, A] be the gauge anomaly A = SB In Det M"[A]on 4-dimensional infinite hypercubic lattice, defined from a certain fermionic determi-nant DetM". Suppose that A depends smoothly and locally on the gauge potential A^and on the ghost fields c and that it reproduces for smooth field configurations the gaugeanomaly in the continuum theory (2.28) in the classical continuum limit. Then if the anom-aly in the corresponding continuum theory cancels, tr#_^ Ta{Th,Tc} = 0 etc., A is alwaysBRS trivial, i.e., A = SBB[A], where the functional B depends smoothly and locally on thegauge potential A^. This statement holds to all orders of powers of the gauge potential A^.The explicit form of the nontrivial anomaly A^ SBB is given in eq. (7.50).

Therefore, to all orders of powers of the gauge potential, the anomaly cancellation inthe continuum theory guarantees that of the lattice theory. This seems remarkable but isnot entirely unexpected. Let us recall the expression in the classical continuum limit (2.28).The expression in fact holds to all orders of powers of the lattice spacing a in the classicalcontinuum limit a —> 0 (we assume that the Lorentz covariance restores in this limit). Inthe classical continuum limit, each coefficient of the expansion with respect to a is a localfunctional of the gauge potential and the ghost field. Then the uniqueness theorem of

*** Several variations of this theorem with weaker assumptions are possible. But this form of theoremseems practically reasonable.

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nontrivial anomalies in the continuum theory [16,17] can be appealed and one concludesthat the anomaly (2.28) is the unique possibility (up to contributions of local counterterms).Therefore the anomaly cancellation tr#_^ Ta{Tb,Tc} = 0 guarantees that A = SBB to allorders of powers of the lattice spacing a in the classical continuum limit a —> 0. Of coursethe expansion with respect to a is (presumably at most) asymptotic and this does not provethe anomaly cancellation for a ^ 0. Nevertheless, this argument makes the content of theabove theorem quite plausible. Finally, we emphasize that the above theorems themselvesdo not assume the Ginsparg-Wilson relation and they are applicable to any formulation ifthe prerequisites of the theorems are fulfilled.

4. Preliminaries in the abelian theory

4 . 1 . NONCOMMUTATIVE DIFFERENTIAL CALCULUS

To determine general nontrivial local solutions to the consistency condition (1.3), weneed De Rham and BRS cohomological information, as in the continuum theory [16,17]. Todiscuss De Rham cohomology on infinite lattice, the technique of noncommutative differentialcalculus [28-31] is very useful, because it makes the standard Leibniz rule of the exteriorderivative valid even on lattice. In fact, this technique was applied successfully [32,33] toan algebraic proof of the higher dimensional extension of the Liischer's theorem of ref. [19](that is basically equivalent to eq. (1.1)). Here we recapitulate its basic setup.

The bases of the 1-form on D-dimensional infinite hypercubic lattice are defined asobjects which satisfy the Grassmann algebra

dxlldxv =—dxvdx^. (4.1)

A generic p-form is defined by

f\n) = — J ^ - ^ i r ^ d x ^ •••dxlip, (4.2)

where the summation of repeated indices is understood. The exterior derivative is thendefined by the forward difference operator as

df(n) = — AAJAtl...A(p(n) dx^dx^ • • • dx^. (4.3)

The nilpotency of the exterior derivative d2 = 0 follows from this definition. The essence ofthe noncommutative differential calculus on infinite lattice is

dxpfin) = f(n + //) dx^, (4.4)

where f(n) is a 0-form (i.e., function). Namely, a function on lattice and the basis of 1-form do not simply commute. The argument of the function is shifted along //-direction by

13

one unit when commuting these two objects. The remarkable fact, which follows from thenoncommutativity (4.4), is that the standard Leibniz rule of the exterior derivative d holds.With eqs. (4.3) and (4.4), one can easily confirm that

d[f(n)g{n)} = df[n)g{n) + (-l)pf(n)dg(n), (4.5)

for arbitrary forms f(n) and g(n) (here f(n) is a p-form). The validity of this Leibniz ruleis quite helpful for following analyses.

We also introduce the abelian gauge potential 1-form and the abelian field strength2-form by

Aa(n) = Al(n) dx^ Fa(n) = ^ F ° » dx»dxv = dAa(n). (4.6)

Note that the Bianchi identity takes the form dFa{n) = 0. We never use the symbol Fa

or F for nonabelian field strength 2-form.

4.2. ABELIAN BRS TRANSFORMATION

From eqs. (1.2) and (2.6), the BRS transformation for the gauge potential and for theghost field in the abelian theory is defined by

SBA^(n) = A M c » , SBca(n) = 0. (4.7)

The BRS transformation is nilpotent 8B = 0 and the abelian field strength is BRS invari-ant 6BF^V{TI) = 0. We also introduce the Grassmann coordinate 9 [57-59] and define theBRS exterior derivative by

s = SB • d6. (4.8)

The usual 1-form dx^ and the BRS 1-form d6 anticommute with each other dx^dO = —dOdx^and the BRS 1-form d9 commutes with itself d0d9 ^ 0. Therefore, for a Grassmann-even(-odd) p-form / (n) , we have

def(n) = ±(-l)Pf(n)dO. (4.9)

We also have

s2 = { s , 4 = 0, (4.10)

where the first relation follows from SB = 0. Finally, we introduce the ghost 1-form by

Ca{n) = ca(n)d6. (4.11)

In terms of these forms, the BRS transformation in the abelian theory (4.7) is expressed as

sAa(n) = -dCa(n), sCa(n) = 0, sFa(n) = 0. (4.12)

The noncommutative rule (4.4) will be always assumed in expressions written in terms offorms.

14

5. Basic lemmas in the abelian theory

With the tools introduced in the preceding section, we establish in this section severallemmas which provide cohomological data. The algebraic Poincare lemma and the covariantPoincare lemma concern the De Rham cohomology, including nontrivial information aboutdependences on the gauge potential and on the ghost field. The Poincare lemma on infinitelattice [19] is the foundation of these lemmas. We also determine the BRS cohomology inthe abelian theory G = U{1)N.

5.1. ALGEBRAIC POINCARE LEMMA

The algebraic Poincare lemma asserts that any d-closed form on D-dimensional infinitelattice is always d-exact up to a constant form; D-forms are exceptional because any D-form is d-closed. Moreover, the lemma asserts that the locality of dependences is preservedbetween the original form and its "kernel."

Algebraic Poincare lemma. Let r\ be a p-form on D-dimensional infinite lattice thatdepends smoothly and locally on the gauge potential Aa and on the ghost Geld ca. Then

dr](n) = 0 & 77(71) = dx(n) + £(n) dDx + B, (5.1)

where B is a constant p-form and the (p — l)-form x and the function £ depend smoothlyand locally on the gauge potential and on the ghost field. The function £ satisfies

) + 0, (5.2)

for a certain local variation S of the gauge potential and the ghost Reid.

Note. The term £ dDx in eq. (5.1) represents a non-topological part in the D-form r\. In otherwords, a D-form 7/top. that is topological, ^2n 5f]toP. (n) = 0 for an arbitrary local variation,is always d-exact up to a constant form.

Proof. We define r\t by rescaling fields as A^ —>• tA^ and c° —>• tca. Then since 77 dependssmoothly on Aa and on ca,

rj(n) =r](n)t=o+ / dt

ri

• The present algebraic Poincare lemma is somewhat different from that of ref. [32]. Practically, thepresent form is more convenient.

15

where

Eq. (5.4) implies

d&^in', n) = dKa{ri, n) = 0, (5.5)

because dr\ = 0 for arbitrary configurations. Moreover, since r\ depends locally on A^ andon ca, 9^(n',n) and Ka(n',n) decay exponentially as \n — n'\ —>• oo. This allows us to applyLiischer's Poincare lemma [19] for p < D to eq. (5.5) which asserts that there exist forms©^ and Ka such that

0«(n', n) = d@l{ri, n), Ka(n', n) = dKa(n', n). (5.6)

These forms O^(n,n') and Ka(n,n') also decay exponentially as \n — n'\ —> oo [19]. Substi-tuting this into eq. (5.3), we have TJ = dx + B where B = rjt=o and

X(n) = V \AaJn')eaJri, n) + ca(n')Ka(n', n)l. (5.7)

From the locality property of O^(n,n') and of Ka(n,n') [19], one can easily see [32] thatx(n) is a local field. Also the smoothness is preserved in the construction (5.7). In this way,the lemma (5.1) is established for p < D.

