###### Abstract

The families index theory for the overlap lattice Dirac operator is applied to derive topological features of the space of SU() lattice gauge fields on the 4-torus: The topological sectors, specified by the fermionic topological charge, are shown to contain noncontractible even-dimensional spheres when , and noncontractible circles in the case. We describe how certain obstructions to the existence of gauge fixings without the Gribov problem in the continuum setting correspond on the lattice to obstructions to the contractibility of these spheres and circles. We also point out a canonical connection on the space of lattice gauge fields with monopole-like singularities associated with the spheres.

Gauge fixing, families index theory, and topological features of the space of lattice gauge fields

David H. Adams

Physics dept., National Taiwan University, Taipei, Taiwan 106, R.O.C.

and

Physics dept., Duke University, Durham, NC 27708, U.S.A.^{1}^{1}1Current
address

email:

## 1 Introduction

Lattice gauge fields on the 4-torus have a fermionic topological
charge which arises in the overlap formalism [1] and can be expressed
as the index of the overlap Dirac operator [2]. It determines a
decomposition of the space of lattice gauge fields
(with a given unitary gauge
group) into topological sectors after excluding a measure-zero subspace
on which the fermionic topological charge is
ill-defined. In this paper we consider the following natural question:
what do these topological sectors look like? Their topology has been
worked out in the U(1) case (with a certain admissibility condition
imposed on the gauge fields) by Lüscher in [3].
This was a key part of the existence proof
in [3] for gauge invariant abelian chiral gauge theory on the
lattice when anomalies cancel.^{2}^{2}2This result, which has also
been established by different
means in the noncompact U(1) case [4],
generalises to U(1) chiral gauge theory on arbitrary even-dimensional torus
[5], and to the electro-weak U(1)SU(2) case
[6]. In the general
nonabelian case the existence of chiral gauge theory with exact gauge
invariance on the lattice has been shown at the perturbative level in
[7, 8]. However, despite the progress in [9],
there is at present no nonperturbative existence proof (except for the special
case where the fermion representation is real [10]).
In this paper
we derive first results on the topology of the sectors of in the
nonabelian SU() case. Besides copies of SU(), which are
present because the lattice gauge fields take values in the gauge group,
we show that the sectors of contain certain noncontractible
even-dimensional spheres in the case, and noncontractible circles
in the case. These are a direct consequence of excluding the
lattice gauge fields for which the fermionic topological charge is
ill-defined.

The presence of the noncontractible -spheres in is derived from results on the orbit space of obtained via the families index theory for the overlap Dirac operator in [11, 12]. The classical continuum limit result Theorem 2 of [11] reveals the presence of certain noncontractible -spheres in the orbit space in the case. Similarly, the classical continuum limit result of Ref. [13] on the lattice version of Witten’s global anomaly implies the presence of noncontractible circles in the orbit space of in the SU(2) case. We show that these give rise to noncontractible -spheres/circles in itself by exploiting the known fact that, in contrast to the continuum situation, gauge fixings without the Gribov problem exist on the lattice (e.g., the maximal tree gauges [14, 15]). This is done in Section 2. The argument is quite implicit though, and our aim in the rest of the paper is to give a more explicit and illuminating construction of noncontractible -spheres/circles in , and explain how the obstructions to their contractibility are the lattice counterparts of certain obstructions to the existence of gauge fixings without the Gribov problem in the continuum setting. We begin in Section 3 by discussing how the gauge fixing issue and topology of the gauge field sectors get related, both in the continuum and on the lattice, when considering the gauge invariance issue for the chiral fermion determinant. In Section 4, we give a fermionic description of certain obstructions to the existence of Gribov problem-free gauge fixings in the continuum SU() gauge theory, based on families index theory for the Dirac operator, and Witten’s global anomaly, in the and cases, respectively. The lattice version of these considerations in Section 5 leads instead to obstructions to the contractibility of -spheres/circles in . We also discuss a connection on the space of lattice gauge fields, with monopole-like singularities associated with the -spheres, which arises naturally in this context. The results of the paper are summarised in Section 6. A property of gauge transformations used in the text is derived in an appendix.

