# Anomaly inflow for dipole symmetry and higher form foliated field theories

Hiromi Ebisu1 1Yukawa Institute for Theoretical Physics, Kyoto University,
Kyoto 606-8502, Japan  Masazumi Honda2 1Yukawa Institute for Theoretical
Physics, Kyoto University, Kyoto 606-8502, Japan  Taiichi Nakanishi2,1 1Yukawa
Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

###### Abstract

In accordance with recent progress of fracton topological phases, unusual
topological phases of matter hosting fractionalized quasiparticle excitations
with mobility constraints, new type of symmetry is studied – multipole
symmetry, associated with conservation of multipoles. Based on algebraic
relation between dipole and global charges, we introduce a series of
$(d+1)$-dimensional BF theories with $p$-form gauge fields, which admit dipole
of spatially extended excitations, and study their physical properties. We
elucidate that gauge invariant loops have unusual form, containing linear
function of the spatial coordinate, which leads to the position dependent
braiding statistics and unusual ground state degeneracy dependence on the
system size. We also show that the theories exhibit a mixed ’t Hooft anomaly
between $p$-form and $(d-p)$-form dipole symmetries, which is canceled by an
invertible theory defined in one dimensional higher via anomaly inflow
mechanism.

###### Contents

  1. 1 Introduction
  2. 2 Dipole algebra
  3. 3 Dipole BF theories
    1. 3.1 Model
      1. 3.1.1 Notations
      2. 3.1.2 Integer BF theory
    2. 3.2 Example: $d=3$, $p=1$
      1. 3.2.1 Loops and braiding statistics
      2. 3.2.2 Ground state degeneracy
  4. 4 Anomaly inflow
    1. 4.1 Review of $1$-form gauging in $(2+1)$d BF theory
    2. 4.2 Anomaly inflow for dipole symmetries
  5. 5 Conclusion
  6. A Ground state degeneracy for general $p$ and $d$
  7. B Construction of lattice spin models with dipole symmetry
    1. B.1 $p=1$ and $d=3$
    2. B.2 $p=2$ and $d=3$
  8. C Higher form subsystem BF theory and anomaly inflow
    1. C.1 Model
    2. C.2 Anomaly inflow

##  1 Introduction

Symmetry plays a prominent role in physics, and have been a guiding principle
for many researches in various contexts. Here we list some examples:
prediction of the form of energy spectrum, classifying phases of matter based
on symmetry breaking, understanding of physics in the vicinity of fixed points
of critical phenomena, and many more. In recent years, a plethora of efforts
have been devoted to update the concept of symmetries. One such example is
generalized symmetry, a.k.a., higher form symmetry [1, 2, 3], which is
associated with symmetry of extended objects. Fruitful interdisciplinary
advances have been made for decades by studying this symmetry, allowing us to
make better understanding of problems not only in the context of high energy
physics but also in view of condensed matter physics, such as classifying
topological phases of matter.111See, e.g., recent pedagogical reviews [4, 5]
and references therein.

Newly proposed fracton topological phases [6, 7, 8] have motivated one to
explore other types of symmetries – multipole symmetries. To see how, let us
first briefly recall the property of the fracton topological phases.
Distinctive feature of these phases is that mobility constraints are imposed
on fractionalized quasiparticle excitations, giving rise to subextensive
ground state degeneracy (GSD) on a torus geometry [9, 10]. Due to this
property, one faces an issue with the UV/IR mixing, implying incapability to
resort to preexisting theoretical frameworks, such as topological field
theories. To handle this problem, new types of symmetries have been
introduced. One of such symmetries is multipole symmetry [11, 12, 13, 14, 15,
16, 17], associated with conservation of multipole moments, such as dipole,
quadrupole, and octopole, e.t.c. Previously, it was argued that the multipole
symmetries give rise to mobility constraints on excitations, which play a
crucial role in understanding of fracton topological phases (see, for
instance, [14, 18, 19, 20, 21]). Furthermore, based on the algebraic relations
between global and dipole charges, a series of topological models have been
constructed [22, 23]. Despite several attempts, full understanding of these
symmetries remains elusive.

In this paper, we discuss a class of field theories with dipole symmetry,
which is the simplest example of the multipole symmetries, and their
anomalies. Furthermore, to aim for establishing a theoretical framework which
contains broader scope, incorporating dipole symmetries and other types of
symmetries in a unified way, we explore the interplay between the dipole
symmetries and higher form symmetries. To this end, we deal with two types of
symmetries on the same footing by introducing higher form dipole symmetries,
and study gauge theories associated with them.

We argue that the algebraic relation between dipole and global charges
naturally yields foliated BF theories [24, 25], BF theories defined on layers
of submanifolds, which were studied in the context of fracton physics. There
has been much progress particularly in $(2+1)$-dimensions [$(2+1)$d], such as
relation between foliated BF theories and another exotic symmetry, called
subsystem symmetry (i.e., symmetry on a submanifold) [26, 27, 28, 29].
Although a few foliated BF theories with dipole symmetries have been studied
in $(2+1)$ dimensions [22], theories defined in higher dimensions (i.e.,
$d>2$) have been much less explored. In this work, we introduce foliated BF
theories with $p$-form dipole symmetries in any dimension, which host dipole
of spatially extended excitations (e.g., membranes when $p=2$) and study their
physical properties and ’t Hooft anomalies of the dipole symmetries. We find
that due to the algebraic relation between global and dipole charges, gauge
invariant loops have unusual form, containing linear function of the spatial
coordinate. This feature leads to the position dependent braiding statistics
and unusual GSD dependence on the system size, i.e., the greatest common
divisor between the charge $N$ and the system size. Note that such GSD
dependence is the manifestation of the UV/IR mixing, stemming from the
conservation of the dipole. The UV/IR mixing that our theories exhibit is
distinct from the one found in previous field theories with subsystem
symmetry, such as [27] where the GSD becomes subextensive.

We also show that the theories exhibit a mixed ’t Hooft anomaly between
$p$-form and $(d-p)$-form dipole symmetries, which is canceled by an
invertible theory defined in one dimensional higher via anomaly inflow
mechanism [30]. The bulk theory has a similar form as the one which was
introduced in the study of the anomaly inflow when gauging $1$-form symmetries
in $(2+1)$d BF theory [3], yet crucial difference from the previous cases is
that we need to appropriately impose the flatness condition of the gauge
fields as they are associated with the dipole symmetries. Our consideration
can also be applied to the cases with foliated BF theories with subsystem
symmetries [24, 25, 31], whose in-depth discussion is provided in
appendix.222See also e.g., [32, 33, 34, 35] for discussion on anomaly inflow
for subsystem symmetries in other theories, and [36] for anomaly inflow for a
$(1+1)d$ scalar theory with the second order spatial derivatives based on the
so-called wire construction. This paper would contribute to not only better
understanding of fracton physics, but also to achieving the ultimate goal to
construct a theoretical framework, incorporating various kinds of symmetries.

The rest of this paper is structured as follows. In Sec. 2, we introduce
algebraic relations between global charges and dipole moments, which play a
crucial role throughout this work. In Sec. 3, we introduce BF theories which
respect higher-form dipole symmetries. We also discuss their physical
properties, such as braiding statistics between fractional excitations and the
GSD on a torus. In Sec. 4, we study the ’t Hooft anomaly and its anomaly
inflow for the higher form dipole symmetries. Finally, in Sec. 5 we conclude
our work with a few remarks. Some technical details, including construction of
the UV spin models that correspond to the dipole BF theories, and thorough
analysis on the BF theories with subsystem symmetry and their anomaly, are
provided in Appendixes.

##  2 Dipole algebra

To systematically introduce the foliated BF theories, we start with
introducing multipole algebra, which describes the multipole symmetries. In
the present work, we focus on dipole algebra, which is the simplest algebraic
relations between global charges and dipole moments. Note that the (zero-form)
dipole algebra was introduced in the previous studies [17, 22]. In what
follows, we are going to generalize such an algebra to higher form, which is
necessary to investigate the BF theories and their anomaly.

Suppose that we have a theory in $(d+1)$-dimensions with conserved
$(p-1)$-form charges associated with global U(1) and dipole symmetries,
defined on $(d+1-p)$-dimensional spatial submanifold, $\Sigma_{d+1-p}$.333Here
we take $1\leq p\leq d$. The $p=1$ case corresponds to the ordinary global
symmetry. We denote the global charge by $Q[\Sigma_{d+1-p}]$ and dipole charge
by $Q_{I}[\Sigma_{d+1-p}]$, where the index $I=1,\cdots,d$ denotes the dipole
degrees of freedom in the $I$-th spatial direction. Also, throughout this
work, we interchangeably represent the spatial direction $I=1,2,3,\cdots,$ as
$I=x,y,z,\cdots,$ depending on the context.

While the charge $Q$ follows the relation

$[iP_{I},Q]=0,$ (1)

the dipole charges are subject to the following relation:

$[iP_{I},Q_{J}]=\delta_{IJ}Q.$ (2)

An intuition behind this relation can be understood by associating the global
and dipole charges with $\rho$, $x_{J}\rho$, where $\rho$ denotes the density
of the U(1) charge, and $x_{J}$ as the $J$-th spatial coordinate,
respectively, and thinking of shifting them by a constant in the $I$-th
direction ($I,J=1,\cdots,d$) [22]. For instance, if we shift the dipole moment
$x_{I}\rho$ by a constant in the $I$-th direction, then the change of dipole
moment under the shift gives $(x_{I}+\Delta x_{I})\rho-x_{I}\rho=(\Delta
x_{I})\rho$, where $\Delta x_{I}$ is constant, corresponding to the nontrivial
commutation relation between the transnational operator and the dipole. Such
an intuitive understanding of the relation (2) will be useful in the
discussion on the BF theories presented in the next section.

We write the charges $Q$ and $Q_{I}$ via integral expression using the
$p$-form conserved currents as

$Q[\Sigma_{d+1-p}]=\int_{\Sigma_{d+1-p}}*j^{(p)},\quad
Q_{I}[\Sigma_{d+1-p}]=\int_{\Sigma_{d+1-p}}*K_{I}^{(p)}.$

To reproduce the relation (2), we demand that

$*K_{I}^{(p)}=*k_{I}^{(p)}-x_{I}*j^{(p)}$ (3)

with $k_{I}^{({p})}$ being a local (non-conserved) current. One can verify the
relation (2) by (3). With the dipole algebra (2) with (3), we gauge these
symmetries. To do so, we introduce $U(1)$ $p$-form gauge fields $a^{({p})}$,
$A^{I({p})}$ with the coupling term being defined by444As discussed in [22],
one can regard the gauge group as U(1), taking the fact that quantization
condition of the dipole gauge field depends on the length of the dipole into
consideration. As we will see soon below, if we define the theory on a
discrete lattice, it can be understood by that the dipole charge is quantized
since the lattice spacing becomes a minimal unit of distance. We set such a
length to be 1 throughout this work.

$S_{c}=\int_{V_{d+1}}\left(a^{({p})}\wedge*j^{({p})}+\sum_{I=1}^{d}A^{I({p})}\wedge*k_{I}^{({p})}\right),$
(4)

where $V_{d+1}$ denotes the spacetime. We need to have a proper gauge
transformation in such a way that the condition of the coupling term being
gauge invariant yields the conservation law of the higher form currents. It
turns out that the following gauge transformation does this job:

$a^{({p})}\to
a^{({p})}+d\Lambda^{({p-1})}+(-1)^{p-1}\sum_{I}\sigma^{I({p-1})}\wedge
dx_{I},\quad A^{I({p})}\to A^{I({p})}+d\sigma^{I({p-1})}.$ (5)

Here, $\Lambda^{({p-1})}$ and $\sigma^{I({p-1})}$ denote the $(p-1)$-form
gauge parameters. Indeed, one can verify that the gauge invariance of the
coupling term $S_{c}$ under the gauge transformation (5) yields

$d*j^{({p})}=0,\quad d(*k_{I}^{({p})}-x_{I}*j^{({p})})=d*K_{I}^{({p})}=0.$

In what follows, following the terminology in fracton physics [24, 25], we
interpret $dx_{I}$ as $1$-form field, that we dub foliation field:

$e^{I}\vcentcolon=dx_{I}.$ (6)

In the context of fracton topological phases, such a field was introduced so
that along the foliation field, layers of submanifolds are stacked.

For later purposes, we also define the dipole algebra with hierarchy structure
being inverted, which we call dual dipole algebra. To do this, instead of (1)
and (2), we think of $d$ global charges $\widehat{Q}_{I}$ and one dipole
charge $\widehat{Q}$ with relation

$[iP_{I},\widehat{Q}_{J}]=0,\quad[iP_{I},\widehat{Q}]=-\widehat{Q}_{I}.$ (7)

Analogous to the argument below (2), this relation can be intuitively
understood by interpreting the dipole and $d$ global charges as
$\widehat{\eta}=-\sum_{I=1}^{d}x_{I}\rho_{I}$, and $\rho_{I}$, respectively,
and thinking of the translation operator that acts on it. For example, by
shifting the dipole $\widehat{\eta}$ in the $I$-th direction, one obtains the
second relation in (7). Following the similar argument presented around
(3)-(5), we define gauge fields associated with the global and dipole charges
(7) as $\widehat{a}^{I({p})}$ and $\widehat{A}^{({p})}$ respectively with the
following gauge transformations:

$\displaystyle\widehat{a}^{I({p})}\to\widehat{a}^{I({p})}+d\widehat{\chi}^{I({p-1})}-(-1)^{p-1}\widehat{\sigma}^{({p-1})}\wedge
e^{I},\quad\widehat{A}^{({p})}\to\widehat{A}^{({p})}+d\widehat{\sigma}^{({p-1})}.$
(8)

Here, $\widehat{\chi}^{I({p-1})}$ and $\widehat{\sigma}^{({p-1})}$ the are the
$(p-1)$-form gauge parameters.

The dipole algebra (2) is related to the dual one (7) by inverting the
hierarchy structure of the algebra. Stated symbolically,

$\begin{Bmatrix}Q_{1}&Q_{2}&\cdots&Q_{d}\\\
&Q&&\end{Bmatrix}\quad\leftrightarrow\quad\begin{Bmatrix}&\widehat{Q}&&\\\
\widehat{Q}_{1}&\widehat{Q}_{2}&\cdots&\widehat{Q}_{d}\end{Bmatrix}.$ (9)

These algebras put different mobility constraints on charges. In the case of
the dipole algebra (2), a single charge is immobile as dipole moment is
conserved in any spatial direction. On the contrary, in the case of the dual
dipole algebra (7), which consists of $d$ charges (labeled by $I=1,\cdots,d$)
and one dipole, the $I$-th charge, $\widehat{Q}_{I}$, is mobile in the
direction perpendicular to the $I$-th direction, yet it is immobile in the
$I$-th direction. Such mobility constraints play a crucial role in
understanding physics of the dipole BF theory that we study in the next
section. In what follows, we will discuss ’t Hooft anomalies in the BF
theories, relegating the discussion on the subsystem BF theories to Appendix.
C.