For p = D, dt] = 0 is a trivial statement and thus we decompose r\ as

'I — '/top. ' •*-a -^i W"° /

where ^n Srjtop,(n) = 0 for an arbitrary local variation. Then 8^ and Ka in eq. (5.4) defined

from ?]top. satisfy

a(n',n) = 0. (5.9)

Then Liischer's Poincare lemma for p = D [19] asserts that there exist 0" and Ka whichsatisfy eq. (5.6). The rest is the same as for p < D and we have r/tOp. = dx + B. •

16

5.2. ABELIAN BRS COHOMOLOGY

Abelian BRS cohomology. Let X be a form on infinite hypercubic lattice that dependssmoothly and locally on the gauge potential and on the ghost Geld. Then,

sX(n) = 0^X{n) = Cai(n) • • • Caa(n)X[Q1'"aa][{Fi}; n] + sY{n), (5.10)

where the form XQ (n) depends smoothly and locally only on the abelian fieldstrength F^v. The form Y(n) depends smoothly and locally on the gauge potential andon the ghost field, fn particular, differences of the ghost field can appear only in the BRStrivial part sY.

Note. The form XQ""13' is totally antisymmetric on the upper indices because ghost 1-

forms Ca simply anticommute with each other. XQ ag\n) depends only on the field

strength F^(n) and its differences, such as AMF"p(n), A^F"p(n), A/UAZ/F^7(n) and so on;

obviously XQ '"a^ is gauge invariant. In what follows, we denote as XQ " [{Fi}\ to indi-cate this particular dependence on the field strength, including smoothness and locality ofthe dependence.

Proof. The proof of the abelian BRS cohomology for a single U(l) case [32] can be repeatedby simply supplementing another index a to the gauge potential A^ and to the ghost field c.Thus we do not reproduce it here to save the space. •

5.3. COVARIANT POINCARE LEMMA

As in the continuum theory [16], the following covariant Poincare lemma is crucial todetermine general nontrivial local solutions to the consistency condition. This lemma for asingle U(l) case G = U(l) was given in ref. [32]. It turns out that, however, its extensionto multi-f/(l) cases is not trivial, due to the reason which will be explained after the proof.In fact, we have at present only the following cumbersome proof that works only for 4- orlower dimensional lattice.

Covariant Poincare lemma. On 4-dimensional infinite hypercubic lattice, ifp-form ap[{F{}]is d-closed for p < 4, or if 0:4 [{Fj}] = dx3 + B4 where B4 is a constant 4-form, then ap is ofthe structure

ap[{Ft}; n] = dap_i[{F?}; n] + Bp + Fa(n)Bap_2 + Fa{n)Fh{n)B{

pa% (5.11)

where Fa is the field strength 2-form and B 's are constant forms.

Note. Here all the expressions are written in terms of the noncommutative differentialcalculus.

17

Proof. We prove the lemma step by step from 0-form p = 0 until 4-form p = 4.

For p = 0. The lemma trivially holds by the algebraic Poincare lemma (5.1). Namely, thed-closed 0-form «o must be a constant ao = BQ.

For p = 1. By the algebraic Poincare lemma, the d-closed 1-form a\ is d-exact up to aconstant 1-form. Also a.\ is s-closed because it is a function of the field strength. Namely,

Oll=dxl + Blj sai = 0. (5.12)

Since these equations imply sa\ = SCIXQ = — dsx^ = 0, the algebraic Poincare lemma assertsthat

sX°o = O, (5.13)

where we have used the fact that the right hand side cannot be a constant. The solution tothis equation is given by the abelian BRS cohomology (5.10) for g = 0 case:

and thus eq. (5.12) shows that the lemma holds for p = 1:

a1=duj0[{Fi}] + B1. (5.15)

For p = 2. In this case, from the algebraic Poincare lemma, we have

a2 = dxi + B1, sa2 = 0, (5.16)

and, in a similar way as the p = 1 case, these lead to the following descent equations

sx°i = dxl sxl = 0. (5.17)

The general solution to the last equation is given by the abelian BRS cohomology

xh = CaUo[{Fi}] + *Po. (5-18)

We may however absorb (3Q in the redefinition of xl a n ( i x\i

Xl^xl + sf3o, Xi^Xi+df3o, (5.19)

without changing a2- We can therefore take xl = Ca(jQ. Then the first equation in eq. (5.17)

18

reads

A1 ° ° 5.20Cad%

Now consider a special configuration of the ghost field ca(n) —>• ca = const. Then theconsistency of eq. (5.20) requires

du% = 0, s(x°1+Aa^) = 0, (5.21)

because s (nomething) is proportional to differences of the ghost fields such as ca(n + //) —ca(n). Note that U)Q does not depend on the ghost field. The solution to the first equationof eq. (5.21) is given by the present lemma for p = 0, which we have shown above:

u$ = B$ (const.), (5.22)

and then the second relation of eq. (5.21) implies

by the BRS cohomology. Going back to the original relation (5.16), we have

a2 = -FaB* + du^Fi}] + B2, (5.24)

because dAa = Fa. This shows the lemma for p = 2.

For p = 3. In this case, the counterparts of eqs. (5.16) and (5.17) are

a3 = dX°2 + B3, sa3 = 0, (5.25)

and

0. (5.26)

The solution to the last equation is (we have absorbed the BRS trivial part as eq. (5.19))

xl = CaChJ«h\{m (5-27)

where UQ is antisymmetric on a <H> b. For following arguments, it is quite helpful tointroduce the symmetrization symbol, that is defined by

sym(X1X2 ...XN) = YJjf[ ^Xa(l)Xa{2) • • • Xa{N), (5.28)a

where the summation is taken over all the permutations a. The sign factor ea is defined asthe signature arising when the product X\ • • • X^ is made to the order X(T^X(T^2) • • • ^CT(JV)

19

by regarding all JQ's are ordinary (i.e., the form basis dx^ simply commutes with functions)forms. With use of the symmetrization symbol, eq. (5.27) is trivially written as

x2 = s y m ( c a C 6 ) 4 ° 6 ] , (5.29)

and then the second relation of eq. (5.26) reads

sX\ = 2sym(dCaCb)J^b] + sym(CaC6)d4a6]

= s [-2 sym(AaCb)J^ + sym(CaCb)d< at] (5-3°)

We consider the special configuration ca(n) —>• const. As eq. (5.20), the consistencyof eq. (5.30) requires

duly J = 0, s\Xi + 2sym(AC )LUQ J = 0. (5.31)

The general solution to the first equation is UQ = BQ and then the second equation implies

(by the BRS cohomology) x\ = -2sym(AaCb)B[Qb] + Cauf[{Fi}]. Substituting these into

the first relation of eq. (5.26), we have

sX°2 = s [sym(AaAh)Bl*b] - AaLO^ - 2 sym(FaCb)Bl°b] - Cadu)\. (5.32)

We again consider the configuration ca(n) —>• const. Then eq. (5.32) requires

2FaB[f[ + duj\ = 0, s [x°2 ~ sym(AaAb)Bl*b] + Aau^ = 0. (5.33)

In deriving the first relation, we have noted the fact that the constant ghost form Cb andthe 2-form Fa simply commute and thus Cb can be factored out from the equation. We nextconsider a configuration F®v{n) —> const. Since UJ\ depends only on the field strength, we

see that the constant BQ must vanish for the consistency of eq. (5.33) and thus

dul = 0 => ul = du%[{Fi}] + Bl, (5.34)

by the present lemma for p = 1. Substituting this into the second relation of eq. (5.33) andthen by using the first equation (5.25), we have

o?3 = —F dujQ — F B^ + duj2 + -B3

A( T?a, ,a 1 1 1 "\ Z?a R a _L R

= d{-t UQ+u)2)-r B1 + B3,

where we have used the Bianchi identity dFa = 0. This shows the lemma for p = 3.Later, we apply the covariant Poincare lemma to the case II above, by regarding components of thenonabelian gauge potential A^n) = J2aA^(n)Ta as if the abelian gauge potential. In this case, itis impossible to make F^v(n) = const, while keeping the range of A^(n) as eq. (2.9). However, it ispossible to make F^v{n) = const. = 0(1/R) inside of a block of the size R. The term djj\ then behavesas ~ exp(—<xR) because the dependence oiu\ is local. Since the first relation of eq. (5.33) holds for anarbitrary R, this implies that each terms have to vanish separately.

