Lattice construction of exotic invertible topological phases

Ryohei Kobayashi

Institute for Solid State Physics,
---
University of Tokyo, Kashiwa, Chiba 277-8583, Japan

In this paper, we provide state sum path integral definitions of exotic
invertible topological phases proposed in the recent paper by Hsin, Ji, and
Jian. The exotic phase has time reversal ($T$) symmetry, and depends on a
choice of the spacetime structure called the Wu structure. The exotic phase
cannot be captured by the classification of any bosonic or fermionic
topological phases, and thus gives a novel class of invertible topological
phases. When the $T$ symmetry defect admits a spin structure, our construction
reduces to a sort of the decorated domain wall construction, in terms of a
bosonic theory with $T$ symmetry defects decorated with a fermionic phase that
depends on a spin structure of the $T$ symmetry defect. By utilizing our path
integral, we propose a lattice construction for the exotic phase that
generates the $\mathbb{Z}_{8}$ classification of the (3+1)d invertible phase
based on the Wu structure. This generalizes the $\mathbb{Z}_{8}$
classification of the $T$-symmetric (1+1)d topological superconductor proposed
by Fidkowski and Kitaev. On oriented spacetime, this (3+1)d invertible phase
with a specific choice of Wu structure reduces to a bosonic Crane-Yetter TQFT
which has a topological ordered state with a semion on its boundary. Moreover,
we propose a subclass of $G$-SPT phases based on the Wu structure labeled by a
pair of cohomological data in generic spacetime dimensions. This generalizes
the Gu-Wen subclass of fermionic SPT phases.

###### Contents

  1. 1 Introduction
  2. 2 Review on Wu structure
  3. 3 Path integral of an exotic phase
    1. 3.1 Oriented case
    2. 3.2 Unoriented case
  4. 4 (3+1)d exotic invertible phase: $\mathbb{Z}_{8}$ classification
    1. 4.1 Arf-Brown-Kervaire invariant: Kitaev type phase
    2. 4.2 Relation to the Crane-Yetter model in oriented case
  5. 5 Gu-Wen type phases
  6. 6 Discussion
  7. A Review on higher cup product
  8. B The obstruction class for Wu structure
  9. C Bulk-boundary Grassmann integral

##  1 Introduction

The classification of gapped phases is an important problem in condensed
matter physics. Though the classification problem is difficult and unsolved in
general, one can simplify the problem by considering a simplified class of
systems with a unique gapped ground state on arbitrary closed spatial
manifolds. Such phases are called invertible topological phases. In the
presence of a global symmetry $G$, invertible topological phases are also
called symmetry protected topological (SPT) phases protected by $G$.
111Precisely speaking, it is more common to define SPT phases as invertible
phases which become trivial when we forget the global symmetry, following [1].
For example, the (1+1)-dimensional topological superconductor (Kitaev wire) is
not counted as an SPT phase based on such a definition, though it is an
invertible topological phase. In this thesis, we do not make a careful
distinction between the two concepts, namely invertible phases and SPT phases.

In the case of free fermions, a complete classification of SPT phases has been
obtained using K-theory [2, 3]. In the case of intrinsically interacting
systems, the classification in general differs from the free phases [4], and
we have to use a completely different method to perform the classification [1,
5, 6, 7, 8, 9, 10, 11, 12]. For example, a large class of interacting bosonic
SPT phases with the global symmetry can be classified by utilizing group
cohomology [1]. For the case of fermionic systems, the SPT phases have a
richer classification than the bosonic phases; for example, Gu and Wen found
that a subclass of fermionic SPT phases is classified by a pair of
cohomological data [5], generalizing the classification of bosonic phases.
These fermionic SPT phases are called Gu-Wen phases or super-cohomology
phases. Later, a comprehensive classification scheme of SPT phases utilizing
cobordism group is proposed in [6], which is thought to classify invertible
field theories which effectively describe SPT phases. The cobordism group
provides a generic and powerful framework that correctly predicts the
classification of interacting invertible topological phases, based on onsite
(0-form) global symmetries, time reversal symmetry, or higher-form symmetries
[6, 13, 14, 15, 16].

So far, almost all the invertible topological phases discussed in literature
has been either bosonic or fermionic. This implies that the effective field
theories describing these topological phases require either oriented or spin
structure (or these spacetime structures twisted by other symmetry groups such
as pin+ or pin-) of the spacetime. 222Here we assume that the effective field
theories are Lorentz invariant, and thereby oriented $SO(d)$ or spin group
$Spin(d)$ contains the Lorentz group. Recently, there are also classes of
topological phases whose effective field theories lack the Lorentz invariance,
such as fractons [17, 18, 19, 20] or those realized by foliated field theories
[21]. These theories are beyond the scope of the present paper. As such, the
classification of invertible topological phases based on the cobordism groups
has been performed for the corresponding field theories with (possibly
twisted) oriented or spin structure.

Recently, it was proposed in [22] that there are nontrivial invertible
topological field theories based on the spacetime structure called Wu
structure, which is inequivalent to any (possibly twisted) oriented or spin
structure previously discussed in literature, and thus gives a new class of
invertible field theories which are phrased neither as bosonic nor fermionic.
The topological phases that depend on the spacetime Wu structure are called
exotic topological phases. Wu structure in $d$ spacetime dimensions
corresponds to the global symmetry given by a specific nontrivial mixture of
the spacetime Lorentz symmetry $O(d)$ and the 1-form $\mathbb{Z}_{2}$
symmetry. Since the Lorentz group is taken as $O(d)$, the exotic topological
phases possess the time reversal symmetry. Mathematically, the symmetry is
described by a specific 2-group that corresponds to a sort of the extension of
$O(d)$ by the 1-form $\mathbb{Z}_{2}$ symmetry. This generalizes spin/pin
structure required for fermionic systems, where one extends the Lorentz
symmetry $SO(d)$ or $O(d)$ by the ordinary (0-form) $\mathbb{Z}_{2}$ symmetry
that corresponds to the $\mathbb{Z}_{2}$ fermion parity.

In this paper, we explore a lattice realization of exotic topological phases.
We define topologically invariant path integral constructions of exotic
invertible topological phases in and more than (3+1)d in terms of a state sum
on a lattice. In particular, the existence of an explicit state sum for the
topological path integral of exotic invertible topological phases suggests
that there should also exist corresponding commuting projector Hamiltonians,
tensor network descriptions, and explicit quantum circuits that prepare these
exotic topological phases. We leave it to future work to explicitly develop
these descriptions.

### Summary of the main results

Here we summarize the main results of the paper. First, we propose a state sum
path integral for a specific theory $z(\eta,a)$ on a spacetime $d$-manifold
$M$ equipped with a triangulation, coupled with the Wu structure $\eta$ and
the $(d-3)$-form $\mathbb{Z}_{2}$ symmetry whose background gauge field is
given by $a\in Z^{d-2}(M,\mathbb{Z}_{2})$. The partition function $z(\eta,a)$
is not topologically invariant, but has an ’t Hooft anomaly of the
$(d-2)$-form $\mathbb{Z}_{2}$ symmetry characterized by the
$(d+1)$-dimensional response theory

$\displaystyle(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{3}(a)}=(-1)^{\int
a\cup_{d-5}a},$ (1.1)

where $\mathop{\mathrm{Sq}}\nolimits^{i}$ denotes the Steenrod operation of
cohomology groups [23] and $\cup_{i}$ is higher cup product, see Appendix A
for their review. That is, if one considers gauge transformation and the re-
triangulation of the spacetime, the partition function transforms as

$\displaystyle
z(M^{\prime};\eta^{\prime},a^{\prime})=(-1)^{\int_{\widetilde{M}}\mathop{\mathrm{Sq}}\nolimits^{3}(a)}z(M;\eta,a),$
(1.2)

where $\widetilde{M}=M\times[0,1]$ is a $(d+1)$-manifold that interpolates $M$
and $M^{\prime}$ equipped with the Wu structure and the $(d-2)$-form gauge
field $a$, which restrict to $(\eta,a)$ and $(\eta^{\prime},a^{\prime})$ on
$M$ and $M^{\prime}$ respectively. $M^{\prime}$ is a $d$-manifold which is the
same as $M$ with a different triangulation and the gauge field $a^{\prime}$,
equivalent to $a$ in cohomology $[a]=[a^{\prime}]$.

One can further show that the partition function becomes a quadratic function
of the $(d-2)$-form gauge field $a\in Z^{d-2}(M,\mathbb{Z}_{2})$ which
satisfies the quadratic property

$\displaystyle z(\eta,a)z(\eta,b)=z(\eta,a+b)(-1)^{\int_{M}a\cup_{d-4}b}.$
(1.3)

The construction of the theory $z(\eta,a)$ is done by a sort of decorated
domain wall construction, namely it involves decorating the symmetry defect
with a topological phase in lower spacetime dimensions. That is, $z(\eta,a)$
is given by decorating the codimension-1 defect of the time-reversal ($T$)
symmetry of a bosonic phase with a specific gapped theory, which is expressed
as a path integral of Grassmann variables supported on the $T$ symmetry
defect. The Grassmann integral on the $T$ symmetry defect is essentially the
same as that utilized to express the partition function of fermionic SPT
phases based on the spin structure by Gu and Wen [5]. In particular, when the
$T$ symmetry defect admits the spin structure, we find that the Wu structure
can be specified by a choice of spin structure on the codimension-1 $T$
symmetry defect. If the Wu structure is given in this manner, our path
integral of $z(\eta,a)$ is precisely given by a decorated domain wall
construction, where we decorate the $T$ symmetry defect of a bosonic
topological phase with the fermionic topological phase based on the spin
structure of the $T$ symmetry defect.

In the case of the spacetime dimension $d=4$, the theory $z(\eta,a)$ gives a
topologically invariant path integral, since the ’t Hooft anomaly (1.1)
vanishes in $d=4$. Actually, when the spacetime is oriented, the Wu structure
can be taken to be trivial, so in that case the exotic topological phase
reduces to a bosonic topological phase. In particular, on oriented 4-manifolds
$z(\eta,a)$ reproduces a (3+1)d bosonic Crane-Yetter TQFT based on a unitary
modular tensor category $\mathcal{C}=\\{1,s\\}$ that has a single nontrivial
anyon $s$ which is a semion.

For the case of $d=4$, one can gauge the 1-form $\mathbb{Z}_{2}$ symmetry of
$z(\eta,a)$, which gives the Arf-Brown-Kervaire (ABK) invariant based on the
quadratic refinement $z(\eta,a)$ of $[a]\in H^{2}(M,\mathbb{Z}_{2})$. The
resulting theory turns out to generate the $\mathbb{Z}_{8}$ classification of
the exotic invertible topological phase [22]. This generalizes the
$\mathbb{Z}_{8}$ classification of the (1+1)d pin- invertible topological
phases that corresponds to the (1+1)d topological superconductor proposed by
Fidkowski and Kitaev [4].