## 2 Topology of the space of lattice gauge fields

We assume familiarity with the lattice formulation of SU() gauge theory on as summarised in [11] and refer to that paper for the definitions and notations used in the following. To begin with, the space of lattice gauge fields is

(2.1) |

(one copy for each lattice link). This space is connected, but decomposes
into disconnected sectors labeled by the fermionic topological charge
after excluding the codimension 1 submanifold of gauge fields for which the
charge is ill-defined. The fermionic topological charge is given by the
index of the overlap Dirac operator
[2]^{3}^{3}3It reduces to the continuum topological charge in the
classical continuum limit [16].,
and the excluded fields are those for which this operator is ill-defined,
i.e. the ones for which the Hermitian Wilson-Dirac operator
(with suitable negative mass term) has
zero-modes. We denote the resulting space of lattice gauge fields by .
A sufficient (although not necessary) condition for a
lattice gauge field to lie in is that its plaquette
variables satisfy the “admissibility condition”,

(2.2) |

where is the product of the ’s around the lattice plaquette . For sufficiently small , this condition guarantees the absence of zero-modes for [17, 18]. When is the lattice transcript of a continuum field , and is the plaquette specified by a lattice site and directions and , then . Hence the lattice transcript of a smooth continuum field (or family of continuum fields) is guaranteed to lie in when the lattice is sufficiently fine (see [16] for further discussion of this point).

Since is a product of copies of SU(), its topology is known. The topology of is more complicated and has not yet been determined in the present SU() case. (It has so far only been worked out in the U(1) case with the admissibility condition (2.2) imposed [3]; we discuss this case further below.) In the following we apply results from the families index theory for the overlap Dirac operator [11] to derive first results on the topology of in the SU() case:

Theorem. When the lattice is sufficiently fine, the topological sectors of contain noncontractible -dimensional spheres for case, and noncontractible circles in the SU(2) case. in the

Remark. Note that, by (2.1), for and , since the same is true for and of . In contrast, by the theorem, for case, and in the SU(2) case. Hence the noncontractible -spheres and circles mentioned in the theorem are all contractible in ; their noncontractibility in reflects the topological consequences of excluding from the fields for which the fermionic topological charge is ill-defined. in the

We will give two proofs of the Theorem in this paper. Both involve studying
as a -bundle over the orbit space , where
is the subgroup of gauge transformations
satisfying the condition
for some arbitrarily chosen basepoint in .
The reason for
this condition is to exclude the (nontrivial) constant gauge transformations;
consequently, acts freely on (unlike the full
group ).^{4}^{4}4For example, the trivial gauge field is
invariant under all constant gauge transformations, and constant
with values in the center of act trivially on all gauge fields.
This is crucial for our arguments in the following. The first, most direct
proof of the theorem is given in the remainder of this Section.
It makes use of the fact that gauge fixings without the Gribov
problem exist on the lattice. Examples of these are the maximal tree gauges
[14, 15] which we review further below.
Gauge fixings which do not have the Gribov problem [19]
are referred to as ‘good’ in the following.
A good gauge fixing picks out a submanifold of which
intersects each orbit precisely once. Since acts freely on
, this determines a decomposition

(2.3) |

The one-to-one correspondence between elements of and then gives

(2.4) |

i.e. a trivialisation of as a -bundle over . Hence the topology of is determined by that of and . The part of the Theorem now follows from results obtained via the families index theory for the overlap Dirac operator in [11]: It was shown there that the topological sectors of contain noncontractible -spheres for

The presence of noncontractible circles in in the SU(2) case can be
derived in a similar way, using the results of Ref. [13] on the lattice
version of Witten’s global anomaly.^{5}^{5}5The presence of the global
anomaly on the lattice had earlier been verified numerically in
[20, 21].
The lattice version of the global anomaly
can be expressed as (mod 2), where is the net number of crossings
of the origin (counted with sign) by the eigenvalues of the overlap Dirac
operator as the background gauge field is smoothly varied along a path
from an initial field to a final field .
The results of [13] show that, when and are the lattice
transcripts of a continuum path and a topologically nontrivial map
, then the lattice anomaly reproduces the continuum one,
i.e. (mod 2) when the lattice is sufficiently fine (for a single
fermion species in the fundamental representation of SU(2)).^{6}^{6}6
The requirement that is inconsequential here since we can
replace to enforce this condition without
affecting the topological properties of .
The path is a circle in , and clearly can only change
by even integers under deformations of this circle. Hence the nonvanishing of
(mod 2) implies noncontractibility of the circle in .
This together with (2.4) implies the presence of noncontractible circles
in itself. This completes our first proof of the Theorem.

