DESY 98-180

Abelian chiral gauge theories on the lattice

with exact gauge invariance

Martin Lüscher

Deutsches Elektronen-Synchrotron DESY

Notkestrasse 85, D-22603 Hamburg, Germany

E-mail:

Abstract

It is shown that U(1) chiral gauge theories with anomaly-free multiplets of Weyl fermions can be put on the lattice without breaking the gauge invariance or violating any other fundamental principle. The Ginsparg-Wilson relation plays a key rôle in this construction, which is non-perturbative and includes all topological sectors of the theory in finite volume. In particular, the cancellation of the gauge anomaly and the absence of global topological obstructions can be established on the basis of this relation and the lattice symmetries alone.

February 2021

1. Introduction

For well-known reasons the formulation of chiral gauge theories on the lattice proves to be difficult and no completely satisfactory solution of the problem has been found so far [?]. One of the propositions that have been made is to put the gauge-fixed theory on the lattice and to include a set of counterterms in the action with coefficients chosen in such a way that the BRS symmetry is restored in the continuum limit [?] (for a review and further references see refs. [?,?]). Using lattices with different lattice spacings for the gauge and the fermion fields is another idea which is being actively pursued [?–?]. The symmetry breaking terms can then be suppressed by choosing the lattice spacing in the fermion sector to be much smaller than the other lattice spacing.

A few years ago an entirely different approach was suggested by Kaplan [?], who noted that fermion modes which are bound to a four-dimensional defect in a five-dimensional lattice are chiral under certain conditions. Later this led to the so-called overlap formulation of chiral gauge theories [?,?], where the fermion partition function is written as a transition matrix element (the “overlap”) between the ground states of two auxiliary Hamilton operators. This development no doubt represents a big step forward, but as in the other cases the gauge symmetry is broken on the lattice. Moreover the locality properties of the theory are not transparent.

In this paper we consider U(1) gauge theories where the gauge field couples to left-handed Weyl fermions with charges satisfying

This is the classical condition for anomaly cancellation and the continuum theory is thus well-defined to all orders of perturbation theory. In particular, the effective gauge field action generated by the fermions is uniquely determined up to finite renormalizations of the gauge coupling [?–?].

The main result obtained here is that these theories can be put on the lattice without breaking the gauge invariance or violating other basic principles such as the requirement of locality. The construction is non-perturbative and one has the right number and type of Weyl fermions from the beginning. Not surprisingly it is technically rather involved and perhaps not as explicit as one would like, particularly in finite volume, where the non-trivial topology of the space of gauge fields gives rise to additional complications. The present paper is hence mainly of theoretical interest, clarifying a question of principle, but it does not provide a formulation of chiral gauge theories on the lattice which would be immediately usable for non-perturbative studies through numerical simulations. One should however note that chiral gauge theories are anyway a difficult case for numerical simulations, because the effective action has a non-zero imaginary part in general.

The starting point in this paper is the recent discovery
that chiral symmetry can be preserved on the lattice without having
to compromise in any other ways [?–?].
One achieves this by choosing a lattice Dirac operator
satisfying
^{†}^{†} For notational convenience the lattice spacing is set to
so that all length scales are given
in numbers of lattice spacings. In particular, the right-hand side
of eq. (1.2) should be multiplied with if physical units are employed

This relation (which is originally due to Ginsparg and Wilson [?]) guarantees that the fermion action is invariant under a group of infinitesimal transformations which may be regarded as a lattice form of the usual chiral symmetries. Moreover the non-invariance of the fermion integration measure under flavour-singlet transformations straightforwardly leads to the expected axial anomaly [?,?].

Having an exact chiral symmetry of the action, it turns out to be relatively easy to introduce left- and right-handed fields [?]. Because of the anomaly the fermion integration measure however does not decompose in a unique way and one ends up with a gauge field dependent phase ambiguity. To fix the phase of the measure so that the gauge symmetry and the locality of the theory are preserved is the principal problem which has to solved if one would like to set up chiral gauge theories along these lines.

All this will be explained in more detail in the next two sections. We then discuss the conditions which an ideal fermion integration measure should fulfil (sect. 4). Whether such measures exist is far from obvious and the rest of the paper is in fact entirely devoted to this question. For clarity the results are first presented in sect. 5 in a concise form, with all proofs and technical details being deferred to sections 6–11. A few concluding remarks are collected in sect. 12.