In addition, by utilizing the lattice construction of $z(\eta,a)$ we propose a
subclass of the exotic invertible topological phases with an onsite (0-form)
symmetry $G$, which contains anti-unitary time-reversal symmetry. In this
paper, we limit ourselves to the case of $G=G_{0}\times\mathbb{Z}_{2}^{T}$
with $\mathbb{Z}_{2}^{T}$ a time-reversal symmetry, and the total symmetry
group is direct product of $G_{0}$ and the 1-form symmetry of the Wu
structure. It should be interesting to consider the nontrivial 2-group that
involves $G$ and the 1-form $\mathbb{Z}_{2}$ symmetry of the Wu structure. We
comment on the expectation about some of twisted cases in the discussion (Sec.
6), and leave the study on such a twisted Wu structure by $G$ for future work.

Then, we propose a subclass of exotic SPT phases labeled by a pair of the
cohomological data

$\displaystyle(\nu_{d},n_{d-2})\in C_{\rho}^{d}(BG,U(1))\times
Z^{d-2}(BG,\mathbb{Z}_{2}),$ (1.4)

which are subject to the constraint

$\displaystyle\delta_{\rho}\nu_{d}=\frac{1}{2}\mathrm{Sq}^{3}(n_{d-2})\mod 1,$
(1.5)

where $\rho$ denotes the twisted $G$-action on $U(1)$ that acts by complex
conjugation. This generalizes the Gu-Wen phase [5] that constitutes the
subclass of the fermionic SPT phases, labeled by $(\nu_{d},n_{d-1})\in
C_{\rho}^{d}(BG,U(1))\times Z^{d-1}(BG,\mathbb{Z}_{2})$ subject to the Gu-Wen
equation which basically replaces $\mathop{\mathrm{Sq}}\nolimits^{3}$ in (1.5)
with $\mathop{\mathrm{Sq}}\nolimits^{2}$.

In general, by utilizing the theory $z(\eta,a)$, one can produce a state sum
path integral of an exotic topological phase, starting with a bosonic theory
with a $(d-3)$-form $\mathbb{Z}_{2}$ symmetry which has a specific ’t Hooft
anomaly. That is, we consider a partition function of a bosonic topological
phase $Z_{b}(a)$ coupled with a $(d-2)$-form $\mathbb{Z}_{2}$ gauge field
$a\in Z^{d-2}(M,\mathbb{Z}_{2})$. Let us assume that the bosonic theory
$Z_{b}(a)$ has an ’t Hooft anomaly characterized by the response action
$(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{3}(a)}$. Then, the partition
function of the exotic invertible phases can be constructed as

$\displaystyle Z(\eta)\propto\sum_{[a]\in
H^{d-2}(M,\mathbb{Z}_{2})}Z_{b}(a)z(\eta,a).$ (1.6)

This construction of exotic topological phases generalizes that of fermionic
topological phases by Gaiotto and Kapustin [24] based on fermion condensation
within the path integral framework.

This paper is organized as follows. After reviewing Wu structure in Sec. 2, we
provide a path integral definition for the theory coupled with the Wu
structure in Sec. 3. In Sec. 4, we study the spacetime dimensions $d=4$, and
construct an invertible topological phase that generates the $\mathbb{Z}_{8}$
classification. We also show that the phase is equivalent to a bosonic Crane-
Yetter TQFT in the oriented case with a specific choice of the Wu structure.
In Sec. 5, we propose a Gu-Wen type subclass of the $G$-SPT phases based on
the Wu structure.

##  2 Review on Wu structure

In this section, we briefly review for Wu structure of the spacetime in
continuum. First of all, let us recall spin structure of the spacetime
required for fermionic phases. Wu structure is then regarded as a natural
generalization of spin structure.

### spin structure

A relativistic quantum field theory in $d$ spacetime dimensions possesses the
Lorentz $SO(d)$ symmetry. However, since fermions are spinors, fermions
transform according to the double cover of $SO(d)$, which is $Spin(d)$. To
define the field theory on a generic spacetime manifold, one needs to consider
a $SO(d)$ bundle $\phi:M\to BSO(d)$, which is the tangent bundle $TM$ of an
oriented triangulated manifold $M$. In order to have fermions, the transition
functions $\phi_{ij}\in SO(d)$ defined on 1-simplices $\braket{ij}$ must be
lifted to $\widetilde{\phi}_{ij}\in Spin(d)$. Since $Spin(d)$ is the group
extension

$\displaystyle\mathbb{Z}_{2}\to Spin(d)\to SO(d)$ (2.1)

whose extension is given by $w_{2}\in H^{2}(BSO(d),\mathbb{Z}_{2})$. Then,
$Spin(d)$ is identified as $SO(d)\times\mathbb{Z}_{2}$ as a set, so we can
express $\widetilde{\phi}_{ij}$ as a pair $(\phi_{ij},\eta_{ij})\in
SO(d)\times\mathbb{Z}_{2}$. The nontrivial group extension is reflected in the
multiplication law of $\mathbb{Z}_{2}$ elements $\eta_{ij}$ twisted by
$w_{2}$. Namely, for transition functions $\widetilde{\phi}_{ij}\in Spin(d)$,
we have the multiplication law

$\displaystyle\widetilde{\phi}_{01}\widetilde{\phi}_{12}=(\phi_{01}\phi_{12},\eta_{01}+\eta_{12}+w_{2}(\phi_{01},\phi_{12})).$
(2.2)

Due to the cocycle condition
$\widetilde{\phi}_{01}\widetilde{\phi}_{12}=\widetilde{\phi}_{02}$, we find on
each 2-simplex $\braket{012}$

$\displaystyle\eta_{01}+\eta_{12}+\eta_{02}=w_{2}(\phi_{01},\phi_{12})$ (2.3)

In coordinate-free notation, this is precisely the equation
$\delta\eta=w_{2}$.

### Wu structure

The Wu structure is a natural generalization of the spin structure illustrated
above. The symmetry groups involved are Lorentz group $O(d)$ that includes
time reversal symmetry, and a 1-form $\mathbb{Z}_{2}$ symmetry denoted as
$\mathbb{Z}_{2}^{[1]}$ here. Then, the total structure of the symmetry
$\mathbb{G}$ is described as a nontrivial mixture called a 2-group between
$O(d)$ and $\mathbb{Z}_{2}^{[1]}$ symmetry,

$\displaystyle\mathbb{Z}_{2}^{[1]}\to\mathbb{G}\to O(d)$ (2.4)

and the 2-group structure involving $O(d)$ and $\mathbb{Z}_{2}^{[1]}$ is
specified by the cohomology class $w_{1}w_{2}\in H^{3}(BO(d),\mathbb{Z}_{2})$
called the Postnikov class. Analogously to the nontrivial group extension for
the spin structure, a nontrivial 2-group has a distinct algebra of symmetry
generators, thereby has a distinct configuration of background gauge field
from that of the direct product $\mathbb{Z}_{2}^{[1]}\times O(d)$.

That is, if we denote the background gauge field of $\mathbb{Z}_{2}^{[1]}$ as
$\eta\in C^{2}(M,\mathbb{Z}_{2})$ and that of $O(d)$ as $\phi:M\to BO(d)$,
then these background gauge fields are correlated on each 3-simplex
$\braket{0123}$ as

$\displaystyle\eta_{012}+\eta_{013}+\eta_{023}+\eta_{123}=w_{1}(\phi_{01})w_{2}(\phi_{12},\phi_{23}).$
(2.5)

In coordinate-free notation, this is precisely the equation
$\delta\eta=w_{1}w_{2}$.

It is worth noting that the background $\mathbb{Z}_{2}^{[1]}$ gauge field
twisted by the Postnikov class $\delta\eta=w_{1}w_{2}$ reflects the algebra
for the symmetry generators of $\mathbb{Z}_{2}^{[1]}$ and $O(d)$ in the
2-group. To see this, we regard the background gauge field of $p$-form
symmetry as an insertion of a $(d-p-1)$-dimensional symmetry defects on the
Poincaré dual of the $(p+1)$-form background gauge field. Then, a 3-simplex is
understood as an associator which shifts the configuration of 0-form $O(d)$
symmetry defects by the $F$-move, see Fig. 1 (a).

During the $F$-move inside a 3-simplex, the three $O(d)$ symmetry defects
$\phi_{01},\phi_{12},\phi_{23}$ meets at the junction in the middle of the
movie, which gives a codimension-3 junction of the $O(d)$ symmetry defects,
see Fig. 1 (b). Then, the 2-group relation $\delta\eta=w_{1}w_{2}$ means that
the $\mathbb{Z}_{2}^{[1]}$ symmetry generator
$w_{1}(\phi_{01})w_{2}(\phi_{12},\phi_{23})$ is sourced from the codimension-3
junction. The 2-group is understood as the modification of the associativity
of the 0-form $O(d)$ symmetry generators by a generator of the 1-form
$\mathbb{Z}_{2}^{[1]}$ symmetry, controlled by the Postnikov class in
$H^{3}(BO(d),\mathbb{Z}_{2})$. See [25, 26, 27, 28] for an introduction of the
2-group.

Figure 1: Wu structure is the nontrivial 2-group between the Lorentz $O(d)$
symmetry and the $\mathbb{Z}_{2}^{[1]}$ 1-form symmetry. The Wu structure is
specified by a choice of a $\mathbb{Z}_{2}$ 2-cochain $\eta$ with
$\delta\eta=w_{1}w_{2}$, where $\eta$ gives a background gauge field of the
$\mathbb{Z}_{2}^{[1]}$ symmetry. This is understood as sourcing the
$\mathbb{Z}_{2}^{[1]}$ symmetry defect from the codimension-3 junction of
$O(d)$ symmetry defects on each 3-simplex.

##  3 Path integral of an exotic phase

Now we construct a path integral definition for $z(\eta,a)$ on a triangulated
$d$-manifold $M$ that depends on the Wu structure specified by a choice of
$\eta\in C^{2}(M,\mathbb{Z}_{2})$ with $\delta\eta=w_{1}w_{2}$, and a
$(d-2)$-form gauge field $[a]\in H^{d-2}(M,\mathbb{Z}_{2})$. The partition
function $z(\eta,a)$ is valued in $\\{\pm 1,\pm i\\}$. Then, the main
properties of $z(\eta,a)$ is summarized as follows:

  1. 1.

The quadratic property

$\displaystyle z(\eta,a)z(\eta,b)=z(\eta,a+b)(-1)^{\int_{M}a\cup_{d-4}b}.$
(3.1)

  2. 2.

The change of $z(\eta,a)$ under the gauge transformation $a\to a+\delta\chi$
or under the change of the triangulation is controlled by ’t Hooft anomaly
whose response action is given by

$\displaystyle(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{3}(a)}.$ (3.2)

The description of the Wu structure used for $z(\eta,a)$ is based on a
specific chain $S\in Z_{d-3}(M,\mathbb{Z}_{2})$ that represents the Poincaré
dual of the obstruction class $w_{1}w_{2}$ for the Wu structure. Then, the Wu
structure is specified by a choice of a $(d-2)$-chain $E\in
C_{d-2}(M,\mathbb{Z}_{2})$ with $\partial E=S$, which is Poincaré dual to
$\eta\in C^{2}(M,\mathbb{Z}_{2})$. As discussed in Appendix B, we prepare $S$
based the choices of the cochain and chain representative of $w_{1}$ and
$w_{2}$ respectively, and then take the pairing of them. In particular, when
the spacetime manifold $M$ is oriented, we take the representative of $w_{1}$
as zero and thereby $S=0$. So, for oriented spacetime manifolds $\eta$ is
closed and one can specify the Wu structure by $[\eta]\in
H^{2}(M,\mathbb{Z}_{2})$. As a warm-up, we start with the description of the
simplest case $z(\eta,a)$ on oriented manifolds.