The preceding argument gives more information on besides the fact that it is nonvanishing for -spheres in the continuum orbit space discussed above can have arbitrary integer values; hence for any given integer , will contain a -sphere with topological charge when the lattice is sufficiently fine. Since -spheres in with different topological charges represent different elements in , it follows that the number of different elements in can be made arbitrarily large by taking the lattice to be sufficiently fine, and becomes infinite in the continuum limit. By (2.4), the same is true for . On the other hand, the argument in the SU(2) case does not indicate more than one nontrivial element in or since there is only one equivalence class of topologically nontrivial maps . : The topological charges of the

We remark that imposing the admissibility condition (2.2) does not change the situation regarding the topological features of derived above. The -spheres/circles in , from which the -spheres/circles in arise, are the lattice transcripts of certain -spheres/circles in which come from -dimensional balls/line segments in (see [11]), and the lattice transcripts of such families of continuum gauge fields are guaranteed to satisfy the admissibility condition for any when the lattice is sufficiently fine, cf. the discussion following (2.2) above. Then the arguments above go through unchanged.

The decomposition (2.3), which was crucial for the above proof of the Theorem, relies on the fact that good gauge fixings exist on the lattice. Such gauge fixings can be obtained from maximal trees [14, 15] as we now discuss. A tree in the lattice is a collection of lattice links from which no closed loops can be formed. The tree is called maximal if it is not possible to add another lattice link without getting a closed loop. An example of a maximal tree in a 2-dimensional lattice is given in Fig. 1.

It was pointed out by Creutz [14] (see also [15])
that, given a maximal tree, any lattice gauge field can be gauge transformed
into a field whose link variables along the tree are all equal to 1 in a way
that is unique up to constant gauge transformations. Furthermore, any two
gauge fields in the same gauge orbit get transformed into fields which
coincide up to a constant gauge transformation. Thus a maximal tree determines
a lattice gauge fixing which, modulo constant gauge transformations, is free
of Gribov ambiguities.^{7}^{7}7As discussed in [22],
the residual symmetry under constant gauge transformations is easily handled.
(General methods for dealing with theories with a residual global symmetry
have also been discussed in [23].) However, finding
physically acceptable lattice gauge fixings which can be implemented in
Monte Carlo simulations without Gribov copies arising is a nontrivial
problem which has yet to be completely resolved. For reviews of this issue
see, e.g., [24, 25].
We now point out that maximal trees determine good
gauge fixings. Indeed, for each lattice gauge field there is a unique
gauge transformation satisfying which transforms it to a field
whose link variables along the tree are all equal to 1. The key observation
now is that any two gauge fields in the same orbit get sent to
the same gauge-fixed field under this procedure. This is seen as follows.
For each link not contained in the maximal tree, the link variable of the
gauge-fixed field is the product of the original link variables around a
closed loop, starting and ending at , which is formed by adding the link
to the tree. (We leave the straightforward verification of this fact to the
interested reader.) Consequently, under a gauge transformation of the
original field, the nontrivial link variables of the gauge-fixed field
just get conjugated by , i.e. they are unchanged.
Thus the procedure picks out precisely one lattice gauge field in each
orbit, i.e. we have a good gauge fixing.

We note in passing the topological structure of , which can be determined from a maximal tree gauge fixing as follows. The “gauge-fixed” submanifold of picked out by a maximal tree gauge fixing consists of the lattice gauge fields whose link variables are 1 on the links of the tree. I.e. is a product of copies of SU() with one copy for each lattice link not contained in the tree. A general property of maximal trees (whose straightforward verification we again leave to the reader) is that the number of links making up the tree is , where is the number of lattice sites. Thus is the product of copies of SU(). Since , this reveals the topology of . In particular, it follows that the noncontractible -spheres/circles in discussed above are all contractible in . As a small consistency check, we also note that the decomposition

(2.5) |

reproduces (2.1), since is the product of copies of SU() (one for each lattice site except ).

In the U(1) case, the topology of when the admissibility condition (2.2) is imposed was completely worked out by Lüscher in [3]. In this case, the topological sectors are labeled by topological fluxes rather than a topological charge. Using a good gauge fixing (different from the maximal tree gauges; it is a lattice version of the Landau gauge supplemented with an additional condition to exclude Gribov copies), Lüscher showed that the topological structure of each topological sector is

(2.6) |

where is a contractible submanifold in . The factor in the gauge-fixed submanifold is a remnant of the product of copies of U(1) making up . Thus, in the U(1) case, decomposing into topological sectors by imposing the admissibility condition (2.2) has the effect of removing some of the initial topological structure (by “breaking open” some of the copies of U(1) to get the contractible subspace ), but does not produce any new nontrivial topological structure. In contrast, the result of this section shows that, in the SU() cases, new nontrivial structures are in fact produced (i.e. the noncontractible -spheres/circles).

