Solving the local cohomology problem in U(1) chiral gauge theories within a finite lattice

Abstract

In the gauge-invariant construction of abelian chiral gauge theories on the lattice based on the Ginsparg-Wilson relation, the gauge anomaly is topological and its cohomologically trivial part plays the role of the local counter term. We give a prescription to solve the local cohomology problem within a finite lattice by reformulating the Poincar\'e lemma so that it holds true on the finite lattice up to exponentially small corrections. We then argue that the path-integral measure of Weyl fermions can be constructed directly from the quantities defined on the finite lattice. Comment: revised version, 35pages, using JHEP3.cls

Full text

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arXiv:hep-lat/0309022v2 14 Jan 2005
Preprint typeset in JHEP style - HYPER VERSION
hep-lat/0309022
Solving the local cohomology problem in U(1) chiral
gauge theories within a finite lattice
Daisuke Kadoh, Yoshio Kikukawa and Yoichi Nakayama
Department of Physics, Nagoya University, Nagoya 464-8602, Japan
E-mail: kadoh@eken.phys.nagoya-u.ac.jp, kikukawa@eken.phys.nagoya-u.ac.jp
Abstract: In the gauge-invariant construction of abelian chiral gauge theories on the
lattice based on the Ginsparg-Wilson relation, the gauge anomaly is topological and its
cohomologically trivial part plays the role of the local counter term. We give a prescription
to solve the local cohomology problem within a finite lattice by reformulating the Poincar´ e
lemma so that it holds true on the finite lattice up to exponentially small corrections. We
then argue that the path-integral measure of Weyl fermions can be constructed directly
from the quantities defined on the finite lattice.
Keywords: Lattice Gauge Theory, Chiral Symmetry, the Ginsparg-Wilson relation.