###  3.1 Oriented case

Here let $M$ be an oriented manifold. In that case $\eta$ is closed
$\delta\eta=0$, and $z(\eta,0)$ is given by

$\displaystyle z(\eta,a)=e^{2\pi i\int q(a)}(-1)^{\int_{E}a},$ (3.3)

where $E\in Z_{d-2}(M,\mathbb{Z}_{2})$ is the Poincaré dual of $\eta$, and

$\displaystyle
q(a)=\frac{1}{4}(\widehat{a}\cup_{d-4}\widehat{a}+\widehat{a}\cup_{d-3}\delta\widehat{a}),$
(3.4)

where $\widehat{a}\in C^{d-2}(M,\mathbb{Z}_{4})$ is the $\mathbb{Z}_{4}$ lift
of $a$. When $d=4$, the above $q(a)$ gives a cohomology operation
$q:H^{2}(M,\mathbb{Z}_{2})\to H^{4}(M,\mathbb{Z}_{4})$ known as the Pontryagin
square [29, 30, 31], and thus $z(\eta=0,a)$ is topologically invariant.
However, the action $z(\eta,a)$ is not topologically invariant for $d>4$,
because

$\displaystyle\delta(q(a))=\frac{1}{2}a\cup_{d-5}a=\frac{1}{2}\mathop{\mathrm{Sq}}\nolimits^{3}(a)\mod
1.$ (3.5)

This demonstrates the ’t Hooft anomaly of $z(\eta=0,a)$ represented in (3.2).
This can be seen by using the Leibniz rule for higher cup product

$\displaystyle\delta(u\cup_{i}v)=(-1)^{p+q-i}u\cup_{i-1}v+(-1)^{pq+p+q}v\cup_{i-1}u+\delta
u\cup_{i}v+(-1)^{p}u\cup_{i}\delta v,$ (3.6)

with $u\in C^{p},v\in C^{q}$ cochains. According to the Leibniz rule we have

$\displaystyle\begin{split}\delta(\widehat{a}\cup_{d-4}\widehat{a})&=2(-1)^{d}\widehat{a}\cup_{d-5}\widehat{a}+\delta\widehat{a}\cup_{d-4}\widehat{a}+(-1)^{d}\widehat{a}\cup_{d-4}\delta\widehat{a},\\\
\delta(\widehat{a}\cup_{d-3}\delta\widehat{a})&=(-1)^{d}\widehat{a}\cup_{d-4}\delta\widehat{a}-\delta\widehat{a}\cup_{d-4}\widehat{a}+\delta\widehat{a}\cup_{d-3}\delta\widehat{a}.\end{split}$
(3.7)

Putting these expressions into $\delta(q(a))$ shows (3.5). Next, let us
consider a quadratic property of $z(\eta,a)$ in (3.1). We have

$\displaystyle\begin{split}q(a)+q(b)-q(a+b)&=\frac{1}{4}(\widehat{a}\cup_{d-4}\widehat{b}+\widehat{b}\cup_{d-4}\widehat{a}+\widehat{a}\cup_{d-3}\delta\widehat{b}+\widehat{b}\cup_{d-3}\delta\widehat{a}).\end{split}$
(3.8)

Here, the rhs can be rewritten as

$\displaystyle
q(a)+q(b)-q(a+b)=\frac{1}{2}{a}\cup_{d-4}{b}+\frac{1}{4}\delta(\widehat{a}\cup_{d-3}\widehat{b}+\delta\widehat{a}\cup_{d-2}\widehat{b})\mod
1.$ (3.9)

To see this, we note that

$\displaystyle\begin{split}\frac{1}{4}\delta(\widehat{a}\cup_{d-3}\widehat{b})&=\frac{1}{4}(\delta\widehat{a}\cup_{d-3}\widehat{b}+(-1)^{d}\widehat{a}\cup_{d-3}\delta\widehat{b}+(-1)^{d-1}\widehat{a}\cup_{d-4}\widehat{b}+(-1)^{d}\widehat{b}\cup_{d-4}\widehat{a})\\\
&=\frac{1}{4}((-1)^{d-1}\widehat{b}\cup_{d-3}\delta\widehat{a}+(-1)^{d-1}\delta(\delta\widehat{a}\cup_{d-2}\widehat{b})-\delta\widehat{a}\cup_{d-2}\delta\widehat{b}\\\
&\quad+(-1)^{d}\widehat{a}\cup_{d-3}\delta\widehat{b}+(-1)^{d-1}\widehat{a}\cup_{d-4}\widehat{b}+(-1)^{d}\widehat{b}\cup_{d-4}\widehat{a}).\end{split}$
(3.10)

Here we again used the Leibniz rule (3.6). Since
$\frac{1}{4}\delta\widehat{a}\cup_{d-2}\delta\widehat{b}=0\mod 1$, we have

$\displaystyle\begin{split}\frac{1}{4}\delta(\widehat{a}\cup_{d-3}\widehat{b})&=\frac{1}{4}((-1)^{d-1}\widehat{b}\cup_{d-3}\delta\widehat{a}+(-1)^{d-1}\delta(\delta\widehat{a}\cup_{d-2}\widehat{b})\\\
&\quad+(-1)^{d}\widehat{a}\cup_{d-3}\delta\widehat{b}+(-1)^{d-1}\widehat{a}\cup_{d-4}\widehat{b}+(-1)^{d}\widehat{b}\cup_{d-4}\widehat{a})\mod
1.\end{split}$ (3.11)

Hence,

$\displaystyle\begin{split}\frac{1}{2}&(\widehat{a}\cup_{d-4}\widehat{b})+\frac{1}{4}\delta(\widehat{a}\cup_{d-3}\widehat{b}+\delta\widehat{a}\cup_{d-2}\widehat{b})\\\
&=\frac{(-1)^{d}}{4}(\widehat{a}\cup_{d-4}\widehat{b}+\widehat{b}\cup_{d-4}\widehat{a}+\widehat{a}\cup_{d-3}\delta\widehat{b}+\widehat{b}\cup_{d-3}\delta\widehat{a})\mod
1.\end{split}$ (3.12)

Thus we obtain (3.9). When the spacetime manifold $M$ is oriented, only the
first term of the rhs of (3.9) contributes mod 1 to the integral on $M$. Thus
we obtain the quadratic property as

$\displaystyle z(\eta,a)z(\eta,b)=z(\eta,a+b)(-1)^{\int_{M}a\cup_{d-4}b}.$
(3.13)

This is what we want.

###  3.2 Unoriented case

Now let us construct the Grassmann integral $z(\eta,a)$ on a $d$-manifold $M$
which might be unoriented. We construct an unoriented manifold by picking
locally oriented patches, and then gluing them along codimension one loci by
transition functions. The locus where the transition functions are orientation
reversing, constitutes a representative of the dual of first Stiefel-Whitney
class $w_{1}$. We will sometimes call the locus an orientation reversing wall,
represented as a $(d-1)$-cycle $W$ Poincaré dual to $w_{1}$. For convenience,
we endow $M$ with a barycentric subdivision for the triangulation of $M$.
Namely, each $d$-simplex in the initial triangulation of $M$ is subdivided
into $(d+1)!$ simplices, whose vertices are barycenters of the subsets of
vertices in the $d$-simplex. We then assign a local ordering to vertices of
the barycentric subdivision, such that a vertex on the barycenter of $i$
vertices is labeled as $i$. Each simplex can then be either a $+$ simplex or a
$-$ simplex, depending on whether the ordering agrees with the orientation or
not.

Figure 2: The $\pm$ signs of $d$-simplices near the orientation reversing wall
$W$ for $d=2$, which is represented as a red line. The $d$-simplices are
barycentric-subdivided.

Then, one can explicitly obtain a $(d-3)$-cycle $S\in
Z_{d-3}(M,\mathbb{Z}_{2})$ that represents the Poincaré dual of $w_{1}w_{2}$
based on the barycentric subdivision. That is, $S$ is given by the set of all
$(d-3)$-simplices of $W$. Since it is known that the set of all
$(D-i)$-simplices in a $D$-dimensional closed manifold gives the
representative of $w_{i}$ [32, 33], one can see that $S$ also gives the
representative for the Poincaré dual of $w_{2}$ in $W$. The proof that $S$
represents the Poincaré dual of $w_{1}w_{2}$ of $M$ is given in Appendix B.

Analogously to what we did in the oriented case, we would like to define
$z(\eta,a)$ in terms of the action $e^{2\pi iq(a)}$ with $q(a)$ defined in
(3.4). However, the action $e^{2\pi iq(a)}$ does not correctly produce the
quadratic property (3.1); according to the quadratic property of $q(a)$
obtained in (3.9), we instead have

$\displaystyle e^{2\pi i\int q(a)}e^{2\pi i\int q(b)}=e^{2\pi i\int
q(a+b)}\cdot(-1)^{\int_{M}a\cup_{d-4}b}(-1)^{\int_{W}a\cup_{d-3}b}.$ (3.14)

This is because the integration of the coboundary term
$\frac{1}{4}\int\delta(\widehat{a}\cup_{d-3}\widehat{b}+\delta\widehat{a}\cup_{d-2}\widehat{b})$
in (3.9) evaluates non-trivially on the orientation-reversing wall, as
$\frac{1}{2}\int_{W}a\cup_{d-3}b\mod 1$. Hence, in order to obtain the desired
quadratic property of $z(\eta,a)$, we introduce an additional term
$\sigma(W;a)$ on the orientation-reversing wall, and define

$\displaystyle z(\eta,a)=e^{2\pi iq(a)}\sigma(W;a)(-1)^{\int_{E}a},$ (3.15)

where $\sigma(W;a)$ is a $(d-1)$-dimensional action supported on $W$, and
$E\in C_{d-2}(M,\mathbb{Z}_{2})$ is the Wu structure that satisfies $\partial
E=S$. If $W$ admits a spin structure, $E$ can be taken as the element of
$C_{d-2}(W,\mathbb{Z}_{2})$ since $S$ represents the Poincaré dual of $w_{2}$
on $W$. Then the Wu structure of $M$ is specified by the spin structure of
$W$. In that case the action (3.15) is regarded as a decorated domain wall
construction, where one decorates the $T$ symmetry defect of a bosonic theory
with a spin (fermionic) theory $\sigma(W;a)(-1)^{\int_{E}a}$.

$\sigma(W;a)$ has the two main properties summarized as follows:

  1. 1.