The existence proof for the noncontractible -spheres/circles in given in this section was rather implicit. In Section 5 we give an alternative, more illuminating proof, involving a different, more explicit construction of -spheres/circles. In that approach, the obstructions to contractibility are seen to be the lattice counterparts of certain obstructions to the existence of good gauge fixings in the smooth continuum setting (discussed in Section 4). But first, in the next section, we describe a simpler version of this correspondence that arises naturally when considering the gauge invariance issue for the chiral fermion determinant.

## 3 Relating gauge fixing and topology via the chiral fermion determinant

An important issue in lattice chiral gauge theory is
whether a smooth, gauge invariant phase choice exists for the (left- or
right-handed) overlap chiral fermion determinant [1].
The latter can be
expressed as , where is the overlap Dirac operator
[26].
Existence of good gauge fixings has implications for this:
If a smooth phase choice for exists on the gauge-fixed
submanifold picked out by the gauge fixing, then a smooth
gauge invariant (under )
phase choice on the whole of is obtained by simply setting
, where
denotes the action of a gauge transformation on a gauge field
.^{8}^{8}8The “smoothness” parts of these statements
break down if is replaced by , since the latter does not act
freely on the space of gauge fields.

On the other hand it is known, both in the smooth continuum setting
[27] and in the lattice setting [28, 29, 30], that
there are topological obstructions to gauge invariance of the chiral fermion
determinant in the U(1) and SU() cases (with fermion in the fundamental
representation, or, more generally, when anomalies don’t
cancel).^{9}^{9}9The fact that we are restricting the gauge transformations
to does not change this situation, cf. the Appendix.
It follows that, in these cases, either no good gauge fixing exists,
or, if one does exist, then no smooth phase choice for the
chiral fermion determinant exists on the submanifold picked out
by the gauge fixing.

Recall that the chiral fermion determinant is really a section in a U(1)
determinant line bundle over the space of gauge fields (cf. the final section
of [27] in the continuum, and [1, 28, 3, 29]
in the lattice setting). A smooth phase choice for the determinant on a
submanifold of the space
of gauge fields is equivalent to a trivialisation of the determinant line
bundle over the submanifold. Therefore, the question of whether such a phase
choice exists is intimately related to the topology of the space of gauge
fields. In the continuum setting, the topological sectors of the space
of smooth continuum gauge fields have no nontrivial topological structure –
they are just infinite-dimensional affine vectorspaces. Consequently, by
a standard mathematical fact, the determinant line bundle is trivialisable
over (and any submanifold of ). Combining this with the preceding
observations, we conclude that, in the continuum setting, obstructions to
gauge invariance of the chiral fermion determinant are also
obstructions to the existence of good gauge fixings.
In particular, such gauge fixings cannot exist in the U(1) and SU()
continuum gauge theories.^{10}^{10}10There are more direct ways to see
these obstructions to gauge fixings, which do not involve the chiral
fermion determinant; these are described in the SU() case in Section 3.

On the other hand, in the lattice setting we already know that good gauge fixings exist. Then the preceding observations lead to the conclusion that, when obstructions to gauge invariance are present (e.g., in the U(1) and SU() cases), the submanifold picked out by the gauge fixing is noncontractible in . For if was contractible, then, by a standard mathematical fact, the determinant line bundle would be trivialisable over , i.e. a smooth phase choice for the chiral fermion determinant would exist on , and we could then obtain a -gauge invariant phase choice on the whole of in the way described earlier.

The preceding gives a first demonstration of how obstructions to the existence of good gauge fixings in the continuum correspond on the lattice to obstructions to contractibility of certain submanifolds in . Actually, the arguments above only establish this correspondence for the trivial topological sector, since in the other sectors the chiral fermion determinant vanishes. The obstructions to gauge invariance of the chiral fermion determinant have a natural description in the context of families index theory: they are the topological charges of the Dirac index bundle over 2-spheres in the gauge field orbit space (cf. [11] and the last section of [27] for the lattice and continuum settings, respectively). In the following sections we use the families index theory to to show more precisely the correspondence between gauge fixing obstructions in the continuum and topological structure in the sectors of the space of lattice gauge fields; the derived results hold in general and not just for the topologically trivial sector.

