# Heat Kernel, Spectral Functions and Anomalies in Weyl Semimetals

A. V. Ivanov∗, M. A. Kurkov† and D. V. Vassilevich‡
∗St. Petersburg Department of Steklov Mathematical Institute of RAS,
27 Fontanka, St. Petersburg 191023, Russia
∗Leonhard Euler International Mathematical Institute,
10 Pesochnaya nab., St. Petersburg 197022, Russia
E-mail<EMAIL_ADDRESS>
† Dipartimento di Fisica “E. Pancini”, Università di Napoli Federico II,
Complesso Universitario di Monte S. Angelo Edificio 6, via Cintia, 80126
Napoli, Italy.
† INFN-Sezione di Napoli, Complesso Universitario di Monte S. Angelo Edificio
6,
via Cintia, 80126 Napoli, Italy.
E-mail<EMAIL_ADDRESS>
‡ Center of Mathematics, Computation and Cognition, Universidade Federal do
ABC
09210-580, Santo André, SP, Brazil
E-mail<EMAIL_ADDRESS>
###### Abstract

Being motivated by applications to the physics of Weyl semimetals we study
spectral geometry of Dirac operator with an abelian gauge field and an axial
vector field. We impose chiral bag boundary conditions with variable chiral
phase $\theta$ on the fermions. We establish main properties of the spectral
functions which ensure applicability of the $\zeta$ function regularization
and of the usual heat kernel formulae for chiral and parity anomalies. We
develop computational methods, including a perturbation expansion for the heat
kernel. We show that the terms in both anomalies which include electromagnetic
potential are independent of $\theta$.

##  1 Introduction and the statement of results

Weyl semimetals [1] are without doubt among the most exciting new materials.
The spectrum of quasiparticles in these materials is described by a Dirac
operator containing a constant axial vector field $b$. This field is
responsible for the existence of topologically protected boundary states with
quite unusual dispersion relations – the so called Fermi arcs. In the massless
case, the field $b$ can be removed by a chiral transformation at the expense
of including an anomaly term in the action. One of the manifestations of such
anomaly term is the chiral magnetic effect [2, 3] (see also [4, 5]) consisting
in the appearance of an electric current in the direction of external magnetic
field.

The importance of boundaries for physics of Weyl semimetals is very well
appreciated. There have been many quantum computations for these materials in
the presence of boundaries, though almost all of them were done in a
simplified model where all boundary contributions were taken into account
through an effective $(2+1)$-dimensional model describing boundary states.
Such models do not always provide a good approximation to full quantum
effective action as has been demonstrated recently in [6, 7]. There were a few
calculations done for full quantum systems [8, 9, 10] for specific geometries
with boundaries or compact dimensions.

Quantum anomalies are very important objects of Quantum Field Theory. They can
be (almost entirely) expressed through integrals of local densities and thus
carry information about generic geometries and field configurations. Though
the anomalies do not describe the quantum effective action completely, the
chiral anomaly tells us how this action changes when the field $b$ is turned
off by means of a chiral transformation. Such an important quantity as the
Hall conductivity at zero external magnetic field is defined by the so-called
parity anomaly. The aim of the present work is to study these two anomalies on
4-dimensional manifolds with boundaries by means of the heat kernel methods
(see [11] for a physicist oriented introduction). As we shall see in a moment,
a consistent formulation of this problem requires chiral bag boundary
conditions with a varying chiral phase $\theta$ on $\partial\mathcal{M}$. Very
little is known about the spectral geometry of Dirac operators with these
conditions. (We will give a short literature survey below in this Section.)
Therefore, before starting calculations, we shall have to establish basic
properties of main spectral functions of the Dirac operator with these
boundary conditions. To be able to use the full power of spectral theory of
elliptic operators, we will work in Euclidean signature, assuming a kind of
Wick rotation in quantum effective action.

In this paper, we work on a smooth 4-dimensional Riemannian manifold
$\mathcal{M}$ with a smooth boundary $\partial\mathcal{M}$. This manifold is
equipped with a spin structure, a $U(1)$ gauge connection, and an axial vector
field $b$. We shall _not_ assume that $b$ is constant in most of this work.
The corresponding vector bundle over $\mathcal{M}$ will be denoted by
$\mathcal{S}$. Locally, the Dirac operator acting on sections of $\mathcal{S}$
has the form

$\not{D}=\mathrm{i}\gamma^{\mu}(D_{\mu}+\gamma_{5}b_{\mu}),$ (1)

where $\gamma^{\mu}$ denotes Dirac gamma matrices,
$\gamma^{\mu}\gamma^{\nu}+\gamma^{\nu}\gamma^{\mu}=2g^{\mu\nu}$. The covariant
derivative $D_{\mu}$ contains an electromagnetic potential $A$ and a spin
connection $w$. In components,
$D_{\mu}=\partial_{\mu}+\tfrac{1}{8}[\gamma_{\rho},\gamma_{\sigma}]w_{\mu}^{\rho\sigma}+\mathrm{i}A_{\mu}$.
$D$ is compatible with the Clifford structure,
$D_{\mu}\gamma^{\nu}=\gamma^{\nu}D_{\mu}$. The chirality matrix $\gamma_{5}$
is defined as

$\gamma_{5}=\tfrac{1}{4!}\epsilon^{\mu\nu\rho\sigma}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}\gamma_{\sigma}\,,$
(2)

where $\epsilon^{\mu\nu\rho\sigma}$ is the totally antisymmetric Levi-Civita
tensor.

In the applications to condensed matter problems, the Dirac operator contains
the Fermi velocity $v_{F}$. The operator (1) is written under the assumption
$v_{F}=1$. The dependence of quantum effective action on $v_{F}$ can be
restored by using a suitable modification of procedure [6].

The classical action

$S=\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{g}\psi^{\dagger}(x)\not{D}\psi(x)$
(3)

is invariant under local chiral transformations

$\not{D}\to\not{D}_{\phi}=e^{\gamma_{5}\phi}\not{D}e^{\gamma_{5}\phi},\qquad\psi\to\psi_{\phi}=e^{-\gamma_{5}\phi}\psi,$
(4)

where $\phi$ is a smooth function. The transformation of $\not{D}$ in (4) is
equivalent to a gradient transformation of $b$,

$b_{\mu}\to b_{\mu}+\partial_{\mu}\phi.$ (5)

Thus, if $b$ is a constant vector or a gradient of a scalar function, it can
be completely removed from the action. If $\partial\mathcal{M}$ is empty, the
whole effect is the very well known anomaly term. If there is a boundary, the
situation is much more complicated. One has to find a suitable set of boundary
conditions which respects chiral symmetry and compute boundary contributions
to the anomaly for such boundary conditions.

A minimal set of local boundary conditions which is closed under chiral
transformations is the so called chiral bag boundary conditions

$\Pi_{-}(\theta)\psi|_{\partial\mathcal{M}}=0,\qquad\Pi_{-}(\theta)=\tfrac{1}{2}\bigl{(}1-\mathrm{i}\varepsilon_{\alpha}\gamma_{5}e^{\gamma_{5}\theta}\gamma^{\mathbf{n}}\bigr{)}.$
(6)

Here $\gamma^{\mathbf{n}}=\mathbf{n}_{\mu}\gamma^{\mu}$ where $\mathbf{n}$ is
an inward pointing unit normal vector to $\partial\mathcal{M}$, and $\theta$
is a smooth function on $\partial\mathcal{M}$. $\varepsilon_{\alpha}$ is a
real parameter, $\varepsilon_{\alpha}=\pm 1$, which is constant on each of the
connected components $\partial\mathcal{M}_{\alpha}$ of the boundary
$\partial\mathcal{M}$. These conditions originally appeared in the context of
bag model of hadrons [12] (see also [13]), thus the name. With these
conditions, the normal component of fermion current
$\psi^{\dagger}\gamma^{\mathbf{n}}\psi$ vanishes identically on
$\partial\mathcal{M}$ making the Dirac operator symmetric. One can easily
verify, that

$\Pi_{-}(\theta-2\phi)\psi_{\phi}|_{\partial\mathcal{M}}=0,$ (7)

so that the conditions (6) indeed form an orbit of the group of chiral
transformations. Various spectral functions of the Dirac operator with chiral
bag boundary conditions were discussed in [14, 15, 16, 17, 18, 19, 20] – but
only for $\theta=const$, which is not sufficient to incorporate local chiral
transformations.