The quadratic property

$\displaystyle\sigma(W;a)\sigma(W;a)=\sigma(W;a+b)(-1)^{\int_{W}a\cup_{d-3}b}.$
(3.16)

  2. 2.

if one considers gauge transformation and the re-triangulation of the
spacetime, the partition function transforms as

$\displaystyle\sigma(W^{\prime};a^{\prime})=(-1)^{\int_{\widetilde{W}}\mathop{\mathrm{Sq}}\nolimits^{2}(a)}(-1)^{\int_{S_{\widetilde{W}}}a}\sigma(W;a).$
(3.17)

where $\widetilde{W}=W\times[0,1]$ is a $d$-manifold that interpolates $W$ and
$W^{\prime}$ equipped with the $(d-2)$-form gauge field $a$, which restrict to
$a$ and $a^{\prime}$ on $W$ and $W^{\prime}$ respectively. $W^{\prime}$ is a
$(d-1)$-manifold which is the same as $W$ with a different triangulation and
the gauge field $a^{\prime}$, equivalent to $a$ in cohomology
$[a]=[a^{\prime}]$. $S_{\widetilde{W}}$ is a set of all $(d-2)$-simplices of
$\widetilde{W}$.

Based on the quadratic property of $\sigma(W;a)$, one can immediately check
that the combined action (3.15) satisfies the desired quadratic property
(3.1), since the extra factor $(-1)^{\int_{W}a\cup_{d-3}b}$ for the quadratic
property of $e^{2\pi iq(a)}$ in (3.14) is precisely canceled by the quadratic
property of $\sigma(W;a)$ in (3.16).

One can further check the invariance of $z(\eta,a)$ under moving the
orientation-reversing wall $W$, which guarantees the invariance of the path
integral under shifting the $T$ symmetry defect. To see this, suppose we
initially have the orientation-reversing wall $W$ which is moved to the final
configuration $W^{\prime}$. Let $\widetilde{W}$ be a $d$-manifold that
interpolates $W$ and $W^{\prime}$,
$\partial\widetilde{W}=W\sqcup\overline{W}^{\prime}$. According to the
property of the Grassmann integral (3.17), we have

$\displaystyle\sigma(W^{\prime};a|_{W^{\prime}})=(-1)^{\int_{\widetilde{W}}\mathop{\mathrm{Sq}}\nolimits^{2}(a)}(-1)^{\int_{S_{\widetilde{W}}}a}\sigma(W;a|_{W}).$
(3.18)

Meanwhile, since the action $e^{2\pi iq(a)}$ is complex conjugated under the
orientation reversal, which means when the orientation is reversed on a single
$d$-simplex

$\displaystyle e^{2\pi iq(a)}\to e^{-2\pi iq(a)}=e^{2\pi
iq(a)}\cdot(-1)^{a\cup_{d-4}a},$ (3.19)

so the action $e^{2\pi i\int q(a)}$ gets shifted by
$(-1)^{\int_{\widetilde{W}}a\cup_{d-4}a}$. In addition, the factor
$(-1)^{\int_{E}a}$ gets shifted by $(-1)^{\int_{S_{\widetilde{W}}}a}$. This is
shown by noting that $\partial S_{\widetilde{W}}=S_{W}+S_{W^{\prime}}$, where
$S_{W}$ (resp. $S_{W^{\prime}}$) is the set of all $(d-3)$-simplices of $W$
(resp. $W^{\prime}$), see Appendix C for the derivation. This means that
$\partial(S_{\widetilde{W}}+E+E^{\prime})=0$, with $E$ (resp. $E^{\prime}$) a
choice of the Wu structure that satisfies $\partial E=S_{W}$ (resp. $\partial
E^{\prime}=S_{W^{\prime}}$). This shows that

$\displaystyle(-1)^{\int_{E}a}=(-1)^{\int_{S_{\widetilde{W}}}a}(-1)^{\int_{E^{\prime}}a}.$
(3.20)

Thus, one can see that the combined action $z(\eta,a)$ in (3.15) is completely
invariant under the shift of the $T$ defect, since the variation of each term
is precisely canceled with each other.

This shows that the ’t Hooft anomaly of $z(\eta,a)$ is the same as the
oriented case, since one can freely move the $T$ symmetry defect without
shifting the partition function. Thus, the ’t Hooft anomaly is given by the
response action (3.2),

$\displaystyle(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{3}(a)}.$ (3.21)

#### The definition of $\sigma(W;a)$: Gu-Wen Grassmann integral

Now we provide the definition of $\sigma(W;a)$, and demonstrate the properties
(3.16) and (3.17). $\sigma(W;a)$ is realized by a theory called the Gu-Wen
Grassmann integral [5, 24], expressed as a path integral of Grassmann
variables supported on $W$. We note that $W$ is oriented, and the sign of each
$(d-1)$-simplex of $W$ is defined by taking the sign of the neighboring
$d$-simplex of $M$.

To construct the Grassmann integral $\sigma(W;a)$, we assign a pair of
Grassmann variables $\theta_{e},\overline{\theta}_{e}$ on each $(d-2)$-simplex
$e$ of $W$ such that $a(e)=1$, we associate $\theta_{e}$ on one side of $e$
contained in one of $(d-1)$-simplices of $W$ neighboring $e$ (which will be
specified later), $\overline{\theta}_{e}$ on the other side. Then, we define
$\sigma(W;a)$ as

$\sigma(W;a)=\int\prod_{e|a(e)=1}d\theta_{e}d\overline{\theta}_{e}\prod_{t}u(t),$
(3.22)

where $t$ denotes a $(d-1)$-simplex of $W$, and $u(t)$ is the product of
Grassmann variables contained in $t$. For instance, for $(d-1)=2$, $u(t)$ on
$t=(012)$ is the product of
$\vartheta_{12}^{a(12)},\vartheta_{01}^{a(01)},\vartheta_{02}^{a(02)}$. Here,
$\vartheta$ denotes $\theta$ or $\overline{\theta}$ depending on the choice of
the assigning rule, which will be introduced later. The order of Grassmann
variables in $u(t)$ will also be defined shortly. We note that $u(t)$ is
ensured to be Grassmann-even since $a$ is closed.

Due to the fermionic sign of Grassmann variables, $\sigma(W;a)$ becomes a
quadratic function, whose quadratic property depends on the order of Grassmann
variables in $u(t)$. We will adopt the order used in Gaiotto-Kapustin [24],
which is defined as follows.

  * •

For $t=(01\dots d-1)$, we label a $(d-2)$-simplex $(01\dots\widehat{i}\dots
d-1)$ (i.e., a $(d-1)$-simplex given by omitting a vertex $i$) simply as
$\widehat{i}$.

  * •

Then, the order of $\vartheta_{i}$ for $+$ $(d-1)$-simplex $t$ is defined by
first assigning even $(d-2)$-simplices in ascending order, then odd simplices
in ascending order again:

$\widehat{0}\to\widehat{2}\to\widehat{4}\to\dots\to\widehat{1}\to\widehat{3}\to\widehat{5}\to\dots$
(3.23)

  * •

For $-$ $(d-1)$-simplices, the order is defined in opposite way:

$\dots\to\widehat{5}\to\widehat{3}\to\widehat{1}\to\dots\to\widehat{4}\to\widehat{2}\to\widehat{0}.$
(3.24)

For example, for $(d-1)=2$,
$u(012)=\vartheta_{12}^{a(12)}\vartheta_{01}^{a(01)}\vartheta_{02}^{a(02)}$
when $(012)$ is a $+$ triangle, and
$u(012)=\vartheta_{02}^{a(02)}\vartheta_{01}^{a(01)}\vartheta_{12}^{a(12)}$
for a $-$ triangle. Then, we choose the assignment of $\theta$ and
$\overline{\theta}$ on each $e$ such that, if $t$ is a $+$ (resp. $-$)
simplex, $u(t)$ includes $\overline{\theta}_{e}$ when $e$ is labeled by an odd
(resp. even) number, see Fig. 3.

Figure 3: Assignment of Grassmann variables on 1-simplices in the case of
$d=2$. $\theta$ (resp. $\overline{\theta}$) is represented as a black (resp.
white) dot.

Based on the above definition of $u(t)$, the quadratic property of $u(t)$ is
given by (3.16),

$\sigma(W;a)\sigma(W;a^{\prime})=\sigma(W;a+a^{\prime})(-1)^{\int_{W}a\cup_{d-3}a^{\prime}},$
(3.25)

for closed $a,a^{\prime}$. To see this, we just have to bring the product of
two Grassmann integrals

$\sigma(W;a)\sigma(W;a^{\prime})=\int\prod_{e|a(e)=1}d\theta_{e}d\overline{\theta}_{e}\prod_{e|a^{\prime}(e)=1}d\theta_{e}d\overline{\theta}_{e}\prod_{t}u(t)[a]\prod_{t}u(t)[a^{\prime}]$
(3.26)

into the form of $\sigma(W;a+a^{\prime})$ by permuting Grassmann variables,
and count the net fermionic sign. First of all, each path integral measure on
$e$ picks up a sign $(-1)^{a(e)a^{\prime}(e)}$ by permuting
$d\overline{\theta}_{e}^{a(e)}$ and $d\theta_{e}^{a^{\prime}(e)}$. For
integrands, $u(t)$ on different $(d-1)$-simplices commute with each other for
closed $a$, so nontrivial signs occur only by reordering
$u(t)[a]u(t)[a^{\prime}]$ to $u(t)[a+a^{\prime}]$ on a single $(d-1)$-simplex.
The sign on $t$ is explicitly written as

$(-1)^{\sum_{e,e^{\prime}\in t}^{e>e^{\prime}}a(e)a^{\prime}(e^{\prime})},$
(3.27)

where the order $e>e^{\prime}$ is determined by $u(t)$. Hence, the net
fermionic sign is given by

$\sigma(W;a)\sigma(W;a^{\prime})=\sigma(W;a+a^{\prime})\prod_{t}(-1)^{\epsilon[t,a,a^{\prime}]},$
(3.28)

with

$\epsilon[t,a,a^{\prime}]=\sum_{e,e^{\prime}\in
t,e>e^{\prime}}a(e)a^{\prime}(e^{\prime})+\sum_{e\in t,e>0}a(e)a^{\prime}(e),$
(3.29)

where $e>0$ if $u[t]$ includes a $\overline{\theta}_{e}$ variable. The sign
$\epsilon[t,a,a^{\prime}]$ turns out to have a neat expression in terms of the
higher cup product.

At a $+$ simplex, after some efforts we can rewrite $\epsilon[t,a,a^{\prime}]$
as

$\begin{split}\epsilon[t,a,a^{\prime}]&=\sum_{i}a_{2i+1}\cdot\delta
a^{\prime}(t)+\sum_{i<j}a_{2i+1}a^{\prime}_{2j+1}+\sum_{i>j}a_{2i}a^{\prime}_{2j}\\\
&=a\cup_{d-3}a^{\prime}+a\cup_{d-2}\delta a^{\prime}.\end{split}$ (3.30)

At a $-$ simplex, similarly we have

$\begin{split}\epsilon[t,a,a^{\prime}]&=\sum_{i}a_{2i}\cdot\delta
a^{\prime}(t)+\sum_{i<j}a_{2i+1}a^{\prime}_{2j+1}+\sum_{i>j}a_{2i}a^{\prime}_{2j}\\\
&=\delta a(t)\delta a^{\prime}(t)+a\cup_{d-3}a^{\prime}+a\cup_{d-2}\delta
a^{\prime}.\end{split}$ (3.31)

We can see the quadratic property (3.25) when $a,a^{\prime}$ are closed.