2. Lattice fields and functional integral

The lattice theories constructed in this paper are defined in the traditional manner, where one begins by specifying the space of fields and the lattice action. Expectation values of arbitrary products of the fields are then obtained as usual from the functional integral. The definition of the integration measure for Weyl fermions is non-trivial, however, and there are further technical details which need to be discussed carefully. A summary of notational conventions is included in appendix A.

2.1 Gauge fields

We choose lattice units and construct the theory on a finite lattice of size with periodic boundary conditions. U(1) gauge fields on such a lattice may be represented through periodic link fields,

on the infinite lattice. The independent degrees of freedom are then the link variables at the points in the block

( is assumed to be an integer). Under gauge transformations

the periodicity of the field will be preserved if is periodic. This is not the most general possibility, but the convention is here adopted that only periodic functions are referred to as gauge transformations.

For the gauge field action we take a somewhat unusual expression which effectively imposes an upper bound on the lattice field tensor. The reasons for this will become clear later. As in the case of the standard Wilson action we write

with being the bare coupling. The plaquette action is however taken to be of the more complicated form

where is a fixed number in the range and the field tensor is defined through

Note that the Boltzmann factor is a smooth function of the link variables with this choice of action. In particular, the functional integral can be set up in the usual way with the standard integration measure for U(1) lattice gauge fields.

This concludes the definition of the pure gauge part of the theory. There are a few remarks which should be added here.

(a) The Boltzmann factor is a product of local factors, one for each plaquette on the lattice. The locality of the theory is thus guaranteed. Moreover since it is differentiable, no special precautions are required when performing partial integrations in the functional integral (such as those needed when deriving the field equations).

(b) As already mentioned, our choice of action is such that the functional integral is effectively restricted to the space of fields satisfying

Gauge fields of this type will be referred to as admissible in the following.

(c) When physical units are employed, the parameter should be replaced by where denotes the lattice spacing. It is then immediately clear that the curly bracket in the definition (2.6) of the action and the bound (2.9) are irrelevant in the classical continuum limit. As far as the weak coupling phase is concerned, there is in fact little doubt that the lattice theory defined here is in the same universality class as the standard lattice theory. In particular, it is a valid lattice regularization of the free U(1) gauge theory.

2.2 Magnetic flux sectors

Before proceeding to the fermion fields, we briefly discuss the topology of the space of admissible fields. Proofs and further details will be given in sect. 7. One might expect that there is no interesting topological structure in this simple theory, but this is not so. The key observation is that the magnetic flux

through the –planes of the lattice is conserved and quantized. In other words, for any admissible field the associated flux satisfies

where is an integer tensor independent of .

Evidently the flux quantum numbers cannot change when the gauge field is continuously deformed. The field space is thus a disjoint union of the sectors of all admissible fields with a given magnetic flux. Moreover it can be shown that each of these sectors has the topology a multi-dimensional torus times a convex space.

2.3 Lattice Dirac operator

We first consider Dirac fermions and discuss the projection to the left-handed components in the next subsection. Dirac fields on the lattice carry a Dirac index and a flavour index . As in the case of the gauge field it is convenient to assume that the fermion fields are defined on the infinite lattice. Periodic boundary conditions are then imposed through the requirement that

Other types of periodic boundary conditions could be admitted here with little change in the following.

Under gauge transformations the fermion fields transform according to the representation

where is the charge of the fermion with flavour . Throughout the paper we take it for granted that the condition for anomaly cancellation, eq. (1.1), is satisfied. A simple example of an acceptable charge assignment is thus given by

Taking pairs of charges with opposite sign is another possibility, but in the present context this is a less interesting case, because one ends up with a chiral theory which is effectively vector-like.

The proper choice of the lattice Dirac operator is of central importance in the following. Apart from being a solution of the Ginsparg-Wilson relation (1.2), the operator should fulfil a number of technical requirements. In particular, it should be local, gauge covariant and differentiable in the gauge field. The complete list of requirements is given in appendix B.

