Topological Charge of Lattice Abelian Gauge Theory
789
Progress of Theoretical Physics, Vol. 105, No. 5, May 2001
Topological Charge of Lattice Abelian Gauge Theory
Takanori Fujiwara, Hiroshi Suzuki∗,∗) and Ke Wu∗∗)
Department of Mathematical Sciences, Ibaraki University, Mito 310-8512, Japan
∗ High Energy Group, Abdus Salam ICTP, Trieste, 34014, Italy
(Received November 15, 2000)
§1.
Introduction
In a recent paper 1) Lüscher investigated generic structures of the chiral anomaly
for abelian gauge theory on the lattice. His work was extended to arbitrary higher
dimensions in subsequent papers, 2) where the topological part of the axial anomaly
is shown to be interpretable as a lattice generalization of the Chern character within
the framework of noncommutative differential calculus. 3) In the continuum theory,
the Chern character gives an integer topological winding number when integrated
over the base manifold, and it coincides with the index of the Dirac operator. 4) The
chiral anomaly on the lattice may also be related with the index 5) of the GinspargWilson Dirac operator. 6), 7) It is therefore very natural to seek an extension from
the continuum to the lattice of the index theorem relating the analytical index of
the Dirac operator with the topological invariant of the manifold on which the Dirac
operator is defined. In this respect, it is, however, not clear in the constructions of
Refs. 1) and 2) whether the lattice analogue of the Chern character can be related
to some topology of the gauge theory on the lattice.
One might think that it makes no sense to study the topological configurations
on the lattice, since any lattice field can be continuously deformed into a trivial configuration, and hence no nontrivial topological invariants can be constructed. But
this is not the case. As argued in Refs. 8)–11), it is indeed possible to define a smooth
fiber bundle, and hence a topological winding number, for a given lattice field configuration if it contains no exceptional link variables. 8), 9), 11) In the case of abelian
theories, Lüscher has shown in Ref. 12) that the configuration space of the link
variables satisfying the admissibility condition has a rich topological structure. The
admissibility condition can be considered to be a kind of smoothness condition for
∗)
Permanent address: Department of Mathematical Sciences, Ibaraki University, Mito 310-8512,
Japan.
∗∗)
Permanent address: Institute of Theoretical Physics, Academia Sinica, P. O. Box 2735, Beijing
100080, China.
The configuration space of abelian gauge theory on a periodic lattice becomes topologically disconnected when exceptional gauge field configurations are removed. It is possible
to define a U (1)-bundle from the nonexceptional link variables by a smooth interpolation
of the transition functions. The lattice analogue of the Chern character obtained using a
cohomological technique based on noncommutative differential calculus is shown to give a
topological charge related to the topological winding number of the U (1)-bundle.
790
T. Fujiwara, H. Suzuki and K. Wu
∆µ f (n) = f (n + µ̂) − f (n),
∆∗µ f (n) = f (n) − f (n − µ̂).
(1.1)
Then the theorem is extended to the lattice Λ as follows:
Theorem : Let q be a gauge invariant and smooth ultralocal function of the abelian
gauge potentials Aµ on a locally hypercubic regular lattice Λ of dimension D without
boundaries that satisfies the topological invariance
δq(n) = 0
(1.2)
n∈Λ
for arbitrary local variations of the gauge potentials Aµ → Aµ + δAµ . Then q(n) for
arbitrary n ∈ Λ takes the form
[D/2]
q(n) =
l=0
βµ1 ν1 ···µl νl Fµ1 ν1 (n)Fµ2 ν2 (n + µ̂1 + ν̂1 )
× · · · × Fµl νl (n + µ̂1 + ν̂1 + · · · + µ̂l−1 + ν̂l−1 ) + ∆∗µ kµ (n),
(1.3)
where Fµν (n) = ∆µ Aν (n) − ∆ν Aµ (n) is the field strength, the coefficient βµ1 ν1 ···µn νn
is antisymmetric in its indices, and the current kµ can be chosen to be gauge invariant
and ultralocal in the gauge potential.
For functions q on the infinite lattice ZD this theorem holds. Since Λ is assumed to be locally hypercubic and regular, the same identity should also follow for
ultralocal functions q. The point here is that the gauge invariant current kµ can be
the gauge field configuration, ensuring the existence of gauge potentials continuously
parameterizing the link variables. The essential point here is that the configuration
space of the admissible link variables is topologically disconnected.
In this paper we investigate the topological charge of abelian gauge theory on a
periodic lattice in an arbitrary number of even dimensions and argue that the lattice
analogue of Chern character obtained in Refs. 1) and 2) indeed gives an integervalued topological invariant through its relation to the topological winding number of
a U (1)-bundle constructed from the lattice gauge fields by the interpolation method
of Refs. 8) and 11).
We should add a brief argument concerning the theorem given in Refs. 1) and 2),
where an infinite hypercubic regular lattice is assumed. We can extend this theorem
to topologically nontrivial lattices Λ without boundaries by restricting the functions
on Λ to ultralocal functions. We assume that Λ is locally hypercubic and regular.
This implies that for any point n ∈ Λ one can find a set Un of lattice points and
links with a hypercubic regular lattice structure of the same dimension. Hypercubic
regular lattices with periodic boundary conditions, which we consider, are examples
of Λ. We call functions f on Λ ultralocal if f (n) for any n ∈ Λ depends only on the
gauge potentials associated with links within the subset Un of Λ. The abelian gauge
potentials on the lattice are treated in detail in §3. (See also Refs. 13), 14), 1), 12).)
Throughout this paper we assume a lattice spacing a = 1. The forward and backward
difference operators ∆µ and ∆∗µ are then defined by
Topological Charge of Lattice Abelian Gauge Theory
791
§2.
Topological charge of abelian gauge theory on T D
Fiber bundles over a manifold are topologically classified by the equivalence class
of transition functions. Our first main concern is to give a formula for the topological
winding number of the fiber bundle in terms of transition functions. In this section
we consider U (1)-bundles over a torus T D of dimension D = 2N defined by the
identification
x ∼ x + Lµ̂
for x ∈ RD , µ = 1, · · · , D
(2.1)
where µ̂ denotes the unit vector in the µ-th direction, and the period L of the torus
is assumed to be a positive integer. A hypercubic periodic lattice Λ of dimension D
is defined as the set of integral lattice points in T D .
We divide T D into LD hypercubes c(n) (n ∈ Λ) defined by
D
c(n) = x ∈ T |x = n +
D
yµ µ̂, 0 ≤ yµ ≤ 1 .
(2.2)
µ=1
We assume that L is sufficiently large that any restricted bundle over c(n) is trivial.
Mathematically, this can be achieved for L ≥ 2. For later convenience, let us denote
the intersection of c(n) and c(n − µ̂) by p(n, µ) and the common boundary of p(n, µ),
p(n, ν), · · ·, p(n, σ) by p(n, µ, ν, · · · , σ).
chosen to be ultralocal, (...truncated)