20

For p = 4. Similarly as the above cases, we have

a4 = dxl + B4, sa4 = 0, (5.36)

and

\ l l l l 0. (5.37)

The solution to the last equation is given by %g = syra(CaCbCc)u)Q [{Fi}] a n ( i then thethird relation of eq. (5.37) reads

sX\ = s [-3 sym(AaCbCc)Jvbc]} - sym(CaCbCc)dJQbc]. (5.38)

The consistency for ca(n) —>• const, requires UJQ = BQ C' (const.) and thus

x\ = -3sym(AaChCc)Bl°bc] + sym(CaCh)Jf][{Fi}l (5.39)

and the second equation of eq. (5.37) becomes

sX\ = s[-3Sym(AaAhCc)Bl°bc] - 2Sym(AaCh)J*

-3sym(FaCbCc)BlQbc] + sym(CaCb)dJ"b].

Setting ca(n) —>• const, in this equation and then setting F®v{n) —> const., we see that

BQ C = 0 and doj® = 0. The present lemma for p = 1 then asserts that u® = dur^ [{Fi}] +

nf '. The general structure of x\ is therefore given by

, .1 o c , r T V 1 1 Aa/~ib\ ( J ,[°&] _i_ p [ a & ] \ _i_ / ^ a ,a\S T?W (tr. A I \

X2 — — sym^/i o )\OLUJQ -\-JD-^ ) -\- u (-"- Ll Or I" /

Substituting this into the first relation of eq. (5.37), we have

sX°3 = s [sym(AaAb){dutb] + B[?b]) - A°UJ%\L J (5.42)

9 OVTVI f TPa(~i^\ (At X \ _i_ Di a"J \ (~iadi ) a

The consistency for ca(n) —>• const, requires

and the consistency for F®v(n) —>• const.,

5[a6] = 0> 2 F a d 4 H + ^ 2 = 0. (5.44)

The last equation can be written as d(u2 — 2FbujQ ) = 0 and then the present lemma

21

for p = 2 asserts that

bJ*b] B% + FbBf. (5.45)

Note that BQ6 is not necessarily symmetric under a <H> b at this stage. From this, it is notdifficult to see that eq. (5.42) yields

sd \Sym(AaAb)J"b]] + s \-Aa{dut + Ba2 + FhB

(5 46)

We now arrived at the final stage which requires a special consideration. In eq. (5.46),the last term of the right hand side is not manifestly s-exact. So define

<p\ = 2 [sym(FaCh) + CaFh~\ J*b] = (CaFh - FbCa)J^]. (5.47)

In the context of ordinary differential calculus, ip\ identically vanishes because Ca and Fcommute with each other. However we cannot simply throw away ip\ in the context ofnoncommutative differential calculus. We first note sip\ = 0. Also, when ca(n) —>• const.,Ca and Fh commute and ip\ = 0 as noted above. Therefore ip\ oc AMca. But these factscombined with the BRS cohomology (5.10) show that y>\ is s-trivial, y>\ = SY2 (actually,otherwise eq. (5.46) becomes inconsistent). In fact, by noting the noncommutative rule (4.4),one finds

$ = -[A^ca{n) + A , c » + AliAv<?{n)]d9Fb(n)ulfb] = sY2, (5.48)

where

Y2 = -[Al(n)+Al(n) + AvAl(n)]Fb(n)Jfb]. (5.49)

Therefore eq. (5.46) gives

X°3 = -Aa{dur( + B\ + FhBf) + u3[{Fi}] + d[sym(AaA6)4a6] - V-j] > (5-50)

and from the first equation (5.36), we have

a4 = d(-Fau)l + u3) - FaBl - FaFhBf + B4. (5.51)

Finally, we have to show that the term proportional to the antisymmetric part of _BQ6

under a <->• b, which again vanishes in the ordinary differential calculus, can be expressed

22

as dcj3({Fi}). We first note that FaFbB[Qb] = dcp3 where

^3 = \{AaFh - FbAa)B[*b] - ^dY2, (5.52)

and Y2 in the second term is defined by UJQ —>• BQ in eq. (5.49). The last term —dY2/2 of

course does dot contribute to F°F6_Bg , but it makes ^3 gauge invariant. In fact,

s(p3 = --(dCaFh - FhdCa)B[Qh] + -dsY2 = 0, (5.53)

where use of eqs. (5.48) and (5.47) has been made. More explicitly, after some calculation,we have

()(5.54)

5) + F^(n + j)Fb^(n + a)] dxa dxp dx7B[«b].

This establishes the lemma for p = 4. D

If one repeats the above argument for p = 5 (assuming that the dimension of lattice isgreater than 4), the treatment becomes much more involved due to the noncommutativity offorms. Because of this, we could not find an iterative formula for ap with general p, unlikethe treatment in the continuum theory [16]. This fact suggests that our noncommutativedifferential calculus is not powerful enough and there exists another hidden algebraic struc-ture. This is an interesting problem although we do not investigate it here. Of course, theproof in this subsection is sufficient for applications in 4-dimensional lattice.

5.4 . TOPOLOGICAL FIELDS IN THE ABELIAN THEORY

Once the above three lemmas are established, it is straightforward to show the followingtheorem which generalizes the theorem of ref. [19] to multi-[/(l) cases.

Theorem. Let q(n) be a gauge invariant Geld on 4-dimensional inGnite hypercubic latticethat depends smoothly and locally on the abelian gauge potential A^. Suppose that q(n) istopological, namely

) = 0, (5.55)

for an arbitrary local variation of the gauge potential. Then q{n) is of the form

q(n) = a + P^F^n) + 7 M w C ( ^ > + £ + ?) + AJfc n), (5.56)

where the current k^in) depends smoothly and locally only on the Geld strength and thus isgauge invariant.

23

Proof. We multiply the volume form d4x to q(n) and define the 4-form Q4 = qd4x. Q4 isgauge invariant sQ^ = 0 and thus, from the BRS cohomology (5.10), Q4 = Q4({Fi}). Fromthe algebraic Poincare lemma (5.1), on the other hand, Q4 = dx3 + E>4 because the 4-form Q4is topological ^nSQ^n) = 0 from the assumption (5.55). From these, we can apply thecovariant Poincare lemma (5.11) to Q4 which yields

q{n)d4x = Q4(n) = BA + Fa{n)B% + Fa(n)Fh(n)B{Q

ab) + da3(n). (5.57)

Finally we factor out the volume form d4x from the both sides of this equation. Noting thenoncommutative rule (4.4), we have eq. (5.56). •

Note. The third term of eq. (5.56) is a total difference on lattice and thus in fact satisfiesthe topological property (5.55):

e»vp<TF°v{n)Fhpo(n + £ + ^) = ^Vpa\ [A*(n)ApAha(n + V)]. (5.58)

This relation can easily be derived from the relation FaF = d(AaF ) valid in the contextof noncommutative differential calculus. See also ref. [32].

6. Nontrivial anomalies in the abelian theory

Because of the nilpotency SB = 0, any functional of the form A = SBB is a solutionto eq. (1.3). If the functional B is local, such an anomaly can be removed by the redefinitionof the effective action In Det M' —>• In Det M' — B which does not change the physical contentof the theory. Therefore the solution to the consistency condition (1.3) of the form A = 5BBwith a local functional B will be referred as trivial or BRS trivial.

6.1. NONTRIVIAL LOCAL SOLUTIONS

In this subsection, we study the structure of local solutions to the consistency condi-tion (1.3) in the abelian theory G = U(1)N. The BRS transformation is given by eq. (4.7).The ghost number of the solution is not restricted. We will find a very close analogue to thesolutions in the continuum theory [16].

We seek the solution A by regarding A as a smooth and local functional of the gaugepotential A^ and the ghost field c°. Since eq. (1.3) must hold for arbitrary configurationsof A^ and c°, we have the variational equation

UBA = £ { M j ( B ) f c ^ - &,(„, [,BJM- + A ; ^ ] } = 0, (6.1)ft I I

where SSBA^ = A^Sc0 and 5§BCa = 0 have been used. The coefficients of the variations

24

5Aa (n) and 6ca(n) have to vanish separately:

dA dA .„ dAA*.^r——- = 0.