## 4 Continuum considerations

In this section we describe obstructions to good gauge fixings
in the space of smooth continuum SU() gauge fields on .
For simplicity we restrict our considerations to the topologically
trivial sector. Then consists of the smooth maps
and is the subgroup with .
The most direct way to see the obstructions is via the approach
of I. Singer in
[31] (see also [32], where Singer’s argument for SU(2)
gauge fields on is generalised to gauge groups including general
SU() and spacetimes including ):^{11}^{11}11Note that our setup is
different from that of [31, 32]: There the considerations are
restricted to the irreducible gauge fields, which are acted freely upon
by (where is the center of SU()). The
obstructions are then given by and are different from
the ones in our case. A drawback of that setup is that the trivial
gauge field is excluded; hence one cannot consider, e.g., the free
gluon propagator. Our setup, where the space of gauge fields is unrestricted
and the gauge transformations are instead restricted to , avoids this.
Regarding as a -bundle over ,
a good gauge fixing is equivalent to a trivialisation

(4.1) |

Since , existence of a trivialisation implies that and for all . Nonexistence of good gauge fixings now follows from the fact that there are always nonvanishing ’s, which can be seen as follows. Since the topological structure of SU() is essentially modulo a finite set of equivalence relations, there are smooth maps with nonvanishing degree for is imposed (see the Appendix). In the latter case, corresponds to a map , where , i.e. can be regarded as an element in . This element is clearly nonzero since has nonvanishing degree. It follows that for . Such maps still exist when a condition

These considerations do not have an immediate lattice counterpart because for the space of lattice gauge fields. Therefore, we now present another perspective on the above obstructions to good gauge fixings by showing that they are the obstructions to trivialising the -bundle over certain -spheres in in the case, and the obstruction to trivialising over certain circles in in the case.

A smooth map with
, together with a
gauge field , determines an -sphere in as follows.
Define the family in by
; then an -family in is
obtained by setting . Extend this to
a family by setting . Here
denotes the -dimensional unit ball and is the radial coordinate.
Since the ’s are all gauge equivalent, the family
descends to an family in the orbit space, i.e. an -sphere
in which we denote by .^{12}^{12}12We are assuming that
no two interior points in the family are gauge
equivalent, which will be true in the generic case.
In the case, is to be regarded as the boundary
of , i.e. the disjoint union of two points. In this case,
consists of two maps .

Proposition. The -bundle is trivialisable over if and only if can be extended to a smooth map with .

Corollary. (i) is not trivialisable over when the degree of is nonzero. Thus for there are -spheres in over which is nontrivialisable. (ii) In the SU(2) case there are circles in over which is nontrivialisable.

Proof. Part (i) of the corollary follows from the proposition by noting that an extension of corresponds to a smooth family , given by , which describes a smooth deformation of to a map which is independent of . The degree of such vanishes; it follows that the same must be true for all and in particular for . But we have already noted above that ’s with nonvanishing degree exist when with case of the proposition and the fact that there are maps which cannot be continuously deformed to the identity map: If is taken to consist of such a and the identity map then no extension connecting these exists. . Part (ii) of the corollary follows from the

To prove the Proposition we first show that trivialisability of over implies that an extension of exists. A trivialisation of over ,

(4.2) |

determines a “gauge-fixed” -sphere in , defined as the image of under the trivialisation map (4.2). Let denote the unique element of lying in the gauge orbit through , and let denote the unique gauge transformation relating these by

(4.3) |

Since is independent of , we have , hence is independent of . Set , then so is an extension of with . Conversely, given an extension of with , define the smooth family in by , and define by (4.3). From the definitions, which implies that is independent of and therefore that the family is an -sphere in . It is clear from the constructions that the orbit through intersects at precisely one point, namely , so is a “gauge-fixed” submanifold for determining a trivialisation (4.2).

In preparation for the lattice considerations in the next section we conclude this section with an alternative “fermionic” proof of the Corollary of the Proposition above, based on families index theory for the Dirac operator coupled to gauge fields [33] and Witten’s global SU(2) anomaly [34]. A well-known fact, following from the results of [33], is that the topological charge (integrated Chern character) of the index bundle of the Dirac operator over the above -sphere in equals the degree of . If is contractible in then the topological charge must vanish, so the ’s with nonvanishing degree determine noncontractible -spheres in , in which case . On the other hand, as noted earlier, if a good gauge fixing exists then for all . Thus we see again that the degree of is an obstruction to the existence of good gauge fixings.