Gauge covariant solutions of the Ginsparg-Wilson relation are not easy to find. The “perfect” lattice Dirac operator of refs. [?,?] is one of them and another solution has been derived by Neuberger [?] from the overlap formalism. In this case all properties described in appendix B have been established rigorously [?]. Note that it suffices to define the Dirac operator for all admissible gauge fields since only these contribute to the functional integral. The relevant results of ref. [?] in fact apply for admissible fields only and if is such that for all .

In infinite volume the action of the Dirac operator is given by

where the kernel is a matrix in Dirac and flavour space. For periodic fields eq. (2.15) may be rewritten in the form

i.e. the finite-volume kernel is obtained from the kernel on the infinite lattice by applying the reflection principle. Evidently, since we are dealing with the same operator, the Ginsparg-Wilson relation holds in finite volume too.

From the properties listed in appendix B it follows that is periodic in and separately. Moreover it transforms in the same way as under the gauge and lattice symmetries and from the locality of the operator one infers that

where is the localization range of .

2.4 Weyl fermions

Using the Ginsparg-Wilson relation, the infinitesimal transformation

is easily shown to be a symmetry of the fermion action

Eq. (2.18) is the chiral transformation of ref. [?] except that the fermion and the anti-fermion fields are here treated asymmetrically. The reason for this is that the present formulation allows one to decompose the fields into left- and right-handed components in a natural way. First note that the operator satisfies

So if we define the projectors

it is immediately clear that the left-handed fields
and (and the complementary components)
transform under lattice chiral rotations in the same way as
the corresponding fields in the continuum theory.
In particular, left- and right-handed fields decouple in the action (2.19)
^{†}^{†} That the action can be split this way
has independently been noted
by Hasenfratz and Niedermayer [?,?].
A closely related observation has also been made by Narayanan
in the context of the
overlap formalism [?].

We now eliminate the right-handed components by imposing the constraints

on the fermion fields. An important point to note here is that these conditions are local and gauge-invariant. The same is true for the action (2.19) and we thus have a completely satisfactory definition of the theory at the classical level.

2.5 Fermion integration measure

To set up the quantum theory we also need to specify an integration measure for left-handed fields. The basic difficulty which one has here is that the subspace of left-handed fermion fields depends on the gauge field. As a consequence there is a non-trivial phase ambiguity in the integration measure.

To make this explicit let us suppose that , , is a basis of complex valued, periodic fermion fields such that

(the bracket denotes the obvious scalar product for fermion fields in finite volume). The quantum field may then be expanded according to

where the coefficients generate a Grassmann algebra. They represent the independent degrees of freedom of the field and an integration measure for left-handed fermion fields is thus given by

An important mathematical fact which should be kept in mind in the following is that the measure is independent of the particular basis that has been chosen up to a phase factor. One can quickly see this by noting that a change of basis

implies a change of the measure by the factor which is a pure phase factor since is unitary. On the other hand, the remark shows that one has a phase ambiguity which is cannot be ignored because the basis (and hence the phase of the measure) depends on the gauge field. One can try to fix the ambiguity in some ad hoc manner, but as will become clear in sect. 4 such prescriptions are likely to be unsatisfactory. For the time being we assume that some particular basis has been chosen and proceed with the definition of the theory.

In the case of the anti-fermion fields the subspace of left-handed fields is independent of the gauge field and one can take the same orthonormal basis for all gauge fields. The ambiguity in the integration measure

is then only a constant phase factor.

Fermion expectation values of any product of fields are now obtained as usual through the functional integral

Note that this integral is completely well-defined. The integration variables are the coefficients and in terms of which the action assumes the form

The fermion fields in the product should be expanded similarly and the integral can then be evaluated following the standard rules for Grassmann integration.

In the definition (2.28) a complex factor has been included, which allows one to adjust the relative phase and absolute weight of the topological sectors. The factor only depends on the magnetic flux quantum numbers and we are free to set . One might be tempted to do the same in all other sectors as well, but this may not be the proper choice since the number of integration variables depends on the sector which is being considered (cf. subsect. 3.2).

Full normalized expectation values are finally given by

where the normalization factor is defined through the requirement that and denotes the usual integration measure for U(1) lattice gauge fields.

3. Correlation functions and effective action

Apart from the fact that we have not fixed the phase of the fermion integration measure, the theory is completely defined at this point and one can begin to study its properties. In the following paragraphs we work out a few quantities and address some of the basic questions which one may have in order to demonstrate the consistency of the approach.

