YITP-SB-2022-19

Higher Berry Phase of Fermions and Index Theorem

Yichul Choi1,2 and Kantaro Ohmori3,

1 | C. N. Yang Institute for Theoretical Physics,
---|---
| SUNY, Stony Brook, NY 11794, USA
2 | Simons Center for Geometry and Physics,
| SUNY, Stony Brook, NY 11794, USA
3 | Faculty of Science, University of Tokyo
| Hongo 7-3-1, Bunkyo, Tokyo, Japan

Abstract

When a quantum field theory is trivially gapped, its infrared fixed point is
an invertible field theory. The partition function of the invertible field
theory records the response to various background fields in the long-distance
limit. The set of background fields can include spacetime-dependent coupling
constants, in which case we call the corresponding invertible theory a
parameterized invertible field theory. We study such parameterized invertible
field theories arising from free Dirac fermions with spacetime-dependent mass
parameters using the Atiyah-Patodi-Singer index theorem for superconnections.
In particular, the response to an infinitesimal modulation of the mass is
encoded into a higher analog of the Berry curvature, for which we provide a
general formula. When the Berry curvature vanishes, the invertible theory can
still be nontrivial if there is a remaining torsional Berry phase, for which
we list some computable examples.

###### Contents

  1. 1 Introduction and summary
    1. 1.1 Overview
    2. 1.2 Summary of results
    3. 1.3 Organization
  2. 2 Massive fermion partition functions
    1. 2.1 Phase of the partition function as $\eta$-invariant
    2. 2.2 Reducing the partition function to an interface
    3. 2.3 Generalized cohomology perspectives
  3. 3 Berry curvature
    1. 3.1 Superconnection
    2. 3.2 Berry curvature from Chern character of superconnection
    3. 3.3 Examples
      1. 3.3.1 Single Dirac fermion in even dimensions
      2. 3.3.2 Odd dimensional examples
      3. 3.3.3 Symmetry breaking mass term
  4. 4 Torsional Berry phase
    1. 4.1 Reducing onto zero locus
    2. 4.2 Examples
      1. 4.2.1 Two-dimensional Dirac fermion
      2. 4.2.2 Effective action from symmetry breaking mass
  5. A When $d$ is even
  6. B APS index theorem and superconnection
  7. C Explicit torus partition functions

##  1 Introduction and summary

###  1.1 Overview

A physical system often comes with parameters affecting the dynamics of
concern. In such cases, we can study its response to adiabatic deformations of
the parameters. In quantum mechanics, it is well-known that when one
adiabatically deforms a quantum system along a cyclic path in the space of
parameters, the state of the system after the process can be different from
the initial state, where the difference is captured by the Berry phase
[Ber84].

When the physical system considered is defined on a space with a positive
number of dimensions, which can be either of a lattice or a continuous space,
the parameters can be modulated both in the space and the time directions. In
particular, in a relativistic quantum field theory (QFT), it is natural to let
the parameters to modulate arbitrarily in the spacetime. The response to such
general deformations is encoded into the higher Berry phase [KS20a, KS20b,
HKT20, KS22], which in this paper we often simply call the Berry phase. To be
precise, the Berry phase is definable when the parameters vary within a locus
in the parameter space along which the system is trivially gapped,111 A system
is said to be trivially gapped when it has a single gapped vacuum with no
topological excitations like anyons.  which is a generalization of the
familiar fact from quantum mechanics that the Berry phase can be defined only
if there does not occur a level-crossing along the adiabatic deformation. The
IR fixed point of such a trivially gapped theory coupled to the modulated
parameters is called a _parameterized invertible field theory_.

An easy example of a higher Berry phase is that of a single free massive Dirac
fermion in 1+1-dimensions. The theory has a single complex mass parameter $m$,
and the mass term preserves the vector $U(1)$ global symmetry. We can rotate
the phase of the mass parameter by the axial symmetry action on the fermion,
making $m$ real. This however generates a complex phase because of a mixed
anomaly when the vector $U(1)$ background gauge field $A$ is turned on. If we
write the mass parameter as $m=|m|e^{\mathrm{i}\alpha}$, the generated phase
in the fermion partition function $\mathcal{Z}$ is

$\mathcal{Z}[m=|m|e^{\mathrm{i}\alpha}]=\mathcal{Z}[m=|m|]\exp\left(-\frac{2\pi\mathrm{i}}{(2\pi)^{2}}\int\alpha\mathrm{d}A\right)\,.$
(1.1)

This formula applies even when $\alpha$ modulates over the spacetime up to
terms suppressed by $|m|^{-1}$. While this simple example can be thoroughly
explained by the conventional ’t Hooft anomaly, it is more direct to regard
this phase as an analog of the Berry phase, because the axial symmetry does
not exist when the mass is turned on.

The nontrivial Berry phase in particular means nontrivial physics on the
boundary. In the example of the above 1+1d single Dirac fermion, this can be
seen as follows. We prepare an interface between the fermion theory with a
constant positive mass $m_{0}$ and the fermion theory with a mass
$m_{\alpha}=m_{0}e^{\mathrm{i}\alpha}$ by linearly interpolating between the
two mass values. We regard the mass $m_{0}$ as the reference point and the
interface as the boundary of the invertible theory with the parameter
$m_{\alpha}$. While the boundary is trivially gapped with a generic value of
$\alpha$, it has a degeneracy when $\alpha=\pi$ because there occurs a zero-
mode of a massless complex fermion localized along the boundary. In other
words, the boundary cannot be trivially gapped uniformly over the parameter
space. This is an “anomaly in the space of coupling constants” on the boundary
[CFLS19a, CFLS19b].222This is also referred to as a “global inconsistency.”
See [GKSW14, GKKS17, TK17, KSTZ17, KT17, Tho17, TS18, HHN+19, KK19, STU20,
Sha20, Ü20, HT20, TU22] for developments on this.

From this viewpoint, the Berry phase corresponds to the “inflow action” in the
conventional anomaly inflow language [CJH85]. In the anomaly inflow context,
it is convenient to consider the formal derivative of the inflow action or the
anomaly polynomial. For an anomaly in the space of couplings, the anomaly
polynomial can be regarded as a natural generalization of the Berry curvature
called the higher Berry curvature [KS20a, KS20b, HKT20, KS22]. In the example
of (1.1), it is

$\mathcal{B}=-\frac{1}{(2\pi)^{2}}\mathrm{d}\alpha\mathrm{d}A\,.$ (1.2)

When there are multiple fermions, the form of the Berry phase becomes
complicated. The purpose of this paper is to provide new methods to calculate
the Berry phases and the Berry curvatures for more general theories of massive
Dirac fermions. This will be done by first relating the Berry phase of a
$d$-dimensional fermion system to a certain $(d+1)$-dimensional
$\eta$-invariant, and then invoking the Atiyah-Patodi-Singer (APS) index
theorem for superconnections [Qui85, Kah11, KS21, GY21]. In particular, we
provide an explicit formula computing the Berry curvature for such theories.
The index theorem necessary for the generalization to Majorana fermions is
studied in [GY21].

Some special cases of the Berry curvature of fermions with modulated mass
parameters were studied in the past in [GW81, AW99], and also in [HHY21] for
the purpose of studying crystalline symmetry protected topological phases. See
also [TK10, Shi21, WQB+21, AWH22] for other studies of parameterized
invertible theories in the condensed matter physics literature. Fermions with
modulated mass parameters were also studied in [GL20]. On the phenomenological
side, integrating out fermions coupled to various gauge fields as well as
Higgs fields was recently discussed in [EQV+20, AH20, QSV21]. Our closed-form
formula for the Berry curvature may help streamline some portion of such
calculations, since the modulated mass parameters can be thought of as
background Higgs fields.

###  1.2 Summary of results

As stated, in this paper, we examine the class of parameterized invertible
theories arising from the low-energy limit of massive free Dirac fermions in
$d$ spacetime dimensions. The parameter space is the space of all possible
non-degenerate mass matrices denoted as $P$, and the problem boils down to
studying partition functions of free fermions with spacetime-dependent
modulated mass parameters (in addition to other symmetry backgrounds). The
phase of the (regularized) partition function is identified with the Berry
phase introduced above.

Specifically, we put an arbitrary number $N_{f}$ of Dirac fermions on a
$d$-dimensional spacetime manifold $X$ which is equipped with the various
background fields, that is, the modulated mass parameters as well as
background gauge fields for some global symmetries. The parameter space $P$ is
determined by $d$ and $N_{f}$. Then, it turns out that it is convenient to
consider an auxiliary $(d+1)$-dimensional free fermion system living on a
manifold $Y_{+}$ which bounds $X$. That is, $X=\partial Y_{+}$ and all the
background fields extend into $Y_{+}$. Importantly, we allow the
$d$-dimensional mass matrix to become degenerate inside the
$(d+1)$-dimensional bulk $Y_{+}$. Manifolds $X$ together with the background
fields that do not admit such $Y_{+}$ even when we allow the mass matrix to
become degenerate inside the bulk are needed only for the purpose of detecting
ordinary invertible theories that do not see the parameter space. Because we
are not interested in these ordinary invertible theories in this paper, we can
safely assume that $Y_{+}$ always exists.

In this setup, we can state our main result:

$\left.\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}^{(0)}_{d}(X)}\middle/\left|\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}_{d}^{(0)}(X)}\right|\right.=\exp\left(2\pi\mathrm{i}(\eta_{K}(Y_{+};\mathtt{APS})-\eta_{K^{(0)}}(Y_{+};\mathtt{APS})\right)\,.$
(1.3)

Here, $\mathcal{Z}_{d}(X)$ is the partition function of the $d$-dimensional
theory with a modulated but non-degenerate mass parameters living on $X$, and
$\eta_{K}(Y_{+};\mathtt{APS})$ is the APS $\eta$-invariant for a hermitian
operator $K$ which extends the $d$-dimensional Dirac operator on $X$ to the
bulk of $Y_{+}$ [APS75b]. The quantities with the superscript $(0)$ are for
the regulator fermions. We derive this result by considering
$(d+1)$-dimensional Dirac fermions living in the bulk $Y_{+}$. The original
$d$-dimensional theory is realized as the localized edge modes living on the
boundary $\partial Y_{+}=X$. The result works even when the mass term breaks
certain global symmetries for which we want to turn on the background gauge
fields, by promoting the mass parameters to be an appropriate section of the
associated bundle.

The equation (1.3) connects two different viewpoints. The left-hand side is
the Berry phase of the original system in $d$-dimensions. The right-hand side
is the Berry phase of the $(d+1)$-dimensional fermion system with the
parameter space $\widetilde{P}=P\cup D$ for which the original $d$-dimensional
system behaves as the boundary anomalous theory. Here, $D$ is the space of
degenerate mass matrices (which again is determined by $d$ and $N_{f}$), and
in general $\widetilde{P}=\mathbb{R}^{N}$ for some integer $N$. We will
provide an interpretation of the equation from the generalized cohomology
viewpoint in Section 2.3.

The consequence of the result (1.3) is that we can apply the index theorem for
superconnections to calculate the right-hand side [CFLS19a, CFLS19b, KS21].
The definition of a superconnection will be reviewed in Section 3.1. For the
math literature regarding the superconnection and the relevant index theorem,
see [Qui85, Kah11, GY21]. In particular, this approch enables us to write down
an explicit formula for the Berry curvature $\mathcal{B}$:

$\mathcal{B}=\int_{\widetilde{M}_{0}=0}^{\widetilde{M}_{0}=\infty}\left[\mathop{\mathrm{ch}}\nolimits(\mathcal{F})-\mathop{\mathrm{ch}}\nolimits(\mathcal{F}^{(0)})\right]\wedge\hat{A}\,,$
(1.4)

where $\mathcal{F}$ is the field strength for the superconnection, and
$\mathop{\mathrm{ch}}\nolimits(\mathcal{F})$ is the corresponding Chern
character.333$\mathop{\mathrm{ch}}\nolimits(\mathcal{F}^{(0)})$ is the
contribution from the regulator fermions. $\hat{A}$ is the $\hat{A}$-genus
responsible for the usual gravitational contribution, and $\widetilde{M}_{0}$
is the overall scale of the (renormalized) mass matrix.

We also show some examples where the resulting parameterized invertible theory
matches global anomalies in the parameter space in one lower dimensions. In
these cases, the Berry curvature vanishes and the partition function defines a
bordism invariant. We call such a bordism invariant a _torsional Berry phase_.

###  1.3 Organization

This paper is organized as follows. In Section 2, we derive Eq. (1.3) in two
steps, by relating the partition function of the $d$-dimensional fermion
system of interest to that of an auxiliary $(d+1)$-dimensional system. We
first show that the phase of the partition function of the $(d+1)$-dimensional
system is given by the APS $\eta$-invariant of a hermitian operator $K$ that
we will define, and then show that the regularized $(d+1)$-dimensional
partition function is equal to the regularized $d$-dimensional partition
function that we are after in a certain limit. The essential idea is based on
earlier works by Yonekura [Yon16], and Yonekura and Witten [WY19]. Then, we
interpret Eq. (1.3) from the perspective of a generalized cohomology theory.
In Section 3, we derive Eq. (1.4) and demonstrate it in several examples. In
Section 4, we show examples of parameterized invertible theories with
vanishing Berry curvatures, where the partition function can be computed using
Eq. (1.3). In Appendix B, we derive the APS index theorem in the presence of
the modulated mass parameters, by closely following the derivation given in
[KY21] for the case of massless Dirac operators. It is a generalization of the
theorem derived in [KS21] to the case of manifolds with boundary,444A special
case of this was also treated in [KS21]. and we use as an input their
calculation result which was based on the Fujikawa method [Fuj79, Fuj84].

##  2 Massive fermion partition functions

Our goal in this section is to express (the phase of) the partition function
of massive free Dirac fermions in $d$ spacetime dimensions with the spacetime-
dependent mass parameters in terms of $\eta$-invariants (see Eq. (1.3)). We
denote the number of flavors as $N_{f}$ which is arbitrary. We would like to
achieve this possibly in the presence of other background gauge fields for
some global symmetries and in general on a curved spacetime manifold. The
global symmetries we consider are (subgroups of) $U(N_{f})\times U(N_{f})$ if
$d$ is even, or $U(N_{f})$ if $d$ is odd, as well as the fermion parity
symmetry. Usually, we will not explicitly specify for which subgroup of
$U(N_{f})\times U(N_{f})$ or $U(N_{f})$ the background gauge fields are turned
on, and the general computations are valid for any choice of the subgroup
unless otherwise mentioned. The $d$-dimensional spacetime manifold will be
denoted as $X$, which is assumed to be closed and oriented. All the
discussions will be in the Euclidean signature.

The Lagrangian of the theory is

$\mathcal{L}_{d}=\bar{\psi}(\not{D}+M)\psi\,.$ (2.1)

The flavor and spinor indices are implicit in $\psi$. When $d$ is odd, $M$
takes values in $N_{f}\times N_{f}$ hermitian matrices. When $d$ is even, the
mass matrix has the form of $M=M^{\prime}+\mathrm{i}\gamma M^{\prime\prime}$
where $M^{\prime}$ and $M^{\prime\prime}$ are valued in $N_{f}\times N_{f}$
hermitian matrices and $\gamma$ is the chirality matrix. We let the mass
matrix be a nontrivial function on the spacetime manifold, or a section of a
bundle acted by a global symmetry in case the mass term is not invariant under
the chosen symmetry. However, to ensure that the partition function is nonzero
and thus we can define its phase, we demand that we do not cross the massless
points, i.e., $\mathrm{det}(M)$ is nonzero everywhere on the spacetime
manifold $X$.

We are in particular interested in the long-distance limit of the fermion
theory. This limit can be achieved by sending the mass eigenvalues to
infinity, instead of sending the volume of the spacetime to infinity. In this
limit the fermion theory becomes invertible. The invertible theory at the IR
fixed point naturally couples with the scalar background $M$, which moves in
the space of non-degenerate mass matrices. The partition function of the
invertible theory records the long-distance reaction to a modulation of the
mass $M$ along the spacetime.

To achieve our goal, it helps to prepare the fermion system by a domain-wall
or interface construction [JR76].555We will mostly refer to it as an interface
instead of the commonly-used terminology domain wall, since this interface is
formed by adjusting the background field configurations by hand and is not a
dynamical object of the theory. That is, we will realize our $d$-dimensional
fermion system of interest on an interface of an auxiliary $(d+1)$-dimensional
fermion system by varing the $(d+1)$-dimensional mass parameters
appropriately. A similar construction for massive fermions was considered in
[FOY17, FFM+20a, FFM+20b, OY21].

Concretely, we consider a $(d+1)$-dimensional theory of $N_{f}$ flavors of
free massive Dirac fermions if $d$ is odd, or $2N_{f}$ flavors of them if $d$
is even. We put this theory on a closed manifold
$Y=\overline{Y}_{-}\cup_{\overline{X}}(X\times I)\cup_{X}Y_{+}$ which is shown
in the left panel of Figure 1. The bar over a manifold indicates the
orientation reversal of the manifold. $Y_{+}=Y_{-}$ are the same
$(d+1)$-dimensional manifolds, and they have the boundary $\partial
Y_{\pm}=X$. All the background fields on $X$ extend into $Y_{\pm}$ in the same
way.666We assume that the manifold $X$ equipped with various background gauge
fields and the mass parameters is null-bordant if we allow $\text{det}(M)$ to
vanish inside the null-bordism. $\cup_{X}$ means gluing along the common
boundary $X$ by the identity map, and similarly for $\cup_{\overline{X}}$.
Since we are not interested in detecting ordinary symmetry protected
topological phases that do not depend on the mass parameters, this suffices
for our purposes. $Y_{\pm}$ are assumed to have collar neighborhoods near the
boundary, i.e., near the boundary they look like a product of $X$ and an
interval. The union of the collar neighborhoods of $X$ in $Y_{\pm}$ and the
middle region $X\times I$ forms the collar neighborhood of $X$ in $Y$. We
denote the coordinate transverse to $X$ inside this collar neighborhood as
$y$, and this will be viewed as the Euclidean time direction. Along the collar
neighborhood, the metric factorizes as $ds^{2}_{Y}=dy^{2}+ds^{2}_{X}$ where
$ds^{2}_{X}$ is the metric on $X$, and any background gauge fields are
required to be independent of $y$ and to have no $y$ component. That is,
locally we can write $A=A_{i}(x)dx^{i}$ for the background gauge field $A$
where $x^{i}$ are the coordinates along $X$ for $i=1,\dots,d$.

Figure 1: (Left) The $(d+1)$-dimensional closed manifold
$Y=\overline{Y}_{-}\cup_{\overline{X}}(X\times I)\cup_{X}Y_{+}$ on which the
auxiliary $(d+1)$-dimensional theory is defined. (Right) The profile of
$M_{1}(y)$ along the collar neighborhood of $X$ in $Y$ forms an interface. On
the interface, the original $d$-dimensional theory of interest appears as a
collection of localized modes.

