Chiral anomaly for local boundary conditions

Abstract

It is known that in the zeta function regularization and in the Fujikawa method chiral anomaly is defined through a coefficient in the heat kernel expansion for the Dirac operator. In this paper we apply the heat kernel methods to calculate boundary contributions to the chiral anomaly for local (bag) boundary conditions. As a by-product some new results on the heat trace asymptotics are also obtained.

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arXiv:hep-th/0309019v1 1 Sep 2003
Chiral anomaly for local boundary conditions
Valery N.Marachevsky∗
V. A. Fock Institute of Physics, St. Petersburg University,
198504 St. Petersburg, Russia
Dmitri V.Vassilevich†
Institut f¨ ur Theoretische Physik, Universit¨ at Leipzig,
D-04109 Leipzig, Germany
February 1, 2008
Abstract
It is known that in the zeta function regularization and in the Fujikawa
method chiral anomaly is defined through a coefficient in the heat kernel
expansion for the Dirac operator. In this paper we apply the heat kernel
methods to calculate boundary contributions to the chiral anomaly for
local (bag) boundary conditions. As a by-product some new results on the
heat trace asymptotics are also obtained.
PACS: 11.15.-q, 11.30.-j, 02.40.-k
Keywords: chiral anomaly, heat kernel, local boundary conditions
1 Introduction
Chiral anomaly, which was discovered more than 30 years ago [1], still plays an
important role in particle physics. On smooth manifolds without boundaries
many successful approaches to the anomalies exist1. Modern developments in
theoretical physics require anomaly calculations on branes and domain walls (see,
e.g., [6]). Taking into account also more traditional applications as the bag model
of hadrons [7], we see that understanding the chiral anomaly in the presence of
boundaries or singularities is an important task.
∗email: maraval@mail.ru, root@VM1485.spb.edu
†Also at V. A. Fock Insitute of Physics, St.Petersburg University, Russia;
Dmitri.Vassilevich@itp.uni-leipzig.de
1The reader may consult, for example, ref. [2] or the original papers [3, 4, 5].
email:
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In the case of non-trivial background fields, and especially in the presence
of boundaries or singularities, the heat kernel technique seems to be the most
adequate one for analysing the one-loop effects (see [8] for a recent review). The
heat kernel approach to the anomalies is essentially equivalent to the Fujikawa
approach [4] and to the calculations based on the finite-mode regularization [9],
but it can be more easily extended to complicated geometries.
In the present paper we consider an euclidean version of the bag boundary
conditions [7] (see also [10]). Although these are the most simple and physically
most natural boundary conditions, chiral anomaly for these conditions has not
been calculated so far2. This can be explained by the fact that the calculation is
indeed rather involved. We use the zeta function regularization and our approach
to the anomaly follows closely the paper [11]. Mathematical foundations for
analysing spectral geometry of the Dirac operator with local boundary conditions
were developed in [12]. A simple overview of spectral properties of Dirac operator
on manifolds with boundary can be found in [13].
To calculate the anomaly we employ the following strategy. First we relate the
anomaly to a heat kernel coefficient for the square of the Dirac operator. Then
we generalise the problem and consider the heat kernel for an arbitrary operator
of the Laplace type with mixed boundary conditions and with a matrix valued
smearing function. This generalisation allows us to use the heavy machinery of the
heat kernel expansion for such operators. In particular, important information
follows from two simple examples of boundary value problems in one and two
dimensions. In this way we are able to calculate first five heat kernel coefficients.
Chiral anomaly is then obtained simply by substituting explicit expressions for
the connection, the potential, and other quantities in terms of the background
vector and axial vector fields.
In the next section we remind some basic formulae regarding the zeta function
regularization, chiral anomaly, and local (bag) boundary conditions. Sec. 3 is
devoted to the relationship between Dirac and Laplace operators.
kernel coefficients are calculated in sec. 4, which is the main technical part of this
paper. In sec. 5 we return to the Dirac operator and finally calculate boundary
contributions to chiral anomaly in two and four dimensions. Our results and
their possible extensions are discussed in sec. 6. Appendix A contains the heat
kernel coefficients for mixed boundary conditions with a scalar smearing function.
Appendix B gives details of the particular case calculations used in sec. 4.
The heat
2There exist calculations of global chiral anomaly (i.e. for a position-independent chiral
transformation parameter) for some other types of boundary operators, and also calculation of
the scale anomaly for bag boundary conditions. Literature on these topics is rather large, so
that for uniformity we quote nobody here. A literature survey can be found in [8].
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2 Boundary conditions and chiral anomaly
Let us consider an n-dimensional Riemannian manifold M with boundary ∂M.
Dirac γ-matrices satisfy the Clifford commutation relation
{γµ,γν} = −2gµν.(1)