In what follows, we shall mostly consider an infinitesimal version of the
chiral transformations

$\delta_{\phi}b_{\mu}=\partial_{\mu}(\delta\phi),\qquad\delta_{\phi}\theta=-2(\delta\phi),$
(8)

where $\delta\phi$ is an (infinitesimal) parameter. It is understood that the
second equation in (8) should have contained the restriction of $\delta\phi$
to the boundary. We shall never write this restriction explicitly.

Our purpose is to analyse the anomalies arising in the effective action
obtained by integrating out the fermion field $\psi$. We use the
$\zeta$-function regularization of the effective action [21, 22] (see [23] for
a pedagogical introduction). The $\zeta$ function of $\not{D}$ is defined as

$\zeta(s,\not{D})=\sum_{\lambda>0}\lambda^{-s}+e^{-\mathrm{i}\pi
s}\sum_{\lambda<0}(-\lambda)^{-s},$ (9)

where $\lambda$ denotes the eigenvalues of $\not{D}$. The sums are convergent
if $\Re s$ is sufficiently large. The $\zeta$ function can be extended to the
whole complex plane as a meromorphic function. These properties are valid for
strongly elliptic boundary conditions. We postpone the discussion of this
point until Section 2. The regularized determinant of $\not{D}$ and the
regularized effective action are defined as

$W_{s}=-[\ln\det{(\not{D})}]_{s}\equiv\Gamma(s)\zeta(s,\not{D}).$ (10)

The regularization is removed in the limit $s\to 0$. Next, we represent
$\zeta(s,\mathcal{D})$ as a sum of two expressions, one being symmetric and
the other – antisymmetric with respect to the inversion
$\mathcal{D}\to-\mathcal{D}$. The symmetric part is expressed through
$\zeta(s/2,\not{D}^{2})$ while the antisymmetric is defined through the $\eta$
function

$\eta(s,\not{D})=\sum_{\lambda>0}\lambda^{-s}-\sum_{\lambda<0}(-\lambda)^{-s}.$
(11)

Thus, we have

$W_{s}=\tfrac{1}{2}\Gamma(s)\left[\bigl{(}1+e^{-\mathrm{i}\pi
s}\bigr{)}\zeta(s/2,\not{D}^{2})+\bigl{(}1-e^{-\mathrm{i}\pi
s}\bigr{)}\eta(s,\not{D})\right].$ (12)

Let us assume that both $\zeta(s/2,\not{D}^{2})$ and $\eta(s,\not{D})$ are
regular at $s=0$. By expanding $W_{s}$ near $s=0$ we obtain

$W_{s}=\left(\tfrac{1}{s}+\mbox{f.t.}\right)\zeta(0,\not{D}^{2})+\tfrac{1}{2}\zeta^{\prime}(0,\not{D}^{2})+\frac{\mathrm{i}\pi}{2}\eta(0,\not{D})+\mathcal{O}(s),$
(13)

where f.t. denotes some finite terms and
$\zeta^{\prime}(0,\not{D}^{2}):=\partial_{s}|_{s=0}\zeta(s,\not{D}^{2})$. The
expression (13) is divergent at $s\to 0$. The pole term has to be removed by
some sort of renormalization procedure. There remains a finite renormalization
ambiguity which consists in keeping a finite term proportional to
$\zeta(0,\not{D}^{2})$. Usually, such terms do not influence quantum
anomalies. Let us assume that this is true in our case as well. Therefore,
without losing any essential information we may write the renormalized
effective action as

$W=\tfrac{1}{2}\zeta^{\prime}(0,\not{D}^{2})+\frac{\mathrm{i}\pi}{2}\eta(0,\not{D}).$
(14)

This is a well known formula. The derivation was also rather standard. We
presented it here in full detail to highlight the assumptions which had to be
made. These assumptions will be justified by Theorem 1.1, see below.

Calculation of $W$ for generic choice of $\mathcal{M}$, $A$, and $b$ is a
hopeless task. However, the anomalies associated to $W$ are computable. The
first one is the chiral or Adler–Bell–Jackiw anomaly

$\mathcal{A}=\tfrac{1}{2}\delta_{\phi}\,\zeta^{\prime}(0,\not{D}^{2}).$ (15)

The second one is the whole second term in (14)

$W_{\rm odd}=\frac{\mathrm{i}\pi}{2}\eta(0,\not{D}),$ (16)

which, by somewhat stretching the terminology, we shall call the parity
anomaly. This anomaly was introduced [24, 25, 26] to measure the violation of
spatial reflection symmetry in quantum theories on manifolds without
boundaries. It also measures the spectral asymmetry of Dirac operator.

Very little is known about these anomalies in the presence of boundaries. The
chiral anomaly was calculated for a constant $\theta$ in two dimensions [14]
and for $\theta=0$ in four dimensions [27]. The parity anomaly with $\theta=0$
in four dimensions was calculated in [28, 29].

The anomalies (15) and (16) contain both local and global contributions. Local
contributions are smooth universal functionals of external fields. Global
contributions are in general non-smooth since they are related to zero modes
of the Dirac operator. To separate the local parts we shall assume that
$\not{D}$ _does not have zero modes_.

Our strategy is as follows. Since the spectral theory of Dirac operators
subject to chiral bag boundary conditions with varying chiral parameter
$\theta$ is an essentially uncharted area first we have to prove basic
properties of the spectral problem and spectral functions which have been
announced above. Such properties are usually taken for granted in the physics
literature even though there are numerous counterexamples of boundary
conditions for which they do not hold. Thus, some attention is required to put
quantum field theory computations of firm grounds. General properties of our
spectral problem are summarized in the following

######  Theorem 1.1.

Let $\not{D}$ be a Dirac operator on a smooth four-dimensional manifold
$\mathcal{M}$ with a smooth boundary $\partial\mathcal{M}$ subject to chiral
bag boundary conditions as defined above.
(a) Let $Q\in C^{\infty}(\mathrm{End}(\mathcal{S}))$ be a smooth endomorphism
(a matrix-valued function) with a compact support. Then, there are full
asymptotic expansions at $t\to+0$

$\displaystyle{\mathrm{Tr}\,}\bigl{(}Qe^{-t\not{D}^{2}}\bigr{)}\simeq\sum_{n=0}^{\infty}t^{(n-4)/2}a_{n}(Q,\not{D}^{2}),$
(17)
$\displaystyle{\mathrm{Tr}\,}\bigl{(}Q\not{D}e^{-t\not{D}^{2}}\bigr{)}\simeq\sum_{n=0}^{\infty}t^{(n-5)/2}a_{n}^{\eta}(Q,\not{D}),$
(18)

with locally computable coefficients $a_{n}$ and $a_{n}^{\eta}$.
Let in addition $\mathcal{M}$ be compact. Assume that $\not{D}$ has not zero
eigenvalues. Then we have the following assertions.
(b) The $\eta$ function $\eta(s,\not{D})$ is regular at $s=0$.
(c) The $\zeta$ function $\zeta(0,\not{D}^{2})$ is invariant under chiral
transformations (8).
(d) The chiral anomaly (15) reads

$\mathcal{A}=-2a_{4}(\gamma_{5}(\delta\phi),\not{D}^{2}).$ (19)

(e) Let $\delta\not{D}$ be the variation of $\not{D}$ induced by a smooth
(infinitesimal) variation of $A$ and $b$. Then

$\delta\eta(0,\not{D})=-\frac{2}{\sqrt{\pi}}\,a_{3}(\delta\not{D},\not{D}^{2}).$
(20)

Next, we proceed with actual calculation of the anomalies by using the heat
kernel expansion. The heat kernel coefficients which define the anomalies are
the bulk and boundary integrals of local polynomials in the fields entering
our problem and their derivatives having appropriate symmetry properties and
correct mass dimension. (These two requirements will be explained in detail in
Section 3 below.) The coefficients of these polynomials are functions of
$\theta$. Our task, therefore, is to define all allowed polynomials and to
compute the corresponding functions. Here, we face two problems as compared to
simpler types of boundary conditions.