We will specialize to the case of odd $d$ here, and leave the similar
calculations for the even $d$ case to Appendix A. The final conclusion equally
applies to both cases. The Lagrangian for the $(d+1)$-dimensional theory is
given by

$\mathcal{L}_{d+1}=\bar{\Psi}(\not{D}+M_{1}+\mathrm{i}\Gamma M_{2})\Psi$ (2.2)

where $\Gamma$ is the $(d+1)$-dimensional chirality matrix. We have $N_{f}$
flavors of $(d+1)$-dimensional Dirac fermions, where the flavor indices are
again implicit. A single Dirac fermion in $(d+1)$-dimensions has $2^{\lfloor
d+1\rfloor}$ components, whereas in $d$-dimensions it has $2^{\lfloor
d\rfloor}$ components. Since $d+1$ is even, we have $2\times 2^{\lfloor
d\rfloor}=2^{\lfloor d+1\rfloor}$. The mass matrices $M_{1}$ and $M_{2}$ can
in general be any $N_{f}\times N_{f}$ hermitian matrices. We take the mass
parameter $M_{1}$ to be $M_{1}=M_{1}(y)\mathds{1}$ where $\mathds{1}$ is the
$N_{f}\times N_{f}$ identity matrix, and $M_{1}(y)$ is a smooth function
varying from $+m_{0}$ to $-m_{0}$ along the middle region $X\times I$ where
$y\in I=[y_{+},y_{-}]$, with $m_{0}$ being a positive constant. Inside
$Y_{\pm}$ we will let $M_{1}=\pm m_{0}\mathds{1}$ to be constant. That is, the
configuration of $M_{1}(y)$ forms a smooth interface from
$M_{1}(y_{+})=+m_{0}$ to $M_{1}(y_{-})=-m_{0}$, see the right panel of Figure
1. $M_{2}=M(x)$ is taken to be equal to the mass matrix for the original
$d$-dimensional fermions and is independent of $y$ inside the collar
neighborhood. $M_{2}$ should be suitably extended into the bulk of $Y_{\pm}$,
but the precise form of the extension is not important for us. The only
assumption is that $M_{2}=M$ as well as all the background gauge fields on $X$
are extended to $Y_{+}$ and $Y_{-}$ in the same way, as mentioned earlier.

With this setup, there will appear localized $d$-dimensional massive Dirac
fermions living on the interface $X$ with the Lagrangian given by Eq. (2.1).
This can be justified by considering the Dirac equaion [JR76], and we will
make it more precise later. Intuitively, if we take the limit $m_{0}\to\infty$
while keeping $M$ finite, all the $(d+1)$-dimensional bulk modes decouple and
we will be left with the $d$-dimensional fermions that we are interested in.
Below we will see how this idea is concretely realized and why this procedure
of realizing the $d$-dimensional fermion on an interface of an auxiliary
$(d+1)$-dimensional system is useful.

###  2.1 Phase of the partition function as $\eta$-invariant

First, we would like to show that

$\mathcal{Z}_{d+1}(Y)=|\mathcal{Z}_{d}(X)|\exp\left(2\pi\mathrm{i}\eta_{K}(Y_{+};\mathtt{APS})\right)$
(2.3)

when $m_{0}\to\infty$, by applying a similar analysis as in [Yon16, WY19].
Here, $\mathcal{Z}_{d}(X)$ is the partition function of the $d$-dimensional
theory (2.1) on $X$, and similarly $\mathcal{Z}_{d+1}(Y)$ is the partition
function of the $(d+1)$-dimensional theory (2.2) on $Y$.
$\eta_{K}(Y_{+};\mathtt{APS})$ is the APS $\eta$-invariant of a herimitian
operator $K$ on $Y_{+}$ which we will describe below, with the APS boundary
condition imposed on the boundary $\partial Y_{+}=X$.

Let $Y_{L}\equiv X\times I$ for short. The partition function of the
$(d+1)$-dimensional theory on the manifold $Y$ computes the following
amplitude:

$\mathcal{Z}_{d+1}(Y)=\bra{Y_{-}}\hat{Y}_{L}\ket{Y_{+}}\,.$ (2.4)

The states $\ket{Y_{\pm}}$ are elements in the Hilbert space $\mathcal{H}_{X}$
of the spatial slice $X$, defined by the path integral over the region
$Y_{\pm}$. The operator
$\hat{Y}_{L}:\mathcal{H}_{X}\rightarrow\mathcal{H}_{X}$ corresponds to the
Euclidean time evolution given by the cylindrical region $Y_{L}=X\times I$.

We define the $(d+1)$-dimensional massive Dirac operator $\mathsf{D}_{d+1}$ as

$\mathsf{D}_{d+1}=\Gamma^{\mu}D_{\mu}+M_{1}+\mathrm{i}\Gamma M_{2}\,.$ (2.5)

We pick a basis for the gamma matrices such that

$\displaystyle\begin{split}\Gamma^{i}&=\gamma^{i}\otimes\sigma^{2}\,,\\\
\Gamma^{y}&=\mathbb{I}\otimes\sigma^{3}\,,\\\
\Gamma&=\mathbb{I}\otimes\sigma^{1}\,,\end{split}$ (2.6)

where $\sigma$’s are the Pauli matrices, $\gamma^{i}$’s are the
$d$-dimensional gamma matrices, and $\mathbb{I}$ is the $2^{\lfloor
d\rfloor}\times 2^{\lfloor d\rfloor}$ identity matrix. In this basis, inside
the collar neighborhood of $X$ in $Y$, we have

$\displaystyle\begin{split}\mathsf{D}_{d+1}&=\Gamma^{y}\partial_{y}+\Gamma^{i}D_{i}+M_{1}+\mathrm{i}\Gamma
M_{2}\\\
&=\begin{bmatrix}\partial_{y}+M_{1}&\mathrm{i}\mathsf{D}_{d}^{\dagger}\\\
\mathrm{i}\mathsf{D}_{d}&-\partial_{y}+M_{1}\end{bmatrix}\end{split}$ (2.7)

where

$\mathsf{D}_{d}=\gamma^{i}D_{i}+M$ (2.8)

is the $d$-dimensional massive Dirac operator (recall $M_{2}=M$ along the
collar neighborhood). On the other hand, inside $Y_{\pm}$ where we have
constant $M_{1}=\pm m_{0}$, we can write

$\mathsf{D}_{d+1}=-\mathrm{i}K\pm m_{0}$ (2.9)

where $K=\mathrm{i}(\Gamma^{\mu}D_{\mu}+i\Gamma M_{2})$ is a hermitian
operator on $Y$.

The Hamiltonian on $X$ is given by

$\hat{H}(y)=\int_{X}\Psi^{\dagger}\begin{bmatrix}M_{1}(y)&\mathrm{i}\mathsf{D}_{d}^{\dagger}\\\
-\mathrm{i}\mathsf{D}_{d}&-M_{1}(y)\end{bmatrix}\Psi$ (2.10)

which is dependent on the Euclidean time $y$. Accordingly, the Euclidean time
evolution along $Y_{L}$ is given by

$\hat{Y}_{L}=T\\!\exp\left(-\int_{y_{+}}^{y_{-}}\hat{H}(y)dy\right)\,,$ (2.11)

where $T\\!\exp$ denotes the time-ordered exponential.

For large $m_{0}$, the Euclidean path integral over the collar regions of
$Y_{\pm}$ projects the states $\ket{Y_{\pm}}$ onto the ground states
$\ket{\Omega_{\pm}}$ at $y=y_{\pm}$, respectively, as the energies of the
excited states will scale with $m_{0}$. Thus,
$\ket{Y_{\pm}}\propto\ket{\Omega_{\pm}}$ and the proportionality constants are
just phases if we normalize all the states properly.

Combining, we can rewrite the $(d+1)$-dimensional partition function as

$\displaystyle\begin{split}\mathcal{Z}_{d+1}(Y)&=\bra{Y_{-}}\hat{Y}_{L}\ket{Y_{+}}\\\
&=\braket{Y_{-}}{\Omega_{-}}\bra{\Omega_{-}}\hat{Y}_{L}\ket{\Omega_{+}}\braket{\Omega_{+}}{Y_{+}}\\\
&=\bra{\Omega_{-}}T\\!\exp\left(-\int_{y_{+}}^{y_{-}}\hat{H}(y)dy\right)\ket{\Omega_{+}}\times\frac{\braket{\Omega_{+}}{Y_{+}}}{\braket{\Omega_{-}}{Y_{-}}}\,,\end{split}$
(2.12)

where in the last step we used the fact that $\braket{Y_{-}}{\Omega_{-}}$ is a
pure phase.

To compute various state overlaps, we perform a mode expansion of $\Psi$ on
the spatial slice $X$. Following Fujikawa [Fuj84], consider two hermitian
operators on $X$ given by
$\mathsf{H}^{1}_{d}=\mathsf{D}_{d}^{\dagger}\mathsf{D}_{d}$ and
$\mathsf{H}^{2}_{d}=\mathsf{D}_{d}\mathsf{D}_{d}^{\dagger}$, and let the
eigenvalues and the orthonormal eigenfunctions of $\mathsf{H}^{1}_{d}$ be

$\mathsf{H}^{1}_{d}\psi_{a}=\lambda_{a}^{2}\psi_{a}\,.$ (2.13)

The eigenvalues are all positive, $\lambda_{a}>0$ for all $a$, due to the
assumption that the $d$-dimensional mass matrix is non-degenerate. In this
case, we can choose the orthonormal set of eigenfunctions of
$\mathsf{H}^{2}_{d}$ to be

$\tilde{\psi}_{a}=\frac{1}{\lambda_{a}}\mathsf{D}_{d}\psi_{a}$ (2.14)

so that they satisfy
$\mathsf{H}^{2}_{d}\tilde{\psi}_{a}=\lambda_{a}^{2}\tilde{\psi}_{a}$. The
magnitude of the partition function of the $d$-dimensional fermion (2.1) on
$X$ is formally given by $|\mathcal{Z}_{d}(X)|=\prod_{a}\lambda_{a}$ [Fuj84].

Now, define

$\Psi_{+,a}=\begin{bmatrix}\psi_{a}\\\
0\end{bmatrix}\,,\quad\Psi_{-,a}=\begin{bmatrix}0\\\
-i\tilde{\psi}_{a}\end{bmatrix}\,,$ (2.15)

and perform a mode expansion

$\Psi=\sum_{a}\left(A_{+,a}\Psi_{+,a}+A_{-,a}\Psi_{-,a}\right)\,.$ (2.16)

Using this expansion, we can first identify the ground state $\ket{\Omega(y)}$
for the Hamiltonian $\hat{H}(y)$ at fixed $y$. In terms of the modes
$A_{\pm,a}$, the Hamiltonian reads

$\hat{H}(y)=\sum_{a}A_{a}^{\dagger}\begin{bmatrix}M_{1}(y)&\lambda_{a}\\\
\lambda_{a}&-M_{1}(y)\end{bmatrix}A_{a}$ (2.17)

where $A=(A_{+,a},A_{-,a})^{T}$. Following the conventions in [Yon16, WY19],
we can diagonalize the Hamiltonian as follows. Define the angle
$0<\theta_{a}(y)\leq\pi/2$ by

$\left(\text{cos}2\theta_{a}(y),\text{sin}2\theta_{a}(y)\right)=\frac{(M_{1}(y),\lambda_{a})}{\sqrt{M_{1}(y)^{2}+\lambda_{a}^{2}}}\,,$
(2.18)

and let

$B_{a}(y)=\begin{bmatrix}B_{1,a}(y)\\\
B_{2,a}(y)\end{bmatrix}=\begin{bmatrix}\text{cos}\theta_{a}(y)&\text{sin}\theta_{a}(y)\\\
-\text{sin}\theta_{a}(y)&\text{cos}\theta_{a}(y)\end{bmatrix}A_{a}\,.$ (2.19)

In terms of these new modes, the Hamiltonian is diagonalized,

$\hat{H}(y)=\sum_{a}\sqrt{\lambda_{a}^{2}+M_{1}(y)^{2}}\left(B_{1,a}^{\dagger}(y)B_{1,a}(y)-B_{2,a}^{\dagger}(y)B_{2,a}(y)\right)\,.$
(2.20)

We can now easily see that the ground state $\ket{\Omega(y)}$ at fixed $y$ is
the state that is annihilated by all $B_{1,a}(y)$ and all
$B_{2,a}^{\dagger}(y)$. In particular,
$\ket{\Omega(y_{\pm})}\equiv\ket{\Omega_{\pm}}$.

Going back to Eq. (2.12), we will now argue that the fraction
$\frac{\braket{\Omega_{+}}{Y_{+}}}{\braket{\Omega_{-}}{Y_{-}}}$ gives us the
exponentiated $\eta$-invariant of the operator $K$. The operator $K$ that we
defined in Eq. (2.9) is a hermitian operator on a closed manifold $Y$, but
when restricted to $Y_{+}$, it requires an appropriate choice of boundary
condition on $\partial Y_{+}=X$ to maintain the hermiticity. One such boundary
condition is given by the APS boundary condition. In our conventions, the
condition is stated as follows. We can write
$K=\mathrm{i}\Gamma^{y}(\partial_{y}+\mathcal{D}_{X})$ inside the collar
neighborhood of $X$, where

$\mathcal{D}_{X}=\begin{bmatrix}0&\mathrm{i}\mathsf{D}_{d}^{\dagger}\\\
-\mathrm{i}\mathsf{D}_{d}&0\end{bmatrix}$ (2.21)

is a hermitian operator on $X$. The eigenvlues of $\mathcal{D}_{X}$ come in
pairs, $\pm\lambda_{a}$, since $\\{\Gamma^{y},\mathcal{D}_{X}\\}=0$. The
corresponding eigenfunctions are $\Psi_{+,a}\pm\Psi_{-,a}$. The APS boundary
condition on $Y_{+}$ is given by

$\mathtt{APS}:\mathcal{P}_{<0}^{\mathcal{D}_{X}}\Psi|_{\partial Y_{+}=X}=0$
(2.22)

where $\mathcal{P}_{<0}^{\mathcal{D}_{X}}$ is the projection operator onto the
subspace spanned by the eigenfunctions with negative eigenvalues of
$\mathcal{D}_{X}$ on $X$. This means in the mode expansion

$\displaystyle\begin{split}\Psi&=\sum_{a}(A_{+,a}\Psi_{+,a}+A_{-,a}\Psi_{-,a})\\\
&=\sum_{a}\left[\frac{1}{2}(A_{+,a}+A_{-,a})(\Psi_{+,a}+\Psi_{-,a})+\frac{1}{2}(A_{+,a}-A_{-,a})(\Psi_{+,a}-\Psi_{-,a})\right]\,,\end{split}$
(2.23)

the coefficients $\frac{1}{2}(A_{+,a}-A_{-,a})$ are constrained to be $0$,
whereas the other coefficients $\frac{1}{2}(A_{+,a}+A_{-,a})$ are
unconstrained. The quantum state corresponding to such a boundary condition is
characterized by [Yon16, WY19]

$\bra{\mathtt{APS}}(A_{+,a}-A_{-,a})=\bra{\mathtt{APS}}(A_{+,a}+A_{-,a})^{\dagger}=0$
(2.24)

for all $a$.

Following [Yon16, WY19], let $\ket{E}$ be a state defined by

$A_{+,a}\ket{E}=A_{-,a}\ket{E}=0$ (2.25)

for all $a$. The states $\ket{\Omega(y)}$ and $\ket{\mathtt{APS}}$ can be
realized as

$\displaystyle\begin{split}\ket{\Omega(y)}&=\prod_{a}B_{2,a}^{\dagger}\ket{E}\,,\\\
\ket{\mathtt{APS}}&=\prod_{a}\frac{(A_{-,a}-A_{+,a})^{\dagger}}{\sqrt{2}}\ket{E}\,.\end{split}$
(2.26)

Then, we claim

$\frac{\braket{\mathtt{APS}}{\Omega_{-}}}{\braket{\mathtt{APS}}{\Omega_{+}}}=\frac{\prod\limits_{a}\left(\text{cos}\theta_{a}(y_{-})+\text{sin}\theta_{a}(y_{-})\right)}{\prod\limits_{a}\left(\text{cos}\theta_{a}(y_{+})+\text{sin}\theta_{a}(y_{+})\right)}=1\,.$
(2.27)

To see this, first notice

$\displaystyle\begin{split}\braket{\mathtt{APS}}{\Omega(y)}&=\frac{1}{\sqrt{2}}\prod_{a,a^{\prime}}\bra{E}\left(A_{-,a}-A_{+,a}\right)B_{2,a^{\prime}}^{\dagger}\ket{\Omega(y)}\\\
&=\frac{1}{\sqrt{2}}\prod_{a,a^{\prime}}\bra{E}\left(A_{-,a}-A_{+,a}\right)\left(-\text{sin}\theta_{a^{\prime}}(y)A_{+,a^{\prime}}^{\dagger}+\text{cos}\theta_{a^{\prime}}(y)A_{-,a^{\prime}}^{\dagger}\right)\ket{\Omega(y)}\\\
&=\frac{1}{\sqrt{2}}\prod_{a}\left(\text{sin}\theta_{a}(y)+\text{cos}\theta_{a}(y)\right)\,,\end{split}$
(2.28)

where the last step follows from the usual anticommutation relations for the
fermionic modes $A_{\pm,a}$ and Eq. (2.25) (we normalize the state $\ket{E}$
so that $\braket{E}{E}=1$). This verifies the first equality sign in (2.27).
Then, at $y=y_{\pm}$, we have $M(y_{\pm})=\pm m_{0}$, and from Eq. (2.18) we
see that $\theta_{a}(y_{+})+\theta_{a}(y_{-})=\pi/2$ for all $a$. From this
the second equality sign of (2.27) follows.

Using the fact that $\ket{Y_{\pm}}\propto\ket{\Omega_{\pm}}$ and Eq. (2.27),
we can rewrite the fraction
$\frac{\braket{\Omega_{+}}{Y_{+}}}{\braket{\Omega_{-}}{Y_{-}}}$ in Eq. (2.12)
as

$\frac{\braket{\Omega_{+}}{Y_{+}}}{\braket{\Omega_{-}}{Y_{-}}}=\frac{\braket{\mathtt{APS}}{Y_{+}}}{\braket{\mathtt{APS}}{Y_{-}}}\frac{\braket{\mathtt{APS}}{\Omega_{-}}}{\braket{\mathtt{APS}}{\Omega_{+}}}=\frac{\braket{\mathtt{APS}}{Y_{+}}}{\braket{\mathtt{APS}}{Y_{-}}}\,.$
(2.29)

In the limit $m_{0}\to\infty$, this last expression is nothing but the
exponentiated $\eta$-invariant for the operator $K$ on $Y_{+}$ with the APS
boundary condition imposed, which can be seen as follows. Since $K$ is
hermitian on $Y_{+}$ with this boundary condition, we may write its complete
set of orthonormal eigenfunctions and the corresponding eigenvalues as

$K\Psi_{j}=\xi_{j}\Psi_{j}\,.$ (2.30)

Each $\Psi_{j}$ satisfies the APS boundary condition,
$\mathcal{P}_{<0}^{\mathcal{D}_{X}}\Psi_{j}|_{X}=0$, and we can expand the
fermion field in the path integral as

$\Psi=\sum_{j}a_{j}\Psi_{j}$ (2.31)

with grassmannian coefficients $a_{j}$. To know what is the valid expansion
for the conjugate field $\bar{\Psi}$, we need to know the boundary condition
imposed on it. If we perform a mode expansion for $\bar{\Psi}$ at the boundary
as

$\bar{\Psi}|_{X}=\sum_{a}\left[B_{1,a}(\Psi_{+,a}+\Psi_{-,a})^{\dagger}+B_{2,a}(\Psi_{+,a}-\Psi_{-,a})^{\dagger}\right]\,,$
(2.32)

then we find that $B_{1,a}$ and $B_{2,a}$ are canonically conjugate to
$\frac{1}{2}(A_{+,a}-A_{-,a})$ and $\frac{1}{2}(A_{+,a}+A_{-,a})$,
respectively. This is because $\Gamma^{y}$ from
$\bar{\Psi}\Gamma^{y}\partial_{y}\Psi$ in the kinetic term permutes
$\Psi_{+,a}+\Psi_{-,a}$ and $\Psi_{+,a}-\Psi_{-,a}$, see Eq. (2.23).