3.1 Fermion propagator

If has no zero-modes it is straightforward to show that

where the fermion propagator is a periodic function satisfying

In other words, is the kernel of the inverse of the Dirac operator in finite volume. Note that there is no dependence on the bases and here since the phase ambiguity of the fermion integration measure cancels in eq. (3.1).

From the above and the definition of the chiral projectors it follows that

for all points in . This expression makes it evident that the propagating fermion modes are chiral. The theory thus describes the right number and type of Weyl fermions and there is little doubt that one recovers the correct Feynman rules in the continuum limit for the propagator in an external field.

3.2 Fermion number violation

A characteristic feature of chiral gauge theories is that fermion number violating processes can take place. This is possible whenever the numbers of left- and right-handed zero-modes of the Dirac operator, and , are not the same.

We can now easily check this in the lattice theory.
First note that the dimensions of the
spaces of left-handed fermion and anti-fermion fields
can be different. Since these spaces are the eigenspaces of
the corresponding chiral projectors, the
difference of their dimensions is given by
^{†}^{†} Here and below
the symbol “” implies a trace over the space of fermion fields
in finite volume, “” the same in infinite volume and “”
a trace over Dirac and flavour indices only

where the second equality follows from the index theorem [?,?]. The index is a topological invariant which assumes a fixed and in general non-zero value in each magnetic flux sector.

In all sectors where the index does not vanish, the matrix which appears in the action (2.29) has thus a rectangular shape. So if we temporarily choose the basis vectors and such that the first of them are the zero-modes, the action becomes

which is a non-degenerate quadratic form in the integration variables and associated with the other modes. The functional integral (2.28) hence vanishes unless is a product of fermion and anti-fermion fields times an arbitrary polynomial in pairs of these fields and the gauge field variables. In other words, has to have a net fermion number equal to and the lattice theory thus complies with the expected selection rules for fermion number violating processes.

3.3 Effective action

In the vacuum sector the dimensions of the spaces of left-handed fermion and anti-fermion fields are the same and the fermion partition function is hence given by

Chiral determinants in the continuum theory are usually studied by computing their variation under infinitesimal deformations of the gauge field [?–?]. We can do the same here and it will soon become clear that this is a useful exercise.

So let us consider a variation

of the gauge field, where is any real periodic vector field. After some algebra the associated variation of the effective action is then found to be given by

One might have expected to end up with the first term only, but since the basis vectors depend on the gauge field one has a second term,

which may be regarded as a contribution of the fermion integration measure.

The current which is defined through

is going to play an important rôle in the following. In particular, it will be shown later that the measure can be reconstructed from the current if certain conditions are fulfilled. Note that the measure term transforms according to

under basis transformations (2.26) and is hence unchanged if the transformation preserves the integration measure. As a consequence the current should be thought of as a quantity which is associated with the measure rather than the basis vectors . It is also immediately clear from this that any two measures with the same current are related to each other by a constant phase factor in each topological sector.

3.4 Integrability condition

The significance of the measure term may be further elucidated by computing the “curvature” . Starting from eq. (3.9), this is easily done and in a few lines one obtains

As expected from the transformation law (3.11), the curvature does not depend on the choice of the basis vectors . In particular, if it is not equal to zero it cannot be made to vanish by adjusting the basis and in these cases the measure term is hence required to ensure the integrability of eq. (3.8).

It is interesting to note in this connection that essentially the same happens in Leutwyler’s construction of the chiral determinant in the continuum theory [?], where a local counterterm has to be added to restore the integrability after applying a finite-part prescription to the variation of the determinant. The analogy will be even more striking after the discussion in the next section, which will lead us to require that the current should be a local expression in the gauge field. The measure term then assumes the form of a local counterterm.

3.5 Gauge anomaly

Although the fermion action and the projection to the left-handed fields are gauge-invariant, the effective action tends to be non-invariant due to the anomaly and the fact that the fermion integration measure depends on the gauge field. To work this out, let us consider a gauge variation

where is any periodic gauge function and the forward difference operator defined in appendix A. If we introduce the generator

of the fermion representation (2.13) of the gauge group, it is then obvious that

and taking eq. (3.8) into account one obtains

for the gauge variation of the effective action. Note that is equal to the sum of the axial anomalies associated with the flavours of fermions in the theory, weighted with their charge [?,?]. In other words, is the anomaly of the current which couples to the gauge field.