(6.2)

Since A is local, dA/dAa(n) is a local field. Then the abelian BRS cohomology (5.10) givesthe general solution to the first equation with the ghost number g,

(6.3)

wherereads,

a depends only on the field strength. The second relation of eq. (6.2) then

dAdca(n)

(n)

(6.4)

x c a i + 1 (n-/})••• ca* (n - $)ufl-ai'"a9] (n)

where from the second line to the third line, we have used ca(n) = ca(n — ju) + SBA^U — ju).Considering the consistency of the above equation under ca(n) —>• const., we have

and then again from the BRS cohomology,

(6.5)

dAdca(n)

= -AlY.?(n) -i = l

cai (n)

X /^-^%—\- 1 / ry-\ i i \ sir-bo t IY~\

C y t l — /J,) • • • C y i ll'-4{n) + 6BYa{n),

(6.6)n

where Xa^ai'"ag~^ depends only on the field strength.

25

Now the functional A can be reconstructed from its variations (6.3) and (6.6). Weintroduce At by rescaling variables as A^ —> tA^ and c° —>• tca. Noting At=o = 0 for g > 0,we have

a f \ " <• Inn'

-fin)- -— . (6.7Jo Uh Jo —L dtA^n) dtca(n)

After substituting eqs. (6.3) and (6.6) and shifting the coordinate s —>• s + yU, this yields

>(n)cai(n)---ca3(n)n ^ '

i=l~ r ••• i (6 8)

where the following abbreviations have been introduced

a[ai-ag] = f1 ^ ] f'

O JO1 r1

r rYa — df Y(\ Ya — dt Ya+1 a — I u b 1 at} 1 — I u b 1 t-

Jo Jo

(6.9)

r I

Note that A ^ a y 1 °sJ = 0 from eq. (6.5) and all these fields are local from the above

construction. In particular, uy 1 and xlai'"aa~\ depend only on the field strength.

Eq. (6.8) provides the most general local solutions to the consistency condition. Yet itcontains trivial solutions in various ways. First, by noting ca°(n + //) = ca°(n) + 5g^4^0(n)

and 5BAa^{n) = ca2(n + //) — c°2(n), it is easy to see that the symmetric part of UJ^on (IQ •<->• ai contributes only to a BRS trivial part:

Af {n)cai (n) • • • ca«(n) - gca°{n + / x ) ^ 1 {n)ca2(n) • • • ca* (n)] ^0(llW"aa (n)

1n(n) ,

• For g = 0, the following expressions hold by simply adding a constant ./U=o to .4.

26

and

(n A n) ^ 1 (n) (n)

(6.11)

is antisymmetric under theTherefore, for nontrivial solutions, we can assume that i

exchange ao O ai, namely, u^ai ag' is totally antisymmetric

solutions.

°s

Henceforth we use the symbol ~ to indicate the equivalence relation modulo BRS trivialparts. The last term of eq. (6.8) is BRS trivial. Also, as noted above, cu^0 a§ is totally anti-symmetric in nontrivial solutions. Then inserting 5BA^ (n) = cOi(n + /i) — caj(n) to eq. (6.8),after some rearrangements, we have the following relatively simple expression

V caicai (n) • • • cah (n)Af° (n)cak+1 (n ca« (n +

lk=o (6.12)

This expression takes a particularly simple form in terms of the noncommutative differentialcalculus. We introduce the dual 3-form of uj,. by

i SJ(n) = - £ (6.13)

Then by using the noncommutative rule (4.4), it is easy to see that

Ad4x{d9)9 ~ Y^ Uym(AaoCa i C a i • • • (6.14)

On the other hand, the divergence-free condition (6.5) becomes

°-°9] = 0. (6.15)

We can now apply the covariant Poincare lemma (5.11) to the 3-form QL°o---ogJ because itdepends only on the field strength. This yields

(6.16)

But the contribution of ajf0 can be absorbed into the second term of eq. (6.14) up to a

27

trivial part, because

4o°COl---Co»)d4O0'"°'']

( - l ) s s y m ( F a ° C O 1 •••Caa)c

n

+ s J2 \(-l)9+1-sym(AaoAaiCa2 • • • Ca°)a[£°'"aB]] •n L -I

Similarly, from the covariant Poincare lemma, we have

Xlai-aa}(l4X

but it is easy to see that a^1 a does not contribute to the nontrivial part.

So, up to this stage, we have obtained

AdAx{d6)9 ~ V \cai •.. ca°rfai~a<>l d4x

4-Cai C°9 (B^1 'a^ + T?bT>[ai"-ag]b , rpb rpc n[ai---ag](bc)\ (6.19)

where ^2n 5jC}ai'"a9^ 0 under a certain local variation of the gauge potential. Formally thisexpression is identical to the list of nontrivial solutions in the continuum theory (see eq. (6.24)of the second reference of ref. [16]). Recall however that eq. (6.19) is an expression in thecontext of noncommutative differential calculus and it is valid for a finite lattice spacing a ^0.

It is easy to see that eq. (6.19) satisfies SBA = 0. Doesn't eq. (6.19) contain BRS trivialparts anymore? SB (something) is always proportional to a difference of the ghost field suchas AMc°. But this does not necessarily imply that all terms of eq. (6.19) are BRS nontrivial.In contrast to the BRS cohomology (5.10), this expression contains the summation y ^ .Therefore, after the "integration by parts", a difference of the ghost field may be resulted.

The term proportional to E>2 in fact contains BRS trivial parts. Namely, by noting dCa =—sAa, we have

Cai • • • ca»FbB[^"a3]b ~ Yl sym(CYai • • • Ca'dAb)Bl^"'ag]b

(6.20)

n

= J2(-l)9gsym(sAaiCa2 • • •Ca°Ab)B[f1'"a9]bJ2(

28

Note that the commutator of Ca and the field strength 2-form Fb is proportional to adifference of the ghost field and thus, from the BRS cohomology, it is BRS trivial. Thereforethe ordering of Ca and Fh is arbitrary in the first expression of eq. (6.20) up to BRS trivialparts. We have used this fact for the first ~ equality. By comparing the second line and thefourth line of the above expression, we see that eq. (6.20) is equivalent to

n n

A comparison with the left hand side of eq. (6.20) shows that the antisymmetric part

of B2ai under a\ <-> b is BRS trivial ~ 0. Therefore B2

ai a3' must be symmetricunder a\ <H> b to contribute nontrivial solutions.

Similarly, we have (suppressing ^n)

ag](bc) ^ / ^ a i £iag pb\pcp>[ai---ag](bc)

° (6-22)fjagpbpc£>[ai---ag](bc) ^ / ^ a i £iag pb\pcp>[ai---ag](bc)

and therefore B^ a must be symmetric under a\ <-> b.

The term proportional to B\ also might contain BRS trivial parts depending on symme-try of indices. However, the noncommutativity prevented us to imitate the procedure in thecontinuum theory [16].

Let us summarize the result: The general structure of local solutions to the consistencycondition eq. (1.3) is given by eq. (6.19). The constant forms B2 and Bo have the followingsymmetries:

R[ai—Og]6 _ r>[ba2---ag]ai R[oi—os](6c) _ R[6o2—Og](oic) t

° ° n n [

The solution (6.19) is nontrivial, i.e., it cannot be written as A = 5BB by using a localfunctional B. The classical continuum limit of eq. (6.19) with eq. (6.23) coincides withthe nontrivial solutions in the continuum theory [16] (with a partial exception for _B|°°mentioned above). Then if eq. (6.19) was BRS trivial, the classical continuum limit of thelocal functional B would counter the nontrivial solutions in the continuum theory. But thiscontradicts with the result of ref. [16].

In this subsection, we obtained the general nontrivial local solutions with an arbitraryghost number. For discussions of the gauge anomaly in the next subsection, knowledge ofghost number one solutions is enough. The solutions with higher ghost number, however,might become relevant in future applications. For example, it might be possible to addressthe commutator anomaly [60] in the context of lattice gauge theory starting with the aboveexpressions.