##  3 Dipole BF theories

In this section, we introduce BF theories with the dipole symmetries that we
dub dipole BF theories. Due to the UV/IR mixing, treatment of the dipole BF
theories in the continuum limit is a bit subtle. To handle this issue, we put
our theory in more appropriate format, that is sometimes called integer BF
theory [15], consisting of integer valued gauge fields on a discrete lattice.
We discuss several physical properties of the model, such as braiding
statistics and GSD on torus geometry before diving into the anomaly inflow of
the higher dipole symmetries.

###  3.1 Model

Using the gauge fields $a^{({p})}$ and $A^{I({p})}$ in (5) associated with the
dipole algebra (2), we introduce the following gauge invariant $(p+1)$-form
fluxes:

$f^{({p+1})}\vcentcolon=da^{({p})}+(-1)^{p}\sum_{I}A^{I({p})}\wedge
e^{I},\quad F^{I({p+1})}\vcentcolon=dA^{I({p})}.$ (10)

In terms of these fluxes, let us consider the following $(d+1)$-dimensional
theory, which we call $p$-form dipole BF theory:555Here we take $1\leq p\leq
d-1$. In the case of $p=d$, the theory describes a spontaneous symmetry
braking phase, which we do not consider in this work.

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{N}{2\pi}\left[b^{({d-p})}\wedge
f^{({p+1})}+\sum_{I}c^{I({d-p})}\wedge F^{I({p+1})}\right]$ (11)
$\displaystyle=$
$\displaystyle\frac{N}{2\pi}\left[b^{({d-p})}\wedge\left(da^{({p})}+(-1)^{p}\sum_{I}A^{I({p})}\wedge
e^{I}\right)+\sum_{I}c^{I({d-p})}\wedge dA^{I({p})}\right],$

where $b^{({d-p})}$ and $c^{I({d-p})}$ denote the $(d-p)$-form gauge fields.
To study physical properties of this theory, such as braiding statistics of
fractional excitations and the GSD on a torus geometry, we face an issue with
the UV/IR mixing — the theory is sensitive to the UV physics, making analysis
on the theory in the continuum limit much more challenging. To remedy this
problem, we make use of argument presented in e.g, [37, 22], where we
introduce integer BF theories, consisting of integer valued gauge fields which
are defined on a discrete lattice. In this formalism, we discuss braiding
statistics and the GSD to get more physical insights from the theory (11)
before studying the anomaly inflow.

####  3.1.1 Notations

To do this, we prepare several notations. In order to investigate the integer
BF theory with dipole symmetry, comprised of integer valued gauge fields, we
define an integer valued $p$-form field, located on a $p$-cell in a
$(d+1)$-dimensional discrete lattice as666The field with a tilde on the top
$\tilde{(\cdot)}$ represents integer valued field throughout this work.

$\tilde{A}^{({p})}_{[\mu_{1}\cdots\mu_{p}]}(\hat{\mathbf{r}}_{A})\vcentcolon=\frac{1}{p!}\sum_{\sigma}Sgn(\sigma)\tilde{A}^{({p})}_{\mu_{\sigma(1)}\cdots\mu_{\sigma(p)}}(\hat{\mathbf{r}}_{A}),$
(12)

where $[\mu_{1}\cdots\mu_{p}]$ denotes $p$-th anti-symmetrized index and
$\sigma$ represents the permutation with its signature denoted by
$Sgn(\sigma)$. Also, we have introduced the coordinate of the field as
$\hat{\mathbf{r}}_{A}=(\hat{x}_{0},\hat{x}_{1},\cdots,\hat{x}_{d})$. Without
bringing any confusion, we omit the lower case indices and the coordinate of
the gauge field on the left hand side of (12), and simply write it as
$\tilde{A}^{({p})}$. We introduce two more notations. We define a differential
operator $\Delta$, mapping a field $\tilde{A}^{({p})}$ to
$\Delta\tilde{A}^{({p})}$ on $(p+1)$-cell, which is given by the oriented sum
of $\tilde{A}^{({p})}$ on the boundary of the $(p+1)$-cell. More explicitly,
it is defined as

$\Delta\tilde{A}^{({p})}_{[\mu_{1}\cdots\mu_{p+1}]}\vcentcolon=\frac{1}{p!}\sum_{\sigma}Sgn(\sigma)\Delta_{\mu_{\sigma(1)}}\tilde{A}^{({p})}_{\mu_{\sigma(2)}\cdots\mu_{\sigma(p+1)}},$
(13)

which is abbreviated as $\Delta\tilde{A}^{({p})}$. We also define the
following integer valued field located on $(p+1)$-cell, which corresponds to
the wedge product between a $p$-form field and a foliation field $e^{I}$ in
the continuum:

$\tilde{A}^{({p})}e^{I}\vcentcolon=\frac{1}{p!}\sum_{\sigma}Sgn(\sigma)\tilde{A}^{({p})}_{\mu_{\sigma(1)}\cdots\mu_{\sigma(p)}}\delta^{I}_{\mu_{\sigma(p+1)}}\quad(I=1,\cdots,d).$
(14)

Here $\delta^{I}_{J}$ represents the Kronecker delta.

####  3.1.2 Integer BF theory

With these preparations, now we are in a good place to discuss the integer BF
theory. The integer BF theory that corresponds to (11), which is defined on a
$(d+1)$-dimensional discrete lattice, reads

$\boxed{\mathcal{L}=\frac{2\pi}{N}\sum_{(p+1)-{\rm
cell}}\left[\tilde{b}^{({d-p})}\left(\Delta\tilde{a}^{({p})}+(-1)^{p}\sum_{I}\tilde{A}^{I({p})}e^{I}\right)+\sum_{I}\tilde{c}^{({d-p})}\Delta\tilde{A}^{I({p})}\right].}$
(15)

Here, the gauge fields $\tilde{a}^{({p})}$ and $\tilde{A}^{I({p})}$ reside on
$p$-cells in the lattice whereas $\tilde{b}^{({d-p})}$ and
$\tilde{c}^{I({d-p})}$ do on dual $(d-p)$-cells. The theory (15) consists of
$d+1$ layers of the $p$-form $\mathbb{Z}_{N}$ BF theories, corresponding to
the first and third terms, and the coupling between the BF theories described
by the second term. In this sense, (15) is the $p$-form analog of the foliated
BF theories [24, 25]. Note that while in the preexisting foliated BF theories
are made of subextensive number of layers of the BF theories, our theory
consists of the finite number of layers. As shown below, the coupling between
the layers brings unusual physical properties such as braiding statistics and
the GSD.

The theory (15) admits the following gauge symmetry:

$\displaystyle\tilde{a}^{({p})}\to\tilde{a}^{({p})}+\Delta\tilde{\chi}^{({p-1})}+(-1)^{p-1}\sum_{I}\tilde{\sigma}^{I({p-1})}e^{I}+N\tilde{k}^{({p})}_{a},\quad\tilde{A}^{I({p})}\to\tilde{A}^{I({p})}+\Delta\sigma^{I({p-1})}+N\tilde{k}_{A}^{I({p})},$
$\displaystyle\tilde{b}^{({d-p})}\to\tilde{b}^{({d-p})}+\Delta\tilde{\eta}^{({d-p})}+N\tilde{k}^{({d-p})}_{b},\quad\tilde{c}^{I({d-p})}\to\tilde{c}^{I({d-p})}+\Delta\tilde{\gamma}^{I({d-p})}-(-1)^{d-p}\tilde{\eta}^{({d-p})}e^{I}+N\tilde{k}^{I({d-p})}_{c}.$
(16)

The integer valued gauge parameters $\tilde{\chi}^{({p-1})}$ and
$\tilde{\sigma}^{I({p-1})}$ are defined on $(p-1)$-cells whereas
$\tilde{\eta}^{({d-p-1})}$ and $\tilde{\gamma}^{I({d-p-1})}$ are introduced on
dual $(d-p-1)$-cells. Also, $\tilde{k}_{a}^{({p})}$ and
$\tilde{k}_{A}^{I({p})}$ [$\tilde{k}_{b}^{({d-p})}$ and
$\tilde{k}_{c}^{I({d-p})}$] are integer valued fields on $p$-cell [dual
$(d-p)$-cell]. Note that while gauge fields $\tilde{a}^{({p})}$ and
$\tilde{A}^{I({p})}$ are subject to the gauge transformation of the dipole
algebra (5), the other gauge fields $\tilde{b}^{({d-p})}$ and
$\tilde{c}^{I({d-p})}$ follow the dual dipole algebra (8). In the following,
to get better handle on the theory (15), we study several physical properties
of the model before discussing the anomaly.

###  3.2 Example: $d=3$, $p=1$

We demonstrate unusual properties of the braiding statistics and the GSD of
the model (15) in a specific case, setting $d=3$ and $p=1$. Argument presented
in this subsection can be generalized to any dimension and higher form
symmetries, which is given in Appendix A.

####  3.2.1 Loops and braiding statistics

We study braiding statistics between gauge invariant loops of the theory (15)
with $d=3$ and $p=1$:

$\mathcal{L}=\frac{2\pi}{N}\sum_{2-{\rm
cell}}\left[\tilde{b}^{({2})}\left(\Delta\tilde{a}^{({1})}-\sum_{I}\tilde{A}^{I({1})}e^{I}\right)+\sum_{I}\tilde{c}^{({2})}\Delta\tilde{A}^{I({1})}\right].$

Analogous to the foliated BF theories of fracton topological phases [24, 25],
the coupling between the BF theories modify the gauge transformations, giving
rise to mobility constraints on fractional charges. From the gauge
transformations (16):

$\displaystyle\tilde{a}^{({1})}\to\tilde{a}^{({1})}+\Delta\tilde{\chi}^{({0})}+\sum_{I}\tilde{\sigma}^{I({0})}e^{I}+N\tilde{k}^{({1})}_{a},\quad\tilde{A}^{I({1})}\to\tilde{A}^{I({1})}+\Delta\sigma^{I({0})}+N\tilde{k}_{A}^{I({1})},$
$\displaystyle\tilde{b}^{({2})}\to\tilde{b}^{({2})}+\Delta\tilde{\eta}^{({2})}+N\tilde{k}^{({2})}_{b},\quad\tilde{c}^{I({2})}\to\tilde{c}^{I({2})}+\Delta\tilde{\gamma}^{I({2})}-\tilde{\eta}^{({2})}e^{I}+N\tilde{k}^{I({2})}_{c},$

we can construct several types of gauge invariant operators. First we have the
gauge invariant Wilson loops in the spatial direction, described by

$\displaystyle W_{a}(C)\vcentcolon=\exp\left[\frac{2\pi
i}{N}\sum_{C}\left(\tilde{a}^{({1})}+\sum_{I}\hat{x}_{I}\tilde{A}^{I({1})}\right)\right],\quad
W^{I}_{A}(C)\vcentcolon=\exp\left[\frac{2\pi
i}{N}\sum_{C}\tilde{A}^{I({1})}\right],\quad(I=x,y,z),$ (17)

where $C$ denotes closed path in subspace of $1$-cells. We also have gauge
invariant surface operators given by

$\displaystyle V_{b}(S^{*})\vcentcolon=\exp\left[\frac{2\pi
i}{N}\sum_{S^{*}}\tilde{b}^{({2})}\right],\quad
V^{I}_{c}(S^{*})\vcentcolon=\begin{cases}\exp\left[\frac{2\pi
i}{N}\sum_{S^{*}}\tilde{c}^{I({2})}\right]&{\rm for}\ S^{*}\perp I\text{-th
direction}\\\ \exp\left[\frac{2\pi
i}{N}\sum_{S^{*}}\left(\tilde{c}^{I({2})}-\hat{x}_{I}\tilde{b}^{({2})}\right)\right]&{\rm
for}\ S^{*}\not\perp I\text{-th direction}\end{cases}.$ (18)

where $S^{*}$ is closed surface defined in subspace of dual $2$-cells. Note
that from the gauge transformation of $\tilde{c}^{I({2})}$ (16), depending on
the direction, there are two types of closed surface operators made of
$\tilde{c}^{I}$: (i) when the surface $S^{*}$ is perpendicular to the $I$-th
direction (i.e., the surface is formed at fixed $\hat{x}_{I}$), we have the
gauge invariant surface operator comprised only of the gauge field
$\tilde{c}^{I({2})}$, corresponding to the first line in (18). (ii) when the
surface $S^{*}$ is not perpendicular to the $I$-th direction, composite of
$\tilde{c}^{I({2})}$ and $\tilde{b}^{({2})}$ is the appropriate gauge
invariant surface operator, corresponding to the second line in (18).

While $1$-form and $2$-form gauge invariant loops are also found in the
conventional $(3+1)$d BF theories (or the $(3+1)$d toric code [38, 39] on the
lattice), in the present case, some of the gauge invariant extended operators
contain linear function of the spatial coordinate $\hat{x}_{I}$, which is a
unique feature of the dipole BF theories [19, 23]. The form of the loops (17)
and (18) can be comprehended by mobility constraints on quasiparticles with
the algebraic relations (2) and (7).777 The mobility constraints on charges in
a system which preserves the dipole moment are also discussed in e.g., [40].
We recall the fact that the gauge fields $\tilde{a}^{({1})}$ and
$\tilde{A}^{I({1})}$ are associated with the dipole algebra (2), corresponding
to conservation of three dipole moments, forming in $x$, $y$, and
$z$-direction, and one global charge. Since the dipoles are conserved, a
single charge is immobile in any direction whereas the dipole moments are free
to move. Correspondingly, a single charge associated with the gauge field
$\tilde{a}^{({1})}$, is not mobile in any of spatial direction. To make it
mobile, the charge must accompany with the dipole moment, associated with the
gauge field $\tilde{A}^{I({1})}$. Also, the dipole moments are free to move –
they are consistent with the form of (17), that is, the form of the Wilson
loop of the gauge field $\tilde{a}^{({1})}$ is accompanied with
$\tilde{A}^{I({1})}$, and that of the gauge field $\tilde{A}^{I({1})}$ may
form in any direction. Analogous line of thoughts shows that the form of the
loops (18) can be understood by the fact that the gauge fields
$\tilde{c}^{I({2})}$ and $\tilde{b}^{({2})}$ follow the dual dipole algebra
(7): three global charges $\rho_{i}\>(i=1,2,3)$ and one dipole moment
$-(x\rho_{1}+y\rho_{2}+z\rho_{3})$. For instance, a single charge
corresponding to gauge field $\tilde{c}^{x(2)}$, is mobile in the $y$ and
$z$-direction and is not in the $x$-direction. To make it mobile in the
$x$-direction, the charge must accompany with the charge associated with the
dipole gauge field $\tilde{b}^{({2})}$, which is consistent with (18).