3.6 Vector-like theories

If the charges come in pairs with opposite sign, the continuum theory is formally equivalent to a vector-like theory, where the gauge field couples to Dirac fermions with positive charges. On the lattice we can choose a basis of left-handed fermion fields such that the basis vectors in the sectors with positive and negative charges are related to each other through

where denotes the charge conjugation matrix. The associated fermion integration measure is the same for any such basis and it is also easy to show that the measure term vanishes.

If the basis of left-handed anti-fermion fields is taken to be of the same type, the partition function (3.6) factorizes and in a few lines one obtains

which is the expected result for a vector-like theory. Up to contact terms and with an appropriate assignment of field components, there is in fact a complete matching between the chiral and the vector theory in the vacuum sector. Presumably this is also the case in the other sectors, but the issue will not be pursued here.

4. Conditions on the fermion integration measure

According to the universality hypothesis, the details of the lattice theory should not influence the continuum limit, apart from finite renormalizations, as long as a few basic principles are respected. One of them is that the theory should be formulated locally with no long-range couplings in the action. Symmetries are also very important and the universality of the continuum limit is more likely to be guaranteed if they are preserved on the lattice.

The conditions on the fermion integration measure listed below have been devised with this in mind. They should be regarded as a maximal set of requirements which one may reasonably hope to fulfil and one can be quite confident that the correct continuum limit will be obtained if they are all satisfied.

(1) Differentiability with respect to the gauge field. The expectation value of arbitrary (finite) products of the fermion fields and the link variables should be smooth functions of the gauge field. This is a somewhat technical requirement, but there are a few instances where the smoothness of the fermion integrals seems to be essential. In particular, the derivation of the field equation discussed below is invalid if this is not guaranteed.

As explained in sect. 8, this condition assumes a simple form in terms of the basis vectors and in the following we shall say that the fermion integration measure is smooth if the chosen basis has the properties stated there.

(2) Locality of the field equations. In euclidean field theory the field equations are linear relations between operator insertions in correlation functions. If the action is local, these operators are local composite fields and the locality properties of the theory are thus directly reflected by the field equations.

This leads us to require that the fermion integration measure should be such that the locality of the field equations is guaranteed. In particular, this should be so for the field equations associated with the gauge field, which one derives from the functional integral (2.30) by calculating the change of the integrand under field variations of the type considered in subsect. 3.3. Relatively little work is required for this if the field product does not involve the fermion fields, because only the sectors with vanishing index contribute in this case and one can then make use of eqs. (3.8) and (3.1) to show that

For local variations the first two terms
in this equation are manifestly local.
To ensure the locality of the field equations we thus require that
the current is a local function of the gauge field
^{†}^{†} The notion of locality used in this paper is the same as in
refs. [?,?,?].
Details are given in appendix B for the case of the Dirac operator.
Note that the term only makes sense if
the lattice size is much larger than
the localization range of the fields that one is interested in
.

If one considers more general field products , the field equations are not quite as easy to derive, but in all cases it turns out that the locality of the current implies the locality of the field equations.

(3) Gauge invariance. To preserve the gauge invariance of the theory we require that is a gauge-invariant function of the gauge field if is a gauge-invariant product of the link variables and the fermion fields. In particular, the partition function should be invariant and from our discussion in subsect. 3.5 it is immediately clear that this condition will be fulfilled if

It is possible to prove that no further conditions arise when one considers arbitrary products of fields, i.e. the gauge invariance of the theory is guaranteed if eq. (4.2) holds. One of the consequences of this equation and the integrability condition (3.12) is, incidentally, that the current itself has to be gauge-invariant.

(4) Lattice symmetries. In the continuum limit the imaginary part of the effective action transforms in a particular way under the space-time symmetries. On the lattice one would like to preserve these symmetries as far as possible so as to reduce any remaining phase ambiguity in the fermion integration measure.

Lattice translations, hyper-cubic rotations, reflections at the lattice planes and charge conjugation will be referred to as the “lattice symmetries” in the following. We now demand that the measure term transforms in the same way under these symmetries as the imaginary part of the first term on the right-hand side of eq. (3.8). This is equivalent to requiring the current to transform like the axial current

in ordinary lattice gauge theories with Wilson-Dirac fermions.