When $a=\delta\lambda$ for some $\lambda\in C^{d-3}(W,\mathbb{Z}_{2})$, the
Grassmann integral can be explicitly computed as

$\displaystyle\sigma(W;\delta\lambda)=(-1)^{\int_{W}\lambda\cup_{d-4}\delta\lambda+\lambda\cup_{d-5}\lambda}(-1)^{\int_{S}\lambda},$
(3.32)

see [24] for its derivation. Now let us show the effect of gauge
transformation and re-triangulation of $\sigma(W;a)$ given in (3.17). To see
this, let us introduce an expression of $\sigma(W;a)$ convenient for our
purpose. Let us assume that $W$ equipped with the background gauge field $a\in
Z^{d-2}(W,\mathbb{Z}_{2})$ is null-bordant, i.e., $W$ is a boundary of some
oriented $d$-manifold $X$ and $a$ is extended to $X$. Then, one can consider
the Wess-Zumino-Witten (WZW) like expression of the Grassmann integral

$\displaystyle\sigma(W;a)=(-1)^{\int_{X}\mathop{\mathrm{Sq}}\nolimits^{2}a}(-1)^{\sum_{S_{X}}a},$
(3.33)

where $S_{X}$ is a set of all $(d-2)$-simplices of $X$. Due to the Wu relation
[34], $\mathop{\mathrm{Sq}}\nolimits^{2}(a)+w_{2}\cup a$ is exact for an
arbitrary oriented $d$-manifold. Hence, the above expression does not depend
on the extending manifold $X$. We can explicitly check that (3.33) satisfies
the properties of the Grassmann integral (3.25), (3.32). First, let us check
the quadratic property of the WZW-like expression,

$\displaystyle\begin{split}\sigma(W;a)\sigma(W;a^{\prime})&=\sigma(W;a+a^{\prime})(-1)^{\int_{X}(a\cup_{d-4}a^{\prime}+a^{\prime}\cup_{d-4}a)}\\\
&=\sigma(W;a+a^{\prime})(-1)^{\int_{W}a\cup_{d-3}a^{\prime}}.\end{split}$
(3.34)

Next, when $a=\delta\lambda$ for some $\lambda\in C^{d-3}(X,\mathbb{Z}_{2})$,
we have

$\displaystyle\begin{split}\sigma(W;\delta\lambda)&=(-1)^{\int_{X}\mathop{\mathrm{Sq}}\nolimits^{2}\delta\lambda}(-1)^{\sum_{S_{X}}\delta\lambda}\\\
&=(-1)^{\int_{W}\lambda\cup_{d-4}\delta\lambda+\lambda\cup_{d-5}\lambda}(-1)^{\sum_{S_{X}}\lambda},\end{split}$
(3.35)

where we used $\partial S_{X}=S$, namely the boundary of $S_{X}$ gives the
dual of $w_{2}$ on $W$. See Appendix C for the derivation of $\partial
S_{X}=S$.

Since the WZW-like expression satisfies the key properties (3.25), (3.32), one
can identify the original definition of $\sigma(W;a)$ as the WZW-like
expression up to gauge invariant counterterms, i.e., the Grassmann integral is
expressed in the form of

$\displaystyle\sigma(W;a)=(-1)^{\int_{X}\mathop{\mathrm{Sq}}\nolimits^{2}a}(-1)^{\sum_{S_{X}}a}\epsilon(a),$
(3.36)

where $\epsilon:H^{d-2}(W,\mathbb{Z}_{2})\to\\{\pm 1\\}$ is gauge invariant.
The additional term $\epsilon(a)$ does not affect on the response to gauge
transformation or re-triangulation, so we can identify $\sigma(W;a)$ as the
WZW-like expression for a practical purpose.

Based on the WZW-like expression, we immediately know the effect of re-
triangulation as follows. Suppose we have two configurations of $a$ and
triangulations on $W\times\\{0\\}$ and $W\times\\{1\\}$ interpolated by
$\widetilde{W}=W\times[0,1]$. Then, according to the WZW-like expression for
${\sigma}_{d-1}(W\times\\{0\\})\sigma(W\times\\{1\\})$, up to gauge invariant
counterterms $\sigma(W\times\\{0\\})$ is given by

$\sigma(W\times\\{0\\})=(-1)^{\int_{\widetilde{W}}\mathop{\mathrm{Sq}}\nolimits^{2}a}(-1)^{\sum_{S_{\widetilde{W}}}a}\cdot\sigma(W\times\\{1\\}),$
(3.37)

where $a$ on $W\times\\{0\\}$, $W\times\\{1\\}$ is extended to
$\widetilde{W}$. This expression directly shows that the effect of gauge
transformation and re-triangulation of $\sigma(W,a)$ is controlled by the bulk
response action

$\displaystyle(-1)^{\int_{W}\mathop{\mathrm{Sq}}\nolimits^{2}a}(-1)^{\sum_{S_{\widetilde{W}}}a}.$
(3.38)

This proves (3.17).

##  4 (3+1)d exotic invertible phase: $\mathbb{Z}_{8}$ classification

###  4.1 Arf-Brown-Kervaire invariant: Kitaev type phase

In the spacetime dimensions $d=4$, the partition function $z(\eta,a)$ is free
of ’t Hooft anomaly and gives a topologically invariant path integral, with a
quadratic property for $[a]\in H^{2}(M,\mathbb{Z}_{2})$ given by

$\displaystyle z(\eta,a)z(\eta,b)=z(\eta,a+b)(-1)^{\int_{M}a\cup b}.$ (4.1)

Then the partition function for the exotic invertible phase is obtained by
gauging the $\mathbb{Z}_{2}$ 1-form symmetry

$\displaystyle
Z_{\mathrm{ABK}}(\eta)=\frac{1}{\sqrt{|H^{2}(M,\mathbb{Z}_{2})|}}\sum_{a\in
H^{2}(M,\mathbb{Z}_{2})}z(\eta,a).$ (4.2)

This is the Arf-Brown-Kervaire (ABK) invariant based on the quadratic function
$z(\eta,a)$, valued in 8th root of unity. The theory generates the
$\mathbb{Z}_{8}$ classification of the exotic invertible topological phase
proposed in [22]. This generalizes the $\mathbb{Z}_{8}$ classification of the
(1+1)d pin- invertible topological phases that corresponds to the (1+1)d
topological superconductor [35] proposed by Fidkowski and Kitaev [4], whose
partition function is given by the ABK invariant based on the quadratic
refinement of $H^{1}(M,\mathbb{Z}_{2})$, see [36, 37, 38].

###  4.2 Relation to the Crane-Yetter model in oriented case

When the spacetime 4-manifold is oriented, the above theory
$Z_{\mathrm{ABK}}(\eta)$ with a specific choice of the Wu structure $\eta=0$
is identical to the Crane-Yetter TQFT [39, 40] based on a unitary modular
tensor category (UMTC) whose objects are $\\{1,s\\}$, with $s$ a semion with
$s\times s=1,\theta_{s}=i$. The above ABK invariant on oriented spacetime
4-manifolds has been discussed in [41, 42], and known to be the signature
$\sigma$ of the oriented 4-manifold mod 8 when the Wu structure is chosen as
$\eta=0$,

$\displaystyle Z_{\mathrm{ABK}}(\eta=0)=\exp\left(2\pi
i\cdot\frac{\sigma}{8}\right).$ (4.3)

Meanwhile, the partition function of the Crane-Yetter TQFT is also a
signature,

$\displaystyle Z_{\mathrm{CY}}=\exp\left(2\pi
ic_{-}\cdot\frac{\sigma}{8}\right),$ (4.4)

where $c_{-}$ is the topological central charge (i.e., framing anomaly) mod 8
of the given UMTC. Since we have $c_{-}=1$ mod 8 for the UMTC $\\{1,s\\}$, the
partition function is identical $Z_{\mathrm{ABK}}(\eta=0)=Z_{\mathrm{CY}}$. In
the rest of this section, let us demonstrate the relation between the above
invariant $Z_{\mathrm{ABK}}(\eta)$ in (4.2) and the Crane-Yetter TQFT at the
level of the microscopic action.

The Crane-Yetter TQFT takes as input a UMTC $\mathcal{C}$ describing the
(2+1)d topological order. It associates a partition function
$Z_{\mathrm{CY}}(M)$ to a triangulated 4-manifold $M$ with branching
structure. See [43, 44] for a review of UMTC.

The state sum consists of a summation over all possible assignments of the
following data:

  * •

To each 2-simplex $(ijk)$, assign a simple object (anyon)
$a_{ijk}\in\mathcal{C}$

  * •

To each 3-simplex $\braket{ijkl}$, assign an anyon $b_{ijkl}\in\mathcal{C}$
and an element of the fusion and splitting space $V^{b}_{ijl,jkl}\otimes
V_{b}^{ikl,{ijk}}$ .

For convenience, we will often ignore the distinction between a simplex and
the anyon or group element data assigned to it, i.e. simply write $a_{ijk}$ as
$ijk$, or $b_{ijkl}$ as $ijkl$. We then assign an amplitude
$Z_{b}^{\epsilon(\Delta_{4})}(\Delta_{4})$ to each 4-simplex $\Delta_{4}$ of
$M$, with $\epsilon(\Delta_{4})$ a sign of the $4$-simplex determined by the
branching structure. This amplitude is given diagrammatically in Fig. 4, with
a normalization factor

$\mathcal{N}_{01234}=\sqrt{\frac{\prod_{\Delta_{3}\in\text{3-simplices}}d_{b_{\Delta_{3}}}}{\prod_{\Delta_{2}\in\text{2-simplices}}d_{a_{\Delta_{2}}}}}.$
(4.5)

The 15j symbol is explicitly given by

$\displaystyle Z^{+}(01234)=$
$\displaystyle\sum_{d,a\in\mathcal{C}}F^{024,234,012}_{d,0234,a}R_{a}^{012,234}(F^{024,012,234}_{d})^{-1}_{a,0124}F^{014,124,234}_{d,0124,1234}\times$
$\displaystyle(F^{014,134,123}_{d})^{-1}_{1234,0134}F^{034,013,123}_{d,0134,0123}\times(F^{034,023,012}_{d})^{-1}_{0123,0234}$
(4.6) $\displaystyle Z^{-}(01234)=$
$\displaystyle\sum_{d,a\in\mathcal{C}}(F^{024,234,012}_{d})^{-1}_{a,0234}(R_{a}^{012,234})^{-1}F^{024,012,234}_{d,0124,a}\left(F^{014,124,234}_{d}\right)^{-1}_{1234,0124}\times$
$\displaystyle
F^{014,134,123}_{d,0134,1234}\left(F^{034,013,123}_{d}\right)^{-1}_{0123,0134}\times
F^{034,023,012}_{d,0234,0123}$ (4.7)

on a $+$ and $-$ 4-simplex, respectively. Then we can define the full path
integral of the Crane-Yetter TQFT. Let $\chi$ be the Euler characteristic of
$M$, and let $T^{k}$ be the set of $k$-simplices of $M$. Then define

$Z_{\mathrm{CY}}(M)=\mathcal{D}^{2(N_{0}-N_{1})-\chi}\sum_{\\{a,b\\}}\frac{\prod_{\Delta_{2}\in
T^{2}}d_{a_{\Delta_{2}}}\prod_{\Delta_{4}\in
T^{4}}Z^{\epsilon(\Delta_{4})}(\Delta_{4})}{\prod_{\Delta_{3}\in
T^{3}}d_{b_{\Delta_{3}}}}$ (4.8)

where $N_{k}$ is the number of $k$-simplices of $M$, $d_{a}$ is quantum
dimension of an anyon $a$, and
$\mathcal{D}=\sqrt{\sum_{a\in\mathcal{C}}d_{a}^{2}}$ is the total dimension.