The information about these becomes important [16] when one explicitly computes the higher ordersequence At with t > 4 for the nonabelian anomaly, although we do not pursue this in this paper. Seethe discussion in sec. 7.2.

29

6.2. GAUGE ANOMALY IN ABELIAN THEORY

If we restrict solutions with the ghost number unity, eq. (6.19) tells us that

c » [aa

where we have used the noncommutative rule (4.4) and the symmetry of indices (6.23). Inthis expression, the function Ca(n) satisfies ^2nSCa(n) 7 0 for a certain local variation.

Eq. (6.24) provides the general candidate of nontrivial local gauge anomalies in theabelian theory G = U(1)N. However, depending on the situation, we may further restrictthe coefficients in various ways.

(1) When G = U(l), the last line vanishes due to the anti-symmetrization of indices.Eq. (6.24) then reproduces Liischer's result (1.1) except the "non-topological term" Ca.

(2) The non-topological term Ca and the term proportional to fj? never appear in the gaugeanomaly of the actual system. In the actual system, the gauge anomaly is defined from afermionic determinant by A = SB

m Det M'[A]. Since SBA^U) —>• 0 for ca{n) —>• const, in

the abelian theory, A vanishes as ca(n) —>• const. This limit might be dangerous but itis enough to assume that SA[c, A] = 0 for ca(n) —>• const., where 5 is an arbitrary localvariation of the gauge potential. It is obvious that the non-topological term Ca and the termproportional to /^ do not satisfy this criterion.

(3) If the couplings of the Weyl fermion to gauge fields have the same structure for all U(l)factors except coupling constants (practically this is always the case), then all the coefficients

are independent of group indices and we have aa —>• a, 0, I —>• / S ^ ] , ^ahc> —>• 7, /jf ' —>• 0,

gf^ -+ 0.

(4) From the dimensional counting, all the terms except Ca and the term proportionalto j(-a"c) have negative powers of the lattice spacing as the overall coefficient. Therefore,if the classical continuum limit a —> 0 of A is finite (for a smooth background), all theterms except Ca and ^ahc"> must be absent. In particular, if lima_).o A reproduces the gaugeanomaly in the continuum theory, j(abc^ = — e#/(96?r2) for a single Weyl fermion.

Physically, we are always interested in cases for which (2) and (4) hold. So let us acceptthese. Then we have a content of the theorem for the abelian gauge theory which we statedin section 3.

f In fact, when the lattice volume is finite, the index is obtained in this limit as A —>• ca x (index) up tothe proportionality constant.

30

7. Nonabelian extension

In the section, we study the gauge anomaly for a general (compact) gauge group G =Yla Got, where Ga is a simple group or a U(l) factor. The candidate of the anomaly is givenby the ghost number one solution to the consistency condition (1.3). The general solutionto eq. (1.3) is expressed as

A = A + 6BB, (7.1)

where A ^ 5BB is the BRS nontrivial part (B is a local functional). The BRS transformationis given by eq. (1.2) and it takes the following form in terms of the gauge potential

SBA^U) = -Ap{n) A coth -A^n) A AMc(n) + c(n) + c(n + ju) , (7.2)

where X A Y = [X, Y], X2 A Y = [X, [X, Y]] and so on, and 1 A Y = Y is understood. Weshall use both the matrix notation A^ri) and c(n), and the component notation A^ri) =EsA«(n)T« and c(n) = E ac aHT«.

To make our problem tractable, we set the following assumptions on the anomaly A.

(I) A is defined from a certain effective action as A = 5B lnDet M".

(II) A is a smooth and local functional of the gauge potential A^ and the ghost field c.

(III) The classical continuum limit of A reproduces the anomaly in the continuum theory,eq. (2.28).

Under these assumptions, in section 7.2, we show that the anomaly A, if it exists,is unique (up to the BRS trivial part) to all orders of the gauge potential. The uniqueanomaly is proportional to the gauge anomaly in the continuum theory and this establishesthe theorem for nonabelian theories, stated in section 3. In section 7.3, we show that such asolution in fact exists. As a preparation for sec. 7.2, we need the following lemma.

7.1. BASIC LEMMA: ADJOINT INVARIANCE

Adjoint invariance. Without loss of generality, one can assume that a nontrivial localsolution A is invariant under the adjoint transformation

5aA = 0, (7.3)

where the adjoint transformation Sa is defined by

SaU(n,/,) = [T°,tf(n,/i)], <JM*(n) = -if^Afa), 5ac\n) = -zfabccc(n). (7.4)

Note. There is freedom to add a BRS trivial part SBB to a nontrivial solution. The abovelemma asserts that it is always possible to choose B such that A is adjoint invariant. The

31

following relations hold for the adjoint transformation Sa

[5B, Sa] = 0, [A*, Sa] = 0, [5a, Sb] = ifahc5c. (7.5)

Proof. The functional A is local, i.e., the field li{n) in A = X!n^(n) 1S a l°cal field. Weexpress a(n) in terms of the following set of variables, which was introduced in the proof ofthe abelian BRS cohomology in ref. [32]:

At = (Ai)* • • • (A.fMJin), F? = (Ai)* • • • (ADY°F%,(n), (7.6)

for the gauge potential (D is the dimension of the lattice) and

co(n), and ca(n), (7.7)

for the ghost field, here So is the abelian BRS transformation, 5t)Aa(n) = AIIca(n)

and 5oca(n) = 0. In these expressions, the symbol (AM)P (p is a nonzero integer) has beendefined by

for p > 0,

', for p < 0.

It can be shown [32] that these variables, A®, F?, c" and ca(n) span a (over)complete set,i.e., the field H{n) can be expressed as a function of these variables. A little thought showsthat the relation

r a i= 0, (7.9)

holds on these variables.

Since the nonabelian BRS transformation (1.2) or (7.2) has the structure,

5BA*(n) = tfabcAl(n)cc{n) + (terms proportional to AMca),

SBca(n) = -Uf^cb(n)cc(n),

we have

Aa

F? } = -cb(n)Sb { F? } + (terms proportional to cf), (7.11)

and thus

Sa = — I SR I (7 12)\ dca{n)y y ' ;

on the variables Aa{,Ff, cf and ca(n) (for ca(n) this follows from eq. (7.10)).

32

The remaining argument to show the lemma is almost identical to that of ref. [16]. Weintroduce the Casimir operator:

OK = g°i-°MK)§ai . . . SaMK) ^ (7.13)

where gav"a™(K) _ s t r T a i • • -Tam(-K^ are totally symmetric constants and K runs from 1 tothe rank of the semisimple part of the group G [61]. Using the completeness of eigenfunctionsof OK, we decompose 7i{n) according to the representation A, 7i{n) = 2\ax(n), where

OKax{n) = k(K, X)ax{n), (7.14)

and k{K,X) is the eigenvalue. Since SBA = ^2n$Ba(n) = 0, the dual of the algebraicPoincare lemma (5.1) (p = D) shows that

5Ba(n) = A * ^ ( n ) , (7.15)

where X^in) is a local field. We again apply to this equation the decomposition similar toeq. (7.14):

SBax(n) = A;Xx(n), (7.16)

where use of relations (7.5) has been made.

Now suppose that there exist K and A such that k(K,X) ^ 0 in eq. (7.14). Then, byusing eqs. (7.13), (7.12), (7.5) and (7.16), we have

x(n) = 6B ~ nai-a"(*)(r"(*) S a 2 a xax(n) = 6B, n a i - a"(*)(r"(*) • • -Sa2-—~-ax{n)

- 1 B ( 7 " 1 7 )

ai--a(K)za{K) . . xo 2 ^ XX(n)

Namely, A contains a BRS trivial part ^2nax(n) which we remove by using SBB. After

repeating this procedure, all the eigenvalues k(K,X) in eq. (7.14) are made to vanish andthis implies that A is the singlet representation. Therefore, we can always assume that aBRS nontrivial solution is adjoint invariant SaA = 0. •

We next derive a constraint for A following from the assumptions (I) made above andthe lemma (7.3). Set ca{n) —> ca = const. Since the ghost number of A is unity, we canwrite as

A = cakaXXx[A], (7.18)

where kaX are constants and A labels linearly independent functional XA[A]. We have tothen consider the following two cases separately:

33

(1) When the index a of the ghost field in eq. (7.18) is belonging to a U(l) factor group U(l)a.In this case, we have

A = cu{1)aX[A], (7.19)

but since SBA^ (n) = A^c11^01 = 0, we have SBB = 0 in eq. (7.1) and, from the assump-tion (I), A = A = SB\nDetM"[A] = 0 in the limit cu^a(n) ->• const. This limit might bedangerous as noted at the end of section 6, but the condition 5A = 0, where 8 is an arbitrarylocal variation of the gauge potential, is enough for the following argument.