The first problem is that the number of allowed structures is enormous. To
keep combinatorial complexity under the control we have to impose some
restrictions. First of all, we assume that the Riemannian metric on
$\mathcal{M}$ is flat and that the boundary $\partial\mathcal{M}$ is totally
geodesic (i.e., the extrinsic curvature of $\partial\mathcal{M}$ vanishes).
The first assumption is rather natural if we keep in mind the condensed matter
applications even though a non-flat effective metric may appear due to
deformations or due to position dependence of the Fermi velocity. The second
assumption means that $\partial\mathcal{M}$ is sufficiently smooth so that the
inverse curvature radii are small compared to other characteristic scales of
the model. Even with these restrictions, the number of relevant invariants is
very large. We shall explicitly compute all the terms containing the
electromagnetic potential $A$ since they are most interesting from the
physical point of view. We shall also present a universal method which allows
to obtain closed expressions for all remaining invariants and compute some of
them.

The second problem is that many standard methods of calculation of the heat
kernel coefficients [30] do not work for chiral bag boundary conditions. For
example, the (very important) Lemma 3.1.6 [30] is not applicable since the
boundary conditions cannot be presented in the required (product) form.
However, the Wess–Zumino consistency conditions which are consequences of the
Lie algebra composition law for the generators of anomalous symmetries are
still very useful. These conditions allow to reduce considerably the number of
unknown functions by using relatively simple algebraic methods. The rest of
the computations have to be done with the help of perturbative expansions for
the heat kernel. Our main technical tool will be a version of the heat kernel
derived in [17] for cylindrical geometries and a Dyson-type expansion obtained
below.

From the physical point of view, our principal result means that the most
relevant parts of both anomalies are stable against variations of $\theta$. To
proceed, we need to introduce some geometric notations. As above, the
subscript (superscript) $\mathbf{n}$ will denote normal components of vectors
and tensors on the boundary while Latin indices $i,j,k$ will denote the
tangential components. The Levi-Civita tensor on the boundary is defined as

$\epsilon^{ijk}\equiv\epsilon^{\mathbf{n}ijk}.$ (21)

Let semicolon denote the covariant derivative computed with the Christoffel
symbol $\Gamma^{\mu}_{\nu\rho}$ corresponding to the bulk metric $g$, e.g.
$b_{\mu;\nu}=\partial_{\nu}b_{\mu}-\Gamma^{\rho}_{\mu\nu}b_{\rho}$. The same
quantity corresponding to induced metric $h$ on $\partial\mathcal{M}$ will be
denoted by double dot.

######  Theorem 1.2 (Partial stability of the anomalies).

Let $\not{D}$ be a Dirac operator (1) without zero eigenvalues subject to
chiral bag boundary conditions on a compact manifold $\mathcal{M}$ as
described above. Then the terms containing electromagnetic potential in chiral
and parity anomalies do not depend on $\theta$ and read

$\displaystyle\mathcal{A}(A)=\frac{1}{16\pi^{2}}\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{g}(\delta\phi)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma},$
(22) $\displaystyle
W_{\mathrm{odd}}(A)=-\,\frac{\mathrm{i}}{16\pi}\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\,{\sqrt{h}}\,\varepsilon_{\alpha}\,\epsilon^{\mathbf{n}ijk}A_{i}A_{k:j}.$
(23)

The absence of $\theta$-dependent terms which are quadratic in $A$ can be
easily obtained on symmetry grounds, see Section 3. Therefore, the main
message of Theorem 1.2 is the absence of terms which are linear in $A$. Such
contributions would generate linear (tadpole) terms in the effective action
and mean an instability of Weyl semimetals against emission of photons. No
such effects have been detected so far111There is a related effect of emission
of photons by quasiparticles in Weyl semimetals predicted in [31]. This effect
is very tiny and exists already at the tree level.. Our present work gives a
theoretical explanation for the absence of instabilities in anomalous parts of
the effective action.

One is of course curious whether the stability statement is valid for the
whole anomalies, including also the terms independent of $A$. The answer is
negative, and to show this we calculate a couple of terms in the anomalies.

This paper is organized as follows. In the next section we establish
regularity of the $\eta$ function and derive variational properties of $\zeta$
function. Section 3 studies general form of the heat kernel coefficients
appearing in the anomalies and restrictions imposed on these coefficients by
various symmetry properties. Perturbation theory for the heat kernel is
derived in Section 4 and used for actual calculations in Section 5. Concluding
remarks can be found in the last section.

##  2 Main properties of the spectral functions

Since we are going to consider the spectral problem for $\not{D}^{2}$, which
is a second-order differential operator, we need a second boundary condition
in addition to (6). It reads

$\Pi_{-}(\theta)\not{D}\psi|_{\partial\mathcal{M}}=0.$ (24)

The heat kernel $K_{\not{D}^{2}}(x,y;t)$ of $\not{D}^{2}$ satisfies the heat
equation

$\bigl{(}\partial_{t}+\not{D}_{x}^{2}\bigr{)}K_{\not{D}^{2}}(x,y;t)=0,$ (25)

the initial condition

$K_{\not{D}^{2}}(x,y;0)=\delta(x,y)$ (26)

with $\delta(x,y)$ being the kernel of unit operator, and the boundary
conditions

$\Pi_{-}(\theta(x))K_{\not{D}^{2}}(x,y;t)=0,\qquad\Pi_{-}(\theta(x))\not{D}_{x}K_{\not{D}^{2}}(x,y;t)=0,$
(27)

when $x\in\partial\mathcal{M}$. If $\mathcal{M}$ is non-compact, the boundary
conditions (27) must be supplemented by suitable falloff conditions at
infinity.

Both $\not{D}$ and $\not{D}^{2}$ are elliptic differential (rather than
pseudo-differential) operators. The strong ellipticity of spectral problems
for $\not{D}$ and $\not{D}^{2}$ with chiral bag boundary conditions and a
constant $\theta$ was demonstrated in [17]. Since the proof is local with
respect to the points on the boundary, it is also valid for a varying $\theta$
without changes. Besides, the boundary conditions (6) and (24) are local.
Thus, according to general theory (see [30, Theorem 1.4.5]), the heat kernel
expansion is local and does not contain logarithmic term, which is exactly
assertion (a) of Theorem 1.1. Locality means that the coefficients $a_{n}$ and
$a_{n}^{\eta}$ can be expressed as bulk and boundary integrals of local
invariant polynomials depending on the fields entering $\not{D}$ and boundary
conditions. We will discuss the structure of heat kernel coefficients in more
detail in the next section. Our results for the $\zeta$ and $\eta$ functions
and for anomalies are formulated on a compact $\mathcal{M}$. Thus, the
assumption of $Q$ having a compact support is redundant. However, the local
integral formulae for heat kernel coefficients are valid in the non-compact
case as well. In particular, for the sake of convenience, we shall analyze the
relevant heat kernel coefficients on a non-compact manifold in Section 4.
There, the assumption regarding the support of $Q$ becomes essential.

There are the following integral representations for generalized $\zeta$ and
$\eta$ functions containing a smooth endomorphism $Q$

$\displaystyle\zeta(s,\not{D}^{2};Q)={\mathrm{Tr}\,}\bigl{(}Q(\not{D}^{2})^{-s}\bigr{)}=\frac{1}{\Gamma(s)}\int_{0}^{+\infty}dt\,t^{s-1}{\mathrm{Tr}\,}\bigl{(}Qe^{-t\not{D}^{2}}\bigr{)},$
(28)
$\displaystyle\eta(s,\not{D};Q)={\mathrm{Tr}\,}\bigl{(}Q\not{D}(\not{D}^{2})^{-\frac{s+1}{2}}\bigr{)}=\frac{1}{\Gamma\left(\frac{s+1}{2}\right)}\int_{0}^{+\infty}dt\,t^{\frac{s-1}{2}}{\mathrm{Tr}\,}\bigl{(}Q\not{D}e^{-t\not{D}^{2}}\bigr{)}.$
(29)

These integral representations together with asymptotic expansions (17) and
(18) define the pole structure of $\zeta(s,\not{D}^{2};Q)$ and
$\eta(s,\not{D};Q)$. They have only simple poles with

$\displaystyle
a_{k}(Q,\not{D}^{2})=\mbox{Res}_{s=\frac{4-k}{2}}\left(\Gamma(s)\zeta(s,\not{D}^{2};Q)\right),$
(30) $\displaystyle
a_{k}^{\eta}(Q,\not{D})=\frac{1}{2}\mbox{Res}_{s={4-k}}\left(\Gamma\left(\frac{s+1}{2}\right)\eta(s,\not{D};Q)\right),$
(31)

(see e.g. [19]).