5. Statement of results

In the remainder of this paper we shall show that fermion integration measures satisfying conditions (1)–(4) exist in all topological sectors provided

where denotes the number of fermion flavours with . This includes the multiplet (2.14) and all cases with only even charges. In the vacuum sector there is actually no restriction on the charge assignment apart from the anomaly cancellation condition and it is currently not clear whether the constraint (5.1) reflects a fundamental limitation in finite volume or just a temporary technical difficulty.

For clarity the main steps of the construction are presented below in the form of three theorems together with some key formulae. All proofs are postponed to the later sections which should be consulted for full details.

5.1 Reconstruction theorem

While the fermion integration measure is a relatively complicated object, requiring the specification of a basis of left-handed fields modulo measure preserving basis transformations, the associated current is invariant under such transformations and is clearly much more tractable. The following theorem says that the measure can be reconstructed from the current under certain conditions.

Theorem 5.1.Suppose is a given current with the following properties.

(a) is defined for all admissible gauge fields and depends smoothly on the link variables. (b) is gauge-invariant and transforms as an axial vector current under the lattice symmetries, as described in sect. 4. (c) The linear functional satisfies the integrability condition (3.12). (d) The anomalous conservation law holds.

Then there exists a smooth fermion integration measure in the vacuum sector such that the associated current coincides with . The same is true in all other sectors if the charges satisfy the constraint (5.1). In each case the measure is uniquely determined up to a constant phase factor.

We are thus left with the problem to find a local current with the properties listed above. Since the notion of locality which is being adopted here allows for exponentially decaying tails (with a fixed localization range in lattice units), the current can have non-local contributions that are of this order in the lattice size . In the following our strategy will be to provide an explicit expression for the current in infinite volume and to prove that a solution in finite volume can be obtained by adding an exponentially small correction.

5.2 Anomaly cancellation

Before proceeding with the construction of the current it is however useful to discuss the significance of the anomaly cancellation condition (1.1) in the present framework. For simplicity we consider the theory in infinite volume in this subsection. The properties of the Dirac operator listed in appendix B then imply that the anomaly

is a gauge-invariant local field. Moreover, using the Ginsparg-Wilson relation, the anomaly can be shown to be a topological field satisfying

for any local deformation of the gauge field. It follows from this and a general theorem established in ref. [?] that

where is a constant and a gauge-invariant local current.

We now show that by noting that the Dirac operator is equal to the same analytic expression for each fermion flavour , with the link variables replaced by . The field tensor scales with the charge and there is another power of the charge coming from the generator in eq. (5.2). The contribution to the constant of the fermion with flavour is hence proportional to and after summing over all flavours one gets zero because of eq. (1.1).

The anomaly thus cancels up to a divergence term. At first sight one might think that this is not enough to achieve the gauge invariance of the theory, but we only need to satisfy eq. (4.2) for this and it is then conceivable that the gauge field dependence of the measure exactly compensates for the divergence term. The important point to note here is that one would be unable to cancel the term proportional to in this way. The construction of a fermion integration measure complying with conditions (1)–(4) is hence only possible for anomaly-free fermion multiplets.

5.3 Solution of the integrability condition in infinite volume

One of the technical advantages which one has in infinite volume is that the gauge fields can be represented in a natural way through vector fields. The relevant lemma has been proved in ref. [?] and is quoted here for the reader’s convenience.

###### Lemma 5.2

Suppose is an admissible gauge field on the infinite lattice. Then there exists a vector field such that

Moreover, any other field with these properties is equal to , where the gauge function takes values that are integer multiples of .

The idea is now to construct a solution of the integrability condition first in terms of the vector field. So let us assume that is any given field representing an admissible gauge field as in lemma 5.2. The curve

contracts this field to the classical vacuum configuration in such a way that the field tensor remains bounded by for all . For any variation of the gauge potential with compact support, a linear functional may thus be defined through

where is any gauge-invariant local current, which transforms as an axial vector field under the lattice symmetries and which satisfies . An example of such a field is obtained by averaging the current introduced in subsect. 5.2 over the lattice symmetries, with the appropriate weights so as to project to the axial vector component. Note that an explicit although very complicated expression for in terms of the first and second variations of the anomaly has been derived in ref. [?]. The existence of a current with the required properties is thus guaranteed.