The loops (17) and (18) have several unusual properties compared with the ones
in the conventional BF theories. To see this, we introduce a lattice
translation operator $T_{\hat{x}_{I}}$ which translates fields by a unit of
the lattice spacing in the $I$-th direction. Acting the translation operator
on the Wilson loop, we find

$\displaystyle\frac{T_{\hat{x}_{I}}W_{a}(C)T^{-1}_{\hat{x}_{I}}}{W_{a}(T_{\hat{x}_{I}}(C))}=W_{A}^{I}(T_{\hat{x}_{I}}(C)),$
(19)

where $T_{\hat{x}_{I}}(C)$ is a contour obtained by shifting $C$ by one
lattice spacing in the $I$-th direction. Similarly, the surface operator not
perpendicular to the $I$-th direction satisfies

$\displaystyle\frac{T_{\hat{x}_{I}}V^{I}_{c}(S^{*})T^{-1}_{\hat{x}_{I}}}{V^{I}_{c}(T_{\hat{x}_{I}}(S^{*}))}=V_{b}(T_{\hat{x}_{I}}(S^{*}))\quad(S^{*}\not\perp
I\text{-th direction}).$ (20)

where $T_{\hat{x}_{I}}(S^{\ast})$ is a surface obtained by shifting $S^{\ast}$
by one lattice spacing in the $I$-th direction. These relations remind us of
the dual dipole algebra (7) and dipole algebra (2) that we have seen in the
previous argument of the dipole symmetry888See also [22] for the related
relations found in lattice spin models with dipole symmetry. , namely,
translating a loop by a constant in the $I$-th direction yields another type
of loop, which is in line with the dipole and dual dipole algebra where
translating a dipole moment gives rise to a charge.

We also discuss braiding statistics of the loops (17) and (18). Similar to the
conventional BF theories, we think of a torus geometry and study the braiding
statistics between the loops that wind around the torus in the spatial
directions. To this end, we impose the periodic boundary conditions on the
lattice and set the system size as $L_{x}\times L_{y}\times L_{z}$. Also, we
focus on the braiding the $1$-form loops that go around the torus in the
$x$-direction and $2$-form loops that run along $yz$-plane. Braiding of the
loops in the other spatial directions can be analogously discussed. The
noncontratcible loops of the gauge fields $\tilde{a}^{({p})}$ and
$\tilde{A}^{I({p})}$ that wind around the torus in the $x$-direction are given
by

$\displaystyle W_{a:x}(\hat{y},\hat{z}):=W_{a}(C_{x})=\exp\left[\frac{2\pi
i}{N}\alpha_{x}\sum_{C_{x}}\left(\tilde{a}^{({1})}+\hat{x}\tilde{A}^{x}+\hat{y}\tilde{A}^{y}+\hat{z}\tilde{A}^{z}\right)\right],$
$\displaystyle
W^{I}_{A:x}(\hat{y},\hat{z}):=W^{I}_{A}(C_{x})=\exp\left[\frac{2\pi
i}{N}\sum_{C_{x}}\tilde{A}^{I({1})}\right],$ (21)

where $C_{x}$ represents noncontractible paths around the torus in the
$x$-direction, and

$\alpha_{I}\vcentcolon=\frac{N}{\gcd(N,L_{I})}\quad(I=x,y,z),$ (22)

with $\gcd$ standing for the greatest common divisor. An intuition behind the
first loop (21) is that the argument of the exponent has linear function of
the spatial coordinate $\hat{x}$, hence the loop has to wind around the torus
multiple times in order for it to be consistent with the periodic boundary
condition.999See e.g., [19, 22] for the relevant discussion in a different
topological model with dipole symmetry. The noncontractible loops of the gauge
fields $\tilde{b}^{({d-p})}$ and $\tilde{c}^{I({d-p})}$ that are defined in
dual $2$-cells on a $yz$-plane are described by

$\displaystyle
V_{b:yz}\left(\hat{x}+1/2\right):=V_{b}(S^{*}_{yz})=\exp\left[\frac{2\pi
i}{N}\sum_{S^{*}_{yz}}\tilde{b}^{({2})}\right],$ $\displaystyle
V^{I}_{c:yz}\left(\hat{x}+1/2\right):=V^{I}_{c}(S^{*}_{yz})=\begin{cases}\exp\left[\frac{2\pi
i}{N}\sum_{S^{*}_{yz}}\tilde{c}^{I({2})}\right]&{\rm for}\ I=x\\\
\exp\left[\frac{2\pi
i}{N}\alpha_{I}\sum_{S^{*}_{yz}}\left(\tilde{c}^{I({2})}-\hat{x}_{I}\tilde{b}^{({2})}\right)\right]&{\rm
for}\ I=y,z,\end{cases}$ (23)

where $S^{*}_{yz}$ denotes surfaces on $yz$-plane of the dual $2$-cells that
go around the torus, forming noncontractible $2$-form loops. Similar to the
first term in (21), we multiply the integer $\alpha_{I}$ with the argument of
the exponent of the loop $V^{I}_{c:yz}$ with $I=y,z$ to ensure that the loop
is compatible with the periodic boundary condition.

The nontrivial braiding statistics between the loops (21) and (23) is given by

$\displaystyle W_{a:x}(\hat{y},\hat{z})V_{b:yz}\left(\hat{x}+1/2\right)$
$\displaystyle=$ $\displaystyle
V_{b:yz}\left(\hat{x}+1/2\right)W_{a:x}(\hat{y},\hat{z})\exp\left[-i\frac{2\pi\alpha_{x}}{N}\right]$
$\displaystyle
W^{I}_{A:x}(\hat{y},\hat{z})V^{J}_{c:yz}\left(\hat{x}+1/2\right)$
$\displaystyle=$ $\displaystyle
V^{J}_{c:yz}\left(\hat{x}+1/2\right)W^{I}_{A:x}(\hat{y},\hat{z})\exp\left[-i\frac{2\pi}{N}\left(\delta_{I,J}+(1-\delta_{I,J})\alpha_{J}\right)\right]$
$\displaystyle W_{a:x}(\hat{y},\hat{z})V^{x}_{c:yz}\left(\hat{x}+1/2\right)$
$\displaystyle=$ $\displaystyle
V^{x}_{c:yz}\left(\hat{x}+1/2\right)W_{a:x}(\hat{y},\hat{z})\exp\left[-i\frac{2\pi\alpha_{x}}{N}\hat{x}\right].$
(24)

Due to the fact that some of the loops (21) (23) contain linear function of
the spatial coordinate, $\hat{x}_{I}$, we have unusual braiding statistics
between the loops. Especially, a phase factor obtained by braiding between the
loops $V_{c:yz}^{x}$ and $W_{a:x}$ depends on the spatial coordinates, as
shown in the third line in (24). We regard it as the feature specific to the
BF theory with the dipole symmetries.101010 Position dependent braiding
statistics is also studied in the Maxwell theory with dipole symmetry in [17]
and the Higgsed phase of a tensor gauge theory in [21].

####  3.2.2 Ground state degeneracy

Due to the dipole symmetries, the model exhibits the unusual behaviour of the
GSD when we place it on torus geometry. To see how the model exhibits the
unusual GSD dependence on the system size, we place the theory on a discrete
lattice with system size $L_{x}\times L_{y}\times L_{z}$ with periodic
boundary conditions. The equations of motions of the BF theory (15) ensure
that local gauge invariant fluxes are trivial. Yet, there are non-local ones
forming noncontractible loops of the gauge fields, contributing to the GSD.
Focusing on such loops of the gauge fields $\tilde{a}^{({1})}_{i}$ and
$\tilde{A}^{({1})}_{i}$, we evaluate the number of distinct configurations of
loops, which amounts to the GSD on torus. The gauge invariant noncontractible
loops of $\tilde{a}^{({1})}_{i}$ and $\tilde{A}^{({1})}_{i}$ are given
by111111Here we explicitly write the components of the gauge fields for the
sake of illustration.

$\displaystyle W_{a:x}(\hat{y},\hat{z})=\exp\left[\frac{2\pi
i}{N}\alpha_{x}\sum_{\hat{x}=1}^{L_{x}}\left(\tilde{a}_{x}^{({1})}+\sum_{I}\hat{x}_{I}\tilde{A}_{x}^{I({1})}\right)\right],\quad
W_{a:y}(\hat{x},\hat{z})=\exp\left[\frac{2\pi
i}{N}\alpha_{y}\sum_{\hat{y}=1}^{L_{y}}\left(\tilde{a}_{y}^{({1})}+\sum_{I}\hat{x}_{I}\tilde{A}_{y}^{I({1})}\right)\right],$
$\displaystyle W_{a:z}(\hat{x},\hat{y})=\exp\left[\frac{2\pi
i}{N}\alpha_{z}\sum_{\hat{z}=1}^{L_{z}}\left(\tilde{a}_{z}^{({1})}+\sum_{I}\hat{x}_{I}\tilde{A}_{z}^{I({1})}\right)\right],$
(25) $\displaystyle W^{I}_{A:x}(\hat{y},\hat{z})=\exp\left[\frac{2\pi
i}{N}\sum_{\hat{x}=1}^{L_{x}}\tilde{A}^{I({1})}_{x}\right],\quad
W^{I}_{A:y}=\exp\left[\frac{2\pi
i}{N}\sum_{\hat{y}=1}^{L_{y}}\tilde{A}^{I({1})}_{y}\right],\quad
W^{I}_{A:z}=\exp\left[\frac{2\pi
i}{N}\sum_{\hat{z}=1}^{L_{z}}\tilde{A}^{I({1})}_{z}\right].$ (26)

Note that the loops $W_{a:x}(\hat{y},\hat{z})$ and
$W_{A:x}^{I}(\hat{y},\hat{z})$ have already appeared in the previous
discussion on the braiding statistics (21). To properly count the number of
distinct noncontractible loops of the gauge fields, we need to check several
constraints imposed on the loops. Indeed, from (27), it follows that the loop
$W_{a:x}(\hat{y},\hat{z})$ is deformable in the $y$\- and $z$-directions,
indicating that the loop does not depend on the coordinate
$(\hat{y},\hat{z})$. By the same token, one can show that the other loops in
(25) and (26) do not depend on the coordinates. This stems from the fact that

$\displaystyle\Delta_{i}\left(\tilde{a}_{j}^{({1})}+\sum_{I}\hat{x}_{I}A^{I({1})}_{j}\right)-\Delta_{j}\left(\tilde{a}_{i}^{({1})}+\sum_{I}\hat{x}_{I}A^{I({1})}_{i}\right)=0\quad(i,j=x,y,z),$
(27)

where $\hat{x}_{I}$ represents the $I$-th spatial coordinate of the lattice.
This relation is verified from the equations of motions for
$\tilde{b}^{({2})}$ and $\tilde{c}^{I({2})}$ in the BF theory (15):

$\displaystyle\Delta_{i}\tilde{a}^{({1})}_{j}-\Delta_{j}\tilde{a}^{({1})}_{i}-\tilde{A}^{j(1)}_{i}+\tilde{A}^{i(1)}_{j}=0,\quad\Delta_{i}\tilde{A}^{I({1})}_{j}-\Delta_{j}\tilde{A}^{I({1})}_{i}=0.$
(28)

Indeed, we rewrite the left hand side of (27) as

$\displaystyle\left(\Delta_{i}\tilde{a}_{j}^{({1})}+\sum_{I}\Delta_{i}(\hat{x}_{I})A^{I({1})}_{j}\right)+\sum_{I}\hat{x}_{I}\Delta_{i}\tilde{A}^{I({1})}_{j}-\left(\Delta_{j}\tilde{a}_{i}^{({1})}+\sum_{I}\Delta_{j}(\hat{x}_{I})A^{I({1})}_{i}\right)-\sum_{I}\hat{x}_{I}\Delta_{j}\tilde{A}^{I({1})}_{i}$
(29) $\displaystyle=$
$\displaystyle\Delta_{i}\tilde{a}^{({1})}_{j}-\Delta_{j}\tilde{a}^{({1})}_{i}-\tilde{A}^{j(1)}_{i}+\tilde{A}^{i(1)}_{j}+\sum_{I}\hat{x}_{I}(\Delta_{i}\tilde{A}^{I({1})}_{j}-\Delta_{j}\tilde{A}^{I({1})}_{i})=0.$

Here we have used the fact that $\Delta_{i}(\hat{x}_{I})=\delta^{i}_{I}$.

Also, due to the periodic boundary condition, we have

$\displaystyle W_{a:x}(\hat{y}+L_{y},\hat{z})=W_{a:x}(\hat{y},\hat{z}),\quad
W_{a:y}(\hat{x}+L_{x},\hat{z})=W_{a:y}(\hat{x},\hat{z}),$ (30)

which is rewritten as

$\displaystyle(W^{y}_{A:x})^{\alpha_{x}L_{y}}=(W^{x}_{A:y})^{\alpha_{y}L_{x}}=1.$
(31)

Furthermore, setting $i=x$ and $j=y$, and summing over $\hat{x}$ and $\hat{y}$
in (27) yields

$\displaystyle
L_{x}\sum_{\hat{y}=1}^{L_{y}}\tilde{A}_{y}^{x}=L_{y}\sum_{\hat{x}=1}^{L_{x}}\tilde{A}_{x}^{y}$

from which we have

$\displaystyle(W^{y}_{A:x})^{L_{x}}=(W^{x}_{A:y})^{L_{y}}.$ (32)

From the two conditions (31) and (32), it follows that there are
$N\times\gcd(N,L_{x},L_{y})$ distinct loops of $W^{y}_{A:x}$ and
$W^{x}_{A:y}$. Analogous line of thought shows that there are
$N\times\gcd(N,L_{y},L_{z})$ [resp. $N\times\gcd(N,L_{x},L_{z})$] distinct
loops of $W^{y}_{A:z}$ and $W^{z}_{A:y}$ [resp. $W^{x}_{A:z}$ and
$W^{x}_{A:z}$]. There is no constraint imposed on the three loops,
$W^{x}_{A:x}$, $W^{y}_{A:y}$, and $W^{z}_{A:z}$.