(2) When the index a of the ghost field in eq. (7.18) is belonging to a simple group. TheBRS transformation (1.2) or (7.2) becomes for ca(x) —> ca = const.,

SBA%{n) = -ifahcchA^(n) = -ch5hA*(n),

5Bca = -\ifahcchcc = -SBca- cb5bca,

where 5a is the adjoint transformation. But since the lemma (7.3) asserts that 5aA = 0, theconsistency condition becomes for ca(n) —>• const.,

SBA = -SB(cakaX)Xx[A] = 0. (7.21)

This requires SB(cakaX) = 0. Then the Lie algebra cohomology in ref. [16] asserts thatcak = trc = 0 for a simple group. This shows that *4 = 0 for ca(n) —> const. This limitmight again be dangerous but it is sufficient to assume that 5A = 0, where 5 is an arbitrarylocal variation of the gauge potential.

Combining (1) and (2), we see that the assumption (I) implies

5A = 0, for ca(n) ->• const., (7.22)

where 5 is an arbitrary local variation of the gauge potential. This provides a strong con-straint for the possible form of A as will be seen in the next subsection. In what follows, wedo not need the assumption (I) itself, but use this constraint (7.22) instead.

7.2. UNIQUENESS OF THE NONTRIVIAL ANOMALY

We now expand the anomaly A (7.1) in powers of the gauge potential as

Bt, (7.23)e=i e=i e=i

where £ stands for the number of powers of c and A^ (recall that the ghost number of A isunity). We decompose also the BRS transformation (7.2) according to powers of the fields,

34

= ZXo ^' wnere

SQA^U) = AMc(n), Soc(n) = 0,

$2kc(n) = 0, for A; > 1,

and Bfr is the Bernoulli number and c>2fc+i = 0 for k > 1. Note that, in terms of compo-nents A^ and ca, So has an identical form as the abelian BRS transformation (4.7). Thenilpotency 62

B = 0 implies

fc = 0, for £>0, (7.25)

and the consistency condition (1.3) takes the form

£-1

for £ > 1 . (7.26)k=i

Since

^ J]^-fc, (7-27)

if ^ contains (^-trivial part 5QB'^, B'g can always be absorbed into B^. Therefore we canassume that

At does not contain SQB'J,. (7.28)

The constraint (7.22) has to hold for each order:

5Ai = 0, for ca(n) ->• const., (7.29)

where 5 is an arbitrary local variation of the gauge potential. Also the correct classicalcontinuum limit (III) requires

At a 4 ° O(Ae-1) term of eq. (2.28). (7.30)

We consider the local solution to the consistency condition (7.26) which satisfies theconditions (7.28), (7.29) and (7.30), order by order. The first equation in eq. (7.26) is

50Ai = 0. (7.31)

This equation is completely identical to the consistency condition in the abelian theory.Therefore, from our result in the preceding section, eq. (6.24), the general form of Ai which

35

satisfies eqs. (7.28)-(7.30) is given by

Ax = 0. (7.32)

The next equation in eq. (7.26) is

S0A2 = 0. (7.33)

Again from the result in the abelian theory (6.24), we have

A2 = 0, (7.34)

where use of eqs. (7.28)-(7.30) has been made to conclude this.

The solution to the next equation

S0A3 = 0, (7.35)

has a 5o-nontrivial p a r t . From eq. (6.24), and from the conditions (7.28)-(7.30), we have

13 (n + z

^(n)trAj4Q)(n)Ap4a)(n + P)l (7-36)

(Q) (n) A , [ A ^ (n)ApA^ (n + v)

tr C(Q) (n) AM [^la) (n) A p A ^ (n

where we have used the relation (5.58) to make the property (7.29) manifest. Note that thecondition (7.29) is crucial to eliminate the possibility that the Ca term of eq. (6.24) appears.

The next equation in eq. (7.26) is

5QA± = -<M 3 - (7.37)

The solution to this equation At, if it exists, is unique. If another A4 which also satisfiesthe conditions (7.28)-(7.30) exists,

5o(A4-A 4) = O, (7.38)

and the quantity inside the brackets again satisfies eqs. (7.28) and (7.29). But then eq. (6.24)shows that A'4 — A4 = 0.

36

The above argument can be repeated for higher Ag's. Suppose that a sequence for thenontrivial part, A3, A4, ..., At-i, has been obtained. Then the next term Ag has to satisfyeq. (7.26) and the conditions (7.28)-(7.30). Then the same argument as above shows thatthe solution At, if it exists, is unique.

We have seen that the sequence Ag for the nontrivial anomaly A, which satisfies eq. (7.22)and the assumptions (II) and (HI), if it exists, is unique under the condition (7.28). Thereis no free parameter which can appear in a higher Ag. Moreover, this uniqueness showsthat the anomaly cancellation in the continuum theory implies that of the lattice theory: Ifthe first nontrivial term A3 (7.36), which is proportional to the anomaly in the continuumtheory ITR-L Ta{Tb, Tc} etc., is canceled among the fermion multiplet, then the subsequentsequence of At for I > 4 are completely canceled. In other words, the possible nontrivial localanomaly on lattice A, under the assumptions (I) (III), is always proportional to the anomalyin the continuum theory, to all orders of powers of the gauge potential. This establishes ourtheorem for nonabelian theories, stated in section 3.

The existence of the nontrivial sequence At for £ > 4 might be examined by repeatedlysolving eq. (7.26). However, eq. (7.24) suggests that the explicit form of higher Ags willbecome quite complicated as £ increases. In the next subsection, instead of this analysis,we will give a "compact" form of a nontrivial solution which manifestly satisfies eq. (7.22)and the assumptions (II) (at least for the perturbative region (2.9)) and (III). This explicitlyshows the existence of the nontrivial sequence, Ag with £ > 4. To write down the compactsolution, however, we need the interpolation technique of lattice fields with which the BRStransformation takes a quite simple form. Therefore, we give a quick summary of the methodof ref. [34] in the first part of the next subsection.

7.3. COMPACT FORM OF THE NONTRIVIAL ANOMALY

First we recapitulate essence of the interpolation method of lattice fields in ref. [34](simply extended for infinite lattice). For our argument, the interpolation method has topossess several properties which we will verify. To distinguish from the fields defined onlattice sites n, we use the continuous coordinate x to indicate use of interpolated fields.

The method of ref. [34] consists of the following two steps.

Step 1. One first constructs the interpolated gauge potential A"™ '(x) within each hyper-cube h(m), here m stands for the origin of the hypercube, such that the gauge potentials inneighboring hypercubes h(m — //) and h(m) are related by the transition function vmjlI(x)of the Liischer's principal fiber bundle [40],

dx + A{™]{x)]vm^{x)-\ (7.39)

on the intersection of the two hypercubes x G h(m — //) l~l h(m) (that is a 3-dimensionalcube). For the transition function vm^(x) to be well-defined, the gauge field configuration

• Under the same conditions we assume, the method of ref. [35] might be adopted as well.

37

must be "non-exceptional" [40]. As already noted, it can be shown that if e in eq. (2.1) issufficiently small, the gauge field configuration is non-exceptional. So we assume that e hasbeen chosen as this holds. The gauge potential which satisfies eq. (7.39) can be constructed,starting with a special gauge A^1'(x) = 0 at x ~ m. The explicit expression of A-^1'\x) interms of vmyfJl(x), which is eventually expressed by link variables U [40], is given in ref. [34].We do not reproduce it here because it is rather involved and we do not need the explicitform in what follows. The interesting property of A™ (x) is [34]

Pexp ^ dtA^\n + (1 - *)£)] = < ^ . (7.40)

Namely, the Wilson line constructed from the interpolated gauge potential A^'fa) coincideswith the link variable in the complete axial gauge of ref. [40].