As we explained above, the boundary condition on $\Psi$ is such that at the
boundary $A_{+,a}-A_{-,a}$ vanishes for all $a$, whereas $A_{+,a}+A_{-,a}$ is
unconstrained for all $a$. Correspondingly, at the boundary we have to require
$B_{2,a}$ which is conjugate to $A_{+,a}-A_{-,a}$ to be unconstrained, whereas
$B_{1,a}$ which is conjugate to $A_{+,a}+A_{-,a}$ to vanish. This is the
boundary condition imposed on $\bar{\Psi}$.

This boundary condition is precisely satisfied by $\Psi_{j}^{\dagger}$, thus
we can expand inside the path integral

$\bar{\Psi}=\sum_{j}b_{j}\Psi_{j}^{\dagger}$ (2.33)

with the grassmannian coefficients $b_{j}$. Combined with Eq. (2.31), these
expansions diagonalize the action, $S=\int_{Y_{\pm}}\bar{\Psi}(-\mathrm{i}K\pm
m_{0})\Psi=\sum\limits_{j}(-\mathrm{i}\xi_{j}\pm m_{0})b_{j}a_{j}$. Thus, the
path integral gives us

$\frac{\braket{\mathtt{APS}}{Y_{+}}}{\braket{\mathtt{APS}}{Y_{-}}}=\prod_{j}\frac{-\mathrm{i}\xi_{j}+m_{0}}{-\mathrm{i}\xi_{j}-m_{0}}\to\text{exp}(2\pi\mathrm{i}\eta_{K}(Y_{+};\mathtt{APS}))$
(2.34)

in the limit $m_{0}\to\infty$. Here,

$\eta_{K}(Y_{+};\mathtt{APS})=\frac{1}{2}\sum_{j}\text{sgn}(\xi_{j})$ (2.35)

is the desired $\eta$-invariant of $K$ with the APS boundary condition imposed
on the boundary $\partial Y_{+}=X$. We have defined $\text{sgn}(0)\equiv+1$.
In general, the infinite sum in Eq. (2.35) should be appropriately
regularized. For instance, a common choice is the heat kernel regularization
which is given by

$\eta_{K}(Y_{+};\mathtt{APS})=\lim_{\mu\to\infty}\left[\frac{1}{2}\sum_{j}\text{sgn}(\xi_{j})\text{exp}\left(-\frac{\xi_{j}^{2}}{\mu^{2}}\right)\right]\,.$
(2.36)

An important property of the APS boundary condition that we will exploit in
the next section is that it can be replaced by an infinitely long half
cylinder attached to the boundary. That is, if we consider the manifold
$\widetilde{Y}\equiv Y_{+}\cup_{X}\left(X\times[0,\infty)\right)$, we have

$\eta_{K}(Y_{+};\mathtt{APS})=\eta_{K}(\widetilde{Y})\,.$ (2.37)

All the background gauge fields are continuously glued along the boundary $X$,
and they must be extended into the half-infinite cylinder in such a way they
are independent of the direction transverse to $X$. The manifold
$\widetilde{Y}$ does not have a boundary (and thus we do not need to worry
about boundary conditions) but is now non-compact, see Figure 2. Eq. (2.37)
holds due to the fact that the spectrum of normalizable eigenfunctions of $K$
on $\widetilde{Y}$ is the same as that on $Y_{+}$ with the APS boundary
condition imposed [APS75b]. Alternatively, this can be understood from Eq.
(2.29), which tells us that the APS boundary condition can be replaced by the
ground states $\ket{\Omega_{\pm}}$. Doing so is equivalent to attaching an
infinitely long half cylinder since the Euclidean path integral along the
cylinder prepares the ground states.

Figure 2: The APS boundary condition can be replaced by an infinite half
cylinder. All the mass parameters and the background gauge fields are
independent of the longitudinal direction along the half cylinder.

Next, we need to compute the amplitude
$\bra{\Omega_{-}}T[{\exp\\{-\int_{y_{+}}^{y_{-}}\hat{H}(y)dy\\}}]\ket{\Omega_{+}}$.
For our purposes, it suffices to take the limit where the length of the middle
region $X\times I$ becomes infinitesimal (while maintaining
$|y_{+}-y_{-}|m_{0}\gg 1$), in which case we can simply approximate

$\displaystyle\begin{split}\bra{\Omega_{-}}T[{\text{exp}\\{-\int_{y_{+}}^{y_{-}}\hat{H}(y)dy\\}}]\ket{\Omega_{+}}&\approx\braket{\Omega_{-}}{\Omega_{+}}\\\
&=\prod_{a}\left(\text{cos}\theta_{a}(y_{-})\text{cos}\theta_{a}(y_{+})+\text{sin}\theta_{a}(y_{-})\text{sin}\theta_{a}(y_{+})\right)\\\
&=\prod_{a}\frac{\lambda_{a}}{\sqrt{m_{0}^{2}+\lambda_{a}^{2}}}=|\mathcal{Z}_{d}(X)|_{\text{Reg.}}\,.\end{split}$
(2.38)

Here, the overlap $\braket{\Omega_{-}}{\Omega_{+}}$ is computed using the
presentation of the ground states given in Eq. (2.26). The resulting
expression is essentially the product of all $\lambda_{a}$’s, but the
eigenvalues much larger than $m_{0}$ are regularized and do not contribute.
This corresponds to the regularized magnitude of the $d$-dimensional partition
function [WY19], which we denote as $|\mathcal{Z}_{d}(X)|_{\text{Reg.}}$.

Now, we combine Eqs. (2.12), (2.29), and (2.38). This gives us Eq.(2.3) as
desired. Moreover, we may also consider the ratio

$\frac{\mathcal{Z}_{d+1}(Y)}{\mathcal{Z}_{d+1}^{(0)}(Y)}=\left|\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}_{d}^{(0)}(X)}\right|\times\text{exp}(2\pi\mathrm{i}(\eta_{K}(Y_{+};\mathtt{APS})-\eta_{K^{(0)}}(Y_{+};\mathtt{APS}))$
(2.39)

where the superscript $(0)$ means setting $M$ to be a nonzero constant, which
acts as the Pauli-Villars regulator for the $d$-dimensional fermions. The
reason for doing this is because in the end we want to obtain a purely
$d$-dimensional statement, whereas the regulator $m_{0}$ in Eq. (2.38) has the
$(d+1)$-dimensional origin.

###  2.2 Reducing the partition function to an interface

Here, we will argue that

$\frac{\mathcal{Z}_{d+1}(Y)}{\mathcal{Z}^{(0)}_{d+1}(Y)}=\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}_{d}^{(0)}(X)}$
(2.40)

when $m_{0}\to\infty$. Define two hermitian operators on $Y$,
$\mathsf{H}_{d+1}^{1}=\mathsf{D}^{\dagger}_{d+1}\mathsf{D}_{d+1}$ and
$\mathsf{H}_{d+1}^{2}=\mathsf{D}_{d+1}\mathsf{D}_{d+1}^{\dagger}$. Let the
eigenvalues and eigenfunctions of $\mathsf{H}_{d+1}^{1}$ be777We use the greek
letters $\alpha,\beta,\cdots$ to label $(d+1)$-dimensional eigenfunctions
$\Psi_{\alpha}$, and the latin letters $a,b,\cdots$ to label $d$-dimensional
eigenfunctions $\psi_{a}$.

$\mathsf{H}_{d+1}^{1}\Psi_{\alpha}=\Lambda_{\alpha}^{2}\Psi_{\alpha}\,.$
(2.41)

Again, $\Lambda_{\alpha}>0$ for all $\alpha$ due to the fact that
$\text{det}(M_{1})$ and $\text{det}(M_{2})$ are never simultaneously zero on
$Y$ in our construction. Then, as eigenfunctions of $\mathsf{H}_{d+1}^{2}$, we
can take

$\tilde{\Psi}_{\alpha}=\frac{1}{\Lambda_{\alpha}}\mathsf{D}_{d+1}\Psi_{\alpha}\,,$
(2.42)

similar to Eq. (2.14). Now, the (regularized) fermion partition function can
be written in terms of these data following [Fuj84],

$\frac{\mathcal{Z}_{d+1}(Y)}{\mathcal{Z}^{(0)}_{d+1}(Y)}=\text{Det}(\mathcal{J})\text{Det}(\widetilde{\mathcal{J}}^{\dagger})\frac{\prod\limits_{\alpha}\Lambda_{\alpha}}{\prod\limits_{\alpha}\Lambda_{\alpha}^{(0)}}\,.$
(2.43)

Here, $\mathcal{J}$ and $\widetilde{\mathcal{J}}$ are formally unitary
infinite-dimensional Jacobian matrices with entries

$(\mathcal{J})_{\alpha\beta}=\int_{Y}\Psi^{\dagger}_{\alpha}\Psi^{(0)}_{\beta}\,,\quad(\widetilde{\mathcal{J}})_{\alpha\beta}=\int_{Y}\tilde{\Psi}^{\dagger}_{\alpha}\tilde{\Psi}^{(0)}_{\beta}\,.$
(2.44)

The quantities with superscript $(0)$ are defined in the same way as the
corresponding quantities without the superscript, except for that $M$ is fixed
to be a nonzero constant. Similarly, the $d$-dimensional (regularized) fermion
partition function can be written as

$\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}^{(0)}_{d}(X)}=\text{Det}(\mathcal{K})\text{Det}(\widetilde{\mathcal{K}}^{\dagger})\frac{\prod\limits_{a}\lambda_{a}}{\prod\limits_{a}\lambda_{a}^{(0)}}\,,$
(2.45)

with

$(\mathcal{K})_{ab}=\int_{X}\psi^{\dagger}_{a}\psi^{(0)}_{b}\,,\quad(\widetilde{\mathcal{K}})_{ab}=\int_{X}\tilde{\psi}^{\dagger}_{a}\tilde{\psi}^{(0)}_{b}\,.$
(2.46)

The spectrum of the operators $\mathsf{H}_{d+1}^{1,2}$ contains that of the
operators $\mathsf{H}_{d}^{1,2}$ when $m_{0}$ is much larger than the inverse
length of the collar neighborhood of $X$ in $Y$, due to the existence of the
exponentially localized modes. Explicitly, consider an ansatz, labeled by the
index $a$,

$\Psi_{a}=\mathcal{N}(\psi_{a}\otimes
v^{+})\exp\left(-\int_{y^{*}}^{y}M_{1}(y^{\prime})dy^{\prime}\right)$ (2.47)

where $y^{*}$ is a point inside the interval $[y_{+},y_{-}]$, say the
midpoint, and $v^{+}$ is the eigenvector $\sigma^{3}v^{+}=+v^{+}$ (recall our
conventions for the gamma matrices (2.6)). $\mathcal{N}$ is a normalization
constant. The ansatz satisfies

$\displaystyle\begin{split}\mathsf{H}_{d+1}^{1}\Psi_{a}&=(\mathsf{H}_{d}^{1}\psi_{a})\otimes
v^{+}\exp\left(-\int_{y^{*}}^{y}M_{1}(y^{\prime})dy^{\prime}\right)\\\
&=\lambda_{a}^{2}\Psi_{a}\,.\end{split}$ (2.48)

The normalization constant $\mathcal{N}$ needs to be fixed such that the
ansatzes satisfy the orthonormality condition

$\int_{Y}\Psi_{a}^{\dagger}\Psi_{b}=\delta_{ab}\,.$ (2.49)

This can be explicitly done if $m_{0}\gg|y_{-}-y_{+}|^{-1}$, by choosing

$\mathcal{N}=\left[\int_{y_{+}}^{y_{-}}dy\,\exp\left(-2\int_{y^{*}}^{y}M_{1}(y^{\prime})dy^{\prime}\right)\right]^{-1/2}$
(2.50)

with which the orthonormality condition is satisfied modulo corrections that
are exponentially suppressed by $m_{0}|y_{-}-y_{+}|\gg 1$. In the same manner
we define the ansatz $\tilde{\Psi}_{a}$ satisfying
$\mathsf{H}_{d+1}^{2}\tilde{\Psi}_{a}=\lambda_{a}^{2}\tilde{\Psi}_{a}$. Note
that $\mathcal{N}$ is a number which depends only on the fixed interface
configuration of $M_{1}(y)$ and does not depend on the configuration of
$M_{2}$. Indeed, modulo exponentially small corrections, we have

$\int_{Y}\Psi_{a}^{\dagger}\Psi^{(0)}_{b}=\int_{X}\psi_{a}^{\dagger}\psi^{(0)}_{b}\,,\quad\int_{Y}\tilde{\Psi}_{a}^{\dagger}\tilde{\Psi}^{(0)}_{b}=\int_{X}\tilde{\psi}_{a}^{\dagger}\tilde{\psi}^{(0)}_{b}\,.$
(2.51)

The expression (2.43) is obtained by expanding the fermion fields $\Psi$ and
$\bar{\Psi}$ in terms of the eigenfunctions $\Psi_{\alpha}$ and
$\tilde{\Psi}_{\alpha}$, respectively. Instead, we can perform a partial
expansion with respect to the localized ansatzes,

$\Psi=\sum_{a}(c_{a}\Psi_{a}+\Psi^{\prime})\,,\quad\bar{\Psi}=\sum_{a}(d_{a}\tilde{\Psi}_{a}^{\dagger}+\bar{\Psi}^{\prime})\,,$
(2.52)

where $c_{a}$ and $d_{a}$ are expansion coefficients which are grassmannian.
$\Psi^{\prime}$ belongs to the orthogonal complement of the subspace spanned
by the localized modes $\Psi_{a}$. Similarly, $\bar{\Psi}^{\prime}$ is in the
orthogonal complement of the subspace spanned by $\tilde{\Psi}_{a}^{\dagger}$.

Inserting the partial expansions (2.52), the fermion path integral becomes

$\displaystyle\frac{\mathcal{Z}_{d+1}(Y)}{\mathcal{Z}^{(0)}_{d+1}(Y)}$
$\displaystyle=\text{Det}(\mathcal{K})\text{Det}(\widetilde{\mathcal{K}}^{\dagger})\frac{\prod_{a}\lambda_{a}}{\prod_{a}\lambda_{a}^{(0)}}\times\frac{\int[D\bar{\Psi}^{\prime}][D\Psi^{\prime}]\text{exp}(-\int_{Y}\bar{\Psi}^{\prime}\mathsf{D}_{d+1}\Psi^{\prime})}{\int[D\bar{\Psi}^{\prime}][D\Psi^{\prime}]\text{exp}(-\int_{Y}\bar{\Psi}^{\prime}\mathsf{D}_{d+1}^{(0)}\Psi^{\prime})}$
$\displaystyle=\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}^{(0)}_{d}(X)}\times\frac{\int[D\bar{\Psi}^{\prime}][D\Psi^{\prime}]\text{exp}(-\int_{Y}\bar{\Psi}^{\prime}\mathsf{D}_{d+1}\Psi^{\prime})}{\int[D\bar{\Psi}^{\prime}][D\Psi^{\prime}]\text{exp}(-\int_{Y}\bar{\Psi}^{\prime}\mathsf{D}_{d+1}^{(0)}\Psi^{\prime})}$
(2.53)

where in the first line, we used Eq. (2.51), and in the second line, we
substituted Eq. (2.45). Then, for $m_{0}\to\infty$, we have

$\int_{Y}\bar{\Psi}^{\prime}\mathsf{D}_{d+1}\Psi^{\prime}\approx\int_{Y}\bar{\Psi}^{\prime}M_{1}(y)\Psi^{\prime}\,.$
(2.54)

Recall that $M_{1}=\pm m_{0}$ inside $Y_{\pm}$, whereas in the middle region
$Y_{L}$ it is a smooth interpolating function between the two values $\pm
m_{0}$. Eq. (2.54) is valid because we first factored out the localized modes.
For a localized mode, its variation along the $y$ direction will cancel the
$m_{0}$ mass term and we cannot ignore other terms in the action even in the
limit $m_{0}\to\infty$.

Now, since Eq. (2.54) holds independent of the configuration of the other mass
parameter $M_{2}$, we can deduce that

$\frac{\int[D\bar{\Psi}^{\prime}][D\Psi^{\prime}]\text{exp}(-\int_{Y}\bar{\Psi}^{\prime}\mathsf{D}_{d+1}\Psi^{\prime})}{\int[D\bar{\Psi}^{\prime}][D\Psi^{\prime}]\text{exp}(-\int_{Y}\bar{\Psi}^{\prime}\mathsf{D}_{d+1}^{(0)}\Psi^{\prime})}\rightarrow
1$ (2.55)

as $m_{0}\to\infty$. Thus, in this limit, we conclude

$\frac{\mathcal{Z}_{d+1}(Y)}{\mathcal{Z}^{(0)}_{d+1}(Y)}=\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}^{(0)}_{d}(X)}$
(2.56)

as desired. Combining Eq. (2.56) with Eq. (2.39) from the previous subsection,
we arrive at our central claim:

$\left.\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}^{(0)}_{d}(X)}\middle/\left|\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}_{d}^{(0)}(X)}\right|\right.=\text{exp}(2\pi\mathrm{i}(\eta_{K}(Y_{+};\mathtt{APS})-\eta_{K^{(0)}}(Y_{+};\mathtt{APS}))\,.$
(2.57)

In words, we claim that the phase of the Euclidean partition function of free
fermions coupled to arbitrary background gauge fields and spacetime-dependent
mass parameters is in general given by the APS $\eta$-invariant of the
hermitian operator $K$ that extends the $d$-dimensional Dirac operator to a
$(d+1)$-dimensional manifold $Y_{+}$ which bounds the $d$-dimensional
spacetime manifold $X$. If the $d$-dimensional mass matrix $M$ becomes
infinitely large, one expects to have
$\left|\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}_{d}^{(0)}(X)}\right|\rightarrow
1$. Then, Eq. (2.57) defines the partition function of a _parameterized_
invertible field theory given by the low-energy limit of the $d$-dimensional
fermions with modulated mass parameters.

As a remark, we mention that both the actual fermion system with the partition
function $\mathcal{Z}_{d}(X)$ that we are interested in and the regulator
fermions given by the partition function $\mathcal{Z}_{d}^{(0)}(X)$ are on the
same footing at the technical level. Although we have required the mass
parameters for the regulator fermions to be constant-valued for simplicity,
everything that we said is still valid even if we allow complicated mass
configurations for the regulator fermions. What this tells us is that
precisely speaking, only the difference between two fermion systems is a well-
defined invertible theory, and in that sense, the regulator fermions are there
to specify a “reference point.” This indeed reflects a familiar fact that a
QFT usually has an ambiguity in how one fixes the counterterms that depend
only on the backgrounds fields, where different choices of the counterterms
correspond to different regularization schemes. Still, we will keep making a
conceptual distinction between the physical system that we are interested in
and the regulator system, while exploiting the freedom to choose a convenient
“reference point” when needed.

###  2.3 Generalized cohomology perspectives

We have related the $d$-dimensional fermion system with modulated mass
parameters with a $(d+1)$-dimensional fermion system. It would be useful to
rephrase the statement in a more general perspective using the language of the
generalized cohomology theory.