In summary, there are $\prod_{i=x,y,z}\gcd(N,L_{i})$ distinct noncontractible
loops in (25) whereas there are

$\displaystyle\left(N\times\gcd(N,L_{x},L_{y})\right)\times\left(N\times\gcd(N,L_{y},L_{z})\right)\times\left(N\times\gcd(N,L_{x},L_{z})\right)\times
N^{3}$

distinct loops in (26). Therefore, the GSD, which amounts to the total number
of distinct configurations of the noncontractible loops of the gauge fields
$\tilde{a}^{({1})}_{i}$ and $\tilde{A}^{I({1})}_{i}$, is given by

$GSD=N^{6}\times\gcd(N,L_{x})\times\gcd(N,L_{y})\times\gcd(N,L_{z})\times\gcd(N,L_{x},L_{y})\times\gcd(N,L_{y},L_{z})\times\gcd(N,L_{z},L_{x}).$
(33)

The GSD dependence on the greatest common divisor between charge $N$ and
system size is the manifestation of the UV/IR mixing. Such UV/IR mixing is
distinct from the one in previous theories with subsystem symmetry where the
GSD becomes subextensive.

By the analogous line of thoughts, one can evaluate the GSD for general $p$
and $d$ on torus geometry. Relegating the details to Appendix A, the GSD for
the system size $L_{1}\times\cdots\times L_{d}$ is found to be

$\boxed{GSD=N^{K(d,p)}\times\prod_{1\leq i_{1}<i_{2}\cdots<i_{p}\leq
d}\gcd(N,L_{i_{1}},L_{i_{2}},\cdots,L_{i_{p}})\times\prod_{1\leq
i_{1}<i_{2}\cdots<i_{p+1}\leq
d}\gcd(N,L_{i_{1}},L_{i_{2}},\cdots,L_{i_{p+1}})}$ (34)

with

$K(d,p)\vcentcolon=p\times\binom{d+1}{p+1}.$ (35)

In Appendix B, we demonstrate concrete UV spin models corresponding to the BF
theory with dipole symmetry (15) which exhibit the unusual behavior of the GSD
(34).

##  4 Anomaly inflow

In this section, we argue that the dipole BF theory (15) has a mixed ’t Hooft
anomaly between the higher form dipole symmetries. We also find a bulk theory
in one higher dimensions which cancels the ’t Hooft anomaly by anomaly inflow
mechanism [30].

###  4.1 Review of $1$-form gauging in $(2+1)$d BF theory

Before discussing anomaly inflow in the foliated BF theories (15), we
succinctly review how anomaly inflow mechanism works in the case of
conventional $(2+1)$d BF theory [3]. Readers who are already familiar with
this argument may skip this subsection.

We start with the following BF theory

$\mathcal{L}=\frac{N}{2\pi}a^{(1)}\wedge db^{(1)}.$ (36)

To gauge $1$-form global symmetries, we introduce a background $2$-form field
$\beta^{({2})}$ to gauge one of the $1$-form symmetries. Then we rewrite the
theory as

$\mathcal{L}=\frac{N}{2\pi}a^{(1)}\wedge(db^{(1)}-\beta^{({2})}),$ (37)

which respects the $1$-form background gauge symmetry:

$b^{({1})}\to
b^{({1})}+\lambda^{({1})},\quad\beta^{({2})}\to\beta^{({2})}+d\lambda^{({1})}.$
(38)

Likewise, one could gauge other $1$-form symmetry by introducing a background
$2$-form gauge field $\alpha^{({2})}$ via

$\mathcal{L}=\frac{N}{2\pi}b^{(1)}\wedge(da^{(1)}-\alpha^{({2})})$ (39)

which has the following background gauge symmetry

$a^{({1})}\to
a^{({1})}+\sigma^{({1})},\quad\alpha^{({2})}\to\alpha^{({2})}+d\sigma^{({1})}.$
(40)

Now we gauge both $1$-form symmetries simultaneously, namely

$\mathcal{L}=\frac{N}{2\pi}a^{(1)}\wedge(db^{(1)}-\beta^{({2})})-\frac{N}{2\pi}b^{({1})}\wedge\alpha^{({2})}.$
(41)

However, the theory (41) is not invariant under the gauge transformations (38)
and (40), signaling a ’t Hooft anomaly. Indeed, under the gauge
transformations, we find

$\delta\mathcal{L}=-\frac{N}{2\pi}\left[\sigma^{({1})}\wedge\beta^{({2})}+\lambda^{({1})}\wedge\alpha^{({2})}+\lambda^{({1})}\wedge
d\sigma^{({1})}\right].$ (42)

This cannot be removed by adding counter terms consisting of the background
gauge fields $\alpha^{({2})}$ and $\beta^{({2})}$ to $\mathcal{L}$. Therefore
(42) implies that the theory has the mixed ’t Hooft anomaly between the two
$1$-form symmetries, which is an obstruction to gauge the both symmetries
simultaneously.

It is known that this anomaly can be canceled by combining the theory with an
appropriate bulk theory in $(3+1)$ dimensions:

$S_{\rm inflow}=\frac{N}{2\pi}\int_{M_{3+1}}\Bigl{[}G^{({1})}\wedge
d\alpha^{({2})}+K^{({1})}\wedge
d\beta^{({2})}-\alpha^{({2})}\wedge\beta^{({2})}\Bigr{]}.$ (43)

Here, we have introduced the $1$-form fields $G^{({1})}$ and $K^{({1})}$ to
ensure the flatness condition of the background gauge fields $\alpha^{({2})}$
and $\beta^{({2})}$. To see this inflow term actually cancels the anomaly
(42), we first note that the theory respects the following background gauge
symmetry

$\displaystyle\alpha^{({2})}\to\alpha^{({2})}+d\sigma^{({1})},\quad
K^{({1})}\to K^{({1})}-\sigma^{({1})},$
$\displaystyle\beta^{({2})}\to\beta^{({2})}+d\lambda^{({1})},\quad
G^{({1})}\to G^{({1})}-\lambda^{({1})},$ (44)

up to total derivative. Then, under this transformation, variant of the theory
(43) is given by

$\delta S_{\rm
inflow}=-\int_{M_{3+1}}d\left[\sigma^{({1})}\wedge\beta^{({2})}+\lambda^{({1})}\wedge\alpha^{({2})}+\lambda^{({1})}\wedge
d\sigma^{({1})}\right].$ (45)

Hence, if we have the theory (43) with the boundary in the $z$-direction, say
at $z=0$, (i.e., the theory is defined in $z\geq 0$), the mixed ’t Hooft
anomaly (42) is canceled, which is the well-known anomaly inflow of $1$-form
symmetries. Below, we are going to apply the similar logic to our integer
dipole BF theories (15) to discuss the anomaly inflow when gauging the dipole
symmetries.

###  4.2 Anomaly inflow for dipole symmetries

Now we turn to anomaly inflow for the dipole BF theory (15). Introducing
integer valued background gauge fields $\tilde{\alpha}^{({p+1})}$ and
$\tilde{\Gamma}^{I({p+1})}$, which reside on $(p+1)$-cell, we first gauge the
$p$-form symmetries via

$\displaystyle\mathcal{L}=\frac{2\pi}{N}\sum_{(p+1)-{\rm
cell}}\left[\tilde{b}^{({d-p})}\left(\Delta\tilde{a}^{({p})}+(-1)^{p}\sum_{I}\tilde{A}^{I({p})}e^{I}-\tilde{\alpha}^{({p+1})}\right)+\sum_{I}\tilde{c}^{({d-p})}\left(\Delta\tilde{A}^{I({p})}-\tilde{\Gamma}^{I({p+1})}\right)\right],$
(46)

The theory admits the following background gauge symmetry121212Throughout
subsection, $\tilde{\lambda}_{*}^{*}$ denotes a gauge parameter.:

$\displaystyle\tilde{a}^{({p})}\to\tilde{a}^{({p})}+\tilde{\lambda}_{a}^{({p})},\quad\tilde{A}^{I({p})}\to\tilde{A}^{I({p})}+\tilde{\lambda}_{A}^{I({p})},$
$\displaystyle\tilde{\alpha}^{({p+1})}\to\tilde{\alpha}^{({p+1})}+\Delta\tilde{\lambda}_{a}^{({p})}+(-1)^{p}\sum_{I}\tilde{\lambda}_{A}^{I({p})}e^{I},\quad\tilde{\Gamma}^{I({p+1})}\to\tilde{\Gamma}^{I({p+1})}+\Delta\tilde{\lambda}_{A}^{I({p})}.$
(47)

One can also gauge the $(d-p)$-form symmetries by introducing the integer
valued background gauge fields $\tilde{\beta}^{({d-p+1})}$ and
$\tilde{\Xi}^{I({d-p+1})}$ on dual $(d-p+1)$-cell. It can be accomplished by

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{N}\biggl{[}\sum_{(p+1)-{\rm
cell}}\biggl{\\{}\tilde{b}^{({d-p})}\left(\Delta\tilde{a}^{({p})}+(-1)^{p}\sum_{I}\tilde{A}^{I({p})}e^{I}\right)+\sum_{I}\tilde{c}^{({d-p})}\Delta\tilde{A}^{I({p})}\biggr{\\}}$
(48) $\displaystyle~{}~{}~{}~{}~{}~{}+\sum_{p-{\rm
cell}}\biggl{\\{}-(-1)^{(p+1)(d+1)}\tilde{a}^{({p})}\tilde{\beta}^{({d-p+1})}-(-1)^{(p+1)(d+1)}\sum_{I}\tilde{A}^{I({p})}\tilde{\Xi}^{I({d-p+1})}\biggr{\\}}\biggr{]}.$

The theory (48) respects the following gauge symmetry

$\displaystyle\tilde{b}^{({d-p})}\to\tilde{b}^{({d-p})}+\tilde{\lambda}_{b}^{({d-p})},\quad\tilde{c}^{I({d-p})}\to\tilde{c}^{I({d-p})}+\tilde{\lambda}_{c}^{I({d-p})},$
$\displaystyle\tilde{\Xi}^{I({d-p+1})}\to\tilde{\Xi}^{I({d-p+1})}+\Delta\tilde{\lambda}_{c}^{I({d-p})}-(-1)^{d-p}\tilde{\lambda}_{b}^{({d-p})}e^{I},\quad\tilde{\beta}^{({d-p+1})}\to\tilde{\beta}^{({d-p+1})}+\Delta\tilde{\lambda}_{b}^{({d-p+1})}.$
(49)

We gauge the both $p$-form and $(d-p)$-form symmetries by considering

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{2\pi}{N}\biggl{[}\sum_{(p+1)-{\rm
cell}}\biggl{\\{}\tilde{b}^{({d-p})}\left(\Delta\tilde{a}^{({p})}+(-1)^{p}\sum_{I}\tilde{A}^{I({p})}e^{I}-\tilde{\alpha}^{({p+1})}\right)+\sum_{I}\tilde{c}^{({d-p})}\left(\Delta\tilde{A}^{I({p})}-\tilde{\Gamma}^{I({p+1})}\right)\biggr{\\}}$
(50) $\displaystyle~{}~{}~{}~{}~{}~{}+\sum_{p-{\rm
cell}}\biggl{\\{}-(-1)^{(p+1)(d+1)}\tilde{a}^{({p})}\tilde{\beta}^{({d-p+1})}-(-1)^{(p+1)(d+1)}\sum_{I}\tilde{A}^{I({p})}\tilde{\Xi}^{I({d-p+1})}\biggr{\\}}\biggr{]}.$

However, the theory (50) is not invariant under the background gauge
transformations (47) and (49), signaling the mixed ’t Hooft anomaly. Indeed,
the variant of the action under the transformations reads

$\displaystyle\delta\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{N}\biggl{[}\sum_{(p+1)-{\rm
cell}}\biggl{(}\tilde{\lambda}_{b}^{({d-p})}\tilde{\alpha}^{({p+1})}+\sum_{I}\tilde{\lambda}_{c}^{I({d-p})}\tilde{\Gamma}^{I({p+1})}\biggl{)}$
(51) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{p-{\rm
cell}}\biggl{\\{}(-1)^{(d+1)(p+1)}\tilde{\lambda}_{a}^{({p})}\left(\tilde{\beta}^{({d-p+1})}+\Delta\tilde{\lambda}_{b}^{({d-p})}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+(-1)^{(d+1)(p+1)}\sum_{I}\tilde{\lambda}_{A}^{I({p})}\left(\tilde{\Xi}^{I({d-p+1})}+\Delta\tilde{\lambda}_{c}^{I({d-p})}-(-1)^{d-p}\tilde{\lambda}_{b}^{({d-p})}e^{I}\right)\biggr{\\}}\biggr{]}.$

This variation cannot vanish even if we add any counter term consisting of the
background gauge fields $\tilde{\alpha}^{({p+1})}$,
$\tilde{\Gamma}^{I({p+1})}$, $\tilde{\beta}^{({d-p+1})}$ and
$\tilde{\Xi}^{I({d-p+1})}$.131313Note that cancelling $\delta\mathcal{L}$
needs $(d+1)$-form terms, involving a product of the $(p+1)$-form and
$(d-p+1)$-form background gauge fields. However, we cannot construct such
terms since they must be $(d+2)$-form or higher.  Therefore the dipole BF
theory has the mixed ’t Hooft anomaly between the $p$-form and $(d-p)$-form
symmetries and we cannot gauge them simultaneously.