To analyse chiral variation of the heat trace we use a (modified) technical
trick suggested by Gilkey and Kirsten [19] for boundary conditions with
$\theta=const$. To generalize this trick for arbitrary $\theta(x)$ one should
extend smoothly $\theta$ inside $\mathcal{M}$. Then, one defines a new
operator
$P=e^{-\frac{1}{2}\theta\gamma_{5}}\not{D}e^{\frac{1}{2}\theta\gamma_{5}}$.
The multiplication by $e^{\frac{1}{2}\theta\gamma_{5}}$ changes the boundary
conditions. Thus, for the operator $P$ the boundary condition is
$\Pi_{-}(0)\psi|_{\partial\mathcal{M}}=0$. This boundary condition does not
depend on $\theta$. The heat kernel for $P^{2}$ reads

$K_{P^{2}}(x,y;t)=e^{-\frac{1}{2}\theta(x)\gamma_{5}}K_{\not{D}^{2}}(x,y;t)e^{\frac{1}{2}\theta(y)\gamma_{5}}.$
(32)

One can easily check that this kernel satisfies equations (25), (26), and (27)
with $P$ instead of $\not{D}$ and $\theta=0$. For any smooth endomorphism $Q$,
which commutes with $e^{\frac{1}{2}\theta\gamma_{5}}$, we have

${\mathrm{Tr}\,}\bigl{(}Q\,e^{-tP^{2}}\bigr{)}={\mathrm{Tr}\,}\bigl{(}Q\,e^{-t\not{D}^{2}}\bigr{)}.$
(33)

One can write $P=e^{-\theta\gamma_{5}}\not{D}_{b\to
B=b+\frac{1}{2}\partial\theta}$. Under chiral transformations (8), the field
$B$ does not change. The boundary conditions also remain invariant. Thus, the
whole dependence on chiral phase resides in the prefactor
$e^{-\theta\gamma_{5}}$. Thus, the chiral variation of heat trace of
$\not{D}^{2}$ can be written as

$\displaystyle\delta_{\phi}\,{\mathrm{Tr}\,}\bigl{(}e^{-t\not{D}^{2}}\bigr{)}$
$\displaystyle=\delta_{\phi}\,{\mathrm{Tr}\,}\bigl{(}e^{-tP^{2}}\bigr{)}=-2t\
{\mathrm{Tr}\,}\bigl{(}\delta_{\phi}P\cdot P\cdot e^{-tP^{2}}\bigr{)}=-4t\
{\mathrm{Tr}\,}\bigl{(}(\delta\phi)\gamma_{5}P^{2}\cdot e^{-tP^{2}}\bigr{)}$
$\displaystyle=4t\partial_{t}{\mathrm{Tr}\,}\bigl{(}(\delta\phi)\gamma_{5}e^{-tP^{2}}\bigr{)}=4t\partial_{t}{\mathrm{Tr}\,}\bigl{(}(\delta\phi)\gamma_{5}e^{-t\not{D}^{2}}\bigr{)}.$
(34)

This equation determines chiral variations of main spectral functions.

In the case of constant $\theta$, one can prove [19] that some heat kernel
coefficients do not depend on $\theta$. With a varying $\theta$, there are
similar statements meaning chiral invariance of corresponding quantities. Let
us expand the first and the last terms in the chain of equations (34) in
asymptotic series and equate the terms with identical powers of $t$. This
yields

$\delta_{\phi}\,a_{n}(1,\not{D}^{2})=2(n-4)a_{n}((\delta\phi)\gamma_{5},\not{D}^{2}).$

By setting $n=4$ we obtain

$\delta_{\phi}\,a_{4}(1,\not{D}^{2})=0.$ (35)

The coefficient $a_{4}(1,\not{D}^{2})$ is usually related to the global scale
anomaly. Thus, this anomaly is invariant under local chiral transformations as
in the case of manifold without boundary. By using (30) we get
$a_{4}(1,\not{D}^{2})=\zeta(0,\not{D}^{2})$. Thus, formula (35) yields
assertion (c) of Theorem 1.1.

By similar arguments, we have

$\delta_{\phi}\,{\mathrm{Tr}\,}\bigl{(}\not{D}e^{-t\not{D}^{2}}\bigr{)}=2(1+2t\partial_{t})\,{\mathrm{Tr}\,}\bigl{(}(\delta\phi)\gamma_{5}\not{D}e^{-t\not{D}^{2}}\bigr{)}.$
(36)

An expansion of this equation in asymptotic series yields

$\delta_{\phi}\,a_{n}^{\eta}(1,\not{D})=2(n-4)a_{n}^{\eta}((\delta\phi)\gamma_{5},\not{D}).$

For $n=4$ one gets the following chiral invariance condition

$\delta_{\phi}\,a_{4}^{\eta}(1,\not{D})=0.$ (37)

One can connect by a smooth chiral transformation the operator $\not{D}$ with
boundary conditions (6) to a chirally transformed operator $\not{D}_{\phi}$
with boundary conditions with a constant or even vanishing $\theta$. The
coefficient $a_{4}^{\eta}$ does not change. For the transformed operator this
$a_{4}^{\eta}(1,\not{D}_{\phi})=0$, as was demonstrated by Gilkey and Kirsten
[19]. Therefore, $a_{4}^{\eta}(1,\not{D})=0$ also for arbitrary non-constant
$\theta$. By setting $k=4$ in (31) we obtain

$\mbox{Res}_{s=0}\eta(s,\not{D})=2\pi^{-1/2}a_{4}^{\eta}(1,\not{D}_{\phi})=0.$
(38)

Thus, the eta function $\eta(s,\not{D})$ is regular at $s=0$. This completes
the demonstration of Theorem 1.1(b).

The methods for computing variations of spectral functions can be found in
[32] and [33]. To compute chiral variation of $\zeta(s,\not{D})$ we substitute
(34) in the integral representation (28), integrate by parts, assuming that
$s$ is sufficiently large, and use (28) again to get the identity

$\delta_{\phi}\,\zeta(s,\not{D}^{2})=-4s\zeta(s,\not{D}^{2};(\delta\phi)\gamma_{5}).$
(39)

Since $\zeta(s,\not{D}^{2};(\delta\phi)\gamma_{5})$ is regular at $s=0$, we
obtain

$\delta_{\phi}\,\zeta^{\prime}(0,\not{D}^{2})=-4\zeta(0,\not{D}^{2};(\delta\phi)\gamma_{5})=-4a_{4}((\delta\phi)\gamma_{5},\not{D}^{2}).$
(40)

This yields assertion (d) of Theorem 1.1.

The last assertion (e) of Theorem 1.1 is obtained by using the general methods
[33, 32]. There is nothing specific for our case in this respect. We remark
that the formula (20) is valid only if the number of zero modes does not
change due to the variation. This is a part of our general assumption that
$\not{D}$ does not have any zero modes in the relevant range of parameters.

##  3 General form of the anomalies and the Wess–Zumino conditions

Scaling dimension is an important notion which allows to restrict possible
form of the heat kernel coefficients. Let us take a formal parameter $\Lambda$
and rescale the proper time $t\to\Lambda^{-2}t$. We say, that $t$ has a
canonical dimension $-2$. Canonical dimensions will be denoted by square
brackets, $[t]\equiv-2$. To keep the expression $e^{-t\not{D}^{2}}$ scaling
invariant, we rescale $\not{D}\to\Lambda\not{D}$, so that $[\not{D}]=1$. This
behaviour of the Dirac operator may be achieved by assigning $[b]=[A]=1$ and
$[x^{\mu}]=-[\partial_{\mu}]=-1$. The boundary conditions remain invariant if
$[\theta]=0$. The scaling dimensions defined above coincide with canonical
mass dimensions of physical fields. The scaling dimensions of heat kernel
coefficients can be obtained by comparing the dimensions on both sides of
equation (17). Since $Q$ appears linearly, one can assign to it any dimension.
Thus, we obtain

$[a_{n}(Q,\not{D}^{2})]=n-4+[Q].$ (41)

The heat kernel coefficients are given by sums of bulk and boundary integrals.
Since $[\mathrm{d}^{k}x]=-k$, the canonical dimension of bulk integrands is
$n+[Q]$, while boundary integrands have dimension $n-1+[Q]$.