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Contents
1.Introduction1
2.U(1) gauge fields on the finite lattice4
3.Local composite fields on the finite-volume lattice5
4.The Poincar´ e lemma on the finite lattice8
5.Vector potentials on the finite lattice12
6. Cohomological analysis of topological fields on the finite lattice
6.1Cohomological analysis of topological field on the finite lattice
6.2Relation to the result obtained in the infinite lattice
14
15
21
7.Exact cancellation of gauge anomaly22
8.Discussion: Weyl fermion measure on the finite lattice24
A. The bound on ?g? 25
B. Relation between g and g∞
27
C. A proof of the bounds on q[m](x), φ[m]µν(x) and γ[m,w]
30
1. Introduction
Recently it turned out that lattice gauge theory can provide a framework for non-perturbative
study of chiral gauge theories, despite the well-known problem of the species doubling. The
clue to this development is the construction of gauge-covariant and local lattice Dirac op-
erators satisfying the Ginsparg-Wilson relation[1, 2, 3, 4, 5, 6],
γ5D + Dγ5= 2aDγ5D.(1.1)
An explicit solution of the Ginsparg-Wilson relation was derived from the overlap formalism
based on domain wall fermion and is referred as the overlap Dirac operator.1It has made
possible to realize exact chiral symmetry on the lattice[28] and also opened a route to the
1The overlap formalism proposed by Narayanan and Neuberger[7, 8, 9, 10, 11, 12, 13, 14, 15, 16] gives
a well-defined partition function of Weyl fermions on the lattice, which nicely reproduces the fermion zero
mode and the fermion-number violating observables (’t Hooft vertices)[17, 18, 19]. Through the recent
re-discovery of the Ginsparg-Wilson relation, the meaning of the overlap formula, especially the locality
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gauge invariant construction of anomaly-free chiral gauge theories on the lattice[29, 30, 31,
32, 33].
In the gauge-invariant construction of chiral gauge theories on the lattice, one of the
crucial steps is to establish the exact cancellation of gauge anomaly at a finite lattice
spacing. In abelian chiral gauge theories[30, 34]2, it has been achieved through the coho-
mological classification of the chiral anomaly[29, 36, 37], which is given in terms of lattice
Dirac operator satisfying the Ginsparg-Wilson relation[28, 38, 39, 40, 41, 42],
q(x) = tr{γ5(1 − aD)(x,x)} (1.2)
and is a topological field for the admissible lattice gauge fields satisfying the bound3
? 1 − Pµν(x) ?< ǫ,ǫ <
1
30
(1.3)
Pµν(x) ≡ U(x,µ)U(x + ˆ µ,ν)U(x + ˆ ν,µ)−1U(x,ν)−1. (1.4)
For an anomaly-free multiplet of Weyl fermions satisfying the anomaly cancellation condi-
tion of the U(1) charges,
?
it has been shown that the chiral anomaly is cohomologically trivial,
α
e3
α= 0,(1.5)
?
α
eαqα(x) = ∂∗
µkµ(x),qα(x) = q(x)|U→Ueα, (1.6)
where kµ(x) is a certain gauge-invariant and local current. The cohomologically trivial part
of the chiral anomaly is then used in the gauge-invariant construction of the Weyl fermion
measure. In short, it plays the role of the local counter term in the effective action for the
Weyl fermions.4
For the practical computation of observables in the lattice abelian chiral gauge theories,
it is required to compute the Weyl fermion measure for every admissible configuration.
properties, become clear from the point of view of the path-integral. The gauge-invariant construction
by L¨ uscher[30] based on the Ginsparg-Wilson relation provides a procedure to determine the phase of the
overlap formula in a gauge-invariant manner for anomaly-free chiral gauge theories. For Dirac fermions,
the overlap formalism provides a gauge-covariant and local lattice Dirac operator satisfying the Ginsparg-
Wilson relation[1, 2, 16, 4, 6]. The overlap formula was derived from the five-dimensional approach of
domain wall fermion proposed by Kaplan[20]. In its vector-like formalism[21, 22, 23, 24], the local low
energy effective action of the chiral mode precisely reproduces the overlap Dirac operator [25, 26, 27].
2See also [35] for a gauge-invariant construction of abelian chiral gauge theories in non-compact formu-
lation.
3It has been shown by Neuberger [43] that the constant 1/30 in the above bounds can be improved to
1/6(2 +√2).
4For nonabelian chiral gauge theories, the local cohomology problem can be formulated with the topo-
logical field in 4+2 dimensional space.[44, 31, 32, 33, 45] So far, the exact cancellation of gauge anomaly has
been shown in all orders of the perturbation expansion for generic nonabelian theories[46, 47, 48], and non-
perturbatively for SU(2)×U(1)Y electroweak theory, both in the infinite lattice[49]. In the five-dimensional
approach using the domain wall fermion[20, 21, 22, 23, 24, 26, 27], the local cohomology problem can be
formulated in 5+1 dimensional space[50].
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However it seems difficult to follow the steps given in [30] literally. The first problem is the
use of the infinite lattice in order to make sure the locality property of the cohomologically
trivial part.5As a closely related problem, the vector-potential-representative of the link
variable used in the cohomological analysis is unbounded. The second problem is the use
of the continuous interpolations in the space of the admissible U(1) gauge fields.