Figure 4: The amplitude associated to a $+$ 4-simplex 01234, where the
branching structure induces the order on the labels. Black lines are anyon
lines taking values in the input UMTC $\mathcal{C}$.

Now let us demonstrate the equivalence of the Crane-Yetter path integral to
the ABK invariant in (4.2) for the case that $\mathcal{C}$ is a semion theory
$\mathcal{C}=\\{1,s\\}$. The nontrivial $F$ and $R$ symbols of
$\mathcal{C}=\\{1,s\\}$ are $F^{sss}_{s}=-1,R^{ss}_{1}=i$, otherwise $1$ [45].

Since the $F^{abc}_{d}$ is real and symmetric under the permutation of labels
$a,b,c$ for our UMTC, we can see that

$\displaystyle Z^{+}(01234)=$ $\displaystyle
R_{a}^{012,234}F^{014,124,234}_{d}F^{014,134,123}_{d}F^{034,013,123}_{d}F^{034,023,012}_{d},$
(4.9)

and we have $Z^{-}(01234)=(Z^{+}(01234))^{-1}$. $a,d\in\mathcal{C}$ that
appears in (4.9) is uniquely determined since $\mathcal{C}=\\{1,s\\}$ is
abelian.

Now, let us consider a 2-form $\mathbb{Z}_{2}$ field $a\in
Z^{2}(M,\mathbb{Z}_{2})$ and we take the anyon at a 2-simplex $(ijk)$ $s$
(resp. $1$) if we have $a_{ijk}=1$ (resp. $a_{ijk}=0$). Then, we show that

$\displaystyle Z^{+}(01234)=e^{2\pi iq(a)}.$ (4.10)

To see this, we utilize the hexagon equation for UMTC [43]

$\displaystyle\begin{split}R^{ac}_{e}F^{acb}_{d}R^{bc}_{g}&=F^{cab}_{d}R^{fc}_{d}F^{abc}_{d},\end{split}$
(4.11)

with $d=a\times b\times c,e=a\times c,f=a\times b$. Due to the invariance of
the $F$ symbol under permutation of labels, it simplifies as

$\displaystyle\begin{split}F^{abc}_{d}&=R^{ac}_{e}R^{bc}_{g}(R^{fc}_{d})^{-1}.\end{split}$
(4.12)

By putting the hexagon equations to (4.9) and noting that $F$ is real, and
both $F$ and $R$ symbols are symmetric under permutation of labels, we can see
that

$\displaystyle\begin{split}Z^{+}(01234)&=R^{012,234}(R^{014,234}R^{014,124}(R^{1234,014})^{-1})((R^{014,134})^{-1}(R^{014,123})^{-1}R^{1234,014})\\\
&\times(R^{034,123}R^{034,013}(R^{0123,034})^{-1})((R^{034,023})^{-1}(R^{034,012})^{-1}R^{0123,034})\\\
&=R^{012,234}\times(R^{014,234}R^{014,124}(R^{014,134})^{-1}(R^{014,123})^{-1})\\\
&\times(R^{034,123}R^{034,013}(R^{034,023})^{-1}(R^{034,012})^{-1})\\\
&=\exp\left(2\pi
i\cdot\frac{1}{4}(\widehat{a}\cup\widehat{a}+\widehat{a}\cup_{1}\delta\widehat{a})\right)=e^{2\pi
iq(a)}.\end{split}$ (4.13)

This equation shows that the Crane-Yetter path integral with a fixed anyon
configuration specified by the 2-form field $a$ is identical to $z(\eta,a)$,
since the normalization factor (4.5) is unit for abelian UMTC. So if we
further sum over the configuration $a\in Z^{2}(M,\mathbb{Z}_{2})$, we obtain
the partition function of the Crane-Yetter TQFT as

$\displaystyle\begin{split}Z_{\mathrm{CY}}(M)&=\mathcal{D}^{2(N_{0}-N_{1})-\chi(M)}\sum_{a\in
Z^{2}(M,\mathbb{Z}_{2})}z(\eta=0,a)\\\
&=\sqrt{2}^{-\chi(M)}2^{(N_{0}-N_{1})}|B^{2}(M,\mathbb{Z}_{2})|\sum_{a\in
H^{2}(M,\mathbb{Z}_{2})}z(\eta=0,a)\\\
&=\sqrt{2}^{-\chi(M)}\frac{|H^{0}(M,\mathbb{Z}_{2})|}{|H^{1}(M,\mathbb{Z}_{2})|}\sum_{a\in
H^{2}(M,\mathbb{Z}_{2})}z(\eta=0,a)\\\
&=\frac{1}{\sqrt{|H^{2}(M,\mathbb{Z}_{2})|}}\sum_{a\in
H^{2}(M,\mathbb{Z}_{2})}z(\eta=0,a)\\\ &=Z_{\mathrm{ABK}}(\eta=0).\end{split}$
(4.14)

Thus, on oriented manifolds, the exotic invertible phase
$Z_{\mathrm{ABK}}(\eta=0)$ is identical to the Crane-Yetter TQFT with the
semion UMTC $\\{1,s\\}$ at the level of the microscopic action. We note that
the (3+1)d theory $z(\eta=0,a)=e^{2\pi i\int q(a)}$ in the oriented spacetime
has been studied in [46] as a lattice model that realizes a (3+1)d SPT phase
protected by 1-form $\mathbb{Z}_{2}$ symmetry. Also, a Hamiltonian model (the
Walker-Wang model [40]) for the Crane-Yetter TQFT based on the semion UMTC
$\\{1,s\\}$ has been obtained in [47].

##  5 Gu-Wen type phases

Finally, we propose an analogue of the Gu-Wen SPT phase based on the onsite
(0-form) $G$ symmetry and the Wu structure of the spacetime, labeled by a pair
of cohomological data

$\displaystyle(\nu_{d},n_{d-2})\in C_{\rho}^{d}(BG,U(1))\times
Z^{d-2}(BG,\mathbb{Z}_{2}),$ (5.1)

which are subject to the constraint similar to the Gu-Wen equation,

$\displaystyle\delta_{\rho}\nu_{d}=\frac{1}{2}\mathrm{Sq}^{3}(n_{d-2})\mod 1,$
(5.2)

where $\rho$ denotes the twisted $G$-action on $U(1)$ where the anti-unitary
elements act by complex conjugation.

By utilizing the path integral of $z(\eta,a)$, it is a simple matter to
consider the Gu-Wen type phase coupled with the Wu structure, for a given data
of $(\nu_{d},n_{d-2})\in C_{\rho}^{d}(BG,U(1))\times
Z^{d-2}(BG,\mathbb{Z}_{2})$ with
$\delta_{\rho}\nu_{d}=\frac{1}{2}\mathrm{Sq}^{3}(n_{d-2})\mod 1$. For a given
$G$-gauge field $g:M\to BG$, the action is defined as

$\displaystyle Z(M;g,\eta)=z(\eta,g^{*}n_{d-2})\exp\left(2\pi
i\int_{M}g^{*}\nu_{d}\right).$ (5.3)

Due to the Gu-Wen type equation (5.2), one can see that the ’t Hooft anomaly
is canceled out in the expression of $Z(M;g,\eta)$, and therefore provides a
topologically invariant theory.

##  6 Discussion

In this paper, we studied a state sum path integral that realizes an exotic
invertible topological phase based on Wu structure of the spacetime. In the
study of the Gu-Wen type exotic $G$-SPT phases, we only considered the case of
$G=G_{0}\times\mathbb{Z}_{2}^{T}$ with $\mathbb{Z}_{2}^{T}$ a time-reversal
symmetry, and the total symmetry group is direct product of $G_{0}$ and the
1-form symmetry of the Wu structure. It should be interesting to consider the
case where $G$ is not a direct product between unitary group and
$\mathbb{Z}_{2}^{T}$, or the case of the nontrivial 2-group that involves $G$
and the 1-form $\mathbb{Z}_{2}$ symmetry of the Wu structure.

For instance, one can think of the case $G=G_{0}\times\mathbb{Z}_{2}^{T}$ and
$G_{0}$ has a 2-group structure with respect to the 1-form $\mathbb{Z}_{2}$
symmetry of the Wu structure, characterized by the Postnikov class $\omega\in
H^{3}(BG_{0},\mathbb{Z}_{2})$. In that case, the Wu structure is twisted as
$\delta\eta=w_{1}w_{2}+\omega$. Since $z(\eta,a)$ constructed in this paper
has the dependence on $\eta$ in the form of $(-1)^{\eta\cup a}$, we expect the
extra contribution to the ’t Hooft anomaly given by the response action
$(-1)^{\int\omega\cup a}$ in the twisted case. Hence, $z(\eta,a)$ for the
twisted Wu structure is expected to have the ’t Hooft anomaly characterized by
the response action

$\displaystyle(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{3}(a)+\omega\cup a}.$
(6.1)

Then, to define the Gu-Wen type phase using the pair $(\nu_{d},n_{d-2})\in
C_{\rho}^{d}(BG,U(1))\times Z^{d-2}(BG,\mathbb{Z}_{2})$ in the form of (5.3),
we need a twisted version of the Gu-Wen equation

$\displaystyle\delta_{\rho}\nu_{d}=\frac{1}{2}(\mathrm{Sq}^{3}(n_{d-2})+\omega\cup
n_{d-2})\mod 1.$ (6.2)

It would be interesting to have an explicit lattice construction of these Gu-
Wen type phases based on the twisted Wu structure.

While we focused on the Wu structure based on the 2-group
$\delta\eta=w_{1}w_{2}$, it would also be interesting to consider the Wu
structure realized by a higher $n$-group involving the Lorentz $O(d)$ symmetry
and the $(n-2)$-form $\mathbb{Z}_{2}$ symmetry, given by the trivialization of
the $(n+1)$-th Wu class $\delta\eta=\nu_{n+1}$ [34, 23]. In general, the Wu
class in $H^{n}(N,\mathbb{Z}_{2})$ has a property that
$\mathop{\mathrm{Sq}}\nolimits^{n+1}(x)+\nu_{n+1}\cup x$ is exact on a closed
$(d+1)$-manifold $N$, for any $x\in H^{d-n}(N,\mathbb{Z}_{2})$.