The invertible field theories, which are the IR fixed points of trivially
gapped QFTs, define a generalized cohomology theory [Kit13, Kit15, FH16,
GJF17, YY21, Yam21]. Let $\mathfrak{M}_{d}$ be the space of $d$-dimensional
invertible theories. Then the statement, or conjecture, is that the set of
spaces $\mathfrak{M}_{d}$ is equipped with a particular structure, called the
$\Omega$-spectrum, to define a generalized cohomology theory. The generalized
cohomology theory on a topological space $P$, which we call
$\operatorname{Inv}^{d}(P)$, is defined by

$\operatorname{Inv}^{d}(P)=[P,\mathfrak{M}_{d}]\,,$ (2.58)

where $[P,Q]$ is the set of homotopy classes of continuous maps from $P$ to
$Q$. Moreover, this cohomology theory $\operatorname{Inv}$ is identified with
a cobordism theory [Kap14, KTTW14, FH16, Yon18, YY21, Yam21].888 When we focus
on the invertible theories that are IR fixed points of free fermion theories,
the space $\mathfrak{M}_{d}$ is homotopic to the space $P_{d}$ of non-
degenerate mass matrices for (infinitely many) $d$-dimensional fermions. The
corresponding cohomology theory is isomorphic to the $K$-theory or
$KO$-theory, when the fermions are Dirac or Majorana, respectively[Kit09,
RSFL10]. We can regard a map $\mathcal{T}:P\to\mathfrak{M}_{d}$ as a family of
invertible QFTs parameterized by elements of $P$. In other words, the
cohomology group $\operatorname{Inv}^{d}(P)$ classifies the deformation
classes of parameterized invertible QFTs with parameter space $P$ [FKS17,
CFLS19a, CFLS19b]. In our context, the massive free fermion theory defines an
element of $\operatorname{Inv}^{d}(P)$, where $P$ is the space of non-
degenerate mass matrices for a specified number $N_{f}$ of flavors.999 As said
in the previous footnote, the space $P_{d}$ is the spectrum for the $K$ (or
$KO$) theory itself when $N_{f}=\infty$. In this case the relevant cohomology
group is $K^{d}(P_{d})=[P_{d},P_{d}]$, and the fermion theory corresponds to
the identity map.

The fact that $\operatorname{Inv}$ is a generalized cohomology theory means
that we can apply the axioms that such a theory satisfies. In particular, it
is natural to consider a _relative_ cohomology group $\operatorname{Inv}(P,A)$
where $A$ is a subset of $P$. Physically, the relative cohomology
$\operatorname{Inv}(P,A)$ classifies (the deformation classes) of invertible
QFTs parameterized by $P$, such that it is trivial when the parameter resides
in $A$. Given such a parameterized invertible QFT $\mathcal{I}$, and a
manifold with boundary $(M,\partial M)$, it is natural to consider the
partition function $Z^{\mathcal{I}}(M,p)$, where $p$ is a map $p:M\to P$ whose
restriction on $\partial M$ is contained in $A$ . In particular, for the
torsion part of $\operatorname{Inv}$, the partition function is the invariant
of the relative bordism class of the pair $(M,\partial M)$ in $(P,A)$.

Recall that in Eq. (2.57), the mass matrix has to be non-degenerate on the
boundary of $Y_{+}$ but does not need to be so inside the bulk. This means
that we were considering the pair $(\widetilde{P},P)$, where $\widetilde{P}$
is the space of all (possibly degenerate) mass matrices, and the exponentiated
$\eta$-invariant which appears on the right-hand side of Eq. (2.57) is
understood as the partition function of the invertible QFT parameterized by
the pair $(\widetilde{P},P)$. Therefore, (2.57) is a correspondence between
$\operatorname{Inv}^{d}(P)$ and $\operatorname{Inv}^{d+1}(\widetilde{P},P)$.
Indeed, because $\widetilde{P}$ is contractible, the general axiom of
cohomology tells us that

$\widetilde{\operatorname{Inv}}^{d}(P)\cong\operatorname{Inv}^{d+1}(\widetilde{P},P),$
(2.59)

where
$\widetilde{\operatorname{Inv}}^{d}(P)=\operatorname{Inv}^{d}(P,\text{pt})$ is
the reduced cohomology group.101010 We have the long exact sequence
$\widetilde{\operatorname{Inv}}^{d}(\widetilde{P})\to\widetilde{\operatorname{Inv}}^{d}(P)\to\operatorname{Inv}^{d+1}(\widetilde{P},P)\to\widetilde{\operatorname{Inv}}^{d+1}(\widetilde{P})$
and $\widetilde{\operatorname{Inv}}^{\star}(\widetilde{P})=0$ because
$\widetilde{P}$ is contractible.  The $d$-dimensional parameterized invertibe
theory is an element of the cohomology group
$\widetilde{\operatorname{Inv}}^{d}(P)$ (since we are not interested in purely
gravitational invertible theories, taking the reduced comology group
suffices). Then, Eq. (2.57) really states that the correspondence (2.59) in
the generalized cohomology theory holds at the level of partition functions of
parameterized invertible QFTs coming from free fermions.

##  3 Berry curvature

The group of ordinary invertible field theories without any parameters are
known to be generated by two classes of invertible theories [FH16, YY21,
Yam21]. On the one hand, there are invertible theories which match the
perturbative anomalies of the boundary theories. The partition functions of
such invertible theories are given by an integral of the Chern-Simons forms
for the background gauge fields as well as the curvature of the spacetime
manifold, which locally are differential forms. Such invertible theories are
classified by the free part of the suitable generalized cohomology group. On
the other hand, there are invertible theories which correspond to the non-
perturbative anomalies of the boundary theories. Their partition functions
define bordism invariants, and these topological invertible theories are
classified by the torsion part of the same generalized cohomology group
[Kap14, KTTW14, Yon18]. A general invertible theory is a mixture of these two
parts.

This understanding continues to hold for the parameterized invertible field
theories as well. In particular, our fermion system in the infinite mass limit
defines a parameterized invertible theory as we explained, and thus its
partition function will take the following general form:

$\left.\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}^{(0)}_{d}(X)}\middle/\left|\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}_{d}^{(0)}(X)}\right|\right.\rightarrow\phi(X)\exp\left(2\pi\mathrm{i}\int_{X}\mathcal{W}\right)\in
U(1)$ (3.1)

as mass goes to infinity, $|M|\to\infty$. Here, $\phi(X)\in U(1)$ is the value
of a bordism invariant $\phi$ evaluated on the spacetime manifold $X$
(equipped with the background gauge fields and the modulated mass parameters),
and $\mathcal{W}$ is locally a (real-valued) differential form made out of the
background gauge fields (as well as the curvature of the spacetime manifold)
and the modulated mass parameters.

Following [KS20a, KS20b, HKT20], we call the right-hand side of Eq. (3.1) a
higher Berry phase. The differential form $\mathcal{W}$ can then be thought of
as a higher analogue of the Berry connection in quantum mechanics, and it is
convenient to take the formal derivative of it to define the (higher) Berry
curvature:

$\mathcal{B}=\mathrm{d}\mathcal{W}\,.$ (3.2)

As was advocated in the Introduction, the Berry curvature is the “anomaly
polynomial” for the theories living on the boundary of the $d$-dimensional
fermion system, which will have the anomaly in the space of coupling constants
(i.e., the mass parameter space) [CFLS19a, CFLS19b].

In this section, we will focus on this Berry curvature, and the bordism
invariant part will be discussed in Section 4. For the purpose of studying
only the Berry curvature part, it will suffice to consider the spacetime
manifold $X$ which is null-bordant even while keeping the mass to be non-
degenerate inside the null-bordism. That is, we will assume the existence of a
$(d+1)$-dimensional manifold $Y$ whose boundary is $\partial Y=X$,111111This
$Y$ is not to be confused with the specific manifold $Y$ that we constructed
in Section 2. where the mass matrix is extended into $Y$ in such a way that it
is everywhere non-degenerate.121212Note that this is different from the
general discussion in Section 2, where inside the manifold $Y_{+}$ with
$\partial Y_{+}=X$ the mass matrix was allowed to become degenerate. In this
case, the fermion partition function in the infinite mass limit is given as

$\left.\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}^{(0)}_{d}(X)}\middle/\left|\frac{\mathcal{Z}_{d}(X)}{\mathcal{Z}_{d}^{(0)}(X)}\right|\right.\rightarrow\exp\left(2\pi\mathrm{i}\int_{Y}\mathcal{B}\right)\,.$
(3.3)

We will provide a universal formula for the Berry curvature $\mathcal{B}$.

In view of Eq. (2.57), our strategy to compute the Berry curvature will be by
using the APS index theorem which relates the $\eta$-invariant to other
quantities. In the more familiar case where there are no modulated mass
parameters, it is well-known that the $\eta$-invariant is partly given by the
integral of the Chern-Simons form, which is an immediate consequence of the
APS index theorem. To take into account the modulated mass parameters, one
needs the version of the APS index theorem which treats Dirac operators
coupled to the _superconnection_ , which combines the ordinary connections
(background gauge fields) and the mass parameters. We will review its
definition in the following subsection. The index theorem for Dirac operators
coupled to the superconnection is discussed in Appendix B.

###  3.1 Superconnection

The spacetime-dependent modulated mass parameters and the background gauge
fields for the global symmetries can be combined into what is known as the
superconnection [Qui85, Kah11] (see also [BGV92]). In physics, superconnection
appeared in the context of string theory [KW99, KL00, TTU00, AIO00], and
recently in the study of massive fermions with spacetime-dependent mass terms
[CFLS19a, KS21, GY21]. Our definitions and conventions directly follow the
ones given in [KS21], and we repeat them here for completeness.

##### Superconnection of even type

When $d$ is even and there are $N_{f}$ Dirac fermions, the most general mass
matrix $M$ is valued in complex $N_{f}\times N_{f}$ matrices. In addition, one
can also couple the system to the $U(N_{f})\times U(N_{f})$ background gauge
fields which we denote as $A_{+}$ and $A_{-}$.131313If we want to consider
only a subgroup of $U(N_{f})\times U(N_{f})$, then some of the components of
$A_{\pm}$ need to be appropriately set to zero. Locally, they are one-forms
taking values in hermitian $N_{f}\times N_{f}$ matrices. The background gauge
invariance can be maintained by allowing the mass term to transform in the
bifundamental representation of $U(N_{f})\times U(N_{f})$.

This set of background fields can be packaged into a _superconnection_
$\mathcal{A}$, which is defined as

$\mathcal{A}=\begin{pmatrix}-\mathrm{i}A_{+}&\mathrm{i}\widetilde{M}^{\dagger}\\\
\mathrm{i}\widetilde{M}&-\mathrm{i}A_{-}\end{pmatrix}\,.$ (3.4)

Here, $\widetilde{M}\equiv M/\Lambda$ is a dimensionless mass matrix where
$\Lambda$ is a cutoff scale, which is to be taken to infinity at the end of
calculations [KS21] (see also the discussion around Eq. (B.28)). We call
$\widetilde{M}$ the renormalized mass matrix. The off-diagonal blocks in (3.4)
anticommute with odd degree differential forms. The field strength for the
superconnection is given by

$\mathcal{F}=\mathrm{d}\mathcal{A}+\mathcal{A}^{2}=\begin{pmatrix}-\mathrm{i}F_{+}-\widetilde{M}^{\dagger}\widetilde{M}&\mathrm{i}D\widetilde{M}^{\dagger}\\\
\mathrm{i}D\widetilde{M}&-\mathrm{i}F_{-}-\widetilde{M}\widetilde{M}^{\dagger}\end{pmatrix}$
(3.5)

where $F_{\pm}=\mathrm{d}A_{\pm}-\mathrm{i}A^{2}_{\pm}$ are $U(N_{f})$ field
strengths, and
$D\widetilde{M}=\mathrm{d}\widetilde{M}-\mathrm{i}A_{-}\widetilde{M}+\mathrm{i}\widetilde{M}A_{+}$
and
$D\widetilde{M}^{\dagger}=\mathrm{d}\widetilde{M}^{\dagger}-\mathrm{i}A_{+}\widetilde{M}^{\dagger}+\mathrm{i}\widetilde{M}^{\dagger}A_{-}$
are the covariant derivatives acting on the renormalized mass matrix which is
in the bifundamental representation.

Similar to the ordinary connections, we can define the Chern character for the
superconnection:

$\mathop{\mathrm{ch}}\nolimits(\mathcal{F})=\sum_{k=0}^{\infty}(\frac{\mathrm{i}}{2\pi})^{k/2}[\text{Str}(e^{\mathcal{F}})]_{k}\,.$
(3.6)

Here, $[\cdots]_{k}$ means that we take the degree $k$ part of the
differential form. The supertrace Str is defined by

$\text{Str}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\text{Tr}(a)-\text{Tr}(d)\,.$ (3.7)

If the mass parameters are much smaller than the cutoff scale $\Lambda$, then
we simply have
$\mathop{\mathrm{ch}}\nolimits(\mathcal{F})=\mathop{\mathrm{ch}}\nolimits(F_{+})-\mathop{\mathrm{ch}}\nolimits(F_{-})$
where $\mathop{\mathrm{ch}}\nolimits(F_{\pm})$ is the usual Chern character
for the $U(N_{f})$ gauge field. Importantly, we will see that to relate the
Chern character of the superconnection to the Berry curvature, we need to
consider the mass parameters that are much bigger than the cutoff scale in
which case the mass parameters contribute nontrivially to the Berry curvature.

##### Superconnection of odd type

When $d$ is odd and we have $N_{f}$ Dirac fermions, the mass matrix $M$ takes
values in $N_{f}\times N_{f}$ hermitian matrices. We can further turn on the
background gauge field $A$ for the $U(N_{f})$ global symmetry, if we allow the
mass matrix to transform in the adjoint representation.141414Again, if we want
to consider only a subgroup of $U(N_{f})$, then appropriate components of $A$
should be set to zero.

We can combine the gauge field $A$ and the mass matrix $M$ into the following
superconnection:

$\mathcal{A}=\begin{pmatrix}-\mathrm{i}A&\mathrm{i}\widetilde{M}\\\
\mathrm{i}\widetilde{M}&-\mathrm{i}A\end{pmatrix}\,.$ (3.8)

Again, the off-diagonal blocks anticommute with odd degree differential forms.
The field strength is similarly defined as

$\mathcal{F}=\mathrm{d}\mathcal{A}+\mathcal{A}^{2}=\begin{pmatrix}-\mathrm{i}F-\widetilde{M}^{2}&\mathrm{i}D\widetilde{M}\\\
\mathrm{i}D\widetilde{M}&-\mathrm{i}F-\widetilde{M}^{2}\end{pmatrix}$ (3.9)

where $F=\mathrm{d}A-\mathrm{i}A^{2}$ and
$D\widetilde{M}=\mathrm{d}\widetilde{M}-\mathrm{i}[A,\widetilde{M}]$.

The Chern character of the superconnection is defined by the same equation
(3.6) as in the even case, with the supertrace now given as

$\text{Str}\begin{pmatrix}a&b\\\
c&d\end{pmatrix}=\sqrt{2}\mathrm{i}^{-3/2}\text{Tr}(b)\,.$ (3.10)

###  3.2 Berry curvature from Chern character of superconnection

Having defined the superconnection and the corresponding Chern character, our
goal is now to compute the Berry curvature $\mathcal{B}$ in Eq. (3.3) for the
$d$-dimensional fermions living on the spacetime manifold $X$. For this, we
will use the results from Section 2, in particular Eqs. (2.37) and (2.57), and
also the APS index theorem explained in Appendix B:

$\eta_{K}(\partial
W)=-\operatorname{Ind}Q+\lim_{\Lambda\to\infty}\int_{W}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{M}=M/\Lambda))\hat{A}(TW)\,.$
(3.11)

On the left-hand side, we have the $(d+1)$-dimensional $\eta$-invariant of the
hermitian operator $K$ that we discussed in Section 2 and it depends on the
background gauge fields as well as the modulated mass parameters. Here, $W$ is
a possibly non-compact $(d+2)$-dimensional manifold, and the operator $K$ is
defined on its boundary $\partial W$ (which does not have a boundary itself,
$\partial\partial W=\emptyset$, and so we do not have to worry about the
boundary condition for $K$). On the right-hand side, $Q$ is a hermitian
operator on $W$ which is defined in Eq. (B.2). The precise form of this
operator $Q$ will not be important for the computation of the Berry curvature,
and the details are given in Appendix B, including the definition of the index
$\operatorname{Ind}Q$. The only important part is that $Q$ needs to have a
discrete spectrum so that the integer-valued index can be defined;
$\operatorname{Ind}Q\in\mathbb{Z}$. For this to hold, the absolute value of
mass eigenvalues should become infinitely large in the asymptotic infinity of
$W$ when $W$ is non-compact.

Next, $\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{M}=M/\Lambda))$ is
the Chern character for the superconnection that we defined in the last
subsection, and here we emphasized that the field strength of the
superconnection depends on the mass parameters only through the combination
$\widetilde{M}=M/\Lambda$. $\hat{A}(TW)$ is the $\hat{A}$-genus for the
tangent bundle of $W$ which will be responsible for the gravitational part of
the Berry curvature in the end. The product
$\mathop{\mathrm{ch}}\nolimits(\mathcal{F})\hat{A}(TW)$ is a formal polynomial
of differential forms, and in the above equation we should take the
degree-$(d+2)$ part of $\mathop{\mathrm{ch}}\nolimits(\mathcal{F})\hat{A}(TW)$
and then integrate over $W$. The limit $\Lambda\to\infty$ is supposed to be
taken after the integral is evaluated [KS21].

To relate this to Eq. (2.57), we consider the following concrete setup. Recall
first that we assume in this section that there exists a $(d+1)$-dimensional
manifold $Y$ with $\partial Y=X$ where the $d$-dimensional mass matrix is
extended into $Y$ while being non-degenarate everywhere. Take such a manifold
$Y$, and we will let the $(d+2)$-dimensional manifold on which we will apply
the index theorem (3.11) to be $W=Y\times\mathbb{R}$, whose boundary is
$\partial W=X\times\mathbb{R}$.151515To be very precise, we want to have a
collar neighborhood of $\partial W$ in $W$ to apply the derivation of the
index theorem given in Appendix B. This can be achieved by appropriately
deforming $W$ and the background field configurations, which does not affect
the dicussions in this section. See Figure 3.

Figure 3: We define a $(d+2)$-dimensional non-compact manifold $W$ as
$W=Y\times\mathbb{R}$. On $W$, the $d$-dimensional mass matrix
$M(x)=M_{0}f(x)$ on $X$ is extended to $\mathcal{M}(w,y)$, which has the
specifically chosen asymptotic behavior as $w\to\pm\infty$.

Let $w$ be the cooridinate along the $\mathbb{R}$ direction on $W$, and at the
boundary of the $w=0$ slice our $d$-dimensional system of interest lives. We
will write the modulated $d$-dimensional mass matrix on $X$ as
$M(x)=M_{0}f(x)$ where $x$ denotes the coordinates on $X$, $M_{0}>0$ is a
constant mass, and $f(x)$ is a dimensionless non-degenerate matrix-valued
function. The function $f(x)$ is extended into $Y$ by a non-degenerate matrix-
valued function $\hat{f}(y)$ where $y$ denotes the coordinates on
$Y$.161616The background gauge fields on $X$ also should be suitably extended
into $Y$ as well as into $W$. We will let them extend into $Y$ in an arbitrary
fashion and be independent of $w$. Now, we let this mass matrix to be extended
into the whole $W$ (away from $w=0$) in the following way:171717In Appendix B
and other parts of the paper, we usually denote both the $d$-dimensional mass
matrix and its extension to higher dimensions by the same symbol $M$, whereas
here we use different symbols to be more explicit. We hope this does not cause
any confusion.