###  3.1 Contributions to the chiral anomaly containing electromagnetic
potential

Apart from having correct scaling behaviour, the heat kernel coefficient
$a_{4}(\gamma_{5}\delta\phi,\not{D}^{2})$ has to be gauge invariant. The most
general form of this coefficient reads

$\displaystyle a_{4}(\gamma_{5}\delta\phi,\not{D}^{2})=$
$\displaystyle-\frac{1}{32\pi^{2}}\int_{\mathcal{M}}\mathrm{d}^{4}x\sqrt{g}(\delta\phi)\epsilon^{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}$
$\displaystyle+\frac{1}{16\pi^{2}}\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\sqrt{h}(\delta\phi)(b_{i}F_{jk}\epsilon^{ijk}f_{1}(\theta)+b^{i}F_{\mathbf{n}i}f_{2}(\theta)$
$\displaystyle\qquad\qquad\,\,\,+F_{ij}f_{3}(\theta)_{:k}\epsilon^{ijk}+F^{\mathbf{n}i}f_{4}(\theta)_{:i}+f_{5}(\theta)F^{\mathbf{n}i}_{\
\ :i}\bigr{)}.$ (42)

Since $[\theta]=0$, the field $\theta$ can enter with any power, so that the
right hand side of (42) is defined up to five arbitrary functions
$f_{a}(\theta)$. All these functions vanish at $\theta=0$, see [27]. The
volume term is the celebrated Adler–Bell–Jackiw term. Due to the locality of
the heat kernel expansion the volume term does not depend on $\theta$.

Chiral transformations (8) form an abelian Lie algebra,

$\delta_{\phi_{1}}\circ\delta_{\phi_{2}}-\delta_{\phi_{2}}\circ\delta_{\phi_{1}}=0.$
(43)

By applying this relation to $\zeta^{\prime}(0,\not{D}^{2})$ and using (15)
and (19) we obtain the Wess–Zumino consistency condition

$\delta_{\phi_{1}}a_{4}(\delta\phi_{2}\gamma_{5},\not{D}^{2})-\delta_{\phi_{2}}a_{4}(\delta\phi_{1}\gamma_{5},\not{D}^{2})=0.$
(44)

This condition can be easily solved yielding

$f_{1}(\theta)-2f^{\prime}_{3}(\theta)=0,\qquad
f_{2}(\theta)-2f^{\prime}_{4}(\theta)=0.$ (45)

Thus, it remains to compute only three of five functions $f_{a}(\theta)$.

###  3.2 Parity anomaly

Here we study $A$-dependent terms in the parity anomaly. Let us take a
variation $\delta A_{\mu}$ of the gauge field $A$. Then,
$\delta\not{D}=-\gamma^{\mu}\delta A_{\mu}$. Since $\eta(0,\not{D})$ is gauge
invariant, the variation can be represented in the form

$\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\sqrt{h}\left[(\delta
A_{j})J^{j}+(\delta F_{\mathbf{n}j})K^{j}\right],$ (46)

where $J^{j}$ is a conserved current, $J^{j}_{\ \ :j}=0$, while $K^{j}$ is
gauge invariant. Therefore, we get

$\displaystyle
a_{3}(\delta\not{D},\not{D}^{2})=\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\sqrt{h}\Big{(}(\delta
A_{i})$ $\displaystyle\epsilon^{ijk}(A_{k}G_{0}(\theta))_{:j}+(\delta
A_{i})\epsilon^{ijk}(b_{k}G_{1}(\theta))_{:j}$
$\displaystyle+b^{i}\,\big{(}\delta F_{ni}\big{)}\,G_{2}(\theta)+\big{(}\delta
F^{\mathbf{n}i}\big{)}\,G_{3}(\theta)\,\theta_{:i}\Big{)}.$ (47)

The functions $G_{a}(\theta)$, $a=0,1,2,3$, have to be determined.

Second variation of any functional (local or nonlocal) has to be symmetric.
Thus, we have for two consequent variations $\delta A^{(1)}$ and $\delta
A^{(2)}$ of the gauge field

$0=\bigl{(}\delta_{A^{(2)}}\circ\delta_{A^{(1)}}-\delta_{A^{(1)}}\circ\delta_{A^{(2)}}\bigr{)}\eta(0,\not{D})=-\frac{2}{\sqrt{\pi}}\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\,\sqrt{h}\epsilon^{ijk}(\delta
A^{(1)}_{i})(\delta A^{(2)}_{k})\theta_{:j}G^{\prime}_{0}(\theta).$

Since this condition has to hold for arbitrary variations and an arbitrary
$\theta(x)$, we conclude that $G^{\prime}_{0}=0$. The value of $G_{0}$ at
$\theta=0$ has been computed in [28]. Hence, we have

$G_{0}(\theta)=\frac{\varepsilon_{\alpha}}{8\pi^{3/2}}.$ (48)

The integration of the term with $G_{0}$ in variation (47) yields second
assertion of Theorem 1.2.

In Section 5.1 it will be shown that the functions $G_{1}$, $G_{2}$, and
$G_{3}$ vanish.

#### Contributions to the anomalies not involving $A_{\mu}$.

Let us briefly describe the contributions to anomalies which have not been
included in the previous discussion. Since these terms do not contain $A$,
they are not restricted by the gauge invariance. Thus, many structures are
allowed. For example, the chiral anomaly depends on more that 20 independent
functions of $\theta$. However, about a half of these function can be
expressed through the rest by means of the Wess–Zumino conditions. All terms
in both anomalies can be calculated at least in principle with the
perturbation expansion from the next section, though at higher orders the
calculations become very time consuming. We present explicitly just a single
boundary term in each the heat kernel coefficients corresponding to the
anomalies

$\displaystyle
a_{4}(\gamma_{5}\,\delta\phi\,\not{D}^{2})=\int_{\partial\mathcal{M}}\mathrm{d}^{3}x\sqrt{h}\,f_{6}(\theta)\,\delta\phi_{;\mathbf{n}\mathbf{n}\mathbf{n}}+\cdots,$
(49) $\displaystyle a_{3}(\mathrm{i}\gamma^{\mu}\gamma^{5}(\delta
b_{\mu}),\not{D}^{2})=\int_{\partial\mathcal{M}}\mathrm{d}^{3}z\,\sqrt{h}\,G_{4}(\theta)\,(\delta
b_{\mathbf{n}})_{;\mathbf{n}\mathbf{n}}+\cdots.$ (50)

The volume terms in $a_{4}$ are fixed as in [27]. $f_{6}$ and $G_{4}$ will be
computed in Section 5.2.

##  4 Perturbation theory for the heat kernel

As we have seen above, the relevant heat kernel coefficients are defined up to
several unknown local functions of the boundary chiral angle $\theta$. To
compute these functions, it is enough to consider the simplest one-boundary
geometry $\mathbb{R}^{3}\times\mathbb{R}_{+}$. In this case we can introduce
the following decomposition
$x=(x^{\|},x^{\mathbf{n}})\in\mathbb{R}^{3}\times\mathbb{R}_{+}$, where
$x^{\|}\in\mathbb{R}^{3}$ and $x^{\mathbf{n}}\in\mathbb{R}_{+}$.

The transformation $\psi\to\gamma_{5}\psi$ maps $\not{D}\to-\not{D}$ and
inverts the signs $\varepsilon_{\alpha}\to-\varepsilon_{\alpha}$ in the
boundary conditions. Since $\zeta(s,\not{D}^{2})$ is invariant under the
reflection $\not{D}\to-\not{D}$, it also has to be invariant under
$\varepsilon_{\alpha}\to-\varepsilon_{\alpha}$. Similarly, $\eta(s,\not{D})$
has to change sign under $\varepsilon_{\alpha}\to-\varepsilon_{\alpha}$. Thus,
in the case with a single connected boundary component it is sufficient to
consider a selected value of $\varepsilon$ (we take $\varepsilon=1$). Complete
dependence of spectral functions on this parameter will be restored below by
using the mentioned property of the spectral functions.