Hence, we expect that a generalization of $z(\eta,a)$ with $a\in
Z^{d-n}(M,\mathbb{Z}_{2})$ given in the form of
$z(\eta,a)=z^{\prime}(a)(-1)^{\int\eta\cup a}$ also exists for $n$-th Wu
structure, where $z^{\prime}(a)$ is regarded as a coboundary of a trivial
cochain $\mathop{\mathrm{Sq}}\nolimits^{n+1}(a)+\nu_{n}\cup a$ evaluated on a
spacetime $d$-manifold $M$. Since the $(-1)^{\int\eta\cup a}$ part has an ’t
Hooft anomaly characterized by a response action $(-1)^{\int\nu_{n+1}\cup a}$,
we expect that the generalization of $z(\eta,a)$ for the $n$-group has an ’t
Hooft anomaly with the response action

$\displaystyle(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{n+1}(a)}.$ (6.3)

It would be interesting to consider an explicit lattice construction of the
theory $z(\eta,a)$ based on the higher Wu structure, and Kitaev or Gu-Wen type
phases obtained by utilizing this theory.

## Acknowledgements

The author thanks Yu-An Chen and Srivata Tata for useful discussions and
related collaborations. The author is supported by the Japan Society for the
Promotion of Science (JSPS) through Grant No. 19J20801.

##  Appendix A Review on higher cup product

A branching structure on a triangulation is a local ordering of vertices,
which can be specified by an arrow on each 1-simplex $\braket{ij}$, such there
are no closed loops on any 2-simplices. This defines a total ordering of
vertices on every single $d$-simplex $\braket{0\dots d}$. In this appendix, we
review cochain-level product operation called higher cup product, whose
definitions are based on branching structure of the triangulation. See also
[48] for a reference on higher cup product.

Let $M$ be a triangulated $d$-dimensional manifold. Firstly, the cup product
gives the product of cochains

$\displaystyle-\cup-:C^{k}(M,\mathbb{Z}_{n})\times C^{l}(M,\mathbb{Z}_{n})\to
C^{k+l}(M,\mathbb{Z}_{n}),$ (A.1)

whose explicit form is written as

$\displaystyle(a\cup b)(0,\dots,k+l)=a(0,\dots,k)b(k,\dots,k+l).$ (A.2)

Note that this definition of cup product depends on the branching structure on
the triangulation, where the ordering of vertices on each $(k+l)$-simplex is
specified as $0\to 1\to\dots\to k+l$. The cup product satisfies the Leibniz
rule at the cochain level,

$\displaystyle\delta(a\cup b)=\delta a\cup b+(-1)^{k}a\cup\delta b.$ (A.3)

According to the Leibniz rule, one can show that the cup product defines the
product of cohomologies $H^{k}(M,\mathbb{Z}_{n})\times
H^{l}(M,\mathbb{Z}_{n})\to H^{k+l}(M,\mathbb{Z}_{n})$. Actually, for given
$a\in Z^{k}(M,\mathbb{Z}_{n})$, $b\in Z^{l}(M,\mathbb{Z}_{n})$, the shift of
these cocycles by coboundaries is evaluated as

$\displaystyle(a+\delta A)\cup(b+\delta B)=a\cup b+\delta((-1)^{k}a\cup
B+A\cup b+A\cup\delta B),$ (A.4)

so this also shifts $a\cup b$ by a coboundary, thus defines a map between
cohomologies. Such a product operation defined on cohomologies is called a
cohomology operation.

As a generalization of the cup product, the higher cup product $\cup_{i}$
gives [49]

$\displaystyle-\cup_{i}-:C^{k}(M,\mathbb{Z}_{n})\times
C^{l}(M,\mathbb{Z}_{n})\to C^{k+l-i}(M,\mathbb{Z}_{n}),$ (A.5)

whose explicit form is written as

$\displaystyle(a\cup_{i}b)(0,\dots,k+l-i)=\sum_{0\leq j_{0}<\dots<j_{i}\leq
k+l-i}(-1)^{p}\cdot a(0\to j_{0},j_{1}\to j_{2},\dots)b(j_{0}\to
j_{1},j_{2}\to j_{3},\dots).$ (A.6)

Here, the notation $i\to j$ denotes all vertices from $i$ to $j$,
$\\{i,i+1,\dots,i+j\\}$. $p$ is the number of permutations need to bring the
sequence of vertices

$\displaystyle 0\to j_{0},j_{1}\to j_{2},\dots,j_{0}+1\to j_{1}-1,j_{2}+1\to
j_{3}-1,\dots$ (A.7)

to the sequence

$\displaystyle 0\to k+l-i.$ (A.8)

In particular, $\cup_{0}$ is identified as the cup product $\cup$ defined in
(A.2). The higher cup product is subject to the generalized Leibniz rule,

$\displaystyle\delta(a\cup_{i}b)=(-1)^{k+l-i}a\cup_{i-1}b+(-1)^{kl+k+l}b\cup_{i-1}a+\delta
a\cup_{i}b+(-1)^{k}a\cup_{i}\delta b,$ (A.9)

which is regarded as that the non-commutative property of $\cup_{i-1}$ is
controlled by the $\cup_{i}$ product. According to the above Leibniz rule, for
closed $a$ and $b$, one can see that $a\cup_{i}b$ is not necessarily closed,
$\delta(a\cup_{i}b)=a\cup_{i-1}b+(-1)^{kl+k+l}b\cup_{i-1}a$ for $a\in
Z^{k}(M,\mathbb{Z}_{n}),\beta\in Z^{l}(M,\mathbb{Z}_{n})$. Hence, the product
$\cup_{i}$ doesn’t give a cohomology operation for $i>0$.

However, when we take $\mathbb{Z}_{n}$ as $\mathbb{Z}_{2}$ it turns out that
the map

$\displaystyle\begin{split}\mathop{\mathrm{Sq}}\nolimits^{d-i}(a)&:Z^{k}(M,\mathbb{Z}_{2})\to
Z^{k+d-i}(M,\mathbb{Z}_{2})\\\
\mathop{\mathrm{Sq}}\nolimits^{d-i}(a)&:=a\cup_{i+k-d}a\end{split}$ (A.10)

does give a cohomology operation. Actually, one can check that
$\mathop{\mathrm{Sq}}\nolimits^{d-i}(a+\delta
A)=\mathop{\mathrm{Sq}}\nolimits^{d-i}(a)+\delta(a\cup_{i+k-d}A+A\cup_{i+k-d}a+A\cup_{i+k-d-1}A+A\cup_{i+k-d}\delta
A)$ by using the generalized Leibniz rule. This shows that
$\mathop{\mathrm{Sq}}\nolimits^{d-i}$ defines a map
$H^{k}(M,\mathbb{Z}_{2})\to H^{k+d-i}(M,\mathbb{Z}_{2})$.

For convenience, we extend the definition of the
$\mathop{\mathrm{Sq}}\nolimits^{d-i}$ operation to non-closed cochains. For a
given $\lambda\in C^{k}(M,\mathbb{Z}_{2})$, we define

$\displaystyle\mathop{\mathrm{Sq}}\nolimits^{d-i}(\lambda)$
$\displaystyle:C^{k}(M,\mathbb{Z}_{2})\to C^{k+d-i}(M,\mathbb{Z}_{2})$ (A.11)
$\displaystyle\mathop{\mathrm{Sq}}\nolimits^{d-i}(\lambda)$
$\displaystyle:=\lambda\cup_{i+k-d}\lambda+\delta\lambda\cup_{i+k-d+1}\lambda.$
(A.12)

One can immediately check that $\mathop{\mathrm{Sq}}\nolimits^{d-i}$ commutes
with the coboundary,

$\displaystyle\delta\mathop{\mathrm{Sq}}\nolimits^{d-i}(\lambda)=\mathop{\mathrm{Sq}}\nolimits^{d-i}(\delta\lambda).$
(A.13)

##  Appendix B The obstruction class for Wu structure

Wu structure is specified by a choice of a $(d-2)$-chain $E\in
C_{d-2}(M,\mathbb{Z}_{2})$ with $\partial E=S$, where $S\in
Z_{1}(M^{d},\mathbb{Z}_{2})$ represents the Poincaré dual of the Wu class
$w_{1}w_{2}$. Here we explain how to prepare the chain $S$. We will see that
$S$ is supported on the orientation-reversing wall $W$, and gives the Poincaré
dual of the 2nd Stiefel-Whitney class $w_{2}$ of $W$.

We describe $w_{2}$ as a $(d-2)$-chain $W_{2}\in
Z_{d-2}(M^{d},\mathbb{Z}_{2})$, and $w_{1}$ as a 1-cochain $w_{1}\in
Z^{1}(M^{d},\mathbb{Z}_{2})$. Then $S$ is prepared by $S=W_{2}\cap w_{1}$. The
cap product $\cap$ can be computed explicitly using the formula [50]

$\displaystyle(0\dots j+k)\cap a=(j\dots k)\left(\int_{(0\dots j)}a\right)$
(B.1)

for $a\in C^{j}(M,\mathbb{Z}_{2})$ and $(0\dots j+k)$ is a $(j+k)$-chain. Then
one can extend it by linearity in adding up chains.

Once we take the barycentric subdivision of the triangulation of $M$, we can
prepare $W_{2}$ as the set of all $(d-2)$-simplices of $M$, except for
$(d-2)$-simplices supported on $W$. To see this, we note that the Poincaré
dual of $w_{2}$ is given by the set of all 2-simplices. Meanwhile, since $W$
is oriented, the set of all $(d-2)$-simplices of $W$ that represents $w_{1}$
of $W$ gives a trivial $(d-2)$-chain, so we can take the above $W_{2}$ as the
dual of $w_{2}$.

Then we explain how we prepare the cochain representative for $w_{1}$ of a
$d$-manifold $M^{d}$. This is done by considering a perturbation of a
codimension-1 orientation-reversing wall $W$ described as a yellow object in
Fig. 5 for the case of $d=3$. That is, we think of shifting $W$ along a vector
field expressed as orange vectors in Fig. 5, which are perpendicular to the
tangent of $W$. This vector field is thought of as a section of the normal
bundle of $W$. Since the normal bundle of $W$ can be nontrivial, the vector
field in general has a $(d-2)$-dimensional zero locus in $W$, shown as a red
line in Fig. 5 (a). The zero locus is taken as a $(d-2)$-cycle of $W$.

At the $(d-2)$-dimensional zero locus, we further consider a vector field
tangent to $W$ and perpendicular to the zero locus, shown as a red vector in
Fig. 5 (a). Then, we perturb the yellow object along the red vectors along the
$(d-2)$-dimensional zero locus. The red vector field is regarded as a section
for the normal bundle of the zero locus in $W$, which again can have a
$(d-3)$-dimensional zero locus, as shown in Fig. 5 (b). By repeating the
process of shifting the yellow object along the $j$-dimensional zero locus to
get a $(j-1)$-dimensional zero locus until we perform for $j=0$, we finally
get a perturbation of $W$.