Theorem 5.3.The linear functional

(a) , for arbitrary gauge functions that are polynomially bounded at infinity. is invariant under gauge transformations (b) The current is a local field, which depends smoothly on the gauge field and which transforms as an axial vector current under the lattice symmetries.(c) δ_ηL _ζ-δ_ζL _η =i Tr{^P_- [δ_η^P_-,δ_ζ^P_-]} in infinite volume for all compactly supported variations and . is a solution of the integrability condition (d) The anomalous conservation law

An important consequence of the gauge invariance of rather than the vector field , since the mapping between the two is one-to-one modulo gauge transformations. It can be shown that the locality, differentiability and symmetry properties of the current are the same independently of which point of view is adopted [?]. may be considered to be a function of the gauge field is that the current

5.4 Construction of the current in finite volume

We now return to the theory in finite volume and first note that becomes a gauge-invariant local field on the finite lattice if attention is restricted to periodic gauge fields. As asserted by the following theorem, this current has all the required properties up to exponentially small finite-lattice corrections.

Theorem 5.4.If the lattice is sufficiently large compared to the localization range of the Dirac operator, say , there exists a current which satisfies

and which fulfils conditions (a)–(d) of theorem 5.1. The bound (5.10) holds uniformly in the gauge field, i.e. the constants , and are independent of the field.

Together with theorem 5.1 this result implies that fermion integration measures satisfying conditions (1)–(4) exist on large lattices. Note that the difference between and vanishes exponentially in the continuum limit, because is a fixed number in lattice units while is a physical length scale. The detailed form of these corrections is hence of little interest.

The theorems quoted in this section are not easy to prove. Most of the difficulties can be traced back to the fact that the space of admissible gauge fields is topologically non-trivial in finite volume. Differential geometry and the theory of fibre bundles are the adequate tools to deal with this problem and the reader who wishes to go through the details in sects. 7–11 will be assumed to be familiar with the relevant mathematical terminology.

6. Proof of theorem 5.3

We first remark that the projector has the same locality properties as the Dirac operator. In particular, the kernel of falls off exponentially away from the support of and the trace in eq. (5.8) is hence rapidly convergent in position space. One of the consequences of this technical observation is that . We now establish the other properties of in the order stated in the theorem. is a well-defined and smooth function of the gauge potential

(a) Gauge invariance. Taking the gauge covariance of the projector and the gauge invariance of the current into account, it is easy to show that the change of under gauge transformations is given by

Expanding the commutators and using the identity

the first term can be rewritten in the form

where denotes the anomaly in infinite volume. Recalling and performing a partial summation it is now clear that the two terms in eq. (6.1) cancel each other.

(b) Locality and symmetry properties of . From what has been said at the beginning of this section, and since is a smooth local function of the gauge field, it is evident that the same is true for . Moreover under the lattice symmetries it transforms as an axial vector field. To prove this for space-time reflections one has to take into account that

In all other cases the transformations commute with the projection to the left-handed fields and the covariance of the current is deduced straightforwardly.

(c) Integrability condition. Starting from the definition (5.8) of , one quickly finds that the second term does not contribute to the curvature

can be shown to vanish by inserting and using the fact that anti-commutes with any variation of the projector . Taking this into account, there are only two terms which contribute to the curvature,

and these may be rewritten in the form

After integration one then ends up with eq. (5.9) since the contribution from the lower end of the integration range is equal to zero.

(d) Anomalous conservation law. Setting (where is any lattice function with compact support) the left-hand side of eq. (5.8) becomes

On the other side we insert the identities

and in a few steps obtain a sum of two terms,

Expressing the trace in the first term through the anomaly (5.2), the terms nearly cancel after a partial integration and one is left with the contribution

from the upper end of the integration range. Comparing with eq. (6.8) this shows that the divergence of is equal to the anomaly and thus completes the proof of the theorem.

7. Topology of the field space in finite volume

We now begin with the detailed discussion of the theory in finite volume and first determine the structure of the space of admissible gauge fields. As will be explained in the next section, the existence of smooth fermion integration measures depends on whether a certain U(1) bundle over this space is trivial. Evidently, to be able to address this problem, one needs to know the topology of the base manifold.