$\mathcal{M}(w,y)=\begin{cases}M_{0}^{2}w\hat{f}(y)&\quad\text{for $w\geq
w^{*}$}\,,\\\ M_{0}\hat{f}(y)&\quad\text{at $w=0$}\,,\\\
-M_{0}^{2}w&\quad\text{for $w\leq-w^{*}$}\,,\end{cases}$ (3.12)

where $w^{*}>0$ is some fixed position. The precise way the mass parameters
interpolate in the intermediate region $w\in(-w^{*},w^{*})$ as well as the
actual value of $w^{*}$ will not be important as we will see. We assume that
$\mathcal{M}$ is finite for finite $|w|$, and otherwise only the asymptotic
behavior of $\mathcal{M}(w,y)$ as $w\to\pm\infty$ will be relevant. As
mentioned above, to apply the index theorem, all of the absolute values of
eigenvalues of $\mathcal{M}(w,y)$ should become infinite as $w\to\pm\infty$,
which holds because $\hat{f}(y)$ is non-degenerate. See Figure 3 for the
illustration.

We are then ready to apply the index theorem (3.11) to this configuration.
First, we look at the integral on the right-hand side of Eq. (3.11). The
integral over $W$ factorizes as
$\int_{W}=\int_{Y}\int_{w=-\infty}^{w=+\infty}$, and we will examine the
integral over $w$. The $\hat{A}$-genus does not depend on $w$ in this
geometry: $\hat{A}(TW)=\hat{A}(T(Y\times\mathbb{R}))=\hat{A}(TY)$. We can thus
pull out the $\hat{A}$-genus out of the $w$ integral. We will devide the $w$
integral into three parts,

$\displaystyle\begin{split}&\int_{w=-\infty}^{w=\infty}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{\mathcal{M}}=\mathcal{M}/\Lambda))\\\
=&\int_{w=w^{*}}^{w=\infty}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{\mathcal{M}}=\mathcal{M}/\Lambda))+\int_{w=-\infty}^{w=-w^{*}}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{\mathcal{M}}=\mathcal{M}/\Lambda))+\int_{w=-w^{*}}^{w=w^{*}}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{\mathcal{M}}=\mathcal{M}/\Lambda))\,.\end{split}$
(3.13)

Consider a change of variable: $\widetilde{M}_{0}(w)=M_{0}^{2}|w|/\Lambda$.
$\widetilde{M}_{0}$ is dimensionless, and it determines the overall scale of
the $(d+2)$-dimensional renormalized mass matrix
$\widetilde{\mathcal{M}}(w,y)$. Note that $\widetilde{\mathcal{M}}(w,y)$
depends on $w$ only through the dimensionless combination $\widetilde{M}_{0}$,
at least outside of the intermediate region $w\in(-w^{*},w^{*})$. There, the
change of variable gives us

$\displaystyle\begin{split}\int_{w=w^{*}}^{w=\infty}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{\mathcal{M}}=\mathcal{M}/\Lambda))&=\int_{\widetilde{M}_{0}=M_{0}^{2}w^{*}/\Lambda}^{\widetilde{M}_{0}=\infty}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{M}))\,,\\\
\int_{w=-\infty}^{w=-w^{*}}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{\mathcal{M}}=\mathcal{M}/\Lambda))&=-\int_{\widetilde{M}_{0}=M_{0}^{2}w^{*}/\Lambda}^{\widetilde{M}_{0}=\infty}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{M}_{0}))\,,\end{split}$
(3.14)

where $\widetilde{M}=\widetilde{M}_{0}\hat{f}$. On the right-hand side, now
the $\Lambda$ dependence is only in the lower limit of the integrals, and it
is easy to take the $\Lambda\to\infty$ limit.181818Although we put a tilde on
$\widetilde{M}=\widetilde{M}_{0}\hat{f}$, there is no $\Lambda$ dependence
here anymore since $\widetilde{M}_{0}$ is simply a dummy variable to be
integrated over. What we are effectively doing here is to simply rewrite the
integral on the manifold $W$ along the $w$ direction on the left-hand side as
the integral inside the parameter space on the right-hand side. Moreover, we
can deduce that the integral over the intermediate region $w\in(-w^{*},w^{*})$
should vanish in the limit $\Lambda\to\infty$, since this integral will
correspond to an integral in the parameter space from
$\widetilde{\mathcal{M}}=0$ to $\widetilde{\mathcal{M}}=0$ as $\mathcal{M}$ is
finite for $w\in(-w^{*},w^{*})$. We see here that the precise interpolating
configuration of $\mathcal{M}$ in the intermediate region is irrelevant in the
$\Lambda\to\infty$ limit as long as $\mathcal{M}$ remains finite as we
claimed.

Combining, we obtain

$\lim_{\Lambda\to\infty}\int_{w=-\infty}^{w=\infty}\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{\mathcal{M}}=\mathcal{M}/\Lambda))=\int_{\widetilde{M}_{0}=0}^{\widetilde{M}_{0}=\infty}\left[\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{M}))-\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{M}_{0}))\right]\,.$
(3.15)

From now on, we will simply write
$\mathop{\mathrm{ch}}\nolimits(\mathcal{F})\equiv\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{M}))$
and
$\mathop{\mathrm{ch}}\nolimits(\mathcal{F}^{(0)})\equiv\mathop{\mathrm{ch}}\nolimits(\mathcal{F}(\widetilde{M}_{0}))$
for brevity. Intuitively, $\mathop{\mathrm{ch}}\nolimits(\mathcal{F}^{(0)})$
is the contribution from the regulator fermions living at $w\to-\infty$, for
which we have chosen the mass parameters to be constant. The integral from
$\widetilde{M}_{0}=0$ to $\widetilde{M}_{0}=\infty$ can be thought of as an
integral in the parameter space along the “radial” direction, that is, we
integrate over the overall scale of the mass matrix from zero to infinity.

If we now insert Eq. (3.15) into Eq. (3.11), we obtain

$\eta_{K}(X\times\mathbb{R})=\int_{Y}\int_{\widetilde{M}_{0}=0}^{\widetilde{M}_{0}=\infty}\left[\mathop{\mathrm{ch}}\nolimits(\mathcal{F})-\mathop{\mathrm{ch}}\nolimits(\mathcal{F}^{(0)})\right]\wedge\hat{A}(TY)\quad\text{mod
$\mathbb{Z}$}$ (3.16)

where we have inserted $\partial W=X\times\mathbb{R}$ on the left-hand side.
The mod $\mathbb{Z}$ difference is from the index $\operatorname{Ind}Q$, and
this difference can be ignored for our purpose since in the end we will
exponentiate both sides.

Our remaining task is to relate the left-hand side of Eq. (3.16) to the
partition function of the $d$-dimensional fermion system of interest, using
the results from Section 2. To this end, first, using the fact that the
precise form of the $(d+2)$-dimensional mass matrix $\mathcal{M}(w,y)$ for
finite $w$ is not important, we will deform the configuration (3.12) into the
following new mass profile:

$\mathcal{M}(w,y)=\begin{cases}M_{0}^{2}w\hat{f}(y)&\quad\text{for
$w\to+\infty$}\,,\\\ M_{1}\hat{f}(y)&\quad\text{for $w^{**}\leq w\leq
w^{**}+L$}\,,\\\ M_{1}&\quad\text{for $-w^{**}-L\leq w\leq-w^{**}$}\,,\\\
-M_{0}^{2}w&\quad\text{for $w\to-\infty$}\,,\end{cases}$ (3.17)

where $w^{**}>w^{*}$ is some fixed position, $M_{1}>M_{0}$ is a positive
constant, and $L>0$ is an arbitrary length scale. Compared to Eq. (3.12), we
are not changing the asymptotic behavior of the mass profile at
$w\to\pm\infty$, but making plateaus of length $L$ at $w^{**}\leq w\leq
w^{**}+L$ and $-w^{**}-L\leq w\leq-w^{**}$ along which the mass matrix is
independent of $w$. See Figure 4 for the illustration. Since the asymptotic
behaviors are unchanged, Eq. (3.16) is still valid with the new mass profile.
Importantly, this is independent of the values of $M_{1}$, $w^{**}$, and $L$.

Figure 4: The modified profile of the $(d+2)$-dimensional mass matrix
$\mathcal{M}(w,y)$. Only the $w>0$ region is shown. The asymptotic behavior at
$w\to\infty$ is unchanged, but there is a plateau of length $L$ along which
$\mathcal{M}(w,y)=M_{1}\hat{f}(y)$ does not depend on $w$. Due to the plateau,
the quantum state prepared by the half infinite cylinder given by
$X\times[w^{**}+\epsilon,\infty)$ is exponentially close to the ground state
$\ket{\Omega}_{M_{1}f(x)}$ on $X$ with the mass configuration $M_{1}f(x)$.

Now, consider cutting the infinitely long cylinder $\partial
W=X\times\mathbb{R}$ into three pieces, $X\times(-\infty,-w^{**}-\epsilon]$,
$X\times(-w^{**}-\epsilon,w^{**}+\epsilon)$, and
$X\times[w^{**}+\epsilon,\infty)$, where $\epsilon>0$ is a small positive
constant. We claim

$\eta_{K}(X\times\mathbb{R})=\eta_{K}\left(X\times(-w^{**}-\epsilon,w^{**}+\epsilon);\mathtt{APS}\right)+\mathcal{O}(e^{-LM_{1}})\,.$
(3.18)

This follows from Eq. (2.37) from Section 2 (see also Figure 2), which says
that for the purpose of computing an $\eta$-invariant on a manifold with
boundary, the APS boundary condition can be replaced by an infinitely long
half cylinder. From the QFT point of view, this was equivalent to saying that
the quantum state characterizing the APS boundary condition can be replaced by
the ground state on the boundary. To use this trick to compute
$\eta_{K}\left(X\times(-w^{**}-\epsilon,w^{**}+\epsilon);\mathtt{APS}\right)$
on the right-hand side of (3.18), note that the manifold
$X\times(-w^{**}-\epsilon,w^{**}+\epsilon)$ has two disconnected boundary
components (if $X$ is connected), and attaching half-infinite cylinders to the
two boundary components yields the original manifold $\partial
W=X\times\mathbb{R}$.

However, to replace the APS boundary condition, along the attached half-
infinite cylinder, all the background field configurations including the mass
parameters should remain fixed. With such a configuration, the path integral
over the cylinder prepares the desired ground state at the original APS
boundary. Our deformed mass profile (3.17) almost fulfills this purpose due to
the existence of the plateaus of finite length $L$. For instance, if we let
$\ket{\psi}$ denote the state defined by the path integral over the region
$w>w^{**}$, $\ket{\psi}$ is almost the ground state $\ket{\Omega}_{M_{1}f(x)}$
up to corrections which are suppressed by the factor of $e^{-LM_{1}}$, as
excited states will have the energies of order $M_{1}$. From this, the claimed
equation (3.18) follows.

Lastly, Eq. (2.57) tells us

$\left.\frac{\mathcal{Z}_{d}(X,M_{1}f(x))}{\mathcal{Z}^{(0)}_{d}(X,M_{1})}\middle/\left|\frac{\mathcal{Z}_{d}(X,M_{1}f(x))}{\mathcal{Z}_{d}^{(0)}(X,M_{1})}\right|\right.=\exp\left(2\pi\mathrm{i}\eta_{K}\left(X\times(-w^{**}-\epsilon,w^{**}+\epsilon);\mathtt{APS}\right)\right)\,.$
(3.19)

On the right-hand side, we do not need to include additional regulator
contribution $\eta_{K^{(0)}}$ as in Eq. (2.57) since the configuration already
includes the regulator fermions living on the boundary $w=-w^{**}-\epsilon$
where the mass matrix is fixed to be the constant $M_{1}$. Now, if we take the
limit $M_{1}\to\infty$, then the right-hand side reduces to
$\eta_{K}(X\times\mathbb{R})$ due to Eq. (3.18). Moreover, on the left-hand
side, this limit by definition defines the (exponentiated) Berry curvature
according to Eq. (3.3). We thus have

$\int_{Y}\mathcal{B}=\eta_{K}(X\times\mathbb{R})\quad\text{mod $\mathbb{Z}$}$
(3.20)

where again the mod $\mathbb{Z}$ difference is not important as only the
exponentiated quantities are of interest. Combined with Eq. (3.16), we can
finally read off the form of the Berry curvature:

$\mathcal{B}=\int_{\widetilde{M}_{0}=0}^{\widetilde{M}_{0}=\infty}\left[\mathop{\mathrm{ch}}\nolimits(\mathcal{F})-\mathop{\mathrm{ch}}\nolimits(\mathcal{F}^{(0)})\right]\wedge\hat{A}(TY)\,.$
(3.21)

That is, we obtain the Berry curvature by integrating the Chern character
(times the $\hat{A}$-genus) along the “radial” direction in the mass parameter
space. The resulting Berry curvature is a degree-$(d+1)$ differential form on
$Y$ which bounds the $d$-dimensional spacetime $X$, that is, $\partial Y=X$.
This is the desired universal formula for the Berry curvature of Dirac
fermions with modulated mass paramters. The formula works in general spacetime
dimensions and for arbitrary number of fermion species coupled to arbitrary
background gauge fields and modulated mass parameters.

###  3.3 Examples

We now demonstrate how Eq. (3.21) is used in practice in a few examples. A
relatively simple situation is when the mass parameters are invariant under
all the global symmetries for which background gauge fields are turned on. In
this case, the Chern character of the superconnection factorizes as

$\mathop{\mathrm{ch}}\nolimits(\mathcal{F})=\mathop{\mathrm{ch}}\nolimits(F)\wedge\mathop{\mathrm{ch}}\nolimits(\widetilde{M})$
(3.22)

where $\mathop{\mathrm{ch}}\nolimits(F)$ is the ordinary Chern character for
the background gauge field ($F$ is the field strength) and we wrote
$\mathop{\mathrm{ch}}\nolimits(\widetilde{M})$ to denote the Chern character
of the superconnection which is obtained by setting $A=0$ in Eq. (3.6). We
will first discuss this type of examples, and will also give an example where
the mass parameter breaks the global symmetry for which the background gauge
field is turned on at the end.

####  3.3.1 Single Dirac fermion in even dimensions

One of the simplest cases for which the mass term contributes to the Chern
character is when we have a single Dirac fermion in even number of spacetime
dimensions [CFLS19a]. Although these examples can be studied by the
conventional application of the anomalous axial rotation of the fermions, here
we explicitly see that we can reproduce them. The general mass parameter is a
complex number $M=M_{0}e^{\mathrm{i}\alpha}$, and the Lagrangian is given by

$\mathcal{L}=\bar{\psi}\left(\not{D}+M_{0}\exp(\mathrm{i}\alpha\gamma)\right)\psi\,,$
(3.23)

where $\gamma$ is the chirality matrix. We will let the regulator fermions to
have $\alpha=0$ and they will couple to the other background gauge fields in
the same way as the physical fermions.

If we turn off the background gauge fields, then the Chern character is given
by

$\mathop{\mathrm{ch}}\nolimits(\widetilde{M})=-\frac{\mathrm{i}}{2\pi}e^{-|\widetilde{M}|^{2}}\mathrm{d}\widetilde{M}\mathrm{d}\widetilde{M}^{*}\,.$
(3.24)

Integrating this along the radial direction in the mass parameter space gives
us $-\mathrm{d}\alpha/2\pi$.

From this we can read off the forms of the Berry curvature for various even
dimensional Dirac fermions. For instance, if we have a single flavor of a two-
dimensional Dirac fermion with the modulated complex mass term as well as the
background gauge field for the $U(1)$ vector symmetry turned on, the Berry
curvature for the theory is given by

$\mathcal{B}=-\frac{\mathrm{d}\alpha}{2\pi}\wedge\frac{F}{2\pi}\,.$ (3.25)

This implies that the effective action after we integrate out the fermion
contains the topological term $-\frac{1}{2\pi}\int\alpha F$, reproducing the
example we saw in Sec 1.1.

As another example, we can consider four-dimensional Dirac fermions, coupled
to a vector $U(N_{f})$ background gauge field and with the $U(N_{f})$
invariant diagonal mass term. In four-dimensions, the background geometry also
contributes to the Berry curvature through the $\hat{A}$-genus. We can read
off that the Berry curvature is given by

$\mathcal{B}=-\frac{\mathrm{d}\alpha}{2\pi}\wedge\Big{(}-\frac{N_{f}p_{1}}{24}+\frac{\text{tr}(F\wedge
F)}{8\pi^{2}}\Big{)}$ (3.26)

where $p_{1}=-\frac{1}{8\pi^{2}}\text{tr}(R\wedge R)$ is the first Pontryagin
class of the spacetime tangent bundle and $F$ is the $U(N_{f})$ background
field strength. The effective action then contains axion-coupling-like
topological terms $\sim\int\alpha\,\text{tr}(R\wedge R)$ and
$\sim\int\alpha\,\text{tr}(F\wedge F)$ if we integrate out the fermions.

####  3.3.2 Odd dimensional examples

In odd dimensions, the most general mass term for $N_{f}$ species of Dirac
fermions is given by a $N_{f}\times N_{f}$ hermitian matrix. Consider two
flavors of Dirac fermions in any odd dimensions,

$\mathcal{L}=\bar{\psi}(\not{D}+M)\psi$ (3.27)

where $M$ is valued in $2\times 2$ hermitian matrices. We can write the mass
matrix as $M=M_{0}+M_{1}\tau^{1}+M_{2}\tau^{2}+M_{3}\tau^{3}$ where
$\tau^{i}$’s are Pauli matrices. The degenerate locus is given by the equation
$M_{0}^{2}=M_{1}^{2}+M_{2}^{2}+M_{3}^{2}$, which forms a two copies of three-
dimensional cones meeting at the origin, inside the four-dimensional parameter
space $\widetilde{P}=\mathbb{R}^{4}$ including degenerate locus. This model
was analyzed in [HKT20], and our results are consistent.

The Berry curvature is most concisely written if we use the following
parameterization of the mass matrix. We can locally diagonalize the mass
matrix so that $M=U\Sigma U^{\dagger}$ where

$\Sigma=\begin{bmatrix}\lambda_{1}&0\\\ 0&\lambda_{2}\end{bmatrix}\,,\quad
U=\begin{bmatrix}\text{cos}\frac{\theta}{2}&e^{\mathrm{i}\phi}\text{sin}\frac{\theta}{2}\\\
-e^{-\mathrm{i}\phi}\text{sin}\frac{\theta}{2}&\text{cos}\frac{\theta}{2}\end{bmatrix}\,.$
(3.28)

$\lambda_{1}$ and $\lambda_{2}$ are real eigenvalues of $M$, and $\theta$ and
$\phi$ parameterize a two-sphere $S^{2}$. When $\lambda_{1}$ and $\lambda_{2}$
have opposite signs, this $S^{2}$ is non-contractible in the parameter space
without crossing the degenerate locus, and it sources a nontrivial Berry
curvature as we will see shortly. When the two eigenvalues have the same sign,
the $S^{2}$ is contractible while not hitting any degenerate locus, and it
does not source any Berry curvature. For the regulator fermions, we will set
$\theta=\phi=0$ and $\lambda_{1}=\lambda_{2}$ to be a fixed positive constant.
They will couple to the other background gauge fields in the same way as the
physical fermions.