Figure 5: (a): The figure of the orientation reversing wall for the spacetime
dimension $d=3$. We perturb $W$ by a vector field perpendicular to $W$ (orange
arrows) to obtain a perturbed object (yellow sheet). Then, on the zero locus
(red line) we further perturb it by a vector field tangent to $W$ and normal
to the zero locus (red arrow). (b): At the zero locus of the red vector field,
e further perturb the yellow sheet to make the perturbed object intersects
transversally with 1-simplices of $M^{d}$.

Then, the perturbation of $W$ intersects transversally with 1-simplices of
$M$. Assigning the intersection number mod 2 on each 1-simplex defines a
cocycle representative of $w_{1}$.

Now we compute the cycle representative of $w_{1}w_{2}$ using the
representative of $w_{1}$ and $W_{2}$ defined above. We take a branching
structure of the barycentric-subdivided simplical complex, by labeling the
barycenter of $i$-simplices by $(d-i)$. According to the formula for the cap
product (B.1), $S=W_{2}\cap w_{1}$ is given by a set of $(d-3)$-simplices
$(j_{0}\dots j_{d-3})$, weighted by a number of $(d-2)$-simplices
$(i,j_{0}\dots j_{d-3})$, where $w_{1}(ij_{0})=1$ and $(i,j_{0}\dots
j_{d-3})\in W_{2}$.

This $(d-3)$-chain $S$ can be rephrased as follows. $S$ is given by a set of
$(d-3)$-simplices $(j_{0}\dots j_{d-3})$ of $W$ weighted by a number of
$(d-2)$-simplices $(i,j_{0}\dots j_{d-3})$ whose vertex $i$ lies on a specific
domain separated by $W$, and the domain is fixed by the perturbation of $W$ to
define the cocycle $w_{1}$. The weight on each $(d-3)$-simplex $(j_{0}\dots
j_{d-3})$ on $W$ is regarded as the intersection number between $W_{2}$ and
the $w_{1}$ sheet that occurs “in the vicinity of” $(j_{0}\dots j_{d-3})$, see
Fig. 6.

Figure 6: The situation near the orientation reversing wall for the spacetime
dimension $d=3$. The weight on each 0-simplex of $W$ corresponds to the
intersection number between the perturbed sheet and $W_{2}$ near the
0-simplex. In this figure, the barycenter of a 2-simplex has the weight 1
while that of a 1-simplex has the weight 3.

Then, we can show that the weight for each $(d-3)$-simplex $W$ is always odd.
To see that the weight is odd for a $(d-3)$-simplex $(j_{0}\dots j_{d-3})$, we
take a non-closed $d$-submanifold $N$ as shown in Fig. 7 such that $\partial
N$ is oriented and contains $(j_{0}\dots j_{d-3})$, and $N$ contains all the
$(d-2)$-simplices $(i,j_{0}\dots j_{d-3})$ counted in the weight. $N$ is
contained in a specific domain of $M$ separated by $W$, and is taken to
respect the original triangulation before taking barycentric subdivision,
i.e., all the $(d-1)$-simplices on $\partial N$ are labeled as $(1,2,\dots
d)$.

Let $S_{N}$ be a set of all $(d-2)$-simplices of $N$, except for those of
$\partial N$. Let $S_{\partial N}$ be a set of all $(d-3)$-simplices of
$\partial N$. We can then see that

$\displaystyle\partial S_{N}=S_{\partial N},$ (B.2)

which shows that the weight on $(j_{0}\dots j_{d-3})$ is odd, since $\partial
S_{N}$ contributes to $(j_{0}\dots j_{d-3})$ from $(i,j_{0}\dots j_{d-3})$ in
$S_{N}$ counted in the weight. Its derivation is essentially found in Appendix
C. This proves that $S$ is the set of all $(d-3)$-simplices of $W$.

Figure 7: The configuration of $N$ for $d=3$. Picking a 0-simplex of $W$
represented as a red star, then there are 1-simplices that evaluate nonzero
$w_{1}$ around the 0-simplex, represented as red lines. $N$ contains all red
1-simplices in $N$ and $\partial N$ contains the starred vertex.

##  Appendix C Bulk-boundary Grassmann integral

In this appendix, we prove the following formula for the chains of barycentric
subdivision frequently used in this paper. Let $N$ be a $d$-manifold with a
non-empty boundary $\partial N$. Then we have

$\displaystyle\partial S_{N}=S_{\partial N},$ (C.1)

where $S_{N}$ is a set of all $(d-2)$-simplices of $N$, and $S_{\partial N}$
is a set of all $(d-3)$-simplices of $\partial N$.

To see this, we introduce a theory for a bulk-boundary system $\sigma(\partial
N,N;a,b)$ with $a\in C^{d-2}(\partial N,\mathbb{Z}_{2})$ and $b\in
Z^{d-1}(M,\mathbb{Z}_{2})$, which satisfy $\delta a=b$ on the boundary
$\partial N$. $\sigma(\partial N,N;a,b)$ satisfies the following two
properties:

  1. 1.

When $b$ is a coboundary $b=\delta\lambda$ and $a=\lambda$, we have

$\displaystyle\sigma(\partial
N,N;\lambda,\delta\lambda)=(-1)^{\int_{N}\mathop{\mathrm{Sq}}\nolimits^{2}\lambda}(-1)^{\int_{S_{N}}\lambda}.$
(C.2)

  2. 2.

When $b=0$ and $a$ is a coboundary $a=\delta\chi$, we have

$\displaystyle\sigma(\partial N,N;\delta\chi,0)=(-1)^{\int_{\partial
N}\mathop{\mathrm{Sq}}\nolimits^{2}\chi}(-1)^{\int_{S_{\partial N}}\chi}.$
(C.3)

Then, setting $\lambda=\delta\chi$ in (C.2) gives

$\displaystyle\sigma(\partial N,N;\delta\chi,0)=(-1)^{\int_{\partial
N}\mathop{\mathrm{Sq}}\nolimits^{2}\chi}(-1)^{\int_{\partial S_{N}}\chi},$
(C.4)

where we used
$\delta\mathop{\mathrm{Sq}}\nolimits^{2}(\chi)=\mathop{\mathrm{Sq}}\nolimits^{2}(\delta\chi)$
reviewed in Appendix A. Comparing this expression of $\sigma(\partial
N,N;\delta\chi,0)$ with (C.3) shows $\partial S_{N}=S_{\partial N}$.

The theory $\sigma(\partial N,N;a,b)$ with the above two properties is
realized by a bulk-boundary version of the Grassmann integral constructed in
[51, 38]. Since we deal with the oriented $N,\partial N$ in this paper, we
present the proof only for the oriented case for simplicity. However,
$\partial S_{N}=S_{\partial N}$ is also valid in unoriented cases.

Now let us write down the boundary Gu-Wen integral coupled with bulk; we
simply write the integral by $\sigma(a,b)$. We assign Grassmann variables
$\theta_{e},\overline{\theta}_{e}$ on each $(d-2)$-simplex $e$ of $\partial
N$, and $\theta_{f},\overline{\theta}_{f}$ on each $(d-1)$-simplex $f$ of
$N\setminus M$. We define the Gu-Wen integral as

$\sigma(a,b)=\int\prod_{f|b(f)=1}d\theta_{f}d\overline{\theta}_{f}\int\prod_{e|a(e)=1}d\theta_{e}d\overline{\theta}_{e}\prod_{t}u(t),$
(C.5)

$u(t)$ is a monomial of Grassmann variables defined on a $d$-simplex of $N$.
$u(t)[b]$ is defined in the same fashion as in the case without boundary if
$t$ is away from the boundary, which is introduced in Sec. 3.2 of the main
text. However, its definition gets modified when $t$ shares a $(d-1)$-simplex
with the boundary. For simplicity, we assign an ordering on vertices of such
$t=(01\dots d)$, so that the $(d-1)$-simplex shared with $M$ becomes
$f_{0}=(12\dots d)$; the vertex $0$ is contained in $N\setminus M$. For
instance, we can take a barycentric subdivision on $N$, and assign $0$ to
vertices associated with $d$-simplices. We further define the sign of
$(d-1)$-simplices on $M$, such that $f_{0}$ and $t$ have the same sign.

Then, $u(t)$ neighboring with $M$ is defined by replacing the position of
$\vartheta_{f_{0}}$ in $u(t)[b]$ with the boundary action of the Grassmann
integral on $f_{0}$, $u(f_{0})[a]=\prod_{e\in f_{0}}\vartheta_{e}^{a(e)}$. We
then have: On a $+$ simplex,

$u(t)=u(f_{0})[a]\cdot\prod_{f\in\partial t,f\neq f_{0}}\vartheta_{f}^{b(f)}.$
(C.6)

On a $-$ simplex,

$u(t)=\prod_{f\in\partial t,f\neq f_{0}}\vartheta_{f}^{b(f)}\cdot
u(f_{0})[a].$ (C.7)

One can check that $u(t)$ defined above becomes Grassmann-even. Then, one can
see that the bulk-boundary Grassmann integral satisfies the quadratic property

$\displaystyle\sigma(a+a^{\prime},b+b^{\prime})=\sigma(a,b)\sigma(a^{\prime},b^{\prime})(-1)^{\int_{\partial
N}(a\cup_{d-3}a^{\prime}+a\cup_{d-2}\delta
a^{\prime})+\int_{N}b\cup_{d-2}b^{\prime}}.$ (C.8)

The proof for the quadratic property is found in [51].

Now let us demonstrate (C.2). The quadratic part of
$\sigma(\lambda,\delta\lambda)$ is determined by noting that the quadratic
property

$\displaystyle\sigma(\lambda+\lambda^{\prime},\delta\lambda+\delta\lambda^{\prime})=\sigma(\lambda,\delta\lambda)\sigma(\lambda^{\prime},\delta\lambda^{\prime})(-1)^{\int_{\partial
N}\lambda\cup_{d-3}\lambda^{\prime}}(-1)^{\int_{N}\lambda\cup_{d-3}\delta\lambda^{\prime}+\delta\lambda^{\prime}\cup_{d-3}\lambda},$
(C.9)

is solved by $(-1)^{\int_{N}\mathop{\mathrm{Sq}}\nolimits^{2}\lambda}$ up to a
linear term. So, $\sigma(\lambda,\delta\lambda)$ can be expressed as

$\displaystyle\sigma(\lambda;\delta\lambda)=(-1)^{\int_{N}\mathop{\mathrm{Sq}}\nolimits^{2}\lambda}(-1)^{\sum_{e\in
S^{\prime}}\lambda(e)},$ (C.10)

with $S^{\prime}$ some set of $(d-2)$-simplices $e$ of $N$. The linear term is
fixed by computing $\sigma(\lambda;\delta\lambda)$ explicitly in the simplest
case; $\lambda(e)=1$ on a single $(d-1)$-simplex, otherwise 0. Then we find
that we get $\sigma(\lambda;\delta\lambda)=-1$ for arbitrary choice of a
$(d-1)$-simplex $e$, which shows that $S^{\prime}=S_{N}$.

One can also see (C.3) by noting that $\sigma(\partial N,N;\delta\chi,0)$
reduces to the ordinary Grassmann integral $\sigma(\partial N;\delta\chi)$
supported solely on $\partial N$. Then, (C.3) is equivalent to (3.32) in the
main text by replacing $W$ with $\partial N$.

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