We will use perturbation theory expansions in $A$, $b$, and derivatives of
$\theta$. The unperturbed heat kernel $K_{0}(x,y;t)$ will correspond to $A=0$,
$b=0$, and a constant $\theta$. Also, let us introduce

$\xi=x^{\mathbf{n}}-y^{\mathbf{n}},\quad\eta=x^{\mathbf{n}}+y^{\mathbf{n}},\quad\sigma=\big{|}x^{\|}-y^{\|}\big{|},$
(51)

and

$c=\cosh{(\theta)},\quad s=\sinh{(\theta)},\quad\tau=\tanh{(\theta)}.$ (52)

A useful expression for the heat kernel $K_{0}$ on a semi-infinite cylinder
$\mathcal{N}\times\mathbb{R}_{+}$ with a compact $\mathcal{N}$ was derived in
[17]. We take $\mathcal{N}=T^{3}$. In the limit of infinite radii, one obtains
the following expression for $K_{0}(x,y;t)$ on
$\mathbb{R}^{3}\times\mathbb{R}_{+}$

${K}_{0}(x,y;t)=\mathcal{K}(x,y,t)\,\mathrm{Id}-\frac{1}{\tau}\,\partial_{\mathbf{n}[x]}\mathcal{B}(x,y,t)\,\mathcal{P}_{+}+\partial_{j[x]}\mathcal{B}(x,y,t)\,\gamma_{5}\gamma^{\mathbf{n}}\gamma^{j}\mathcal{P}_{+},$
(53)

with

$\mathcal{K}(x,y,t)=\frac{1}{(4\pi
t)^{2}}\Big{(}e^{-\frac{\xi^{2}+\sigma^{2}}{4t}}-e^{-\frac{\eta^{2}+\sigma^{2}}{4t}}\Big{)},$
(54)

and

$\mathcal{B}(x,y,t)=-\frac{\tau{c}^{2}e^{-\frac{\eta^{2}+\sigma^{2}}{4t}}}{4\pi^{2}t}\,\left(-\frac{1}{\eta}\,U\left(\frac{\sigma\tau}{\eta},\frac{t}{\eta^{2}c^{2}}\right)\,+\left(\frac{\tau\,}{\sigma}\right)V\left(\frac{\sigma\tau}{\eta},\frac{t}{\eta^{2}c^{2}}\right)\right),$
(55)

where $V(x,t)$ and $U(x,t)$ stand for Voigt profiles defined as

$U(u,t)=\frac{1}{\sqrt{4\pi
t}}\int_{-\infty}^{+\infty}\mathrm{d}y\,\,\frac{e^{-\frac{(u-y)^{2}}{4t}}}{y^{2}+1},\quad
V(u,t)=\frac{1}{\sqrt{4\pi
t}}\int_{-\infty}^{+\infty}\mathrm{d}y\,\,\frac{y\,e^{-\frac{(u-y)^{2}}{4t}}}{y^{2}+1}.$
(56)

The Hermitian projectors are defined as

$\mathcal{P}_{+}:=\frac{\Pi_{+}^{\phantom{\dagger}}\Pi_{+}^{\dagger}}{\cosh^{2}{(\theta)}},\quad\mathcal{P}_{-}:=\frac{\Pi_{-}^{\dagger}\Pi_{-}^{\phantom{\dagger}}}{\cosh^{2}{(\theta)}},$

where
$\Pi_{+}:=\tfrac{1}{2}\bigl{(}1+\mathrm{i}\gamma_{5}e^{\gamma_{5}\theta}\gamma^{\mathbf{n}}\bigr{)}$.

It can be also checked directly that the kernel (53) satisfies the heat
equation, the initial condition, and the boundary conditions.

###  4.1 Variation of $\theta$

To make the formulae easier to read we introduce a notation for the right
action of $\not{D}$ on a matrix-valued kernel $\kappa(x,z)$,

$\kappa(x,z)^{\dagger}\overset{\leftarrow}{\not{D}}(z):=(\not{D}(z)\kappa(x,z))^{\dagger},$
(57)

where $\dagger$ denotes Hermitian conjugation of the matrices.

Let $K_{\not{D}^{2},\theta}(x,y;t)$ and
$K_{\not{D}^{2},\theta+\Delta\theta}(x,y;t)$ be the heat kernels for the
operator $\not{D}^{2}$ with the boundary conditions characterized by chiral
angles $\theta$ and $\theta+\Delta\theta$, respectively. Since the symbol of
the operator does not change, we do not include $\not{D}^{2}$ in the notations
of the heat kernel.

We start with a sequence of obvious identities

$\displaystyle
K_{\theta+\Delta\theta}(x,y;t)=K_{\theta}(x,y;t)+\int_{\mathcal{M}}\mathrm{d}^{4}z\,\sqrt{g}\,\left(\delta(x,z)K_{\theta+\Delta\theta}(z,y;t)-K_{\theta}(x,z;t)\delta(z,y)\right)$
$\displaystyle\phantom{K_{\theta+\Delta\theta}(x,y;t)}=K_{\theta}(x,y;t)+\int_{\mathcal{M}}\mathrm{d}^{4}z\,\sqrt{g}\,\left(K_{\theta}(x,z;t-w)K_{\theta+\Delta\theta}(z,y;w)\right)\big{|}_{w=0}^{w=t}$
$\displaystyle\phantom{K_{\theta+\Delta\theta}(x,y;t)}=K_{\theta}(x,y;t)+\int_{0}^{t}\mathrm{d}w\int_{\mathcal{M}}\mathrm{d}^{4}z\,\sqrt{g}\,\left(-\left(\partial_{t}K_{\theta}(x,z;t-w)\right)K_{\theta+\Delta\theta}(z,y;w)\right.$
$\displaystyle\phantom{K_{\theta+\Delta\theta}(x,y;t)}\qquad\qquad\qquad\qquad\qquad\qquad\left.+K_{\theta}(x,z;t-w)\partial_{w}K_{\theta+\Delta\theta}(z,y;w)\right)$
$\displaystyle\phantom{K_{\theta+\Delta\theta}(x,y;t)}=K_{\theta}(x,y;t)+\int_{0}^{t}\mathrm{d}w\int_{\mathcal{M}}\mathrm{d}^{4}z\,\sqrt{g}\,\left(\not{D}^{2}(x)K_{\theta}(x,z;t-w)K_{\theta+\Delta\theta}(z,y;w)\right.$
$\displaystyle\phantom{K_{\theta+\Delta\theta}(x,y;t)}\qquad\qquad\qquad\qquad\qquad\qquad\left.-K_{\theta}(x,z;t-w)\not{D}^{2}(z)K_{\theta+\Delta\theta}(z,y;w)\right).$
(58)

The integration by parts yields

$\displaystyle
K_{\theta+\Delta\theta}(x,y;t)=K_{\theta}(x,y;t)+\int_{0}^{t}\mathrm{d}w\int_{\partial\mathcal{M}}\mathrm{d}^{3}z\,\sqrt{h}\Big{(}K_{\theta}(x,z;t-w)\overset{\leftarrow}{\not{D}}(z)\mathrm{i}\gamma^{\mathbf{n}}\,K_{\theta+\Delta\theta}(z,y;w)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+K_{\theta}(x,z;t-w)\mathrm{i}\gamma^{\mathbf{n}}{\not{D}}(z)K_{\theta+\Delta\theta}(z,y;w)\Big{)}.$
(59)

Since in the last line $z\in\partial\mathcal{M}$, one can use the boundary
conditions and rewrite factors in the form

$\displaystyle
K_{\theta}(x,z;t-w)\overset{\leftarrow}{\not{D}}(z)=K_{\theta}(x,z;t-w)\overset{\leftarrow}{\not{D}}(z)\Pi_{+}^{\dagger}(\theta),$
$\displaystyle
K_{\theta}(x,z;t-w)=K_{\theta}(x,z;t-w)\Pi_{+}^{\dagger}(\theta(z)),$
$\displaystyle{\not{D}}(z)K_{\theta+\Delta\theta}(z,y;w)=\Pi_{+}(\theta(z)+\Delta\theta(z)){\not{D}}(z)K_{\theta+\Delta\theta}(z,y;w),$
$\displaystyle
K_{\theta+\Delta\theta}(z,y;w)=\Pi_{+}(\theta(z)+\Delta\theta(z))K_{\theta+\Delta\theta}(z,y;w),$
(60)

together with the identity

$\Pi_{+}^{\dagger}(\theta)\gamma^{\mathbf{n}}\Pi_{+}(\theta+\Delta\theta)=\Pi_{+}^{\dagger}(\theta)\gamma^{\mathbf{n}}(\Delta\Pi_{+})\Pi_{+}(\theta+\Delta\theta),$
(61)

where $\Delta\Pi_{+}:=\Pi_{+}(\theta+\Delta\theta)-\Pi_{+}(\theta)$.