The γ-matrices defined in this way are anti-hermitian, 㵆= −γµ. We also
need the chirality matrix which will be denoted γ5independently of the dimension
of M. As usual, γ5γµ= −γµγ5and γ5†= γ5. We assume that n is even. We fix
the sign of γ5by choosing
γ5=in(n+1)/2
n!
ǫµν...ργµγν...γρ. (2)
Consider the Dirac operator
?D = γµ
?
∂µ+ Vµ+ iAµγ5−1
8[γρ,γσ]σ[ρσ]
µ
?
(3)
in external vector Vµand axial vector Aµfields. We suppose that Vµand Aµare
anti-hermitian matrices in the space of some representation of the gauge group.
σ[ρσ]
µ
is the spin-connection3.
The Dirac operator transforms covariantly under infinitesimal local gauge
transformations:
δλAµ= [Aµ,λ]
δλVµ= ∂µλ + [Vµ,λ]
?D →?D + [?D,λ]
˜δϕAµ= ∂µϕ + [Vµ,ϕ],
˜δϕVµ= −[Aµ,ϕ],
?D →?D + i{?D,γ5ϕ}.
(4)
and under infinitesimal local chiral transformations:
(5)
The parameters λ and ϕ are anti-hermitian matrices.
We adopt the zeta-function regularization4and write the effective action for
the Dirac fermions as
W = −lndet?D = −1
2lndet?D2=1
2ζ′(0) +1
2ln(µ2)ζ(0),(6)
3The spin-connection must be included even on a flat manifold if the coordinates are not
Cartesian.
4Our approach to the effective action for fermions is close to that in the papers [11, 14].
We refer to these works for a more detailed derivation of chiral anomaly (with somewhat more
accurate treatment of the zero mode problem).
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where
ζ(s) = Tr(?D−2s),(7)
prime denotes differentiation with respect to s, and Tr is the functional trace.
It is easy to show that the effective action (6) is gauge invariant, δλW = 0,
and that the variation of W under an infinitesimal chiral transformation reads
A :=˜δϕW = −2iTr(γ5ϕ?D−2s)|s=0.(8)
Let us define an integrated heat kernel for a second order elliptic partial
differential operator L by the equation:
K(Q,L,t) := Tr(Qexp(−tL)) , (9)
where Q(x) is a matrix valued function. For the boundary conditions we consider
in this paper (see eq. (23) below) there exists an asymptotic expansion [19] as
t → 0:
K(Q,L,t) ≃
k=0
The heat kernel is related to the zeta function by the Mellin transformation:
?∞
In particular5,
A = −2ian(γ5ϕ,?D2).
[4].
We shall need some basic notions from differential geometry. Let Rµνρσ be
the Riemann tensor, and let Rµν = Rσµνσ be the Ricci tensor. With our sign
convention the scalar curvature R = Rµ
does not play any important role in our calculations. However, we shall see below
that curved space offers no complications in our approach compared to the flat
case.
If the manifold M has a boundary, boundary conditions should be imposed
on the spinor field ψ. We need several basic definitions regarding differential
geometry of manifolds with boundary. Let {ej}, j = 1,...,n be a local orthonor-
mal frame for the tangent space to the manifold and let on the boundary enbe
an inward pointing normal vector. Then {ea}, a = 1,...,n − 1 can be identified
with a local orthonormal frame for the tangent space to the boundary. The frame
{ej} will be used to transform curved (world) indices µ,ν,...,σ to “flat” indices
and back. For example, for a vector vµthis transformation reads: vj = eµ
∞
?
ak(Q,L)t(k−n)/2. (10)
Tr(γ5ϕ?D−2s) = Γ(s)−1
0
dtts−1K(γ5ϕ,?D2,t). (11)
(12)
The same expression for the anomaly follows also from the Fujikawa approach
µis +2 on the unit sphere S2. Curvature
jvµ,
5A rather formal way to derive this equation consists in integration of the asymptotic ex-
pansion (10).
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va= eµ
upper and lower indices.
The extrinsic curvature is defined by the equation
avµ, vn = eµ
nvµ. In Euclidean space there is no distinction between flat
Lab= Γn
ab,(13)
where Γ is the Christoffel symbol. For example, on the unit sphere Sn−1which
bounds the unit ball in Rnthe extrinsic curvature is Lab= δab.
We impose local6boundary conditions:
Π−ψ|∂M= 0,Π−=1
2(1 − γ5γn) ,(14)
which are nothing else than a Euclidean version of the MIT bag boundary con-
ditions [7]. For these boundary conditions Π†
of the fermion current ψ†γnψ vanishes on the boundary.
An important comment on chiral transformations of the boundary conditions
(14) is in order. Finite version of the infinitesimal transformation (5) reads:
−= Π−, and the normal component
?D →?Dϕ= eiϕγ5 ?Deiϕγ5. (15)
This relation yields the following transformation law for the boundary projector:
Π−→ Π[ϕ]
−= e−iϕγ5Π−eiϕγ5=1
2
?1 − γ5γne2iϕγ5?,(16)
so that the boundary condition
Π[ϕ]
−ψ|∂M= 0(17)
remains consistent with (14) and (15). Eq. (17) represents an Euclidean version
[16] of chiral bag boundary conditions [17]. The boundary conditions (17) are
considerably more complicated than (14). Even such fundamental property of
(17) as the strong ellipticity (which ensures, for example, existence of only simple
poles of the zeta function) has been established only recently [18]. Fortunately, as
we stay at the level of linear perturbations, the condition (14) is enough for our
purposes. However, already the Wess-Zumino consistency conditions [3] which
imply two consequent chiral transformations require a more general setting of the
chiral bag (17).
3Dirac and Laplace operators
To calculate the anomaly (12) it is convenient to consider a more general problem
of calculation of the coefficients ak(Q,L) (see eq. (10)) for general matrix valued
6Locality means that the projector Π−acts at each point of the boundary independently.
An example of non-local boundary conditions can be found in [15].
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