For simplicity, consider first the case where there is no additional
background gauge fields. The integral of the Chern character along the radial
direction in the parameter space in this can be obtained as

$\int_{\widetilde{M}_{0}=0}^{\widetilde{M}_{0}=\infty}\,\mathop{\mathrm{ch}}\nolimits(\widetilde{M})=\frac{1}{2}(\text{sgn}(\lambda_{1})+\text{sgn}(\lambda_{2}))-\frac{1}{2}(\text{sgn}(\lambda_{1})-\text{sgn}(\lambda_{2}))\,\omega_{2}\,,$
(3.29)

where
$\omega_{2}=\frac{1}{4\pi}\text{sin}\theta\mathrm{d}\theta\mathrm{d}\phi$ is
the volume form on $S^{2}$ normalized to give 1 when integrated over $S^{2}$.
As we anticipated, the 2-sphere $S^{2}$ parameterized by $\theta$ and $\phi$
leads to a nontrivial contribution proportional to the volume form on $S^{2}$,
only when the sphere is not contractible without crossing the degenerate
locus.

For a concrete example, consider a three-dimensional theory of $2N_{f}$ Dirac
fermions. The mass term is given by $M\otimes I$ where $M$ is the above
$2\times 2$ hermitian matrix valued function and $I$ is the $N_{f}\times
N_{f}$ identity matrix. We couple the fermions to a background $U(N_{f})$
gauge field where the mass matrix is invariant under this $U(N_{f})$. The
Berry curvature 4-form of this system can then be read off,

$\mathcal{B}=\frac{1}{2}(\text{sgn}(\lambda_{1})+\text{sgn}(\lambda_{2})-2)\Big{(}-\frac{N_{f}p_{1}}{24}+\frac{\text{tr}(F\wedge
F)}{8\pi^{2}}\Big{)}-\frac{1}{2}(\text{sgn}(\lambda_{1})-\text{sgn}(\lambda_{2}))\,\omega_{2}\wedge\frac{\text{tr}(F)}{2\pi}\,.$
(3.30)

Here, the first term is nonzero even when the mass parameters are kept
constant (and $-2$ is the contribution from the regulator fermions). It
reflects the familiar fact that the fermion effective action contains the
(gravitational) Chern-Simons term in 3d where the sign of the level depends on
the sign of the mass eigenvalues. The second term is a gauged-WZW-like term,
and its contribution is nontrivial only if the mass parameters are modulated
in spacetime.

####  3.3.3 Symmetry breaking mass term

One can also allow the mass parameters to be charged under some symmetry. For
instance, consider the case of a single flavor of a two-dimensional Dirac
fermion $\psi=(\psi_{L},\,\psi_{R})^{T}$, with a complex mass parameter as
before, but now turn on the background gauge field for the chiral $U(1)_{L}$
symmetry which acts only on the left-handed component. The mass term
explicitly breaks the $U(1)_{L}$ symmetry, but by promoting the mass to be a
background field transforming under this symmetry, one can consistently turn
on the combined background of the modulated mass term plus the gauge field.
The Lagrangian is

$\mathcal{L}=\psi_{L}^{\dagger}(D_{1}+\mathrm{i}D_{2})\psi_{L}+\psi_{R}^{\dagger}(\partial_{1}-\mathrm{i}\partial_{2})\psi_{R}+M_{0}e^{\mathrm{i}\alpha}\psi_{R}^{\dagger}\psi_{L}+M_{0}e^{-\mathrm{i}\alpha}\psi_{L}^{\dagger}\psi_{R}$
(3.31)

where $D_{i}=\partial_{i}-\mathrm{i}A_{i}$. The nontrivial background gauge
transformations are

$\psi_{L}\sim e^{\mathrm{i}\lambda}\psi_{L}\,,\quad A_{i}\sim
A_{i}+\partial_{i}\lambda\,,\quad\alpha\sim\alpha-\lambda\,.$ (3.32)

In this case, we cannot let the regulator fermions to have a constant mass
term since it will not be gauge-invariant. We will set the regulator mass to
be $M_{0}e^{\mathrm{i}\alpha_{0}}$ where $\alpha_{0}\sim\alpha_{0}-\lambda$.
The degree four part of the Chern character that we are after is given by

$[\mathop{\mathrm{ch}}\nolimits(\mathcal{F})]_{4}=\left(\frac{\mathrm{i}}{2\pi}\right)^{2}e^{-\widetilde{M}_{0}^{2}}\left(-\frac{1}{2}F^{2}+\frac{1}{3}F\wedge\widetilde{M}_{0}\mathrm{d}\widetilde{M}_{0}(\mathrm{d}\alpha+A)\right)\,,$
(3.33)

and after integrating from $\widetilde{M}_{0}=0$ to
$\widetilde{M}_{0}=\infty$, we get the following Berry curvature:

$\mathcal{B}=-\frac{1}{8\pi^{2}}F\wedge(d\alpha-d\alpha_{0})\,.$ (3.34)

Note that the Berry curvature is invariant under the background gauge
transformation.191919Similarly to Section 3.3.1, the Eq. (3.34) can also be
deduced from the anomalous $U(1)_{L}$ rotation of $\psi_{L}$.

##  4 Torsional Berry phase

In the last section we studied the (higher) Berry curvature which captures the
response to the continuous deformation of parameters. When the Berry curvature
vanishes, there can still be a nontrivial response against the modulated
parameters $p:X\to P$ that is not continuously deformable to (i.e., not
homotopic to) a constant map. Or, more precisely, the response can be
nontrivial when the pair $(X,p)$ is not null-bordant. Such Berry phases are
classified by the torsion part of $\operatorname{Inv}^{d}(P)$, which is
further identified with the Pontryagin dual of the bordism group [Kap14,
KTTW14, FH16, Yon18]:

$\operatorname{Tors}\operatorname{Inv}^{d}(P)=\mathop{\mathrm{Hom}}\nolimits(\operatorname{Tors}\Omega_{d}^{\text{spin}}(P),U(1))\,.$
(4.1)

In other words, given a $d$-dimensional manifold $X$ and the modulated
parameters $p:X\to P$, the Berry phase defines a bordism invariant of the pair
$(X,p)$ when its Berry curvature vanishes. We will call such a bordism
invariant a torsional Berry phase, and we will calculate it in a few examples
utilizing the result of Section 2.

###  4.1 Reducing onto zero locus

To evaluate the partition function $\mathcal{Z}_{d}(X,M)$ when $\mathcal{B}=0$
but the pair $(X,M)$ is not null-bordant, we again use (2.57). In such cases,
to extend the mass matrix $M$ into $Y_{+}$, we have to let the extended mass
matrix to become degenerate on a locus in $Y_{+}$, which we will call $Y_{D}$.
By definition, $Y_{D}\cap X=\varnothing$ and in particular $Y_{D}$ does not
have a boundary. Then, using the assumption that $\mathcal{B}=0$, we can
reduce the $\eta$-invariant on $Y_{+}$ to the $\eta$-invariant just for the
localized modes along the degenerate mass locus $Y_{D}$.

Specifically, this is accomplished as follows. If $\mathcal{B}=0$, the value
of the $\eta$-invariant on $Y_{+}$ is invariant under any continuous
deformation of the mass parameters. Using this freedom, we will tune the
magnitude of the modulated mass term to be much larger than the regularization
scale $\mu$ for the $\eta$-invariant. That is, consider the regularized
expression of the $\eta$-invariant on $Y_{+}$ that was given in Eq. (2.36),

$\eta_{K}(Y_{+};\mathtt{APS})=\lim_{\mu\to\infty}\left[\frac{1}{2}\sum_{j}\text{sgn}(\xi_{j})\text{exp}\left(-\frac{\xi_{j}^{2}}{\mu^{2}}\right)\right]\,.$
(4.2)

We will let $M_{0}/\mu\to\infty$ where $M(x)=M_{0}f(x)$ as before. When the
mass becomes large, there appear exponentially localized eigenfunctions of the
operator $K$ on the degenerate locus $Y_{D}$, and their eigenvalues are
independent of the value of $M_{0}$. On the other hand, all the other
delocalized eigenfunctions of $K$ have eigenvalues that are proportional to
the overall mass scale $|\xi|\sim M_{0}$. This means that in the limit
$M_{0}/\mu\to\infty$, the $\eta$-invariant (4.2) receives contribution only
from the localized modes because of the regulator in (4.2). If we denote the
hermitian operator that acts on the localized modes on $Y_{D}$ as $K_{D}$,
then according to these observations we conclude that

$\eta_{K}(Y_{+};\mathtt{APS})=\eta_{K_{D}}(Y_{D})$ (4.3)

when the Berry curvature vanishes and the diabolical locus $Y_{D}$ is a
manifold embedded in $Y_{+}$.202020 In general $Y_{D}$ might have self-
intersections or other kinds of singularities, and it is not guaranteed that
one can always smooth out the singularities by deforming the mass parameter
configuration.  We list some examples of this below.

###  4.2 Examples

####  4.2.1 Two-dimensional Dirac fermion

Consider a single flavor of a two-dimensional Dirac fermion,

$\mathcal{L}=\bar{\psi}(\not{\partial}+M_{0}e^{\mathrm{i}\alpha\gamma})\psi\,.$
(4.4)

We do not turn on any additional background gauge fields for internal global
symmetries (other than the spin structure). The space of non-degenerate mass
terms in this case is homotopic to a circle $S^{1}$, and the appropriate
bordism group is

$\widetilde{\Omega}^{\text{Spin}}_{d=2}(S^{1})\cong\mathbb{Z}_{2}\,.$ (4.5)

As a representative of the generator, we can pick a square torus
$T^{2}=S^{1}_{a}\times S^{1}_{b}$. Along the $a$-cycle the mass term winds
around once, i.e., $\oint_{a}d\alpha=2\pi$, and we impose the antiperiodic
(NS) boundary condition for the fermion. Along the $b$-cycle, the mass term
remains constant, and we impose the periodic (R) boundary condition for the
fermion.

The regularized partition function of the Dirac fermion in the infinite mass
limit in this case will determine an element in the Pontryagin dual of the
reduced bordism group which is again $\mathbb{Z}_{2}$. If the theory
corresponds to the trivial (nontrivial) element, then the partition function
on the $T^{2}$ that we constructed will evaluate to 1 ($-1$).

Now, for the auxiliary bulk $Y_{+}$, we can take $D^{2}_{a}\times S^{1}_{b}$
where $D^{2}_{a}$ is a 2-disk bounding the $a$-cycle, and the mass term
vanishes at the origin of the disk. The zero locus of the mass term will then
be $Y_{D}=\\{0\\}\times S^{1}_{b}$ where $0$ denotes the origin of the disk.
According to Eq. (2.57) and Appendix A, the 3d hermitian operator acting on
two flavors of Dirac fermions on $Y_{+}=D^{2}_{a}\times S^{1}_{b}$ is

$K=\mathrm{i}(\tau^{3}\Gamma^{\mu}\partial_{\mu}+\mathrm{i}\tau^{2}M_{0}\text{cos}\alpha-\mathrm{i}\tau^{1}M_{0}\text{sin}\alpha)\,.$
(4.6)

On the other hand, for the localized zero modes on the codimension two circle
$Y_{D}=\\{0\\}\times S^{1}_{b}$, we simply have

$K_{D}=\mathrm{i}\partial\,,$ (4.7)

which is deduced by a simple counting of the number of components in the Dirac
spinors.212121The number of components of the codimension-$p$ localized
fermion is $2^{-p}$ times the number of components for the bulk fermion. That
is, we know that starting from two flavors of Dirac fermions in 3d, we get a
single complex fermion localized on the circle.

It is easy to compute that the exponentiated $\eta$-invariant coming from this
one-dimensional fermion evaluated on the circle with the periodic spin
structure gives us -1. All the nonzero eigenvalues appear in pairs of positive
and negative numbers with the same magnitude, and there is a singe zero mode
given by the constant mode which results in $\eta=1/2$. Thus, we conclude that
the regularized partition function for the original 2d system is

$\lim_{M_{0}\to\infty}\frac{\mathcal{Z}_{2}(T^{2})}{\mathcal{Z}^{(0)}_{2}(T^{2})}=-1\,,$
(4.8)

indicating that this theory defines a nontrivial parameterized invertible
field theory.

We demonstrate that the same result can be obtained by an explicit computation
of the 2d partition functions in Appendix C.

####  4.2.2 Effective action from symmetry breaking mass

As a bit more substantial example, we consider the cases where fermions with a
symmetry breaking mass generates a nontrivial effective action in terms of the
backgrounds for the remaining symmetries. The specific model is what is
considered in [Tho14, WWW18, LOT21].

In the model, we have Weyl fermions in the $(\text{vector, fundamental})$
representation of $SO(2n_{1})\times USp(2n_{2})$ symmetry group, and consider
the mass terms breaking the $USp(2n_{2})$ group into $U(n_{2})$, so that the
fundamental representation of the former splits into the direct sum of
fundamental and anti-fundamental representation of the latter. Becuase this
model has Weyl fermions, we cannot apply the result in this paper regarding
Dirac fermions. However, here we would like to pretend that Eq. (2.57) can be
extended to more general fermion cases, perhaps by utilizing the result of
[GY21]. In addition, the result here is already reported in the references
mentioned above. Their argument is from anomaly matching with the zero modes
along a ’t Hooft-Polyakov monopole configuration, while our method clarifies
that how one would deduce such effective action directly from the fermion
partition function.

The effective action we are aiming for is

$\frac{1}{2}w_{2}(SO(2n_{1}))c_{1}(U(n_{2})),$ (4.9)

where $w_{2}(SO(2n_{1}))$ and $c_{1}(U(n_{2}))$ are the second Stiefel-
Whiteney class and the first Chern class (modulo 2) for the specified symmetry
bundles, respectively. To detect this action, we consider a $d$-dimensional
manifold $X$ with constant mass $M$ breaking $USp(2n_{2})$ to $U(n_{2})$, and
a $(SO(2n_{1})\times U(n_{2}))/\mathbb{Z}_{2}$ bundle $A$ with nonzero
$w_{2}(SO(2n_{1}))c_{1}(U(n_{2}))$. To apply an analog of (2.57) that should
hold for Weyl fermions, we find an extension of $M$ and $A$ into $Y_{+}$. As
the characteristic class $w_{2}(SO(2n_{1}))c_{1}(U(n_{2}))$ is in particular a
bordism invariant, this is impossible with a constant mass $M$ over $Y_{+}$.

To find an extension, we allow $M$ to vary on $Y_{+}$ (but it remains to be
constant on $X=\partial Y_{+}$) and hit $0$ on a locus $Y_{D}$. To be
explicit, we take $X=S^{2}_{(1)}\times S^{2}_{(2)}$ (subscripts are to
distinguish the two factors) and the bundle to be such that
$w_{2}(SO(2n_{1}))$ is dual to the fundamental class of $S^{2}_{(1)}$ and
$c_{1}(U(n_{2})))$ is dual to the fundamental class of $S^{2}_{(2)}$. Then, we
take $Y_{+}=S^{2}_{(1)}\times D^{3}_{(2)}$ where $D^{3}_{(2)}$ is the ball
bounding $S^{2}_{(2)}$. To extend the bundle, we prepare the ’t Hooft-Polyakov
configuration in $D^{3}_{(2)}$. That is, the mass matrix $M$ becomes zero at
the center of $D^{3}_{(2)}$, and the $U(n_{2}))$ bundle is enhanced to a
$USp(2n_{2})$ bundle at the center.222222 Note that in this application, even
though $M$ is constant on $X$, it is important to let it vary inside $Y_{+}$.
The method in the literature considered modulated $M$ on $X$, which we do not
need.

Then, the evaluation of the effective action reduces to evaluating the
$\eta$-invariant of the zero modes localized at the monopole. This zero mode
counting is done in [Tho14, WWW18, LOT21], and it is shown that the state
arising from quantizing the zero modes is in the spinor representation of
$SO(2n_{1})$, which means that the $\eta$-invariant of them is
$\frac{1}{2}w_{2}(SO(2n_{1}))$. Combining all considerations, we conclude that
indeed the fermion with the symmetry breaking mass term generates the
effective action (4.9).

## Acknowledgments

The authors are grateful to Y.-A. Chen, S.-J. Huang, J. Kaidi, Z. Komargodski,
S. Seifnashri, S.-H. Shao, M. Yamashita, and G. Zafrir for helpful
conversations. KO is supported in part by JSPS KAKENHI Grant-in-Aid,
No.22K13969 and in part by Simons Collaboration on Global Categorical
Symmetry.

##  Appendix A When $d$ is even

Here, we repeat the calculations in Section 2 for the case of even $d$. The
goal is again to relate the phase of the partition function of an arbitrary
number $N_{f}$ species of Dirac fermions on a closed $d$-dimensional manifold
$X$ with modulated mass parameters to $(d+1)$-dimensional $\eta$-invariants.
Since $d$ is even, we have the $d$-dimensional chirality matrix $\gamma$, and
the most general $d$-dimensional mass matrix is given by
$M_{1}+\mathrm{i}\gamma M_{2}$ where $M_{1}$ and $M_{2}$ are $N_{f}\times
N_{f}$ hermitian matrices. This system can be realized on an interface, if we
start from $2N_{f}$ species of $(d+1)$-dimensional Dirac fermions on the
manifold $Y$ which we defined in Section 2 (depicted in Figure 1), with the
follwing Dirac operator,

$\mathsf{D}_{d+1}=\Gamma^{\mu}D_{\mu}+\tau^{1}M_{1}+\tau^{2}M_{2}+\tau^{3}M_{3}\,,$
(A.1)

and form an interface by varying the value of $M_{3}$ from $+m_{0}$ to
$-m_{0}$ similar to the odd $d$ case. $\tau$’s are the $2\times 2$ Pauli
matrices. $\tau^{k}M_{k}$ is a shorthand for the tensor product of the two
matrices, for $k=1,2,3$.

We define

$K=\mathrm{i}(\tau^{3}\Gamma^{\mu}D_{\mu}+\mathrm{i}\tau^{2}M_{1}-\mathrm{i}\tau^{1}M_{2})$
(A.2)

which is a hermitian operator on $Y$. When we restrict to $Y_{+}$, we need to
choose an appropriate boundary conidition that makes $K$ hermitian. Along the
collar neighborhood of $X$ in $Y_{+}$, $K$ takes the form
$K=\mathrm{i}\tau^{3}\Gamma^{y}(\partial_{y}+\mathcal{D}_{X})$ where

$\mathcal{D}_{X}=\Gamma^{y}\Gamma^{i}D_{i}+\Gamma^{y}(\tau^{1}M_{1}+\tau^{2}M_{2})$
(A.3)

is a hermitian operator defined on $X$. The nonzero eigenvalues of
$\mathcal{D}_{X}$ come in pairs because of
$\\{\mathcal{D}_{X},\tau^{3}\Gamma^{y}\\}=0$, and $\mathcal{D}_{X}$ does not
have a zero mode since the $d$-dimensional mass matrix is non-degenerate. The
same APS boundary condition as in Eq. (2.22),
$\mathcal{P}_{<0}^{\mathcal{D}_{X}}\Psi|_{X}=0$, makes $K$ a hermitian
operator on $Y_{+}$.