The fermionic proof that the degree of is an obstruction to trivialisability of over (part (i) of the Corollary) is as follows. If such a trivialisation exists then the corresponding “gauge fixed” -sphere in is related to as in (4.3) by a family of gauge transformations. Recalling that the index bundle over is obtained from the index bundle over by identifying the fiber over with the fibre over via the action of on the space of spinor fields, it is easy to see that the family gives an isomorphism between the restriction of the index bundle over to and the restriction of the index bundle over to . Hence the topological charges of these restricted bundles coincide, so the topological charge of the index bundle over is , implying that is noncontractible in if . Since all -spheres in are contractible, this is a fermionic way to see that is not trivialisable over when the degree of is nonzero.

Turning now to the SU(2) case (part (ii) of the Corollary), we give a
fermionic proof that is nontrivialisable over the circle
in coming from the -family
in when is a topologically nontrivial element
in .^{13}^{13}13If necessary we can replace
to get a topologically nontrivial element
in .
Let denote the positive eigenvalues of the Dirac operator
, and the flows of these eigenvalues when the
Dirac operator is coupled to .^{14}^{14}14The argument requires
to be chosen such that the Dirac operator coupled to doesn’t have any
accidental zero-modes. Let denote the net number of crossings of the
origin by these eigenvalues (counted with sign)
as increases from 0 to 1. A trivialisation
of over determines a “gauge-fixed” circle in ,
related to similarly to (4.3) by
for some family of gauge transformations. Then the
Dirac operators coupled to and have the same eigenvalues,
hence the flows of the positive eigenvalues
of the Dirac operator coupled to
coincide with , and the net number
of crossings of the origin by coincides with .
Clearly can only change by an even integer under a deformation of the
circle in . Therefore, if is contractible then
(mod 2), while on the other hand (mod 2) implies is
noncontractible in . Since all circles in are contractible, the
former must hold. It follows that is not trivialisable over
when is topologically nontrivial, since in this case it is known
that (mod 2) —this is Witten’s global SU(2) anomaly
[34].

## 5 Lattice considerations

In this section we describe a lattice version of the continuum considerations of Section 4. A good gauge fixing in the SU() lattice gauge theory is equivalent to a trivialisation

(5.1) |

The ’s have not been determined at present, and can be nonvanishing (as we already know from Section 2). Hence the initial considerations of Section 4 do not have an immediate lattice counterpart. The subsequent parts of Section 4 do have lattice counterparts though, as we now discuss. A map with , together with a gauge field , determine an -family in where , and an -family in . An extension of to a -family then gives an -sphere in . Such an extension need not exist in general; nevertheless it is expected that generic -spheres in will arise in this way. Examples of such are given by the lattice transcripts of the -spheres in discussed in Section 4, i.e. , and are the lattice transcripts of , and ; the resulting family is guaranteed to lie in when the lattice is sufficiently fine, cf. the discussion following (2.2) above.

Now, an argument completely analogous to the proof of the Proposition of Section 4 shows that the -bundle is trivialisable over if and only if can be smoothly extended to a map with . Since trivialisations of (i.e. good gauge fixings) are already known to exist, this implies that the extension is always guaranteed to exist. This is in contrast to the continuum case; the difference is due to the fact that on the lattice the is no requirement that and be smooth in the -variable, since the lattice sites are discrete. (Note that the requirement can always be satisfied in the lattice case by making this part of the definition of .)

The preceding does not mean that every map can by extended to a map . Clearly this is not possible, since, in cases where is even, there are maps , which wrap nontrivially around an -sphere in SU(), and therefore cannot be extended to maps . What the preceding implies is that a necessary condition for the family in , constructed using , to admit an extension is that admits an extension . I.e. the cannot produce an -sphere in in the way described above if it doesn’t admit an extension .

When is the lattice transcript of an -sphere in of the kind discussed in Section 4, i.e. is the lattice transcript of , the existence of an extension for the lattice transcript can be seen explicitly as follows. Consider the element in represented by the map . Since the continuum depends smoothly on , this element is independent of (since is connected). It is zero when (since ), and is therefore zero for all . Hence an extension , exists for each lattice site .