As in the standard BF theory, this anomaly can be canceled by attaching the
dipole BF theory to a bulk theory in one higher dimensions via an extension of
anomaly inflow as follows. Let us consider the following $(d+2)$-dimensional
theory

$\displaystyle\mathcal{L}_{\rm inflow}=\mathcal{L}_{0}+\mathcal{L}_{\rm spt},$
(52)

where

$\displaystyle\mathcal{L}_{0}$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{N}\biggl{[}\sum_{(p+2)-{\rm
cell}}\biggl{\\{}\tilde{M}^{({d-p})}\left(\Delta\tilde{\alpha}^{({p+1})}+(-1)^{p+1}\tilde{\Gamma}^{I({p+1})}e^{I}\right)+\sum_{I}\tilde{N}^{I({d-p})}\Delta\tilde{\Gamma}^{I({p+1})}\biggr{\\}}$
(53) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{p-\rm
cell}\biggl{\\{}\sum_{I}\tilde{O}^{I({p})}\left(\Delta\tilde{\Xi}^{I({d-p+1})}-(-1)^{d-p+1}\tilde{\beta}^{I({d-p+1})}e^{I}\right)+\tilde{P}^{({p})}\Delta\tilde{\beta}^{({d-p+1})}\biggr{\\}}\biggl{]},$

and

$\boxed{\mathcal{L}_{\rm spt}=-(-1)^{(p+1)(d+1)}\frac{2\pi}{N}\sum_{(p+1)-{\rm
cell}}\biggl{[}\tilde{\alpha}^{({p+1})}\tilde{\beta}^{({d-p+1})}+\sum_{I}\tilde{\Gamma}^{I({p+1})}\tilde{\Xi}^{I({d-p+1})}\biggl{]}}.$
(54)

An intuition behind the bulk term (52) is that the term $\mathcal{L}_{0}$ is
what corresponds to the first two terms in (43) to ensure the flatness
condition of the gauge fields, $\tilde{\alpha}^{({p+1})}$,
$\tilde{\Gamma}^{I({p+1})}$, $\tilde{\beta}^{({d-p+1})}$ and
$\tilde{\Xi}^{I({d-p+1})}$, introducing the four auxiliary fields,
$\tilde{M}^{({d-p})}$, $\tilde{N}^{I({d-p})}$, $\tilde{O}^{I({p})}$ and
$\tilde{P}^{I({p})}$. Note that the flatness condition has the different form
compared with usual cases, as we deal with the dipole gauge fields. Also,
analogous to the last term in (43), $\mathcal{L}_{\rm spt}$ describes an
invertible phase comprised of the background gauge fields. To see how the bulk
term (52) works for cancelling the anomaly, we first note that up to the total
derivative, the theory (52) respects the background gauge symmetries in (47)
and (49), jointly with

$\displaystyle\tilde{M}^{({d-p})}\to\tilde{M}^{({d-p})}+(-1)^{d-p+1}\tilde{\lambda}_{b}^{({d-p})},\quad\tilde{N}^{I({d-p})}\to\tilde{N}^{I({d-p})}+(-1)^{d-p+1}\tilde{\lambda}_{c}^{I({d-p})}$
$\displaystyle\tilde{O}^{I({p})}\to\tilde{O}^{I({p})}+(-1)^{d(p+1)}\tilde{\lambda}_{A}^{I({p})},\quad\tilde{P}^{({p})}\to\tilde{P}^{({p})}+(-1)^{d(p+1)}\tilde{\lambda}_{a}^{({p})}.$
(55)

Indeed, under these transformations, one has

$\displaystyle\delta\mathcal{L}_{\rm inflow}$ $\displaystyle=$
$\displaystyle-\frac{2\pi}{N}\Delta\biggl{[}\sum_{(p+1)-{\rm
cell}}\biggl{(}\tilde{\lambda}_{b}^{({d-p})}\tilde{\alpha}^{({p+1})}+\sum_{I}\tilde{\lambda}_{c}^{I({d-p})}\tilde{\Gamma}^{I({p+1})}\biggl{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\sum_{p-{\rm
cell}}\biggl{\\{}(-1)^{(d+1)(p+1)}\tilde{\lambda}_{a}^{({p})}\left(\tilde{\beta}^{({d-p+1})}+\Delta\tilde{\lambda}_{b}^{({d-p})}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(-1)^{(d+1)(p+1)}\sum_{I}\tilde{\lambda}_{A}^{I({p})}\left(\tilde{\Xi}^{I({d-p+1})}+\Delta\tilde{\lambda}_{c}^{I({d-p})}-(-1)^{d-p}\tilde{\lambda}_{b}^{({d-p})}e^{I}\right)\biggr{\\}}\biggr{]}.$

The terms inside the large bracket are identical to (51).

##  5 Conclusion

Construction of unified theoretical scheme which incorporates various kinds of
symmetries is one of the active area of researches for recent years. In this
work, we focus on one of the symmetries that has recently emerged in the
context of fracton topological phases, multipole symmetries, and incorporate
such symmetries into other types of symmetries, higher form symmetries. We
introduce foliated BF theories consisting of layers of higher form BF theories
and couplings between the layers. The model admits dipoles of spatially
extended excitations, leading to unusual behavior; the position dependent
braiding statistics between the loops of the higher form gauge fields, and
system size dependence of the GSD. We also discuss the ’t Hooft anomaly of the
foliated higher form BF theory. Incapability of gauging higher form dipole
symmetries signals the mixed ’t Hooft anomaly, which is canceled by invertible
phases with appropriated terms that ensure the flatness condition of the
dipole gauge fields. Exploration of the higher form dipole BF theories and
their anomaly inflow are central findings in this work, which we believe
contribute to not only better understanding of fracton physics, but also to
achieving the ultimate goal to construct a theoretical framework that unifies
various types of symmetries. In addition, one can also extend our analysis to
the case with the subsystem symmetries, the details of which are given in
Appendix. C.

Before closing this section, let us make a few comments on future perspectives
regarding this work. In this paper, we have seen the dipole and dual dipole
algebra given in (2), (7) and (9). Studying the Maxwell theory with such an
algebra and especially its dualities would be an interesting direction.
Relatedly, it could be intriguing to see if we have duality defect lines or
even more exotic noninvertible duality defects. Investigating whether there is
a $\theta$-term in the Maxwell theory could be an another direction to pursue.
Also, it would be interesting to address whether one can deal with other types
of symmetries in the format that we introduced in this paper. One candidate
would be higher group symmetries, such as $2$-group [41]. When studying
theories with dipole symmetries, we sometimes introduce tensor gauge theories,
gauge theories with higher order spatial derivatives [12]. It could be
interesting to see whether our foliated BF theories (15) are described by such
tensor gauge theories, especially higher form analog of the tensor gauge
theories [42, 43].

Investigating whether one can establish a lattice model of the invertible
field theory that appears in the discussion on the anomaly inflow (54) is an
another interesting direction. In the case of the standard BF theory in
$(2+1)$d (toric code), the mixed anomaly when gauging $1$-form symmetries is
canceled by the invertible theory (43), which has UV lattice counterpart
described by the cluster state (a.k.a. Raussendorf-Bravyi-Harrington cluster
state [44]). Indeed, one can show that on the boundary of such a state yields
the toric code. One naively wonders a similar cluster state can be constructed
which give rise to the topological model corresponding to the dipole BF theory
(15) on the boundary.

In this work, we have seen that the ’t Hooft anomaly is canceled by the
invertible phases consisting of the dipole gauge fields. Investigating whether
such invertible phases are of usefulness in classification of symmetry
protected topological phases with global dipole symmetries would be an
interesting and important direction (see e.g. [45] for relevant discussion in
the case of $(1+1)$d). Since our invertible theories are defined in any
dimension, our consideration may contribute to classifying such phases. We
hopefully leave this issue for future investigations.

Acknowledgment– We thank N. Atlam, B. Han, J.H. Han, Q. Jiang, A. Nayak, Y.
Oh, S. Shimamori, K. Shiozaki,Y. You, for helpful discussion. We also thank
J.H. Han for sharing draft of his work [21] with us. H. E. is supported by
KAKENHI-PROJECT-23H01097. M. H. is supported by MEXT Q-LEAP, JSPS Grant-in-Aid
for Transformative Research Areas (A) “Extreme Universe" JP21H05190 [D01],
JSPS KAKENHI Grant Number 22H01222, JST PRESTO Grant Number JPMJPR2117. T. N.
is supported by JST SPRING, Grant Number JPMJSP2110, and RIKEN Junior Research
Associate Program.

##  Appendix A Ground state degeneracy for general $p$ and $d$

In this appendix, we derive the GSD of the BF theory with the dipole symmetry
in the generic case of $p$ and dimension $d$ given by (34). Since discussion
presented in this section closely parallels the one in the main text (Sec.
3.2), we give the derivation succinctly. To start, we impose the periodic
boundary condition on the lattice and set the system size as
$L_{1}\times\cdots\times L_{d}$. The GSD amounts the number of distinct
configurations of noncontractible $p$-form loops of the gauge fields
$\tilde{a}^{({p})}$ and $\tilde{A}^{I({p})}$ that wind in the spatial
direction of the torus. Defining the indices

$\displaystyle\\{i_{k}~{}(1\leq k\leq p)|1\leq i_{1}<i_{2}<\cdots<i_{p}\leq
d\\},\quad\\{j_{l}~{}(1\leq l\leq d-p)|1\leq j_{1}<j_{2}<\cdots<j_{d-p}\leq
d\\},$

where $\\{i_{k}\\}$ and $\\{j_{l}\\}$ are complement with each other, i.e.,
$\\{i_{k}\\}\cup\\{j_{l}\\}=\\{1,2,\cdots,d\\}$, we find that the
noncontractible $p$-form loops which wind around the torus in the
$i_{1},\cdots,i_{p}$-th spatial directions, have the following forms:

$\displaystyle
W_{a;(i_{1},\cdots,i_{p})}(\hat{x}_{j_{1}},\cdots,\hat{x}_{j_{d-p}})$
$\displaystyle=$ $\displaystyle\exp\left[\frac{2\pi
i}{N}\alpha_{(i_{1},\cdots,i_{p})}\sum_{x_{i_{1}}=1}^{L_{i_{1}}}\cdots\sum_{x_{i_{p}}=1}^{L_{i_{p}}}\left(\tilde{a}_{[i_{1}i_{2}\cdots
i_{p}]}^{({p})}-(-1)^{p}\sum_{I}\hat{x}_{I}\tilde{A}_{[i_{1}i_{2}\cdots
i_{p}]}^{I({p})}\right)\right],$ $\displaystyle
W_{A^{I};(i_{1},\cdots,i_{p})}(\hat{x}_{j_{1}},\cdots,\hat{x}_{j_{d-p}})$
$\displaystyle=$ $\displaystyle\exp\left[\frac{2\pi
i}{N}\sum_{x_{i_{1}}=1}^{L_{i_{1}}}\cdots\sum_{x_{i_{p}}=1}^{L_{i_{p}}}\tilde{A}_{[i_{1}i_{2}\cdots
i_{p}]}^{I({p})}\right].$ (56)

Here, the integer $\alpha_{(i_{1},\cdots,i_{p})}$ is defined by
$\alpha_{(i_{1},\cdots,i_{p})}\vcentcolon=\frac{N}{\gcd(N,L_{i_{1}},\cdots,L_{i_{p}})}$.
Such an integer is multiplied with the argument in the exponent in the first
term of (56) to ensure that the loop, which contains the linear term of the
spatial coordinate, is consistent with the periodic boundary condition [19,
46].

There are several constrains on the loops (56). From equations of motion for
$\tilde{c}^{I({d-p})}$ in (15), one finds that the loop
$W_{A^{I};(i_{1},\cdots,i_{p})}(\hat{x}_{j_{1}},\cdots,\hat{x}_{j_{d-p}})$ is
deformable so that is does not depend on the spatial coordinate
$\hat{x}_{j_{l}}$. Due to the periodic boundary conditions on the loop
$W_{a;(i_{1},\cdots,i_{p})}(\hat{x}_{j_{1}},\cdots,\hat{x}_{j_{d-p}})$, such
as
$W_{a;(i_{1},\cdots,i_{p})}(\hat{x}_{j_{1}}+L_{i_{1}},\cdots,\hat{x}_{j_{d-p}})=W_{a;(i_{1},\cdots,i_{p})}(\hat{x}_{j_{1}},\cdots,\hat{x}_{j_{d-p}})$,
we have

$\displaystyle\left(W_{A^{I};(i_{1},\cdots,i_{p})}\right)^{L_{I}\alpha_{(i_{1},\cdots,i_{p})}}=1\quad(I\neq{i_{1}},\cdots,i_{p}).$
(57)

From the equation of motion for $\tilde{b}^{({d-p})}$ in (15), we obtain

$\left(W_{A^{I};(i_{1},\cdots,i_{p})}\right)^{L_{I}}\times\left(W_{A^{i_{1}};(i_{2},\cdots,i_{p},I)}\right)^{-L_{i_{1}}}\times\left(W_{A^{i_{2}};(i_{2},i_{3},\cdots,i_{p},I,i_{2})}\right)^{L_{i_{2}}}\times\cdots\times\left(W_{A^{i_{p}};(I,i_{1},\cdots,i_{p-1})}\right)^{(-1)^{p}L_{i_{p}}}=1,$
(58)

where the indexes $I,(i_{1},\cdots,i_{p})$ and $[I\neq i_{1},\cdots,i_{p}]$
run cyclically. Further, the equations of motions of (15) ensure that the loop
$W_{a;(i_{1},\cdots,i_{p})}(\hat{x}_{j_{1}},\cdots,\hat{x}_{j_{d-p}})$ is
deformable so that it does not depend on the spatial coordinate
$\hat{x}_{j_{l}}$.

With the constraints (57) and (58), we count the number of distinct
noncontractible loops (56). There are $\binom{d}{p}$ loops which have the form
of $W_{a;(i_{1},\cdots,i_{p})}$, labeled by
$\mathbb{Z}_{\gcd(N,L_{i_{1}},\cdots,L_{i_{p}})}$. Regarding the loops,
$W_{A^{I};(i_{1},\cdots,i_{p})}$, after some algebra, jointly with the
constraints (57) and (58), one finds that there are

$\displaystyle N^{p\binom{d}{p}}\times\prod_{1\leq
i_{1}<i_{2}<\cdots<i_{p+1}\leq
d}\left[N^{p}\gcd(N,L_{i_{1}},L_{i_{2}},\cdots,L_{i_{p+1}})\right]$

distinct noncontractible loops. Therefore, the GSD, which amounts the total
number of distinct configurations of loops (56) is given by

$\displaystyle{\rm GSD}$ $\displaystyle=$ $\displaystyle\prod_{1\leq
i_{1}<i_{2}\cdots<i_{p}\leq
d}\gcd(N,L_{i_{1}},L_{i_{2}},\cdots,L_{i_{p}})\times
N^{p\binom{d}{p}}\times\prod_{1\leq i_{1}<i_{2}<\cdots<i_{p+1}\leq
d}\left[N^{p}\gcd(N,L_{i_{1}},L_{i_{2}},\cdots,L_{i_{p+1}})\right]$ (59)
$\displaystyle=$ $\displaystyle N^{K(d,p)}\times\prod_{1\leq
i_{1}<i_{2}\cdots<i_{p}\leq
d}\gcd(N,L_{i_{1}},L_{i_{2}},\cdots,L_{i_{p}})\times\prod_{1\leq
i_{1}<i_{2}\cdots<i_{p+1}\leq
d}\gcd(N,L_{i_{1}},L_{i_{2}},\cdots,L_{i_{p+1}}),$

which leads to (34) with $K(d,p)$ described by (35).

##  Appendix B Construction of lattice spin models with dipole symmetry

In this appendix, we demonstrate several concrete UV spin models that
correspond to the dipole BF theory (15). We focus on the case of $d=3$ and
$p=1,2$. Generalization to other cases of $d$ and $p$ should be
straightforward. To this end, we follow an approach proposed in [23]. A key
insight to construct the spin models is to recognize the first and third terms
in the BF theory (15) as the $p$-form BF theories whereas the second term as
as the gauged “foliated symmetry protected topological (SPT) phases”, i.e.,
gauged SPT phases [47] stacked in layers. Based on this observation, one can
introduce a spin model in a paramagnet phase and implement control phase gate
operations on the spins, which corresponds to accommodating the SPT phases.
Gauging a global symmetry of the model results in the desired topological spin
model associated with the dipole BF theory (15). The crucial point is to
implement the controlled phase gate operation. Without it, we would end up
with the decoupled layers of the spin model corresponding to the $p$-form BF
theories (i.e., decoupled layers of the toric codes [38]).