7.1 Preliminaries

In the following the lattice is assumed to be finite with periodic boundary conditions as specified in sect. 2. The space of admissible gauge fields is denoted by and the gauge group is taken to be the subset of gauge transformations satisfying at .

For any given gauge field , the Wilson lines winding around the lattice along the coordinate axes are defined by

They are gauge-invariant, but cannot be expressed through the field tensor and thus carry independent information on the gauge field.

###### Lemma 7.1

Any two admissible fields and satisfying

are gauge equivalent.

Proof:If we introduce a new field through

it is obvious that the associated plaquette loops and Wilson lines are all equal to . The product of the link variables along any lattice path from to the origin is hence independent of the chosen path and periodic in . In other words, is an element of the gauge group which transforms to and thus to .

The subspace of all admissible gauge fields with vanishing field tensor contains the pure gauge configurations, but there are also non-trivial configurations with Wilson lines different from . It is straightforward to show, however, that the Wilson lines do not depend on . The gauge-invariant content of such fields is hence encoded in the constant phase factors .

###### Lemma 7.2

The gauge fields with vanishing field tensor are of the form

where is an element of and the field is defined by

for any given set of phase factors . Moreover the representation (7.4) is unique and establishes the isomorphism .

Proof:From the definition (7.5) it is obvious that is a gauge field with vanishing field tensor and Wilson lines equal to . According to lemma 7.1, any other field with these properties is gauge equivalent to . This proves eq. (7.4) and it is now also evident that and are uniquely determined by the gauge field.

7.2 Flux sectors

As already mentioned in subsect. 2.2, the field space is a union of disconnected subspaces labelled by the magnetic flux quantum numbers . We now prove this and provide some further information on the flux sectors.

###### Lemma 7.3

Let be an admissible gauge field and define the associated magnetic flux through eq. (2.10). Then there exists an anti-symmetric integer tensor such that for all .

Proof:If we define a vector potential through

it is straightforward to show that

where takes integer values. Only the second term in this equation contributes to the magnetic flux which is hence an integer multiple of .

The periodicity of the field tensor implies that is independent of the coordinates and . To prove that the flux is also independent of the complementary components of , we note that

This is a straightforward consequence of lemma 5.2 and particularly of eq. (5.6). Using the periodicity of the field tensor again and partial summations, the change of the flux in any direction orthogonal to the –plane is then easily shown to vanish.

As long as only admissible fields are considered, the field tensor is a continuous function of the link variables and the magnetic flux quantum numbers consequently cannot change under continuous deformations of the field. The sectors of all fields with a given set of flux quantum numbers are thus disconnected from each other. There are at most a finite number of sectors since

as one may easily prove by combining eqs. (2.9) and (2.10). Conversely if is any prescribed, anti-symmetric integer tensor satisfying this bound, there exist admissible fields with these flux quantum numbers. An example of such a field is

where the abbreviation has been used. This field is periodic and can be shown to have constant field tensor equal to .

7.3 Topology of

We now determine the structure of the flux sector for any given set of flux quantum numbers . As will be shown below, one of the factors of this manifold consists of the space of all periodic vector potentials satisfying

The index “T” reminds us that these fields are transverse and also serves to distinguish them from the vector potential which has been introduced in sect. 5. Note that is a convex space. In particular, it is contractible and thus topologically trivial.

###### Lemma 7.4

The fields in the sector are of the form

where has vanishing field tensor and is an element of . Moreover this representation is unique and establishes the isomorphism .

Proof:We first prove the uniqueness of the representation (7.13) by noting that the field tensor of is given by

Together with the constraints (7.11) this equation implies that

where denotes the Green function of the lattice laplacian,

In particular, the transverse field is uniquely determined and so are the other factors in eq. (7.13).

To show that any given admissible field with field tensor and flux quantum numbers can be represented in this way, we turn the argument around and define through eq. (7.15). From the properties of the Green function it is then clear that this field satisfies eq. (7.11). Moreover, using eq. (7.8) (which holds for any admissible field) and the fact that the zero-momentum component of is proportional to , it is straightforward to establish eq. (7.14). In particular, is contained in and