The purpose of this paper is to give a prescription to solve the local cohomology
problem within a finite lattice which applies directly to the chiral anomaly on the finite
lattice,
q(x) = tr{γ5(1 − DL)(x,x)},(1.7)
where DL(x,y) is the finite-volume kernel of the Dirac operator. With this method, we will
show that the current kµ(x), which gives the the cohomologically trivial part of the above
chiral anomaly, can be obtained directly from the quantities calculable on the finite lattice.
Then we will argue that the measure of the Weyl fermions can be also constructed directly
from the quantities defined on the finite lattice.6For our purpose, we first examine the
Poincar´ e lemma, which is originally formulated on the infinite lattice[29]. We will show that
the lemma can be reformulated so that it holds true on the finite lattice up to exponentially
small corrections of order O(e−L/2̺), where L is the lattice size and ̺ is the localization
range of the differential forms in consideration.7(Section 4)
We next examine the one-to-one correspondence between the link variable and the
vector potential with a certain good locality property, which is originally derived in the
infinite lattice[29]. We will show that a similar one-to-one correspondence with the desired
locality property can be formulated for the admissible U(1) gauge fields on the finite lattice,
by separating the link variables into the part which is responsible to the magnetic flux (the
constant mode of the field strength) and the part of the local and dynamical degrees of
freedom around the magnetic flux. (Section 5)
Equipped with the modified Poincar´ e lemma and the vector potentials for the admis-
sible U(1) gauge fields on the finite lattice, we will perform the cohomological analysis of
the chiral anomaly directly on the finite lattice. Through this analysis, we will derive a
formula by which the cohomologically trivial part of the chiral anomaly is given in terms
of the quantities defined on the finite lattice. (Section 6)
In this paper, we will focus on how to obtain the cohomologically trivial part (the
local counter term) directly on the finite lattice. We do not claim that the cohomological
classification of the chiral anomaly can be completed within the setup of the finite lattice.
Rather we will use some of the results in the infinite volume as inputs in order to establish
5The cohomologically trivial part is, so far, constructed in two steps: the local cohomology problem is
first solved in the infinite lattice[32] and then the corrections required in the finite lattice are constructed
and added[52] . Since the lattice Dirac operator satisfying the Ginsparg-Wilson relation should have the
exponentially decaying tail[53, 54], the local fields in consideration should have the infinite number of
components. Moreover, the vector potentials used in this analysis are not bounded.
6The continuous interpolation in the space of the admissible gauge fields is also a crucial technique in the
above construction. We will discuss the issue how to implement the continuous interpolation numerically
in the forthcoming paper.
7For the ultra-local tensor fields on a finite lattice, the Poincar´ e lemma has been formulated by Fujiwara
et al in [55].
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the exact cancellation of gauge anomaly on the finite lattice, as we will see in section 6. For
the presentation of our result, we follow closely the convention and the notation adopted
in [29], so that the necessary modification of the cohomological analysis in the finite lattice
becomes clear.
2. U(1) gauge fields on the finite lattice
We consider a finite four-dimensional lattice of size L with periodic boundary conditions
and choose lattice units. The U(1) gauge fields on such a lattice may be represented through
periodic link fields,
U(x,µ) ∈ U(1),x = (x1,··· ,x4) ∈ Z4,(2.1)
U(x + Lˆ ν,µ) = U(x,µ) for all µ,ν = 1,··· ,4,(2.2)
on the infinite lattice. The independent degrees of freedom are then the link variables at
the point x in the block
Γ4[x0]=?x ∈ Z4| − L/2 ≤ xµ− x0µ< L/2?
(2.3)
where x0is a certain reference point. L is assumed to be an even integer. Under gauge
transformations
U(x,µ) → Λ(x)U(x,µ)Λ(x + ˆ µ)−1, (2.4)
the periodicity of the field will be preserved if Λ(x) ∈ U(1) is periodic.
We impose the so-called admissibility condition on the U(1) gauge fields:
|Fµν(x)| < ǫ for all x,µ,ν,(2.5)
where the field tensor Fµν(x) is defined through
Fµν(x) =1
ilnPµν(x),−π < Fµν(x) ≤ π.(2.6)
We require this condition because it ensures that the overlap Dirac operator[2, 4] is a
smooth and local function of the gauge field for ǫ < 1/30 [6].
The admissible U(1) gauge fields on the finite lattice can be classified by the magnetic
fluxes mµν, where
1
2π
s,t=0
mµν=
L−1
?
Fµν(x + sˆ µ + tˆ ν)(2.7)
(integers independent of x). The following field is periodic and can be shown to have
constant field tensor equal to 2πmµν/L2(< ǫ):
V[m](x,µ) = e−2πi
L2[Lδ˜ xµ,L−1
?
ν>µmµν˜ xν+?
ν<µmµν˜ xν],(2.8)
where the abbreviation ˜ xµ= xµmod L has been used. Then any admissible U(1) gauge
field in the topological sector with the magnetic flux mµνmay be expressed as
U(x,µ) =˜U(x,µ)V[m](x,µ).(2.9)
We may regard˜U(x,µ) as the actual local and dynamical degrees of freedom in the given
topological sector.
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