Recalling the fact that the foliation field $e^{I}$, which is a one form field
along which theories defined on submanifold are stacked, as well as that the
form $b^{({d-p})}\wedge A^{I({p})}$ is associated with the topological field
theoretical description of the SPT phase [48] with $(d-p)$\- and $p$-form
global symmetries, the coupling term in the BF theory, $b^{({d-p})}\wedge
A^{I({p})}\wedge e^{I}$ can be interpreted as the stack of gauged
$\mathbb{Z}_{N}^{(d-p)}\times\mathbb{Z}_{N}^{(p)}$ SPT phases141414The index
$\mathbb{Z}_{N}^{(q)}$ denotes $q$-form $\mathbb{Z}_{N}$ symmetry. along the
$I$-th direction.

###  B.1 $p=1$ and $d=3$

To see the construction more explicitly, we envisage a cubic lattice to
realize the spin model, which corresponds to $d=3$ and $p=1$. Further, for
simplicity, we focus on the case with $N=2$. The generalization to the case of
$N>2$ is straightforward by thinking of orientations of the lattice
appropriately. On the cubic lattice, we think of qubits each of which resides
on links. Also, we introduce three types of qubits on each vertex of the dual
cubic lattice (i.e., three qubits at the center of the cube of the original
lattice). Denoting the coordinate of the vertex of the cubic [dual cubic]
lattice as $\mathbf{r}\vcentcolon=(\hat{x},\hat{y},\hat{z})$
$\left[\mathbf{r}^{*}\vcentcolon=(\hat{x},\hat{y},\hat{z})+(\frac{1}{2},\frac{1}{2},\frac{1}{2})\right]$,
and vectors to label the links of the lattice as
$\mathbf{l}_{x}\vcentcolon=(\frac{1}{2},0,0)$,
$\mathbf{l}_{y}\vcentcolon=(0,\frac{1}{2},0)$,
$\mathbf{l}_{z}\vcentcolon=(0,0,\frac{1}{2})$, we write the Pauli operators
acting on these qubits as $\tau^{X}_{0,\mathbf{r}+\mathbf{l}_{I}}$,
$\tau^{Z}_{0,\mathbf{r}+\mathbf{l}_{I}}$ ($I=x,y,z$),
$\tau^{X}_{i,\mathbf{r}^{*}}$, $\tau^{Z}_{i,\mathbf{r}^{*}}$ ($i=1,2,3$),
where the first index of the last two operators distinguishes the three types
of the qubits. We dub the qubit (resp: three qubits) defined on links of the
original cubic lattice (resp: vertex of the dual cubic lattice) as qubits with
type $0$ (resp: qubit with type $i=1,2,3$.).

With these preparations, we introduce the following paramagnet Hamiltonian:

$\displaystyle
H=-\sum_{\mathbf{r},\mathbf{r}^{*}}\left[\sum_{I=x,y,z}\tau^{X}_{0,\mathbf{r}+\mathbf{l}_{I}}+\sum_{i=1,2,3}\tau^{X}_{i,\mathbf{r}^{*}}\right].$
(60)

We implement the controlled Z (CZ) gate on the qubits, which amounts to
accommodating a stack of SPT phases [47].

(a)

(b)

Figure 1: (a) Examples of pairs of qubits on which the CZ gates (62) act. (b)
The configuration of the qubits viewing from the $x$ axis. The examples of the
CZ gates corresponding to left, middle and right of (a) are also shown in
upper left, right, and bottom left, respectively.

The CZ gate operation which acts on a pair of qubits reads

$CZ_{a,b}\ket{a}\ket{b}=(-1)^{ab}\ket{a}\ket{b}\quad(a,b\in\mathbb{Z}_{2}).$
(61)

We think of acting the CZ gate on qubits with type $0$ on the link in the $yz$
plane of the original cubic lattice and the ones with type $1$ at the nodes of
the dual cubic lattice which are adjacent to the $yz$-plane. We show an
example of such pairs of qubits in Fig. (1(a)). We introduce the following CZ
gates:

$\displaystyle\prod_{s,t=\pm
1}CZ_{[1,\mathbf{r}^{*}],[0,\mathbf{f}_{yz}+2\mathbf{l}_{x}+s\mathbf{l}_{y}+t\mathbf{l}_{z}]},$
$\displaystyle
CZ_{[0,\mathbf{r}+\mathbf{l}_{y}],[1,\mathbf{r}^{*}-2\mathbf{l}_{x}]}CZ_{[0,\mathbf{r}+\mathbf{l}_{y}],[1,\mathbf{r}^{*}-2\mathbf{l}_{x}-2\mathbf{l}_{z}]},\quad
CZ_{[0,\mathbf{r}+\mathbf{l}_{z}],[1,\mathbf{r}^{*}-2\mathbf{l}_{x}]}CZ_{[0,\mathbf{r}+\mathbf{l}_{z}],[1,\mathbf{r}^{*}-2\mathbf{l}_{x}-2\mathbf{l}_{y}]},$
(62)

where the first, second and third terms in (62) are portrayed in the left,
middle, and right of Fig. (1(a)). Here, $\mathbf{f}_{yz}$ represents the
coordinate of a plaquette on $yz$ plane, viz
$\mathbf{f}_{yz}\vcentcolon=\mathbf{r}+(0,\frac{1}{2},\frac{1}{2})$. Viewing
from the $x$ axis, the CZ gates in (62) remind us of the cluster states
corresponding to the $\mathbb{Z}_{2}^{(1)}\times\mathbb{Z}_{2}^{(0)}$ SPT
phase (see Fig. 1(b)). Likewise, we implement CZ gates on qubits with type
2(resp: 3) on $zx(\text{resp:}~{}xy)$ plane and the ones with type $0$ at the
nodes adjacent to the plane. By acting the CZ gates, the Hamiltonian (60) is
transformed into

$\displaystyle\widehat{H}$ $\displaystyle=$
$\displaystyle-\sum_{\mathbf{r},\mathbf{r}^{*}}\biggl{[}\tau^{X}_{0,\mathbf{r}+\mathbf{l}_{x}}\tau^{Z}_{2,\mathbf{r}^{*}-2\mathbf{l}_{y}}\tau^{Z}_{2,\mathbf{r}^{*}-2\mathbf{l}_{y}+2\mathbf{l}_{z}}\tau^{Z}_{3,\mathbf{r}^{*}}\tau^{Z}_{3,\mathbf{r}^{*}-2\mathbf{l}_{y}}+\tau^{X}_{0,\mathbf{r}+\mathbf{l}_{y}}\tau^{Z}_{1,\mathbf{r}^{*}-2\mathbf{l}_{x}}\tau^{Z}_{1,\mathbf{r}^{*}-2\mathbf{l}_{x}+2\mathbf{l}_{z}}\tau^{Z}_{3,\mathbf{r}^{*}}\tau^{Z}_{3,\mathbf{r}^{*}-2\mathbf{l}_{x}}$
(63)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\tau^{X}_{0,\mathbf{r}+\mathbf{l}_{z}}\tau^{Z}_{1,\mathbf{r}^{*}-2\mathbf{l}_{x}}\tau^{Z}_{1,\mathbf{r}^{*}-2\mathbf{l}_{x}+2\mathbf{l}_{y}}\tau^{Z}_{2,\mathbf{r}^{*}}\tau^{Z}_{2,\mathbf{r}^{*}-2\mathbf{l}_{x}}+\tau^{X}_{1,\mathbf{r}^{*}-2\mathbf{l}_{x}}\prod_{s,t=\pm
1}\tau^{Z}_{0,\mathbf{f}_{yz}+s\mathbf{l}_{y}+t\mathbf{l}_{z}}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\tau^{X}_{1,\mathbf{r}^{*}-2\mathbf{l}_{y}}\prod_{s,t=\pm
1}\tau^{Z}_{0,\mathbf{f}_{zx}+s\mathbf{l}_{z}+t\mathbf{l}_{x}}+\tau^{X}_{1,\mathbf{r}^{*}-2\mathbf{l}_{z}}\prod_{s,t=\pm
1}\tau^{Z}_{0,\mathbf{f}_{xy}+s\mathbf{l}_{x}+t\mathbf{l}_{y}}\biggr{]},$

where $\mathbf{f}_{zx}$ and $\mathbf{f}_{xy}$ represent the coordinate of a
plaquette on $zx$ and $xy$ plane, respectively, namely
$\mathbf{f}_{zx}\vcentcolon=\mathbf{r}+(\frac{1}{2},0,\frac{1}{2})$,
$\mathbf{f}_{xy}\vcentcolon=\mathbf{r}+(\frac{1}{2},\frac{1}{2},0)$. The
Hamiltonian (63) respects $1$-form and $0$-form global symmetries,
$\mathbb{Z}_{2}^{(1)}\times[\mathbb{Z}_{2}^{(0)}]^{3}$, i.e., it is invariant
under the following spin flips:

$\displaystyle\prod_{l\in
S^{*}}\tau^{X}_{0,l},\quad\prod_{i=1,2,3}\prod_{\mathbf{r}}\tau^{X}_{i,\mathbf{r}}.$
(64)

Here, the first term consists of product of $\tau^{X}_{0}$ on links that form
a closed dual surface, which we abbreviate as $\prod_{l\in
S^{*}}$.151515Example of a such term is given by $\prod_{s,t,u=\pm
1}\tau^{X}_{0,\mathbf{r}+s\mathbf{l}_{x}+t\mathbf{l}_{y}+u\mathbf{l}_{z}}$.

Now we are in a good stage to gauge the global symmetries (64). To do so, we
promote the global symmetries to local ones by introducing an extended Hilbert
space and impose the Gauss law [49, 50]. We define states of qubits on each
face of the cubic lattice and the ones on the links of the dual cubic lattice,
whose Pauli operators are denoted by $X_{0,\mathbf{f}_{ab}}$,
$Z_{0,\mathbf{f}_{ab}}$ ($ab=xy,yz,zx$),
$X_{i,\mathbf{r}^{*}+\mathbf{l}_{I}}$, $Z_{i,\mathbf{l}_{I}}$. Such states are
interpreted as the gauge fields. We introduce the following Gauss law
operators:

$\displaystyle
G_{0,\mathbf{r}+\mathbf{l}_{x}}\vcentcolon=\tau^{X}_{0,\mathbf{r}+\mathbf{l}_{I}}\prod_{\mathbf{f}\supset\mathbf{l}_{I}}X_{0,\mathbf{f}},\quad
G_{i,\mathbf{r}^{*}}\vcentcolon=\tau^{X}_{i,\mathbf{r}^{*}}\prod_{\mathbf{l}^{*}\supset\mathbf{r}^{*}}X_{i,\mathbf{l}^{*}}\quad(i=1,2,3).$
(65)

Here, the product $\prod_{\mathbf{f}\supset\mathbf{l}_{I}}$ stands for
multiplication of the Pauli operators on plaquettes connected with a link
$\mathbf{r}+\mathbf{l}_{I}$. Likewise, we mean the product
$\prod_{\mathbf{l}^{*}\supset\mathbf{r}^{*}}$ by multiplication of the Pauli
operators on dual links which are connected with a vertex on the dual lattice,
$\mathbf{r}^{*}$. Using the operators (65), we impose that the physical state
satisfies

$\displaystyle
G_{0,\mathbf{r}+\mathbf{l}_{x}}\ket{\text{phys}}=G_{i,\mathbf{r}^{*}}\ket{\text{phys}}=\ket{\text{phys}}.$
(66)

Further, we minimally couple the quartet and quadratic spin coupling terms in
the Hamiltonian (63) to the gauge fields via

$\displaystyle\prod_{l\subset\mathbf{f}_{ab}}\tau^{Z}_{0,l}\to
Z_{0,\mathbf{f}_{ab}}\prod_{l\subset\mathbf{f}_{ab}}\tau^{Z}_{0,l},\quad\tau^{Z}_{i,\mathbf{r}^{*}}\tau^{Z}_{i,\mathbf{r}^{*}-2\mathbf{l}_{I}}\to\tau^{Z}_{i,\mathbf{r}^{*}}Z_{i\mathbf{r}^{*}-\mathbf{l}_{I}}\tau^{Z}_{i,\mathbf{r}^{*}-2\mathbf{l}_{I}}$
(67)

so that these terms commute with the Gauss law (65). To ensure the fluxless
condition making the gauge theory dynamically trivial, we add the following
term to the Hamiltonian (63):

$\displaystyle-\sum_{c}\;\prod_{\mathbf{f}_{ab}\subset
c}Z_{0,\mathbf{f}_{ab}}-\sum_{i=1,2,3}\sum_{P^{*}}\prod_{l^{*}\subset
P^{*}}Z_{i,l^{*}},$ (68)

where the product in the first (second) term represents multiplication of the
Pauli operators of faces (dual links) that surround a cube (dual plaquette).
By the Gauss law (66), we set
$\tau^{X}_{0,\mathbf{r}}=\prod_{\mathbf{f}\supset\mathbf{l}_{I}}X_{0,\mathbf{f}},\tau^{X}_{i,\mathbf{r}^{*}}=\prod_{\mathbf{l}^{*}\supset\mathbf{r}^{*}}X_{i,\mathbf{l}^{*}}$.
Also, for the sake of making the figure more visually friendly, we rearrange
the lattice grid so that Pauli operators of the gauge fields are located on
links of the newly defined lattice. Overall, we arrive at the following gauged
Hamiltonian:

$\displaystyle\widehat{H}=-\sum_{\mathbf{f}_{ab},\mathbf{r}}\left(P_{0,\mathbf{f}_{xy}}+P_{0,\mathbf{f}_{yz}}+P_{0,\mathbf{f}_{zx}}+V_{0,\mathbf{r}}\right)-\sum_{i=1,2,3}\sum_{\mathbf{f}_{ab},\mathbf{r}}\left(P_{i,\mathbf{f}_{xy}}+P_{i,\mathbf{f}_{yz}}+P_{i,\mathbf{f}_{zx}}+V_{i,\mathbf{r}}\right)$
(69)

with

$\displaystyle
P_{0,\mathbf{f}_{xy}}\vcentcolon=X_{1,\mathbf{f}_{xy}-\mathbf{l}_{x}}X_{2,\mathbf{f}_{xy}-\mathbf{l}_{y}}\prod_{s,t=\pm
1}X_{0,\mathbf{f}_{yz}+s\mathbf{l}_{x}+t\mathbf{l}_{y}},\quad
P_{0,\mathbf{f}_{yz}}\vcentcolon=X_{2,\mathbf{f}_{yz}-\mathbf{l}_{y}}X_{3,\mathbf{f}_{yz}-\mathbf{l}_{z}}\prod_{s,t=\pm
1}X_{0,\mathbf{f}_{yz}+s\mathbf{l}_{y}+t\mathbf{l}_{z}}$ $\displaystyle
P_{0,\mathbf{f}_{zx}}\vcentcolon=X_{3,\mathbf{f}_{zx}-\mathbf{l}_{z}}X_{1,\mathbf{f}_{zx}-\mathbf{l}_{x}}\prod_{s,t=\pm
1}X_{0,\mathbf{f}_{zx}+s\mathbf{l}_{z}+t\mathbf{l}_{x}},\quad
V_{0,\mathbf{r}}\vcentcolon=\prod_{s,t,u=\pm
1}Z_{0,\mathbf{r}+s\mathbf{l}_{x}+t\mathbf{l}_{y}+u\mathbf{l}_{z}}$ (70)

and

$\displaystyle
V_{1,\mathbf{r}}\vcentcolon=Z_{0,\mathbf{r}+\mathbf{l}_{x}}\prod_{s,t,u=\pm
1}X_{1,\mathbf{r}+s\mathbf{l}_{x}+t\mathbf{l}_{y}+u\mathbf{l}_{z}},\quad
V_{2,\mathbf{r}}\vcentcolon=Z_{0,\mathbf{r}+\mathbf{l}_{y}}\prod_{s,t,u=\pm
1}X_{2,\mathbf{r}+s\mathbf{l}_{x}+t\mathbf{l}_{y}+u\mathbf{l}_{z}}$
$\displaystyle
V_{3,\mathbf{r}}\vcentcolon=Z_{0,\mathbf{r}+\mathbf{l}_{z}}\prod_{s,t,u=\pm
1}X_{3,\mathbf{r}+s\mathbf{l}_{x}+t\mathbf{l}_{y}+u\mathbf{l}_{z}},\quad
P_{i,\mathbf{f}_{ab}}\vcentcolon=\prod_{s,t=\pm
1}Z_{i,\mathbf{f}_{ab}+s\mathbf{l}_{a}+t\mathbf{l}_{b}}~{}(a\neq
b=\\{x,y,z\\})$ (71)

We portray the terms (70) and (71) in Fig. 2. The Hamiltonian (69) has the
similar form as four copies of the $(3+1)$d $\mathbb{Z}_{2}$ toric codes [38]
with crucial difference being that a few Pauli operators are multiplies with
the terms as indicated in the first three terms in (70) and (71), which
correspond to the foliated SPT phases.

Figure 2: The spin model which corresponds to (15) with $N=2$, $d=3$ and
$p=1$, described by the Hamiltonian (69)-(71). The terms given in (70) are
depicted in the first line whereas the ones in (71) are in the second line.

The Hamiltonian (69) is exactly solvable as the individual terms commute with
each other. Hence, the ground state is a state with all of the eigenvalues of
the terms (70) and (71) being one. The role of the gauged foliated SPT phases
is to change the statistics of anyons, giving rise to a topological model with
dipole symmetry [23].

To see the model (69) indeed corresponds to the BF theory (15) with $p=1$ and
$d=3$, we evaluate the GSD on the torus geometry. Similar to the case with the
toric code [38], we count the noncontractible loops or membranes (i.e.,
product Pauli operators, forming a plane) consisting of the Pauli operators
$Z_{0}$, $Z_{i}$, that commute with the Hamiltonian (69) (namely, logical
operators) on the $3D$ torus with system size $L_{x}\times L_{y}\times L_{z}$.
One finds the following membrane operators consisting of $Z_{0}$:

$\displaystyle
V_{0:xy}\left(\hat{z}+\frac{1}{2}\right)=\prod_{\hat{x}=1}^{L_{x}}\prod_{\hat{y}=1}^{L_{y}}Z_{0,\mathbf{r}+\mathbf{l}_{z}},\quad
V_{0:yz}\left(\hat{x}+\frac{1}{2}\right)=\prod_{\hat{y}=1}^{L_{y}}\prod_{\hat{z}=1}^{L_{z}}Z_{0,\mathbf{r}+\mathbf{l}_{x}},\quad
V_{0:zx}\left(\hat{y}+\frac{1}{2}\right)=\prod_{\hat{z}=1}^{L_{z}}\prod_{\hat{x}=1}^{L_{x}}Z_{0,\mathbf{r}+\mathbf{l}_{y}}$
(72)

By multiplying the term $V_{0,\mathbf{r}}$, it can be shown that these loops
are deformable, implying that they do not depend on the spatial coordinate.
The form of these membrane operators are also found in the $(3+1)$d toric
code, yet in the present case there are a few constraints due to the foliated
SPT phases. Indeed, multiplying $V_{1,\mathbf{r}}$ over the entire lattice
gives

$\displaystyle\prod_{\hat{x}=1}^{L_{x}}V_{0:yz}\left(\hat{x}+\frac{1}{2}\right)=\left(V_{0:yz}\right)^{L_{x}}=1,$
(73)

where we have used the fact that the membrane operators (72) do not depend on
the spatial coordinate. From (73), it follows that depending on whether
$L_{x}$ is even or odd, the membrane $W_{0;yz}$ becomes trivial or nontrivial,
which can be succinctly described by that the membrane operator $W_{0;yz}$ is
characterized by $\mathbb{Z}_{\gcd(2,L_{x})}$.Similar reasoning can be applied
to other membranes in (72), indicating that the membranes $W_{0:xy}$ and
$W_{0:zx}$ is labeled by $\mathbb{Z}_{\gcd(2,L_{z})}$, and
$\mathbb{Z}_{\gcd(2,L_{y})}$, respectively.

We turn to evaluation of the loops consisting of string of the Pauli operators
$Z_{i}$. One finds that the model admits the following loop operators:

$\displaystyle
W_{i;x}(\hat{y},\hat{z})=\prod_{\hat{x}=1}^{L_{x}}Z_{i,\mathbf{r}+\mathbf{l}_{x}},\quad
W_{i;y}(\hat{z},\hat{x})=\prod_{\hat{y}=1}^{L_{y}}Z_{i,\mathbf{r}+\mathbf{l}_{y}},\quad
W_{i;z}(\hat{x},\hat{y})=\prod_{\hat{z}=1}^{L_{z}}Z_{i,\mathbf{r}+\mathbf{l}_{z}}\quad(i=1,2,3)$
(74)

One can verify that these loops are deformable so that they do not depend on
the spatial coordinate. Naively, there are $2^{9}$ distinct loops. However, it
is incorrect as we have to take care of several constraints on the loops,
stemming from the foliated SPT phases. One constraint comes from the fact that
multiplying $P_{0:\mathbf{f}_{xy}}$ on a $xy$ plane gives

$\displaystyle\prod_{\hat{x}=1}^{L_{x}}\prod_{\hat{y}=1}^{L_{y}}P_{0:\mathbf{f}_{xy}}=1\quad\Leftrightarrow\quad
W_{1:y}^{L_{x}}\times W_{2:x}^{L_{y}}=1.$ (75)

Another constraint stems from deformation of other types of loops which
consists of string of the Pauli $X_{0}$ operators dressed with $Z_{i}$.
Indeed, one finds that the model admits the following loop operator:

$\displaystyle
W_{0:x}(\hat{y},\hat{z})=\left[\prod_{\hat{x}=1}^{L_{x}}X_{0,\mathbf{r}+\mathbf{l}_{x}}\times(Z_{1,\mathbf{r}+\mathbf{l}_{x}})^{\hat{x}}\right]^{\alpha_{x}},$
(76)

where $\alpha_{x}=\frac{2}{\gcd(2,L_{x})}$. Such a loop was discussed in
different spin models in [23]. Deforming the loop (76) in the $y$ direction
results in

$\displaystyle
W_{0:x}(\hat{y}+1,\hat{z})=W_{0:x}(\hat{y},\hat{z})W_{2:x}^{\alpha_{x}}$ (77)

Iterative use of (77) and taking the periodic boundary condition into account,
we have

$\displaystyle W_{2:x}^{\alpha_{x}L_{y}}=1.$ (78)

By the same token, one can show that

$\displaystyle W_{1:y}^{\alpha_{y}L_{x}}=1,$ (79)

where $\alpha_{y}=\frac{2}{\gcd(2,L_{y})}$. From the conditions (75), (78) and
(79), one finds that there are $2\times\gcd(2,L_{x},L_{y})$ distinct
configurations of the loops, $W_{1,y}$ and $W_{2,x}$. Likewise, one can show
that there are $2\times\gcd(2,L_{y},L_{z})$ [$2\times\gcd(2,L_{z},L_{x})$]
distinct configurations of the loops, $W_{3,y}$ and $W_{2,z}$ [$W_{3,x}$ and
$W_{1,z}$]. There is no constraint on the three loops, $W_{1,x}$, $W_{2:y}$,
and $W_{3:z}$, each of which is labeled by $\mathbb{Z}_{2}$. Overall, the
number of distinct loops (74) is found to be

$\displaystyle
2^{6}\times\gcd(2,L_{x},L_{y})\times\gcd(2,L_{y},L_{z})\times\gcd(2,L_{z},L_{x}).$

Taking the fact that there are
$\gcd(2,L_{x})\times\gcd(2,L_{y})\times\gcd(2,L_{z})$ distinct membrane
operators (72) into consideration, we finally arrive at that the GSD has the
form of (34) in the main text with $N=2$, $p=1$, and $d=3$.

###  B.2 $p=2$ and $d=3$

Similar to the argument in the previous subsection, we can also construct a UV
spin model, corresponding to the BF theory (15) with $p=2$ and $d=3$. Since
the way we construct the model closely parallels the one in the previous
subsection, we outline how to do it succinctly.

Figure 3: The spin model which corresponds to the BF theory (15) with $p=2$,
$d=3$, and $N=2$, described by the Hamiltonian (80)-(82).

In the cubic lattice, we introduce qubits on vertices and three types of
qubits on the vertices of the dual cubic lattice. Defining the paramagnet
Hamiltonian in terms of Pauli $\tau^{X}$ operators, we implement the CZ gates
on it, associated with the foliated SPT phases, respecting global symmetries
$[\mathbb{Z}_{2}^{(1)}]^{3}\times\mathbb{Z}_{2}^{(0)}$. After gauging global
symmetries, $[\mathbb{Z}_{2}^{(1)}]^{3}\times\mathbb{Z}_{2}^{(0)}$, the
Hamiltonian is given by (see Fig. 3)

$\displaystyle\widehat{H}=-\sum_{\mathbf{f}_{ab},\mathbf{r}}\left(P_{0,\mathbf{f}_{xy}}+P_{0,\mathbf{f}_{yz}}+P_{0,\mathbf{f}_{zx}}+V_{0,\mathbf{r}}\right)-\sum_{i=1,2,3}\sum_{\mathbf{f}_{ab},\mathbf{r}}\left(P_{i,\mathbf{f}_{xy}}+P_{i,\mathbf{f}_{yz}}+P_{i,\mathbf{f}_{zx}}+V_{i,\mathbf{r}}\right)$
(80)

with

$\displaystyle
V_{0,\mathbf{r}}=Z_{1,\mathbf{r}+\mathbf{l}_{x}}Z_{2,\mathbf{r}+\mathbf{l}_{y}}Z_{3,\mathbf{r}+\mathbf{l}_{z}}\prod_{s,t,u=\pm
1}X_{0,\mathbf{r}+s\mathbf{l}_{x}+t\mathbf{l}_{y}+u\mathbf{l}_{z}},\quad
P_{0,\mathbf{f}_{ab}}\vcentcolon=\prod_{s,t=\pm
1}Z_{0,\mathbf{f}_{ab}+s\mathbf{l}_{a}+t\mathbf{l}_{b}}~{}(a\neq
b=\\{x,y,z\\})$ (81)

and

$\displaystyle
P_{1,\mathbf{f}_{xy}}\vcentcolon=Z_{0,\mathbf{f}_{xy}-\mathbf{l}_{x}}\prod_{s,t=\pm
1}X_{1,\mathbf{f}_{xy}+s\mathbf{l}_{x}+t\mathbf{l}_{y}},\quad
P_{1,\mathbf{f}_{yz}}\vcentcolon=\prod_{s,t=\pm
1}X_{1,\mathbf{f}_{yz}+s\mathbf{l}_{y}+t\mathbf{l}_{z}},\quad
P_{1,\mathbf{f}_{zx}}\vcentcolon=Z_{0,\mathbf{f}_{zx}-\mathbf{l}_{x}}\prod_{s,t=\pm
1}X_{1,\mathbf{f}_{zx}+s\mathbf{l}_{z}+t\mathbf{l}_{x}}$ $\displaystyle
P_{2,\mathbf{f}_{xy}}\vcentcolon=Z_{0,\mathbf{f}_{xy}-\mathbf{l}_{y}}\prod_{s,t=\pm
1}X_{2,\mathbf{f}_{xy}+s\mathbf{l}_{x}+t\mathbf{l}_{y}},\quad
P_{2,\mathbf{f}_{yz}}\vcentcolon=Z_{0,\mathbf{f}_{yz}-\mathbf{l}_{y}}\prod_{s,t=\pm
1}X_{2,\mathbf{f}_{yz}+s\mathbf{l}_{y}+t\mathbf{l}_{z}},\quad
P_{2,\mathbf{f}_{zx}}\vcentcolon=\prod_{s,t=\pm
1}X_{2,\mathbf{f}_{zx}+s\mathbf{l}_{z}+t\mathbf{l}_{x}}$ $\displaystyle
P_{3,\mathbf{f}_{xy}}\vcentcolon=\prod_{s,t=\pm
1}X_{3,\mathbf{f}_{xy}+s\mathbf{l}_{x}+t\mathbf{l}_{y}},\quad
P_{3,\mathbf{f}_{yz}}\vcentcolon=Z_{0,\mathbf{f}_{yz}-\mathbf{l}_{z}}\prod_{s,t=\pm
1}X_{3,\mathbf{f}_{yz}+s\mathbf{l}_{y}+t\mathbf{l}_{z}},\quad
P_{3,\mathbf{f}_{zx}}\vcentcolon=Z_{0,\mathbf{f}_{zx}-\mathbf{l}_{z}}\prod_{s,t=\pm
1}X_{3,\mathbf{f}_{zx}+s\mathbf{l}_{z}+t\mathbf{l}_{x}}$ $\displaystyle
V_{i,\mathbf{r}}\vcentcolon=\prod_{s,t,u=\pm
1}X_{i,\mathbf{r}+s\mathbf{l}_{x}+t\mathbf{l}_{y}+u\mathbf{l}_{z}}\quad(i=1,2,3).$
(82)

The model has the similar form as the four copies of the toric codes with a
crucial difference being that a few Pauli operators are attached with the
terms, stemming from the foliated SPT phases.