$K_{\theta+\Delta\theta}(x,y;t)=K_{\theta}(x,y;t)+\int_{0}^{t}\mathrm{d}w\int_{\partial\mathcal{M}}\mathrm{d}^{3}z\,\sqrt{h}\,\,K_{\theta}(x,z,t-w)\,\mathcal{P}(z)\,K_{\theta+\Delta\theta}(z,y;w),$
(62)

with

$\mathcal{P}(z):=\overset{\leftarrow}{\not{D}}(z)\mathrm{i}\gamma^{\mathbf{n}}(\Delta\Pi_{+})+\mathrm{i}\gamma^{\mathbf{n}}(\Delta\Pi_{+}){\not{D}}(z).$
(63)

By taking an infinitesimal $\Delta\theta=\delta\theta$ and linearizing the
equation above, we arrive at the following equation for the variation of the
heat kernel

$\displaystyle\delta_{\theta}K_{\not{D}^{2},\theta}(x,y;t)=\frac{1}{2}\int_{0}^{t}\mathrm{d}w\int_{\partial\mathcal{M}}\mathrm{d}^{3}z\,\sqrt{h}(z)\,K_{\not{D}^{2},\theta}(x,z;t-w)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\times\Big{(}\overset{\leftarrow}{\not{D}}(z)\,\mathrm{i}\gamma^{n}\gamma_{5}\,\delta\theta(z)+\mathrm{i}\gamma^{n}\gamma_{5}\,\delta\theta(z)\,{\not{D}}(z)\Big{)}K_{\not{D}^{2},\theta}(z,y;w).$
(64)

###  4.2 Variations of $A$ and $b$

In this subsection we give smooth localized perturbation to $A$ and $b$. The
Dirac operator changes $\not{D}\to\not{D}+\Delta\not{D}$. We assume that
$\theta$ remains unperturbed and introduce the corresponding heat kernels by
$K_{\not{D}^{2}}(x,y;t)$ and $K_{(\not{D}+\Delta\not{D})^{2}}(x,y;t)$,
respectively.

By acting as in the previous subsection, we write

$\displaystyle
K_{(\not{D}+\Delta\not{D})^{2}}(x,y;t)=K_{\not{D}^{2}}(x,y;t)+\int_{0}^{t}$
$\displaystyle\mathrm{d}w\int_{\mathcal{M}}\mathrm{d}^{4}z\,\sqrt{g}\left(\not{D}^{2}(x)K_{\not{D}^{2}}(x,z;t-w)K_{(\not{D}+\Delta\not{D})^{2}}(z,y,w)\right.$
$\displaystyle\left.-K_{\not{D}^{2}}(x,z;t-w)\big{(}\not{D}(z)+\Delta{\not{D}}(z)\big{)}^{2}K_{(\not{D}+\Delta\not{D})^{2}}(z,y;w)\right).$

After integrating by parts twice and using the boundary conditions we arrive
at a Dyson-type equation

$K_{(\not{D}+\Delta\not{D})^{2}}(x,y;t)=K_{\not{D}^{2}}(x,y;t)+\int_{0}^{t}\mathrm{d}w\int_{\mathcal{M}}\mathrm{d}^{4}z\,\sqrt{g}\,\,K_{\not{D}^{2}}(x,z;t-w)\,P(z)\,K_{(\not{D}+\Delta\not{D})^{2}}(z,y;w),$
(65)

where

$P(z)=-\Big{(}\overset{\leftarrow}{\not{D}}(z)\,\Delta\not{D}(z)+\Delta\not{D}(z)\,\not{D}(z)\Big{)}-\big{(}\Delta\not{D}(z)\big{)}^{2}.$
(66)

The variations of $\theta$, $A$, and $b$ can be repeated as many times as
needed. At the end of the day, one arrives at an expansion which can be
represented symbolically as
$K=K_{0}+K_{0}\hat{P}K_{0}+K_{0}\hat{P}K_{0}\hat{P}K_{0}+\dots$, where
$\hat{P}$ is either $P$ or $\mathcal{P}$. Repeated integrations over the
proper time parameter $t$ and over either $\mathcal{M}$ or
$\partial\mathcal{M}$ are understood. The orders of this expansion correspond
to powers of $A$, $b$ or derivatives of $\theta$. Higher order derivatives of
these fields are taken into account through an expansion of the kernel
$K_{0}$, as one will see below. Each coefficient in the heat kernel expansion
requires a finite order of the perturbation series.

##  5 Calculation of the anomalies

###  5.1 Proof of the partial stability

To prove Theorem 1.2 it remains to show that the functions $f_{1}$, $f_{2}$,
$f_{5}$, $G_{1}$, $G_{2}$, and $G_{3}$ vanish. All these functions are
associated with the structures which are linear in the field $A_{\mu}$. There
is a general (though yet non-rigorous) argument which shows that such
structures cannot appear in the heat kernel expansion. Indeed, let us
introduce an antihermitian connection $\mathbf{A}=\mathrm{i}A$ and
antihermitian gamma matrices $\tilde{\gamma}^{\mu}=\mathrm{i}\gamma^{\mu}$.
Then the imaginary unit $\mathrm{i}$ disappears from the Dirac operator and
from the boundary conditions. It does not reappear in the trace rules for
$\tilde{\gamma}$. Since the heat expansion is an expansion over the symbols of
$\not{D}$ and of the boundary operator, the imaginary unit cannot enter the
coefficients. If, after computing the traces over spinor indices we return to
the original field $A$, we observe that all expressions which are linear in
$A$ acquire pure imaginary coefficients. Thus, they have pure imaginary values
on the configurations, which we consider here. Since the heat kernel expansion
of a self-adjoint operator has to be real, we conclude that the coefficient in
front of such expressions vanish.

It appeared quite hard to give precise mathematical meaning to the
construction presented above on arbitrary manifold. However, on
$\mathcal{M}=\mathbb{R}^{3}\times\mathbb{R}_{+}$ the redefinitions described
in the previous paragraph completely remove the imaginary unit from the
unperturbed $K_{0}$ and from the operators $P$ and $\mathcal{P}$. Hence, the
imaginary unit cannot appear at any order of the perturbations series. Since
to define the unknown function in heat kernel coefficients it is sufficient to
perform calculations on this simple background, we conclude that linear terms
in $A$ are not allowed in the anomalies. This completes the proof of Theorem
1.2.

To illustrate the proof, we outline the main steps of perturbative
calculations.

We start with the parity anomaly. To evaluate the functions $G_{1}(\theta)$
and $G_{2}(\theta)$ we need to compute

$\mathrm{Tr}\,\Big{[}(\delta\not{D})e^{-t\not{D}^{2}}\Big{]}$ (67)

on $\mathcal{M}=\mathbb{R}^{3}\times\mathbb{R}_{+}$, where
$\delta\not{D}=-\gamma^{\mu}A_{\mu}$, while $e^{-t\not{D}^{2}}$ has to be
expanded to the linear order in $b$ according to the equation (65). The
relevant expression reads

$\displaystyle\int_{\mathcal{M}}\mathrm{d}^{4}x\,\big{(}\delta
A_{\mu}(x)\big{)}\int_{0}^{t}\mathrm{d}w\int_{\mathcal{M}}\mathrm{d}^{4}z\,\mathrm{tr}\,\Big{[}\gamma^{\mu}K_{0}(x,z;t-w)\,\Big{(}\overset{\leftarrow}{\not{D}}_{0}(z)\,(-\mathrm{i}\gamma_{5}\,\gamma^{\nu}b_{\nu}(z))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+(-\mathrm{i}\gamma_{5}\,\gamma^{\nu}b_{\nu}(z))\,\not{D}_{0}(z)\Big{)}\,K_{0}(z,x;w)\Big{]}.$

By counting the $\gamma$ matrices, one immediately sees that the expression
above is pure imaginary and thus has to vanish. We conclude that

$G_{1}(\theta)=G_{2}(\theta)=0.$ (68)

The expression with $G_{3}$ in (47) depends on the derivative of $\theta$.
Thus, we need to take $\delta\not{D}=-\gamma^{\mu}A_{\mu}$ as above, while
$e^{-t\not{D}^{2}}$ has to be expanded to the linear order in $\delta\theta$
by using (64). One arrives at