Now, the $(d+1)$-dimensional partition function on $Y$ can again be written as

$\displaystyle\mathcal{Z}_{d+1}(Y)$
$\displaystyle=\bra{Y_{-}}\hat{Y}_{L}\ket{Y_{+}}$
$\displaystyle=\bra{\Omega_{-}}T[{\text{exp}\\{-\int_{y_{+}}^{y_{-}}\hat{H}(y)dy\\}}]\ket{\Omega_{+}}\times\frac{\braket{\mathtt{APS}}{Y_{+}}}{\braket{\mathtt{APS}}{Y_{-}}}\times\frac{\braket{\mathtt{APS}}{\Omega_{-}}}{\braket{\mathtt{APS}}{\Omega_{+}}}$
(A.4)

as in Eq. (2.12).

Define
$K^{\prime}=\mathrm{i}(\tau^{3}\Gamma^{\mu}D_{\mu}-\mathrm{i}\tau^{2}M_{1}-\mathrm{i}\tau^{1}M_{2})$
and
$\mathcal{D}_{X}^{\prime}=\Gamma^{y}\Gamma^{i}D_{i}-\Gamma^{y}(\tau^{1}M_{1}+\tau^{2}M_{2})$.
$\mathcal{D}_{X}^{\prime}$ is a hermitian operator on $X$, and under the
boundary condition $\mathcal{P}_{<0}^{\mathcal{D}_{X}^{\prime}}\Psi|_{X}=0$,
$K^{\prime}$ is a hermitian operator on $Y_{+}$.

$\mathcal{D}_{X}$ and $\mathcal{D}_{X}^{\prime}$ share the same spectrum of
eigenvalues, because of the relation
$\tau^{3}\mathcal{D}_{X}=\mathcal{D}_{X}^{\prime}\tau^{3}$. We denote their
eigenfunctions and eigenvalues as
$\mathcal{D}_{X}\psi_{a}=\lambda_{a}\psi_{a}$ and
$\mathcal{D}_{X}^{\prime}\psi_{a}^{\prime}=\lambda_{a}\psi_{a}^{\prime}$,
where $\psi_{a}^{\prime}=\tau^{3}\psi_{a}$. We take the indexing convention
such that $a$ runs over nonzero integers, and $\lambda_{-a}=-\lambda_{a}<0$
for $a>0$. We also let $\psi_{-a}=\tau^{3}\Gamma^{y}\psi_{a}$. (Note the
different notations compared to Section 2.)

Along the collar neighborhood of $X$, we perform the mode expansions

$\Psi(x^{i},y)=\sum_{a}\alpha_{a}(y)\psi_{a}(x^{i})\,,\quad\bar{\Psi}(x^{i},y)=\sum_{a}\beta_{a}(y)(\psi_{a}^{\prime})^{\dagger}(x^{i})=\sum_{a}\beta_{a}(y)\psi_{a}^{\dagger}(x^{i})\tau^{3}\,.$
(A.5)

The APS boundary condition imposed on $\Psi$,
$\mathcal{P}_{<0}^{\mathcal{D}_{X}}\Psi|_{X}=0$, is such that at the boundary
we require all $\alpha_{a<0}$ to be zero, whereas all $\alpha_{a>0}$ are
unconstrained.

If we insert the mode expansions into the action
$S=\int_{Y}\bar{\Psi}\mathsf{D}_{d+1}\Psi=\int_{Y}\bar{\Psi}\tau^{3}(-\mathrm{i}K+m_{0})\Psi$,
we obtain

$S=\int
dy\sum_{a}\left[\beta_{-a}(y)(\partial_{y}+\lambda_{a})\alpha_{a}(y)+M_{3}(y)\beta_{a}(y)\alpha_{a}(y)\right]\,.$
(A.6)

From this, we see that the conjugate variable of $\alpha_{a}$ is $\beta_{-a}$
for each $a$. Thus, the boundary condition to be imposed on $\bar{\Psi}$ is
such that at the boundary all $\beta_{a>0}$ are unconstrained, whereas all
$\beta_{a<0}$ are set to be zero.

$K$ and $K^{\prime}$ also share the same spectrum of eigenvalues because of
$\tau^{3}K=K^{\prime}\tau^{3}$ which is coherent with their respective
boundary conditions we explained above. We denote the eigenfunctions and
eigenvalues of $K$ and $K^{\prime}$ as $K\Psi_{j}=\xi_{j}\Psi_{j}$ and
$K^{\prime}\Psi_{j}^{\prime}=\xi_{j}\Psi_{j}^{\prime}$, respectively, where
$\Psi_{j}^{\prime}=\tau^{3}\Psi_{j}$. Inside the path integral, we will expand
$\Psi=\sum\limits_{j}a_{j}\Psi_{j}$ with grassmannian coefficients $a_{j}$.
Since $(\Psi_{j}^{\prime})^{\dagger}$ satisfies the boundary condition for
$\bar{\Psi}$ (but the naive choice $\Psi_{j}^{\dagger}$ does not), we will
expand the conjugate field as
$\bar{\Psi}=\sum\limits_{j}b_{j}(\Psi_{j}^{\prime})^{\dagger}=\sum\limits_{j}b_{j}\Psi_{j}^{\dagger}\tau^{3}$
with grassmannian coefficients $b_{j}$.

This diagonalizes the action,
$S=\int_{Y_{\pm}}\bar{\Psi}\tau^{3}(-\mathrm{i}K\pm
m_{0})\Psi=\sum\limits_{j}(-\mathrm{i}\xi_{j}\pm m_{0})b_{j}a_{j}$. Thus, we
again obtain

$\frac{\braket{\mathtt{APS}}{Y_{+}}}{\braket{\mathtt{APS}}{Y_{-}}}=\frac{\prod\limits_{j}(-\mathrm{i}\xi_{j}+m_{0})}{\prod\limits_{j}(-\mathrm{i}\xi_{j}-m_{0})}=\text{exp}(2\pi\mathrm{i}\eta_{K}(Y_{+};\mathtt{APS}))$
(A.7)

in the limit $m_{0}\to\infty$, as desired.

To compute the remaining factors in Eq. (A) as in Section 2, we need to write
down the Hamiltonian of the system on $X$. From Eq. (A.6), we read off the
Hamiltonian

$\hat{H}(y)=\sum_{a}\left(\lambda_{a}\beta_{-a}\alpha_{a}+M_{3}(y)\beta_{a}\alpha_{a}\right)$
(A.8)

where $\beta_{-a}=\alpha_{a}^{\dagger}$. Equivalently, we can write

$\hat{H}(y)=\sum_{a>0}[\alpha_{a}^{\dagger}\,\,\,\alpha_{-a}^{\dagger}]\begin{bmatrix}\lambda_{a}&M_{3}(y)\\\
M_{3}(y)&-\lambda_{a}\end{bmatrix}\begin{bmatrix}\alpha_{a}\\\
\alpha_{-a}\end{bmatrix}\,.$ (A.9)

Let

$(\text{cos}2\theta_{a}(y)\,,\text{sin}2\theta_{a}(y))=\frac{(\lambda_{a}\,,M_{3}(y))}{\sqrt{\lambda_{a}^{2}+M_{3}(y)^{2}}}$
(A.10)

where $-\pi/4<\theta_{a}<\pi/4$ for each $a$. If we define, for each $a>0$,

$\begin{bmatrix}A_{1,a}\\\
A_{2,a}\end{bmatrix}=\begin{bmatrix}\text{cos}\theta_{a}&\text{sin}\theta_{a}\\\
-\text{sin}\theta_{a}&\text{cos}\theta_{a}\end{bmatrix}\begin{bmatrix}\alpha_{a}\\\
\alpha_{-a}\end{bmatrix}\,,$ (A.11)

then the modes $A_{a}$ diagonalize the Hamiltonian,

$\hat{H}=\sum_{a>0}\sqrt{\lambda_{a}^{2}+m_{0}^{2}}\left(A_{1,a}^{\dagger}A_{1,a}-A_{2,a}^{\dagger}A_{2,a}\right)\,.$
(A.12)

The state corresponding to the APS boundary condition is characterized by

$\bra{\mathtt{APS}}\alpha_{a<0}=\bra{\mathtt{APS}}\alpha_{a>0}^{\dagger}=0\,.$
(A.13)

On the other hand, the ground state is characterized by

$A_{1,a}(y)\ket{\Omega(y)}=A_{2,a}^{\dagger}(y)\ket{\Omega(y)}=0$ (A.14)

for all $a>0$.

Let $\ket{E}$ be a state satisfying $\alpha_{a}\ket{E}=0$ for all $a$. Then,
the states of interest can be constructed as

$\displaystyle\bra{\mathtt{APS}}$
$\displaystyle=\bra{E}\prod_{a<0}\alpha_{a}\,,$ $\displaystyle\ket{\Omega(y)}$
$\displaystyle=\prod_{a>0}A_{2,a}^{\dagger}(y)\ket{E}\,.$ (A.15)

This gives us
$\braket{\mathtt{APS}}{\Omega(y)}=\prod\limits_{a<0}\text{cos}\theta_{a}(y)$,
and in particular

$\frac{\braket{\mathtt{APS}}{\Omega_{-}}}{\braket{\mathtt{APS}}{\Omega_{+}}}=1\,.$
(A.16)

Furthermore,

$\braket{\Omega_{-}}{\Omega_{+}}=\prod_{a>0}\frac{\lambda_{a}}{\sqrt{\lambda_{a}^{2}+m_{0}^{2}}}=\left[\prod_{a>0}\lambda_{a}\right]_{\text{Reg.}}\,.$
(A.17)

Recall on $X$, we have the hermitian operator
$\mathcal{D}_{X}=\Gamma^{y}\Gamma^{i}D_{i}+\Gamma^{y}(\tau^{1}M_{1}+\tau^{2}M_{2})$
which satisfies $\\{\mathcal{D}_{X},\tau^{3}\Gamma^{y}\\}=0$.
$\tau^{3}\Gamma^{y}$ provides a $\mathbb{Z}_{2}$ grading on the space of
spinors on $X$ on which $\mathcal{D}_{X}$ acts. Let $V_{\pm}$ be the subspace
of spinors on $X$ having the eigenvalue $\tau^{3}\Gamma^{y}=\pm 1$. Consider
the restriction of $\mathcal{D}_{X}$ to the subspace $V_{+}$,
$\mathcal{D}_{X}|_{V_{+}}:V_{+}\rightarrow V_{-}$. $\mathcal{D}_{X}|_{V_{+}}$
can be written as a composition of two maps,
$\mathcal{D}_{X}|_{V_{+}}=(\Gamma^{y}\tau^{1})|_{V_{+}}\circ(\Gamma^{i}\tau^{1}D_{i}+M_{1}+i\tau^{3}M_{2})|_{V_{+}}\equiv(\Gamma^{y}\tau^{1})|_{V_{+}}\circ\mathsf{D}_{d}$,
where $\mathsf{D}_{d}:V_{+}\rightarrow V_{+}$ and
$(\Gamma^{y}\tau^{1})|_{V_{+}}:V_{+}\rightarrow V_{-}$. Also, within the
$V_{+}$ subspace, the matrices $(\Gamma^{y}\tau^{1})|_{V_{+}}$ satisfy the
algebra of $d$-dimensional gamma matrices, and all of them anticommute with
$(\tau^{3})|_{V_{+}}$. Thus, it makes sense to define the $d$-dimensional
gamma matrices $\gamma^{i}\equiv(\Gamma^{y}\tau^{1})|_{V_{+}}$ and the
chirality matrix $\gamma\equiv(\tau^{3})|_{V_{+}}$. Then,
$\mathsf{D}_{d}=\gamma^{i}D_{i}+M_{1}+i\gamma M_{2}$ is the $d$-dimensional
Dirac operator acting on the $N_{f}$ species of $d$-dimensional Dirac fermions
living on $X$, which are precisely the elements of $V_{+}$.

As before, we define two hermitian operators on $X$,
$\mathsf{H}_{d}^{1}\equiv\mathsf{D}_{d}^{\dagger}\mathsf{D}_{d}$ and
$\mathsf{H}_{d}^{2}\equiv\mathsf{D}_{d}\mathsf{D}_{d}^{\dagger}$. We then see
$\mathcal{D}_{X}^{2}|_{V_{+}}=\mathsf{H}_{d}^{1}$, and $\lambda_{a}^{2}$ for
$a>0$ are the eigenvalues of $\mathsf{H}_{d}^{1}$. We conclude
$\prod\limits_{a>0}\lambda_{a}=|\mathcal{Z}_{d}(X)|$, and Eq. (2.39) holds for
even $d$ as well.

Finally, the arguments in Section 2.2 are also still valid. The localized
interface modes analogous to Eq. (2.47) are given by

$\Psi=\mathcal{N}(\psi_{1}\otimes v^{+}_{1}+\psi_{2}\otimes
v^{+}_{2})\exp\left(-\int_{y^{*}}^{y}M_{3}(y^{\prime})dy^{\prime}\right)\,,$
(A.18)

where $\tau^{3}\Gamma^{y}v^{+}_{1,2}=+v^{+}_{1,2}$. $\psi_{1}$ and $\psi_{2}$
are chiral fermions in $d$ dimensions, and combined they form the
$d$-dimensional Dirac fermion $\psi$. If we apply
$\mathsf{H}_{d+1}^{1}\equiv\mathsf{D}_{d+1}^{\dagger}\mathsf{D}_{d+1}$ to this
ansatz, we obtain

$\mathsf{H}_{d+1}^{1}\Psi=\left(\mathsf{H}_{d}^{1}\psi\right)\exp\left(-\int_{y^{*}}^{y}M_{3}(y^{\prime})dy^{\prime}\right)$
(A.19)

and then the remaining steps leading to Eq. (2.40) are unchanged. We conclude
that Eq. (2.57) is valid in any dimensions.

##  Appendix B APS index theorem and superconnection

In this appendix, we give a derivation of the APS index theorem for a Dirac
operator coupled to a superconnection (that is, modulated mass parameters and
background gauge fields) on a manifold with boundary. The derivation of the
APS index theorem for a Dirac operator coupled to an ordinary connection using
path integral was provided in [KY21], and we directly generalize their method.
The case without boundary and also a special case of the APS index theorem for
a cylindrical geometry were discussed in [KS21], and we also use their result
as an input. See [Qui85, Kah11, GY21] for the mathematical literature.

The aim is to relate the $\eta$-invariant of the hermitian operator

$K=\begin{cases}\mathrm{i}(\Gamma^{\mu}D_{\mu}+\mathrm{i}\Gamma M)&\text{for
odd $d$}\,,\\\
\mathrm{i}(\tau^{3}\Gamma^{\mu}D_{\mu}+\mathrm{i}\tau^{2}M_{1}-\mathrm{i}\tau^{1}M_{2})&\text{for
even $d$}\,,\end{cases}$ (B.1)

defined on a closed and null-bordant $(d+1)$-dimensional manifold
$Y$,232323Inside the null-bordism here the mass matrix is allowed to be
degenerate. to the index of a Dirac-like operator $Q$ defined on a
$(d+2)$-dimensional manifold $W$ with boundary $\partial W=Y$, with a suitable
choice of the boundary condition. We will define the operator $Q$ and explain
its relation to the actual $(d+2)$-dimensional Dirac operator below. We assume
there is a collar neighborhood near the boundary $Y$ of $W$, and
$w\in(-\epsilon,0]$ will denote the coordinate transverse to the boundary in
the collar neighborhood, where the boundary is at $w=0$ and $\epsilon>0$. $w$
is to be interpreted as the Euclidean time direction near the
boundary.242424Note that the existence of the collar neighborhood near
boundary was also assumed even in the original mathematical proof in [APS75b,
APS75a, APS76].

For the APS index theorem to be applicable, we need a $\mathbb{Z}_{2}$ grading
in the space of spinors, which anticommutes with the operator $Q$. As will be
explained in more detail, the operator $Q$ of our interest takes the following
form:

$Q=\begin{cases}\mathrm{i}\tau^{1}\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}+\tau^{2}M&\text{for
odd $d$}\,,\\\
\mathrm{i}\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}-\hat{\Gamma}(M_{1}\tau^{1}+M_{2}\tau^{2})&\text{for
even $d$}\,.\end{cases}$ (B.2)

In both cases, the operator $Q$ acts on $2N_{f}$ flavors of
$(d+2)$-dimensional fermions living on $W$. Here, $\hat{\mu}=1,\cdots,d+2$,
and $\hat{\Gamma}^{\hat{\mu}}$ are $(d+2)$-dimensional gamma matrices. For the
case of even $d$, $\hat{\Gamma}$ denotes the $(d+2)$-dimensional chirality
matrix. The $\mathbb{Z}_{2}$-grading that anticommutes with the operator $Q$
is given by $\tau^{3}$ if $d$ is odd, and $\tau^{3}\hat{\Gamma}$ if $d$ is
even. In the zero eigenvalue subspace of the operator $Q$, $Q$ commutes with
the $\mathbb{Z}_{2}$-grading matrix, and they can be simultaneously
diagonalized. The eigenvalue of the $\mathbb{Z}_{2}$-grading matrix can be
either $+1$ or $-1$, and we will call it “chirality.” Then, the index of the
operator is defined as the number of positive chirality zero modes minus the
number of negative chirality zero modes as usual. Equivalently, we may define
an operator $Q_{+}$ to be the operator $Q$ restricted to the subspace of
positive chirality spinors, and then the definition of the index is
$\operatorname{Ind}Q\equiv\operatorname{Ind}Q_{+}=\text{dim}\,\text{ker}\,Q_{+}-\text{dim}\,\text{coker}\,Q_{+}=\text{dim}\,\text{ker}\,Q_{+}-\text{dim}\,\text{ker}\,Q_{-}$,
where $Q_{-}$ is $Q$ restricted to the subspace of negative chirality spinors.
By definition, an operator $Q_{+}$ having finite dimensional kernel and
cokernel is called Fredholm. For us, requiring $Q$ to be hermitian will
suffice to ensure that the index is well-defined. When $Q$ is defined on a
manifold with boundary, which is the case of our interest, we have to impose
non-local boundary conditions on spinors to maintain the hermiticity of $Q$.
We will collectively call such boundary conditions as APS boundary conditions.

We do not require the manifold $W$ to be compact (nor the boundary $\partial
W=Y$), but we do assume that the operator $Q$ with the boundary condition that
we will explain below and the operator $K$ on the boundary have discrete
spectra. This happens if the mass blows up towards the non-compact directions.