Step 2.1. The section of the principal fiber bundle [40]

wm(n) = U{m,iyiU(m,+z1l,2y2U{m+z1l+Z22,3)Z3U(m,+z1l+Z22+z33,4y4 G G, (7.41)

is defined for each lattice site n belonging to the hypercube h(m) where n = m+^2 z^Jl. Thissection is then smoothly interpolated, first on the links, next on the plaquettes, on the cubes,and finally inside the hypercube h(m). At this stage, if the homotopy group TTM-I{G) isnontrivial, the smooth interpolation of the section wm(x) into a M-dimensional (sub)latticemay fail, depending on the configuration of the section wm(x) on a boundary of the M-dimensional (sub)lattice. For example, for G = U(l), TTI(C/(1)) = Z, and if the local windingof wm(x) around a boundary of the plaquette p(m, /i, v),

f){ \ — ^ I m( l"1^ m( ) ('7 491Z7T

does not vanish, then the interpolation of the section wm(x) into the plaquette p(m,/j,,v)

develops a singularity. Similarly, for G = SU(2), Tr^(SU(2)) = Z, and the local winding isgiven by

pW [X)W [X) OaW [X).

dh(m)(7.43)

If Q(m) does not vanish, then the interpolation of wm(x) into the hypercube h(m) developsthe singularity. If these singularities arise, the description in term of the interpolated fields

f The total winding Q = J2mQ(m) on finite periodic lattice is nothing but the Liischer's topologicalcharge [40].

38

becomes inadequate. Fortunately, all local windings are vanishing within the perturbativeregion (2.9), for a sufficiently small e. If e in (2.9) is sufficiently small, the norm of exponentof a product of several link variables is also small and the expression of the interpolatedsection [34] cannot have the "jump" on a boundary of the M-dimensional (sub)lattice. Thisimplies that there is no local winding.

Step 2.2. With the smooth interpolated section wm(x), we define the "global" interpolatedgauge potential by

A,(r) — •7/;mM~1 \^^ + A^(r)] wm(r) (7 441

for x E h(m). The resulting interpolated gauge potential A\(x) is Lie algebra valued.

Now, we need the following properties of the interpolation method to express the non-trivial local solution.

(i) The gauge covariance. This is the most important property for our purpose. Namely,there exists a smooth interpolation of the gauge transformation (in our present context thisbecomes a smooth interpolation of the ghost field) and the lattice gauge (BRS) transfor-mation on the link variables takes an identical form as that of the continuum theory. Thisproperty was shown in ref. [34]. Therefore, the BRS transformation (1.2) induces

5BA°(x) = dllCa(x) + tfabcAl(x)cc(x), 5Bca(x) = -^ifabccb(x)cc(x), (7.45)

on the interpolated fields.

(ii) The transverse continuity. This means the following. The gauge potential A\(x) iscontinuous inside each hypercube and, on the intersection of two neighboring hypercubesx E h(m — ju) l~l h(m), the component transverse to this intersection (namely, A //) iscontinuous across this intersection. We need this property because otherwise integration byparts is not allowed in the following expression. It is easy to see this property, if one notesthat the Liischer's transition function [40] and the interpolated section [34] are related by

vm)li{x) = wm-^(x)wm(x)-1, for xEh{m-j2)n h{m). (7.46)

Then from eqs. (7.39) and (7.44),

w^ix)-1 \dx + Axm-^{x)]wm-^{x) = wm(x)-1 k + Ax

m\x)]wm{x) (7.47)

for x E h(m — ju) l~l h(m). Namely, the interpolated gauge potentials defined from a sideof the hypercube h(m — ju) and defined from a side of h(m) coincide on the intersectionwhen A ^ / i . For A ^ //, the component may jump across the intersection [34], but this

The procedure of ref. [35] can avoid this difficulty.

39

causes no problem for our purpose. The interpolation for the ghost field is obtained bysetting g(n) = exp[Ac(n)] in the interpolation formula for the gauge transformation g(x)in ref. [34]. This gives the smooth ghost field (which is also Lie algebra valued) throughoutthe whole lattice.

(iii) The smoothness and locality. The interpolated gauge potential A\{x) and the ghostfield c(x) are smooth functions of link variables (and of the gauge transformation function)residing nearby the point x. The smoothness (for the perturbative configurations) and thelocality are manifest from the explicit expressions of A™' (x) and of g(x) in ref. [34]. In fact,in this case, the relation is ultra-local.

(iv) The correct continuum limit. In the classical continuum limit, a —>• 0, the interpolatedgauge potential A^x) and the ghost field c(x) reduce to (for smooth configurations) to thegauge potential and the ghost field in the continuum theory. From eq. (7.40), we have

= U(n,fj),

where we have used the definition of the link variable in the complete axial gauge u™n+ [40].This is nothing but the conventional expression that one assumes in the classical continuumlimit (2.21). For the interpolated ghost field, the formula in ref. [34] shows that c(x = n) =c(n).

(v) The constant ghost field. From the formula in ref. [34], it is easy to see that the constantghost field on the sites induces the constant interpolated ghost field,

c{n) = c = const. =>• c(x) = c = const. (7.49)

Now we can write down the nontrivial local solution to the consistency condition (1.3)which satisfies eq. (7.22) and the assumptions (II) and (III) in terms of the interpolated fields.It is

n

4 ^H \ ^ I i4 \ ; (n) i \ o \ A (en) i \ o i (a) i \ , J- i (a) / \ J (a),

ft(n)

+

(x) tr [ 4 a ) (^)5P4Q) (x) + ^ 4 Q ) (x)4Q) (x)4a ) (*)]

tr[c^(x)9M4a)(x)l

(7.50)In this expression, h{n) is the hypercube whose origin is the site n. Note that this is a

• In fact, from the formulas of ref. [34], it can be shown that Afl(x) is constant along the link, Afl(x) =A^n) for a; £ [n,n + 'fl] where U(n,[/,) =expA/J(n).

40

functional of the link variable U(n,(i) and the ghost field c(n), through the interpolationformulas of ref. [34].

It is easy to see that eq. (7.50) satisfies the consistency condition (1.3), because theBRS transformation of the interpolated fields (7.45) has an identical form as that of thecontinuum theory and because eq. (7.50) has formally an identical form as the gauge anomalyin the continuum theory (2.28). More precisely, we need to perform an integration by partswithin each hypercube to show eq. (1.3). Then the transverse continuity (ii) guaranteesthat contributions from a boundary of hypercubes cancel with each other. This solutioneq. (7.50) is moreover ^-nontrivial: From the property (iv) of the interpolation, in theclassical continuum limit (assuming background fields are smooth), eq. (7.50) reproducesthe gauge anomaly in the continuum theory (2.28) that is BRS nontrivial. In other words,if eq. (7.50) was ^-trivial, there exists a local functional B on lattice such that A = SBB.Then the classical continuum limit of B, which is a local functional in the continuum theory,would cancel the gauge anomaly in the continuum theory.

The nontrivial solution (7.50) manifestly fulfills the condition (7.22) from the proper-ties (v) and (ii) of the interpolation. From the above arguments, it is also clear that eq. (7.50)satisfies the assumptions (II) (within the perturbative region (2.9)) and (III).

The existence of the nontrivial solution A (7.50), which satisfies eq. (7.22) and the as-sumptions (II) and (HI), shows the existence of the unique nontrivial sequence At (7.23)that is characterized by eq. (7.28). We first expand A (7.50) in powers of the gauge poten-tial (2.8) as eq. (7.23). Then using eq. (7.27), we can determine Bt order by order such thatthe sequence At satisfies eq. (7.28). This procedure gives the unique sequence At- Note thatwhen the anomaly in the continuum is canceled, the anomaly A (7.50) vanishes. Thereforethe above procedure gives At = 0 for all L This is consistent with the conclusion in thepreceding subsection.

8. Conclusion

In this paper, we have shown that any gauge anomaly A on 4-dimensional infinite latticecan always be removed by local counterterms order by order in powers of the gauge potential,provided that A depends smoothly and locally on the gauge potential and that A reproducesthe gauge anomaly in the continuum theory. The unique exception is proportional to theanomaly in the continuum theory. This implies that the anomaly cancellation condition inlattice gauge theory is identical to that of the continuum theory.