We now give the promised second proof of the Theorem of Section 2. Unlike the first proof, it does not rely on the existence of of good gauge fixings on the lattice, and provides a more explicit description of noncontractible -spheres/circles in . Let be the lattice transcript of a -sphere in of the kind considered in Section 4. For each lattice site choose a smooth extension of the map , to a map , (as discussed in the preceding). In particular, set . This determines a smooth map with and . Now define a -family of lattice gauge transformations by and set

(5.2) |

An easy consequence of the definitions is that independent of . Hence the family is actually a -sphere . Since is gauge equivalent to , this -sphere is guaranteed to be contained in , and to satisfy the admissibility condition (2.2) for any , when the lattice spacing is sufficiently small. In the following we apply the families index theory for the overlap Dirac operator to show that is noncontractible in when is nonvanishing and the lattice is sufficiently fine. Since ’s with nonzero degree exist when , this will establish the first part of the Theorem.

A formula for the topological charge of the index bundle of the
overlap Dirac operator over the -sphere in
was derived in [11]. It reduces in the classical continuum limit to
, the topological charge
of the corresponding -sphere in [11, 12].
Thus when the lattice spacing is sufficiently small.
We now note that, just as in the continuum situation in Section 4,
the relation (5.2) between and the -sphere
in implies that the index bundle over is isomorphic
to the index bundle over , which in turn implies that their
topological charges are the same. To see this explicitly, recall from
[11] that, for , the topological charges of the index bundle
over and coincide with those of the vectorbundle
, given as follows.^{15}^{15}15This bundle was denoted “” in
[11] but we omit the “” subscript here.
The fibre of over is
where is a projection operator
acting on the space of lattice spinor fields, given by

(5.3) |

where is the overlap Dirac operator coupled to . Set , then

(5.4) |

Gauge covariance of implies for all , , so descends to a vector bundle over . The restriction of this bundle to is

(5.5) |

where and the equivalence relation means identify each with via the isomorphism

(5.6) |

For the fibres and are isomorphic via the gauge transformation in (5.2). This extends to a well-defined isomorphism between the fibres at since, as an easy consequence of the definitions, independent of , i.e. respects the equivalence relation in (5.5). Thus and are isomorphic, and therefore have the same topological charge as claimed. It follows that if is contractible in then . Therefore, if then the -sphere constructed above must be noncontractible when the lattice is sufficiently fine. This proves the first part of the theorem.

The proof of the remaining part of the theorem is as follows. Let be the lattice transcript of a circle in of the kind discussed in Section 3, i.e. start with a , pick , then the lattice transcript of the family

(5.7) |

Since , implying , the family
is a circle . Just as in the case of it is
guaranteed to lie in when the lattice is sufficiently fine.
The fermionic argument of Section 3 in the SU(2) case now has the following
lattice version.
The overlap Dirac operator is normal and hence has a complete set of
eigenvectors. The eigenvalues lie on a circle in the complex plane which
passes through the origin, and the nonreal eigenvalues come in complex
conjugate pairs.^{16}^{16}16The convention used in defining the overlap Dirac
operator is that the matrices are hermitian. This corresponds
in the continuum to antihermitian Dirac operator with purely imaginary
eigenvalues, with the nonzero ones coming in complex conjugate pairs.
Let be the eigenvalues of coupled to which have
, and the flows of these eigenvalues
when is coupled to . Let denote the net number of crossings of
the origin (counted with sign) as increases from 0 to 1. Let
denote the analogous number for coupled to . Then, just as in the
continuum setting, the gauge covariance of and the relation (5.7)
imply . Clearly can only change by an even integer under
a deformation of the circle in . Therefore, if is
contractible in then (mod 2) . On the other hand,
the results of [13] show that (mod 2) reproduces Witten’s global
SU(2) anomaly in the classical continuum limit.^{17}^{17}17The presence
of the Witten anomaly on the lattice has also been verified numerically in
[20, 21]. It follows that
is noncontractible when is topologically nontrivial
and the lattice is sufficiently fine.
This completes our second proof of the Theorem of Section 2.