We will first give a detailed derivation of the index theorem assuming $d$ is
odd. Similar calculations can be done for the even dimensional case as well
with minor changes. Let $\Upphi$ be $2N_{f}$ species of $(d+2)$-dimensional
Dirac fermions on $W$, with the Dirac operator given by

$\mathsf{D}_{d+2}=\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}+\tau^{1}m_{1}+\tau^{2}m_{2}+\tau^{3}M\,,$
(B.3)

where $m_{1}$ and $m_{2}$ are constant real numbers, and $\tau^{i}$’s are
Pauli matrices. We will write
$m_{1}+\mathrm{i}m_{2}=m_{0}e^{\mathrm{i}\alpha_{0}}$. $M$ is a general
$N_{f}\times N_{f}$ hermitian matrix-valued function on the boundary $\partial
W=Y$, see Eq. (B.1), and it extends into $W$ in such a way that it is
independent of the coordinate $w$ inside the collar neighborhood. The
Euclidean partition function of this $(d+2)$-dimensional system is

$\mathcal{Z}_{d+2}=\int[D\bar{\Upphi}][D\Upphi]\text{exp}(-\int_{W}\bar{\Upphi}\mathsf{D}_{d+2}\Upphi)\,.$
(B.4)

We may write
$\mathsf{D}_{d+2}=\tau^{1}(-\mathrm{i}Q+m_{0}e^{\mathrm{i}\alpha\tau^{3}})$
where

$\displaystyle\begin{split}Q&=\mathrm{i}\tau^{1}\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}+\tau^{2}M\\\
&=\begin{bmatrix}0&\mathrm{i}\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}-\mathrm{i}M\\\
\mathrm{i}\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}+\mathrm{i}M&0\end{bmatrix}\,.\end{split}$
(B.5)

We need to impose a boundary condition that makes $Q$ hermitian, and
$\tau^{3}$ will be our $\mathbb{Z}_{2}$ grading in this case. We have
$\\{\tau^{3},Q\\}=0$.

On the $(d+1)$-dimensional boundary $\partial W=Y$, we define the
$(d+1)$-dimensional gamma matrices to be

$\Gamma^{\mu}=\mathrm{i}\hat{\Gamma}^{w}\hat{\Gamma}^{\mu}\,\,,\quad\Gamma=\hat{\Gamma}^{w}$
(B.6)

for $\mu=1,\cdots,d+1$ and $\Gamma$ is the chirality matrix in
$(d+1)$-dimensions. The operator
$Q_{+}=\mathrm{i}(\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}+M)$ acting on positive
chirality spinors (bottom left block of Eq.(B.5)) becomes near the boundary

$Q_{+}=\mathrm{i}\hat{\Gamma}^{w}(\partial_{w}-K)$ (B.7)

where $K=\mathrm{i}(\Gamma^{\mu}D_{\mu}+\mathrm{i}\Gamma M)$ is the hermitian
operator on $Y$ that we are after (B.1). If we define another hermitian
operator $K^{\prime}=\mathrm{i}(\Gamma^{\mu}D_{\mu}-\mathrm{i}\Gamma M)$, then
near the boundary we similarly have
$Q_{-}=\mathrm{i}(\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}-M)=\mathrm{i}\hat{\Gamma}^{w}(\partial_{w}-K^{\prime})$
for the upper right block of Eq. (B.5).

For the operator $Q$ to be hermitian, we want the following quantity to
vanish, for any two arbitrary spinors $\Upphi$ and $\Uppsi$,

$\int_{W}\Upphi^{\dagger}Q\Uppsi-\left(\int_{W}\Uppsi^{\dagger}Q\Upphi\right)^{*}=\int_{Y}\left(\Upphi_{+}^{\dagger}\hat{\Gamma}^{w}\Uppsi_{-}+\Upphi_{-}^{\dagger}\hat{\Gamma}^{w}\Uppsi_{+}\right)$
(B.8)

where the $\pm$ subscripts indicate the positive and negative chirality parts
(i.e., eigenvalues of $\tau^{3}$), respectively. We integrated by parts to
obtain the right-hand side. On the boundary $Y$, note that we have the
relation $K\hat{\Gamma}^{w}=-\hat{\Gamma}^{w}K^{\prime}$. If we write the
eigenvalues and eigenfunctions of $K$ as $K\Psi_{j}=\xi_{j}\Psi_{j}$, then
$K^{\prime}$ has the exact opposite spectrum to $K$,
$K^{\prime}\Psi_{j}^{\prime}=-\xi_{j}\Psi_{j}^{\prime}$, where
$\Psi_{j}^{\prime}=\hat{\Gamma}^{w}\Psi_{j}$. Using this fact, we may impose
the following boundary conidition:

$\mathcal{P}^{K}_{\geq 0}\,\Upphi_{+}|_{\partial
W=Y}=0\,\,,\quad\mathcal{P}^{K^{\prime}}_{>0}\,\Upphi_{-}|_{\partial
W=Y}=0\,.$ (B.9)

$\mathcal{P}^{K}_{\geq 0}$ is the projection operator to the subspace of
spinors on $Y$ spanned by the eigenfunctions of $K$ with the eigenvalues
greater than or equal to zero, and similarly $\mathcal{P}^{K^{\prime}}_{>0}$
is the projection operator to the subspace spanned by the eigenfunctions of
$K^{\prime}$ with the eigenvalues strictly greater than zero. Because of
$K\hat{\Gamma}^{w}=-\hat{\Gamma}^{w}K^{\prime}$, we see that for the spinors
satisfying this boundary condition, the boundary term on the right-hand side
of Eq. (B.8) vanishes, and $Q$ is hermitian with this boundary condition.
There also exist more general choices of boundary conditions, but we will not
need them here. See [Yon16, KY21].

To know the corresponding boundary condition to be imposed on $\bar{\Upphi}$
inside the path integral, we consider the follwong mode expansions of fermion
fields near the boundary $\partial W=Y$, in terms of the eigenfunctions of $K$
and $K^{\prime}$:

$\displaystyle\begin{split}\Upphi_{+}(y^{\mu},w)&=\sum_{j}\alpha_{j}^{+}(w)\Psi_{j}(y^{\mu})\,,\\\
\Upphi_{-}(y^{\mu},w)&=\sum_{j}\alpha_{j}^{-}(w)\Psi_{j}^{\prime}(y^{\mu})=\sum_{j}\alpha_{j}^{-}(w)\hat{\Gamma}^{w}\Psi_{j}(y^{\mu})\,,\\\
\bar{\Upphi}_{+}(y^{\mu},w)&=\sum_{j}\beta_{j}^{+}(w)(\Psi_{j}^{\prime})^{\dagger}(y^{\mu})=\sum_{j}\beta_{j}^{+}(w)\Psi_{j}^{\dagger}(y^{\mu})\hat{\Gamma}^{w}\,,\\\
\bar{\Upphi}_{-}(y^{\mu},w)&=\sum_{j}\beta_{j}^{-}(w)\Psi_{j}^{\dagger}(y^{\mu})\,.\end{split}$
(B.10)

In terms of $\alpha^{\pm}_{j}$ and $\beta^{\pm}_{j}$, the action near the
boundary becomes

$S=\int
dw\sum_{j}\left[\beta_{j}^{+}(w)(\partial_{w}-\xi_{j})\alpha_{j}^{+}(w)+\beta_{j}^{-}(w)(\partial+\xi_{j})\alpha_{j}^{-}(w)+\cdots\right]$
(B.11)

where $\cdots$ contains terms proportional to $m_{0}$. We see that
$\beta_{j}^{\pm}$ is conjugate to $\alpha_{j}^{\pm}$ for each $j$. The APS
boundary condition (B.9) imposed on $\Upphi$ corresponds to setting at the
boundary $\alpha^{+}_{j}=0$ if $\xi_{j}\geq 0$, $\alpha_{j}^{-}=0$ if
$\xi_{j}<0$, and leaving the remaining coefficients to be unconstrained. The
corresponding boundary condition for $\bar{\Upphi}$ can then be read off. At
the boundary, we let $\beta^{+}_{j}=0$ if $\xi_{j}<0$, $\beta^{-}_{j}=0$ if
$\xi_{j}\geq 0$, and leave the remaining coefficients to be unconstrained.

The Hamiltonian near the boundary is given by

$\hat{H}=\sum_{j}\xi_{j}\left((\alpha_{j}^{-})^{\dagger}\alpha_{j}^{-}-(\alpha_{j}^{+})^{\dagger}\alpha_{j}^{+}\right)+\cdots$
(B.12)

where we again omitted terms proportional to $m_{0}$. The quantum state
corresponding to the APS boundary condition (B.9) is characterized by the
condition ($\beta^{\pm}_{j}=(\alpha^{\pm}_{j})^{\dagger}$)

$\begin{cases}\bra{\mathtt{APS}}\alpha^{+}_{j}=0&\text{if $\xi_{j}\geq 0$}\\\
\bra{\mathtt{APS}}(\alpha^{+}_{j})^{\dagger}=0&\text{if $\xi_{j}<0$}\\\
\bra{\mathtt{APS}}\alpha^{-}_{j}=0&\text{if $\xi_{j}<0$}\\\
\bra{\mathtt{APS}}(\alpha^{-}_{j})^{\dagger}=0&\text{if $\xi_{j}\geq
0$}\end{cases}$ (B.13)

and we can see that $\ket{\mathtt{APS}}$ is a ground state of the Hamiltonian
(B.12) when $m_{0}=0$. There will be more than one ground states if there are
zero modes $\xi_{j}=0$, and our choice of the boundary condition corresponds
to one of them. We set the ground state energy to be zero.

To derive the index of $Q$ under this boundary condition, our strategy is to
compute the following ratio of the two $(d+2)$\- dimensional partition
functions

$\frac{\mathcal{Z}_{d+2}[m_{0}e^{\mathrm{i}\alpha_{0}}]}{\mathcal{Z}_{d+2}[m_{0}]}=\frac{\int[D\bar{\Upphi}][D\Upphi]\exp\left(-\int_{W}\bar{\Upphi}\tau^{1}(-\mathrm{i}Q+m_{0}e^{\mathrm{i}\alpha_{0}\tau^{3}})\Upphi\right)}{\int[D\bar{\Upphi}][D\Upphi]\exp\left(-\int_{W}\bar{\Upphi}\tau^{1}(-\mathrm{i}Q+m_{0})\Upphi\right)}$
(B.14)

in two different ways, for an arbitrary constant phase $\alpha_{0}$.

Define
$Q^{\prime}=\mathrm{i}\tau^{1}\hat{\Gamma}^{\hat{\mu}}D_{\hat{\mu}}-\mathrm{i}\tau^{2}M$.
For $Q^{\prime}$, we impose different boundary conditions,
$\mathcal{P}^{K^{\prime}}_{>}\Upphi_{+}|_{Y}=0$ and
$\mathcal{P}^{K}_{\geq}\Upphi_{-}|_{Y}=0$. Under this boundary condition,
$Q^{\prime}$ is a hermitian operator on $W$. The two operators are related by
$Q\tau^{1}=-\tau^{1}Q^{\prime}$, which respects the respective boundary
conditions. That is, if we write the eigenvalues and the eigenfunctions of $Q$
as $Q\Upphi_{k}=\upsilon_{k}\Upphi_{k}$, then $Q^{\prime}$ has the exact
opposite spectrum to $Q$,
$Q^{\prime}\Upphi_{k}^{\prime}=-\upsilon_{k}\Upphi_{k}^{\prime}$, with the
eigenfunctions being $\Upphi_{k}^{\prime}=\tau^{1}\Upphi_{k}$.

Boundary conditions are chosen in such a way that we can expand the fermion
fields as

$\Upphi=\sum_{k}a_{k}\Upphi_{k}\,\,,\quad\bar{\Upphi}=\sum_{k}b_{k}(\Upphi_{k}^{\prime})^{\dagger}=\sum_{k}b_{k}\Upphi_{k}^{\dagger}\tau^{1}\,,$
(B.15)

where $a_{k}$ and $b_{k}$ are grassmannian coefficients. The path integral
measure can be defined to be

$[D\bar{\Upphi}][D\Upphi]=\prod_{k}db_{k}da_{k}\,.$ (B.16)

Consider the following change of variables for the path integral,

$\Upphi\rightarrow\Upphi^{\prime}=e^{\mathrm{i}(\alpha/2)\tau^{3}}\Upphi\,\,,\quad\bar{\Upphi}\rightarrow\bar{\Upphi}^{\prime}=\bar{\Upphi}e^{-\mathrm{i}(\alpha/2)\tau^{3}}\,,$
(B.17)

where $\alpha$ can be any arbitrary function which is constant on the
boundary, so that it does not affect the boundary conditions. If we write
$\Upphi^{\prime}=\sum_{k}a_{k}^{\prime}\Upphi_{k}$ and
$\bar{\Upphi}^{\prime}=\sum_{k}b_{k}^{\prime}\Upphi_{k}^{\dagger}\tau^{1}$,
then the coefficients $a_{k}^{\prime}$ and $b_{k}^{\prime}$ are given by

$\displaystyle\begin{split}a_{k}^{\prime}&=\int_{W}\Upphi_{k}^{\dagger}\Upphi^{\prime}=\sum_{l}\left(\int_{W}\Upphi_{k}^{\dagger}e^{\mathrm{i}\alpha\tau^{3}/2}\Upphi_{l}\right)a_{l}\,,\\\
b_{k}^{\prime}&=\int_{W}\bar{\Upphi}^{\prime}\tau^{1}\Upphi_{k}=\sum_{l}b_{l}\left(\int_{W}\Upphi_{l}^{\dagger}e^{\mathrm{i}\alpha\tau^{3}/2}\Upphi_{k}\right)\,.\end{split}$
(B.18)

The path integral measure is potentially non-invariant under this change of
variables,

$\prod_{k}db_{k}da_{k}=\left[\text{Det}(e^{\mathrm{i}\alpha\tau^{3}/2})\right]^{2}\prod_{k}db_{k}^{\prime}da_{k}^{\prime}=\left[\exp(\mathrm{i}\text{Tr}(\alpha\tau^{3}))\right]\prod_{k}db_{k}^{\prime}da_{k}^{\prime}$
(B.19)

where the determinant and the trace is over the whole space of spinors on $W$
spanned by the eigenfunctions $\Upphi_{k}$.

The action also changes under (B.17). For instance, the numerator in Eq.
(B.14) changes as

$\int_{W}\bar{\Upphi}\tau^{1}(-\mathrm{i}Q+m_{0}e^{\mathrm{i}\alpha_{0}\tau^{3}})\Upphi=\int_{W}\left[\bar{\Upphi}^{\prime}\tau^{1}(-\mathrm{i}Q+m_{0}e^{\mathrm{i}(\alpha_{0}-\alpha)\tau^{3}})\Upphi^{\prime}-\frac{\mathrm{i}}{2}(\partial_{\hat{\mu}}\alpha)\bar{\Upphi}^{\prime}\hat{\Gamma}^{\hat{\mu}}\tau^{3}\Upphi^{\prime}\right]\,.$
(B.20)

Now, consider making this change of variables to the numerator in Eq. (B.14)
with constant $\alpha=\alpha_{0}$. Then, the path integral measure picks up
the factor
$\exp(\mathrm{i}\alpha_{0}\text{Tr}(\tau^{3}))=\exp(\mathrm{i}\alpha_{0}\operatorname{Ind}Q)$,
and the remaining path integral is identical to the denominator. Thus, we get

$\frac{\mathcal{Z}_{d+2}[m_{0}e^{\mathrm{i}\alpha_{0}}]}{\mathcal{Z}_{d+2}[m_{0}]}=e^{\mathrm{i}\alpha_{0}\operatorname{Ind}Q}\,.$
(B.21)

Note that this is independent of $m_{0}$, as long as $m_{0}\neq 0$. If
$m_{0}=0$, the partition function vanishes when there exist any zero modes,
making the ratio ill-defined.

Then, following the trick from [KY21], we consider a different change of
variables again to the numerator of Eq. (B.14), where $\alpha$ is given by
$\alpha=\alpha_{0}\theta(w_{0}-w)$ inside the collar neighborhood of the
boundary and remains to be equal to constant $\alpha_{0}$ everywhere inside
the bulk. $w_{0}\in[0,-\epsilon)$ is a fixed Euclidean time inside the collar
neighborhood, and $\theta(w_{0}-w)$ is the Heaviside step function which
vanishes for $w>w_{0}$, i.e., near the boundary. Under this change of
variables, we get

$\frac{\mathcal{Z}_{d+2}[m_{0}e^{\mathrm{i}\alpha_{0}}]}{\mathcal{Z}_{d+2}[m_{0}]}=e^{\mathrm{i}\text{Tr}(\alpha_{0}\theta(w_{0}-w)\tau^{3})}\times\frac{\int[D\bar{\Upphi}][D\Upphi]e^{-\int_{W}\bar{\Upphi}\tau^{1}(-\mathrm{i}Q+m_{0}e^{\mathrm{i}\alpha_{0}\theta(w-w_{0})\tau^{3}})\Upphi}e^{-\frac{\mathrm{i}}{2}\alpha_{0}\mathcal{O}(w_{0})}}{\int[D\bar{\Upphi}][D\Upphi]e^{-\int_{W}\bar{\Upphi}\tau^{1}(-\mathrm{i}Q+m_{0})\Upphi}}$
(B.22)

where
$\mathcal{O}(w_{0})=\int_{Y,w=w_{0}}\bar{\Upphi}\hat{\Gamma}^{w}\tau^{3}\Upphi$
is an “axial” charge operator inserted at $w=w_{0}$ slice inside the collar
neighborhood.

We divide the manifold $W$ into two pieces. Let $W_{1}$ be a cylindrical
region near the boundary $\partial W=Y$ of length $w_{0}\leq w\leq 0$, and
$W_{0}$ be the rest of $W$, such that gluing $W_{1}$ and $W_{0}$ along their
common boundary gives the whole $W$. Path integral over $W_{0}$ defines a
state as usual, $\ket{W_{0}}$, and the path integral over $W_{1}$ gives a time
evolution which we denote as $\hat{W}_{1}$. Then, we have

$\frac{\int[D\bar{\Upphi}][D\Upphi]e^{-\int_{W}\bar{\Upphi}\tau^{1}(-\mathrm{i}Q+m_{0}e^{\mathrm{i}\alpha_{0}\theta(w-w_{0})\tau^{3}})\Upphi}e^{-\frac{\mathrm{i}}{2}\alpha_{0}\mathcal{O}(w_{0})}}{\int[D\bar{\Upphi}][D\Upphi]e^{-\int_{W}\bar{\Upphi}\tau^{1}(-\mathrm{i}Q+m_{0})\Upphi}}=\frac{\braket{\mathtt{APS}}{\hat{W}_{1}(m_{0}e^{\mathrm{i}\alpha_{0}})e^{-\frac{\mathrm{i}}{2}\alpha_{0}\hat{\mathcal{O}}(w_{0})}}{W_{0}}}{\braket{\mathtt{APS}}{\hat{W}_{1}(m_{0})}{W_{0}}}$
(B.23)

where for the time evolution along $W_{1}$, we have
$m_{1}+\mathrm{i}m_{2}=m_{0}e^{\mathrm{i}\alpha_{0}}$ for the numerator,
whereas $m_{1}+\mathrm{i}m_{2}=m_{0}$ for the denominator. $\hat{\mathcal{O}}$
is the quantum operator corresponding to the charge $\mathcal{O}$.

Now, since $\ket{\mathtt{APS}}$ is a ground state on $Y$ when $m_{0}=0$, if we
take the limit $m_{0}\rightarrow 0$ in the above, we obtain

$\frac{\braket{\mathtt{APS}}{\hat{W}_{1}(m_{0}e^{\mathrm{i}\alpha_{0}})e^{-\frac{\mathrm{i}}{2}\alpha_{0}\hat{\mathcal{O}}(w_{0})}}{W_{0}}}{\braket{\mathtt{APS}}{\hat{W}_{1}(m_{0})}{W_{0}}}\rightarrow\frac{\braket{\mathtt{APS}}{e^{-\frac{\mathrm{i}}{2}\alpha_{0}\hat{\mathcal{O}}(w_{0})}}{W_{0}}}{\braket{\mathtt{APS}}{W_{0}}}\,.$
(B.24)

Next, using the mode expansions (B.10), the operator
$\hat{\mathcal{O}}(w_{0})$ can be expressed as