The most important prerequisite for our result is the locality of anomaly A- As wehave shown, the gauge anomaly in the formulation based on the Ginsparg-Wilson Diracoperator is local (for a particular choice of the integration measure) and thus our theorems areapplicable (at least in the perturbative region in which a parameterization of the admissiblespace in terms of the gauge potential is possible). Unfortunately, the gauge anomaly Aappearing in formulations based on the more familiar Wilson Dirac operator or on the Kogut-Susskind Dirac operator is not local, although these Dirac operators themselves are ultra-

41

local. For these operators, the chiral gauge symmetry is broken in the tree level and as aresult the anomaly is given as A = Tr(explicit breaking term) x (propagator). The (massless)propagator in this expression breaks the locality. (In the classical continuum limit, thelocality restores and A reproduces the gauge anomaly in the continuum theory [62,63].)

Let us discuss possible extensions of results in this paper. The most severe limitationof our result for nonabelian theories is that it holds only in an expansion in powers of thegauge potential. An interesting observation relating this is that the expansion of the anomalydensity a(n) (A = Y2n

ain)) a(n) = Yl'eLi a(.{n) m powers of the gauge potential has a finiteradius of convergence. This follows from the smoothness and the locality of the anomaly Awhich we have assumed. If these hold for the admissible configurations (2.1), the radius ofconvergence of this series is given by the right hand side of eq. (2.9). Another interestingpoint is that the compact solution (7.50) is smooth and local at least in the perturbativeregion (2.9). These observations suggest that our result is valid beyond the expansion withrespect to the gauge potential, at least within the perturbative region. What is not clear atpresent is a convergence of the individual series A = Yl^Li A-i and B = YlliLi &e m eQ- (7-23),after imposing eq. (7.28).

By using similar arguments as above, it seems straightforward to classify general topo-logical fields on 4-dimensional infinite lattice (that is a nonabelian analogue of the theo-rem (5.56)) at least to all orders of powers of the gauge potential. According to the resultof ref. [21], this analysis is relevant to see existence of an exactly gauge invariant formu-lation of anomaly-free two-dimensional chiral gauge theories. For /owr-dimensional chiralgauge theories, we have to generalize the covariant Poincare lemma (5.11) to 6-dimensionallattice. This generalization would be straightforward, although the proof may become quitecumbersome.

The restriction to the perturbative region (2.9) for nonabelian theories is due to a compli-cated structure of the admissible space (2.1). If it is possible to parameterize the admissiblespace in terms of the gauge potential, as in the abelian case, the restriction may be relaxed.It is highly plausible that the "rewinding" technique of ref. [35] is useful in this context.

Our results are not yet "realistic" because these are for infinite lattice. For the abeliancase G = U(l), it has been shown [20] that the anomaly cancellation works even for finiteperiodic lattice. To generalize the argument in ref. [20] for nonabelian theories, we haveto first understand the structure of the admissible space on finite lattice (see the aboveremark). Another possible approach to this problem is to clarify De Rham cohomologyon finite (periodic) lattice. This approach corresponds to the argument of ref. [17] in thecontinuum theory.

In this paper, we adopted a "classical" algebraic viewpoint based on the Wess-Zuminoconsistency condition. In the continuum theory, the algebraic approach to the anomaly has aclose relationship to a higher dimensional theory [64]. It seems very important to investigate

However the existence of the global obstruction [50] shows that this must be in general impossible.

42

such a relationship in the context of lattice gauge theory. In fact, there are some indicationsfor such relation [21,45,65].

Finally, let us mention on the physical implication of these analyses. After all, evena local counterterm which makes the effective action gauge invariant exists (for anomaly-free cases), the implementation of gauge invariance requires a fine tuning of parameters inthe counterterm, that is highly unnatural. One might thus be tempting to appeal to themechanism of ref. [66] which dynamically restores the gauge invariance. However, for themechanism of ref. [66] to work, the gauge breaking A (with the ghost field is replaced by alogarithm of the gauge transformation field) has to be "small." In particular, if A ^ 5BBfor a local functional £?, the effective lagrangian for the gauge transformation field wouldbe given by a lattice analogue of the Wess-Zumino lagrangian which modifies the physicalcontent (thus it cannot be regarded "small"). Therefore, study of the gauge anomaly onlattice is important also to examine the necessary condition for the mechanism of ref. [66].In this respect, it seems interesting to study the locality (in a four dimensional sense) ofthe gauge anomaly appearing in the overlap formulation with the Brillouin- Wigner phaseconvention, in connection with the result of ref. [67].

Acknowledgments

The author has greatly benefited from correspondence with T. Fujiwara, Y. Kikukawaand K. Wu and from discussions with P. Hernandez. The author is particularly gratefulto M. Liischer for various helpful remarks, without which this work would not have beencompleted.

APPENDIX A

Here we summarize our notation and the convention. Throughout this paper, we considerthe 4-dimensional infinite lattice Z^. The site of the lattice is denoted by n, m, etc. Thelattice spacing is taken to be unity a = 1 unless otherwise stated. The Greek letters /i,^, etc. denote the Lorentz indices which run from 1 to 4. ft, stands for the unit vector indirection /i. For Lorenz indices, the summation of repeated indices is always understood.The Levi-Civita symbol is defined by epVpa = ^[^pa] and £1234 = 1-

The forward and the backward difference operators are respectively defined by

AM/(n) = f(n + £) - /(n), A*/(n) = f(n) - f(n - £). (A.I)

The symbol d^ is reserved for the standard derivative.

H = R or L stands for the chirality of Weyl fermion and we set e# = +1 and e^ = — 1.

G = Y[a Ga is the (compact) gauge group and here Ga denotes a simple group or a U(l)factor. The Greek indices a, (3, etc. are used to label each factor group. T° stands for the

43

representation matrix of the Lie algebra, [Ta,Tb] = ifabcTc. The summation of repeatedgroup indices a, b, etc. is always understood.

U(n, /i) is the link variable on the link that connects the lattice sites n and n + ju. For theabelian gauge group G = U(1)N, we parameterize the link variable by the gauge potentialas £/a(n,/i) = exp^4"(n). In this case, the superscript a distinguishes each U(l) factor. Theabelian field strength is defined by

P ; » = A M A » - A^«(n) . (A.2)

We never use this symbol F®v to indicate the nonabelian field strength. The plaquettevariable is defined by

P{n, fj,, v) = U{n, fj)U(n + % v)U{n + V, )~lU{n, v)~l. (A.3)

APPENDIX B

In this appendix, we show the calculation of the "Wilson line" W' in eq. (2.17). Fromeqs. (2.11) and (2.17), W is given by

dt J dsTvPt(s)[dsPt(s),dtPt(s)]\ (B.I)

where Pt(s) = PH\u^ut(s) an (l w e explicitly indicated s-dependences defined througheq. (2.13). If we introduce the transporting operator Qt(s) for each s by dtQt{s) =[dtPt{s),Pt{s)]Qt{s) and Q0(s) = 1, we have

Pt(s) = Qt(s)P0(s)Qt(s)\ (B.2)

(note that Qt(s) is unitary). Substituting this into eq. (B.I), after some calculation, we have

dt [ ds [dsTr P0(s)Ql(s)dtQt(s) - dtTv P0(s)Ql(s)dsQt(s)]). (B.3)Jo J

[ [Jo Jo

We then apply the Stokes theorem to this 2-dimensional integration. This yields

W = f dt TrP0(l)Q\(l)dtQt(l) - eH [ ds TrP0(s)Qt(s)d3Qt(s) ^ V (B.4)Jo Jo t=0J

However, from eq. (B.2),

TrPo(l)Ql(l)dtQt{l)=TrPo(l)Ql(l)[dtPt(l),Pt{l)]Qt(l)=O, (B.5)

and dgQo(s) = 0 because Qo(s) = 1. Therefore

W' = exp(-eHJ dsTrP0(s)Q\(s)d8Q1(s)\. (B.6)

Noting Po(s)dsPo(s)Po(s) = 0 and P1(s) = P0(s) (recall that Ui(s) = U0(s)), it can be

44

confirmed that eq. (B.6) is equal to

W = exd-eH J ds Tr [l - P0(s) + P0(s)Q1(s)\ ~'ds [l - P0(s) + P0(s)Q1(s)] j . (B.7)

This shows eq. (2.17).

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