Remarks. (i) The noncontractible -spheres and circles in constructed in the preceding are “rough”: they cannot arise as lattice transcripts of smooth continuum -spheres and circles in since the contractibility of the latter implies contractibility of their lattice transcripts when the lattice is sufficiently fine. The roughness of and originates from the roughness of the of the extensions and (where ); these do not have smooth continuum versions when , and is topologically nontrivial, respectively. (ii) Since the continuum gauge field used as part of the starting point for constructing and in the preceding is smooth and continuous on and therefore has vanishing topological charge, and lie in the trivial topological sector of . However, noncontractible -spheres and circles in the other topological sectors are readily constructed along the same lines as above by starting with a topologically nontrivial in a singular gauge, such that is still continuous on and smooth away from the singularity, and the lattices are restricted to those for which the singularity of doesn’t lie on a lattice link.

Finally, we point out that the topological charge of the overlap
Dirac index bundle over the -spheres is associated with
a monopole interpretation for a certain canonical connection on with
values in the bundle .
The bundle arises as where is to be
interpreted as the trivial bundle over with fibre (the space of
lattice spinor fields on ) and is an orthogonal projection
map whose action on the fibres is given by , defined in (5.3).
Then has the canonical connection , where is the
exterior derivative on .^{18}^{18}18This can be regarded as a generalised
Levi-Civita connection, cf. appendix B of [35]: The Levi-Civita
connection of a riemannian manifold embedded in Euclidean space
can be written as where is the orthogonal
projection of onto the tangent space of at .
On we see from (5.3) that is singular at
the points for which the overlap Dirac operator , and hence the
fermionic topological charge, are ill-defined.
Such singularities are present in the interior of any -ball in
with as its boundary when is
nonvanishing. (Such balls always exist since is contractible in
.)
These singularities of are monopole-like:
the topological charge of on , given by integrating the
Chern character of over , equals .
Indeed, the Chern charater of is a representative for
the Chern character of the bundle , and it was shown in [11]
that the nonzero degree parts of the latter coincide with those of
the overlap Dirac index bundle on .
(In fact the connection was used to derive the
formula (“lattice families index theorem”) for the Chern character of
the lattice index bundle in [11].)

## 6 Summary

When the decomposition of the space of lattice gauge fields into topological sectors is specified by the fermionic topological charge, it is to be expected that fermionic techniques will be required to determine the (topological) structure of the sectors. In this paper we have seen that the lattice families index theory developed in [11, 12] is a useful tool in this regard. Using it, we obtained first results on the topology of the sectors of SU() lattice gauge fields on : For sufficiently fine lattices, the sectors were shown to contain noncontractible -spheres when

Two proofs of this result were given. The first of these, in Section 2, was a short, rather implicit argument using results on the topological structure of the orbit space obtained previously from the families index theory for the overlap Dirac operator in [11], and exploiting the fact that gauge fixings without the Gribov problem exist on the lattice. The second argument, in Section 5, used the lattice families index theory in a more direct way, without relying on the existence of good gauge fixings. It led to noncontractible -spheres in the case, and noncontractible circles , in the case, obtained by a quite explicit prescription as follows: Start with a continuum gauge field and smooth map with and . Then, for each lattice site , extend the map , to a map , . This determines via (5.2). is similarly determined via (5.7) after starting with a topologically nontrivial map and choosing for each lattice site a path from to in SU(2). The noncontractible ’s and ’s are rough – they do not have smooth continuum versions. Nevertheless, the gauge fields contained in them satisfy the admissibility condition (2.2) for any given when the lattice is sufficiently fine.

The arguments furthermore show that the number of elements in for ’s are associated with a monopole interpretation for a certain canonical connection on the space of lattice gauge fields. becomes infinite in the continuum limit, and that the topological charges of the index bundle over the

It would be interesting to to see if further results on the topology of can be extracted via the lattice families index theory or other fermionic techniques. A number of basic questions remain to be answered, for example: Is a given topological sector of path-connected, or can there be more than one connected component? (The decomposition (2.6) shows that the answer is affirmative in the U(1) case, at least when the admissibility condition is imposed.) Determining the topology of would be necessary for extending Lüscher’s existence proof for gauge invariant abelian chiral gauge theory on the lattice [3] to the SU() case.

In the continuum setting, the topological charge (integrated Chern character) of the Dirac index bundle over -spheres in the orbit space was seen to be an obstruction to the existence of good gauge fixings. Explicit examples were described where the obstruction is given by the degree of maps ( case. The Witten global anomaly was seen to be an obstruction to the existence of good gauge fixings in the SU(2) case. In the lattice setting the situation is quite different: Good gauge fixings exist, and the topological charge of the overlap Dirac index bundle over the -spheres ) in the