$\displaystyle-\frac{1}{2}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\delta
A_{\mu}(x)\int_{0}^{t}\mathrm{d}w\int_{\partial\mathcal{M}}\mathrm{d}^{3}z\,\delta\theta(z)\,\mathrm{tr}\,\big{[}\gamma^{\mu}\,K_{0}(x,z;t-w)\Big{(}\overset{\leftarrow}{\not{D}}(z)\,\mathrm{i}\gamma^{\mathbf{n}}\gamma_{5}\,$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\mathrm{i}\gamma^{\mathbf{n}}\gamma_{5}\,{\not{D}}(z)\Big{)}K_{0}(z,x;w)\big{]}.$

This expression is pure imaginary, therefore, we conclude that

$G_{3}(\theta)=0.$ (69)

Now, let us consider the chiral anomaly. The term with $f_{5}$ requires the
first order of the perturbation series for the heat kernel. Thus, the
calculation goes exactly as above. The terms with $f_{1}$ and $f_{2}$ contain
both $A$ and $b$ and thus require the second order of perturbation series in
$\Delta\not{D}$:

$\displaystyle\int_{0}^{t}\mathrm{d}w\int_{0}^{w}\mathrm{d}q\int_{\mathcal{M}}\mathrm{d}^{4}z_{1}\int_{\mathcal{M}}\mathrm{d}^{4}z_{2}\int_{\mathcal{M}}\mathrm{d}^{4}x\,\mathrm{tr}\,\big{[}\gamma_{5}\,\delta\phi(x)\,K_{0}(x,z_{1};t-w)$
$\displaystyle\qquad\qquad\times\Big{(}\overset{\leftarrow}{\not{D}}_{0}(z_{1})\,\Delta\not{D}(z_{1})+\Delta\not{D}(z_{1})\,\not{D}_{0}(z_{1})\Big{)}K_{0}(z_{1},z_{2},w-q)$
$\displaystyle\qquad\qquad\times\Big{(}\overset{\leftarrow}{\not{D}}_{0}(z_{2})\,\Delta\not{D}(z_{2})+\Delta\not{D}(z_{2})\,\not{D}_{0}(z_{2})\Big{)}\,K_{0}(z_{2},x;q)\big{]}.$

One can easily check that all cross terms in the above expression, containing
both $A$ and $b$, are pure imaginary after taking the trace over spinor
indices. Therefore, they must vanish after summing up all individual
contributions.

###  5.2 More terms in the anomalies

Here, we compute the functions $f_{6}(\theta)$ and $G_{4}(\theta)$ defined in
equations (49) and (50), respectively. The term (49) does not depend on $A$
and $b$ and does not contain derivatives of $\theta$. Therefore, it can be
calculated by using the zero order term

$\mathrm{Tr}\,\Big{(}\gamma_{5}\,\delta\phi\,e^{-t\not{D}_{0}^{2}}\Big{)}=\int_{\mathcal{M}}\mathrm{d}^{4}x\,\mathrm{tr}\,\big{(}\gamma_{5}\,\delta\phi(x)\,K_{0}(x,x,t)\big{)}$
(70)

in power series of $t$. The only non-vanishing trace is
$\mathrm{tr}\,\big{(}\gamma_{5}\,\mathcal{P}_{+}\big{)}=2\tau$. At coinciding
arguments $x=y$ the function $\mathcal{B}(x,y,t)$ does not depend on
tangential coordinates. Let us introduce the notation
$\mathcal{C}(x^{\mathbf{n}},t):=\mathcal{B}(x,x,t)$. The expression (70) reads

$\int_{\mathbb{R}^{3}}\mathrm{d}^{3}x^{\|}\,\delta\phi(x^{\|},0)\,\mathcal{C}(x^{\mathbf{n}},t)+\int_{\mathbb{R}^{3}\times{\mathbb{R}_{+}}}\mathrm{d}^{4}x\,\partial_{\mathbf{n}}\delta\phi(x^{\|},x^{\mathbf{n}})\,\mathcal{C}(x^{\mathbf{n}},t),$
(71)

where we have integrated by parts over $x^{\mathbf{n}}$. The function
$\mathcal{C}(x^{\mathbf{n}},t)$ decays exponentially fast at
$x^{\mathbf{n}}\to\infty$ for any positive $t$. To construct an asymptotic
expansion at $t\to+0$ according to the conventional wisdom we introduce a
rescaled coordinate $u$ as $x^{\mathbf{n}}=u\cdot\sqrt{t}$. Then,
${\mathcal{C}(x^{\mathbf{n}},t)=t^{-3/2}\mathcal{C}(u,1)}$. The integrals in
(71) become

$\frac{1}{t^{\frac{3}{2}}}\int_{\mathbb{R}^{3}}\mathrm{d}^{3}x^{\|}\,\delta\phi(x^{\|},0)\,\mathcal{C}(0,1)+\frac{1}{t}\int_{\mathbb{R}^{3}}\mathrm{d}^{3}x^{\|}\int_{0}^{+\infty}\mathrm{d}u\,\partial_{\mathbf{n}}\delta\phi\bigl{(}x^{\|},ut^{\frac{1}{2}}\bigr{)}\mathcal{C}(u,1).$
(72)

We expand $\delta\phi$ in a Taylor series around $x^{\mathbf{n}}=0$. This
yields

$a_{k}(\gamma_{5}\delta\phi,\not{D}_{0}^{2})=c_{k}(\theta)\int_{\mathbb{R}^{3}}\mathrm{d}^{3}x^{\|}\,\partial_{\mathbf{n}}^{k-1}\delta\phi(x^{\|},0),$
(73)

where for $k\geq 2$ we have

$c_{k}(\theta)=\frac{1}{(k-2)!}\int_{0}^{+\infty}\mathrm{d}u\,u^{k-2}\mathcal{C}(u,1).$
(74)

The integration can be performed with the use of Mathematica. We conclude that

$f_{6}(\theta)=c_{4}(\theta)=\frac{1}{16\pi^{2}}\,\coth{(\theta)}\,(\theta\,\coth{(\theta)}-1).$
(75)

The calculation of $G_{4}$ goes exactly the same way. One obtains

$G_{4}(\theta)=-\frac{\varepsilon_{\alpha}c_{3}(\theta)}{\sinh{(\theta)}}=-\frac{\varepsilon_{\alpha}}{16\pi^{\frac{3}{2}}}\,\frac{\cosh{(\theta)}}{1+\cosh{(\theta)}}.$
(76)

Here we restored the dependence of parity anomaly on $\varepsilon_{\alpha}$
according to the rule formulated at the beginning of Section 4.

We see that in general both anomalies contain terms depending on $\theta$.

##  6 Conclusions

Being motivated by applications to Weyl semimetals we studied the heat kernel
expansion, the $\eta$ and $\zeta$ functions, and anomalies for Dirac operators
containing an abelian gauge field and an axial vector field subject to
boundary conditions with arbitrary chiral phase $\theta$. We established main
properties of the spectral functions (like the regularity of $\eta(s,\not{D})$
at $s=0$, for example) which justify the use of $\zeta$ regularization and
usual formulae for the anomalies. We have analysed the restrictions imposed by
symmetries on the heat kernel coefficients. A Dyson-type perturbation theory
for the heat kernel has been derived and used for calculation of the
anomalies. We computed all anomalous contributions containing electromagnetic
field.

The method of pertubative expansion of the heat kernel is about the same
important as our other results. By using this method, one can compute for
example the boundary terms in conformal anomaly which has a lot of
applications. We mention the boundary central charges in Conformal Field
Theories [34, 35, 36] anomaly induced boundary currents [37].

We are planning to study physical effects following from our calculations.
There is, however, a problem to be resolved before doing this. One has to
understand the Wick rotation to Minkowski signature, which is quite a non-
trivial task in the presence of chiral phase $\theta$.

#### Acknowledgements.

The work of D.V.V. was supported in parts by the São Paulo Research Foundation
(FAPESP), project 2016/03319-6, and by the grant 305594/2019-2 of CNPq. The
work of A.V.I. is supported by the grant in the form of subsidies from the
Federal budget for state support of creation and development of world-class
research centers, including international mathematical centers and world-class
research centers that perform research and development on the priorities of
scientific and technological development. The agreement is between MES and
PDMI RAS from ”8” November 2019 number 075-15-2019-1620. Also, A.V.I. is a
winner of the Young Russian Mathematician award and would like to thank its
sponsors and jury.

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