The model (80) is exactly solvable and by analogous lines of thinking to count
the number of distinct noncontractible loops, comprised of $Z_{0}$ or $Z_{i}$,
which is discussed in the previous subsection, one finds that the GSD is given
by

$\displaystyle
GSD=2^{8}\times\gcd(2,L_{x},L_{y})\times\gcd(2,L_{y},L_{z})\times\gcd(2,L_{z},L_{x})\times\gcd(2,L_{x},L_{y},L_{z})$
(83)

which is consistent with (34) with $p=2$, $d=3$, and $N=2$.

##  Appendix C Higher form subsystem BF theory and anomaly inflow

In this appendix, we introduce higher form BF theories with subsystem
symmetry. Such theories are obtained by imposing an additional constraints on
the dipole BF theories (11). We also discuss anomaly inflow for these
theories.

###  C.1 Model

We start by the dipole BF theory (11). Replacing the gauge field
$c^{I({d-p})}$ with $dB^{I({d-p-1})}\wedge e^{I}$, ($I$ is not summed over),
we have

$\boxed{\mathcal{L}=\frac{N}{2\pi}\left[b^{({d-p})}\wedge\left(da^{({p})}+(-1)^{p}\sum_{I}A^{I({p})}\wedge
e^{I}\right)+\sum_{I}B^{I({d-p-1})}\wedge dA^{I({p})}\wedge e^{I}\right]}.$
(84)

Here, we demand that $1\leq p\leq d-1$. Note that in the case of $p=1$ and
$d=3$, the theory (84) becomes the foliated BF theory describing the $X$-cube
model [8, 24, 25]. Compared with the dipole BF theory (15), the theory (84)
has an additional symmetry – subsystem symmetry. Indeed, the theory (84) is
invariant under

$\displaystyle A^{I({p})}\ \to\ A^{I({p})}+\gamma^{I({p-1})}\wedge e^{I}$ (85)

with $\gamma^{I({p-1})}$ being an arbitrary $(p-1)$-form field. As shown
below, the manifestation of the subsystem symmetry can be seen by noticing
that the symmetry (85) puts constraints on the form of gauge invariant
extended operators of the gauge fields $a^{({p})}$ and $A^{I({p})}$ which are
analogue of Wilson loops for the $p$-form gauge fields supported on a closed
$p$-dimensional spatial submanifold. The theory (84) respects the following
gauge symmetry:

$\displaystyle a^{({p})}\to
a^{({p})}+d\Lambda^{({p-1})}+(-1)^{p-1}\sum_{I}\sigma^{I({p-1})}\wedge
e^{I},\quad A^{I({p})}\to
A^{I({p})}+d\sigma^{I({p-1})}+\gamma^{I({p-1})}\wedge e^{I}$ $\displaystyle
b^{({d-p})}\to b^{({d-p})}+d\mu^{({d-p-1})},\quad B^{I({d-p-1})}\to
B^{I({d-p-1})}+d\chi^{I({d-p-2})}+(-1)^{p+1}\mu^{({d-p-1})}+g^{({d-p-2})}\wedge
e^{I}.$

Here, $g^{({d-p-2})}$ denotes an arbitrary $(d-p-2)$-form field. Similar to
the discussion in Sec. 3.2.2, one can evaluate the GSD of (84) on a
discretized torus geometry with system size $L_{1}\times\cdots\times L_{d}$,
which is accomplished by counting a distinct number of noncontractible loops
of the gauge fields $a^{({p})}$ and $A^{I({p})}$ in the spatial direction. We
do not have noncontractible loop consisting of the gauge field
$a^{({p})}$.161616One naively wonders that similar to the case with the dipole
BF theory (15), we could construct the loop of $a^{({p})}$ accompanied by the
gauge fields $A^{I({p})}$, such as the one given in (17). However, due to the
symmetry (85), such a loop is not allowed. As for the loops comprised of
$A^{I({p})}$, we have

$\displaystyle
W_{A^{I}}(\Sigma_{p})(\hat{x}_{I})=\int_{\Sigma_{p}}A^{I({p})}\quad(\Sigma_{p}\perp
I\text{-th direction}).$ (87)

Here, $\Sigma_{p}$ denotes the $p$-form spatial submanifold which has to be
perpendicular to the $I$-th direction due to the symmetry (85). Intuitively,
(87) can be understood by that on each codimension one layer with coordinate
$\hat{x}_{I}$, which is stacked along the $I$-th direction, noncontractible
$p$-form loop is constructed.171717Such a loop is associated with higher form
analog of ”planon”.  The loop (87) is the higher form analog of the Wilson
loop defined on each layers of codimension one submanifold, found in the
$X$-cube model.181818See also [51] for construction of the fracton lattice
models hosting spatially extended excitations.

Naively, there are in total $\binom{d-1}{p}\left(\sum_{i=1}^{d}L_{i}\right)$
such loops, however, there are constraints on these loops. Indeed, equations
of motions for the field $b^{({d-p})}_{[0j_{1}\cdots j_{d-p-1}]}$ gives

$\displaystyle\partial_{[i_{i}}a^{({p})}_{i_{2}\cdots
i_{p+1}]}-(-1)^{p}\sum_{I}A^{I({p})}_{[i_{1}\cdots
i_{p}}\delta^{I}_{i_{p+1}]}=0.$ (88)

Performing integration over spatial directions $i_{1}\cdots i_{p+1}$ of the
torus results in

$\sum_{x_{i_{1}}=1}^{L_{i_{1}}}W_{A^{i_{1}}}(\Sigma_{[i_{2}\cdots
i_{p+1}]})(\hat{x}_{i_{1}})-\sum_{x_{i_{2}}=1}^{L_{i_{2}}}W_{A^{i_{2}}}(\Sigma_{[i_{1}i_{3}\cdots
i_{p+1}]})(\hat{x}_{i_{2}})+\cdots+(-1)^{p+1}\sum_{x_{i_{p+1}}=1}^{L_{i_{p+1}}}W_{A^{i_{p+1}}}(\Sigma_{[i_{1}\cdots
i_{p}]})(\hat{x}_{i_{p+1}})=0.$ (89)

Here, $\Sigma_{[i_{2}\cdots i_{p+1}]}$ in the first term represents spatial
submanifold in the $i_{2}\cdots i_{p+1}$ direction and other terms which have
the form $\Sigma_{[**]}$ are similarly defined. There are $\binom{d}{p+1}$
constraints on the loops (87) described by (89). Hence, the GSD is given by

$\boxed{{\rm GSD}=N^{Q(d,p)}\quad\text{with}\ \
Q(d,p)\vcentcolon=\binom{d-1}{p}\left(\sum_{i=1}^{d}L_{i}\right)-\binom{d}{p+1}}.$
(90)

The theory (84) is the higher form analog of the foliated BF theory of the
fracton models, such as the $X$-cube model. Similar to the $X$-cube model, our
theory also exhibits the subextensive GSD.191919As a consistency check, when
we set $d=3$ and $p=1$, we have $Q(3,1)=2(L_{x}+L_{y}+L_{z})-3$, which appears
in the GSD of the $X$-cube model [8] on torus.

###  C.2 Anomaly inflow

We discuss the anomaly inflow for higher form subsystem symmetry. Analogous to
the previous argument, we gauge the $p$-form symmetries by

$\mathcal{L}=\frac{N}{2\pi}\biggl{[}b^{({d-p})}\wedge\left(da^{({p})}+(-1)^{p}\sum_{I}A^{I({p})}\wedge
e^{I}-\alpha^{({p+1})}\right)+\sum_{I}B^{I({d-p-1})}\wedge\left(dA^{I({p})}-\Gamma^{I({p+1})}\right)\wedge
e^{I}\biggr{]}.$ (91)

The theory has the following gauge symmetry:

$\displaystyle a^{({p})}\to a^{({p})}+\lambda_{a}^{({p})},\quad A^{I({p})}\to
A^{I({p})}+\lambda_{A}^{I({p})}$
$\displaystyle\alpha^{({p+1})}\to\alpha^{({p+1})}+d\lambda_{a}^{({p})}+(-1)^{p}\lambda_{A}^{I({p})}\wedge
e^{I},\quad\Gamma^{I({p+1})}\to\Gamma^{I({p+1})}+d\lambda_{A}^{I({p})}$ (92)

with $\lambda_{a}^{({p})}$ and $\lambda_{A}^{I({p})}$ being the $p$-form gauge
parameters.

Also, one can gauge the $(d-p)$-form symmetries via

$\displaystyle\mathcal{L}=\frac{N}{2\pi}\biggl{[}b^{({d-p})}\wedge\left(da^{({p})}+(-1)^{p}\sum_{I}A^{I({p})}\wedge
e^{I}\right)+\sum_{I}B^{I({d-p-1})}\wedge dA^{I({p})}\wedge e^{I}$
$\displaystyle-(-1)^{(d+1)(p+1)}a^{({p})}\wedge\beta^{({d+1-p})}-(-1)^{d(p+1)}\sum_{I}A^{I({p})}\wedge\Xi^{I({d-p})}\wedge
e^{I}\biggr{]}.$ (93)

This theory has the following gauge symmetry:

$\displaystyle b^{({d-p})}\to b^{({d-p})}+\lambda_{b}^{({d-p})},\quad
B^{I({d-p-1})}\to B^{I({d-p-1})}+\lambda_{B}^{I({d-p-1})},$
$\displaystyle\beta^{({d-p+1})}\to\beta^{({d-p+1})}+d\lambda_{b}^{({d-p})},\quad\Xi^{I({d-p})}\to\Xi^{I({d-p})}+d\lambda_{B}^{I({d-p-1})}+(-1)^{d}\lambda_{b}^{({d-p})}.$
(94)

If we try to gauge both of $p$-form and $(d-p)$-form symmetries by thinking of
the following theory

$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\frac{N}{2\pi}\biggl{[}b^{({d-p})}\wedge\left(da^{({p})}+(-1)^{p}\sum_{I}A^{I({p})}\wedge
e^{I}-\alpha^{({p+1})}\right)+\sum_{I}B^{I({d-p-1})}\wedge\left(dA^{I({p})}-\Gamma^{I({p+1})}\right)\wedge
e^{I}$ (95)
$\displaystyle-(-1)^{(d+1)(p+1)}a^{({p})}\wedge\beta^{({d+1-p})}-(-1)^{d(p+1)}\sum_{I}A^{I({p})}\wedge\Xi^{I({d-p})}\wedge
e^{I}\biggr{]},$

then we have a problem; (95) is not invariant under the gauge transformations
(92) and (94). Indeed, the variant of $\mathcal{L}$ under the transformations
reads

$\displaystyle\delta\mathcal{L}$ $\displaystyle=$
$\displaystyle-\frac{N}{2\pi}\biggl{[}\lambda_{b}^{({d-p})}\wedge\alpha^{({p+1})}+\sum_{I}\lambda_{B}^{I({d-p-1})}\wedge\Gamma^{I}\wedge
e^{I}+(-1)^{(d+1)(p+1)}\lambda_{a}^{({p})}\wedge\left(\beta^{({d-p+1})}+d\lambda_{b}^{({d-p})}\right)$
(96)
$\displaystyle~{}~{}~{}~{}~{}~{}+(-1)^{d(p+1)}\sum_{I}\lambda_{A}^{I}\wedge\left(\Xi^{I({d-p})}+d\lambda_{B}^{I({d-p-1})}+(-1)^{d}\lambda_{b}^{({d-p})}\right)\wedge
e^{I}\biggr{]},$

signaling the mixed ’t Hooft anomaly. To cancel the anomaly, we introduced the
following terms defined in one dimensional higher:

$\displaystyle\mathcal{L}_{\rm inflow}=\mathcal{L}_{0}+\mathcal{L}_{\rm spt}$
(97)

with

$\displaystyle\mathcal{L}_{0}=\frac{N}{2\pi}\biggl{[}M^{({d-p})}\wedge\left(d\alpha^{({p+1})}+(-1)^{p}\sum_{I}\Gamma^{I({p})}\wedge
e^{I}\right)+\sum_{I}N^{I({d-p-1})}\wedge d\Gamma^{I({p})}\wedge e^{I}$
$\displaystyle+O^{({p})}\wedge
d\beta^{({d-p+1})}+\sum_{I}P^{I({p})}\wedge\left(d\Xi^{I({d-p})}+(-1)^{(d-1)}\beta^{({d-p+1})}\right)\wedge
e^{I}\biggl{]},$ (98)

and

$\displaystyle\boxed{\mathcal{L}_{\rm
spt}=-\frac{N}{2\pi}\biggl{[}(-1)^{(p+1)(d+1)}\alpha^{({p+1})}\wedge\beta^{({d-p+1})}+(-1)^{d(p+1)}\sum_{I}\Gamma^{I({p+1})}\wedge\Xi^{I({d-p})}\wedge
e^{I}\biggr{]}}.$ (99)

The bulk terms (97) consist of two types of terms: the terms which ensure the
flatness conditions on the gauge fields (98) introducing auxiliary fields,
$M^{({d-p})}$, $N^{I({d-p-1})}$, $O^{({p})}$, $P^{I({p})}$, and the ones that
correspond to invertible phases (99). Up to total derivative, the bulk terms
(97) respect the gauge symmetries (92) and (94) jointly with

$\displaystyle M^{({d-p})}\to
M^{({d-p})}+(-1)^{d-p}\lambda_{b}^{({d-p})},\quad N^{I({d-p-1})}\to
N^{I({d-p-1})}+(-1)^{d-p-1}\lambda_{B}^{I({d-p-1})}$ $\displaystyle
O^{({p})}\to O^{({p})}+(-1)^{d(p+1)+1}\lambda_{a}^{({p})},\quad P^{I({p})}\to
P^{I({p})}+(-1)^{d(p+1)+d}\lambda_{A}^{I({p})}.$ (100)

More explicitly, under the transformations (92), (94) and (100), (97) gives

$\displaystyle\delta\mathcal{L}_{\rm inflow}$ $\displaystyle=$
$\displaystyle-\frac{N}{2\pi}d\biggl{[}\lambda_{b}^{({d-p})}\wedge\alpha^{({p+1})}+\sum_{I}\lambda_{B}^{I({d-p-1})}\wedge\Gamma^{I}\wedge
e^{I}$ (101)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(-1)^{(d+1)(p+1)}\lambda_{a}^{({p})}\wedge\left(\beta^{({d-p+1})}+d\lambda_{b}^{({d-p})}\right)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+(-1)^{d(p+1)}\sum_{I}\lambda_{A}^{I}\wedge\left(\Xi^{I({d-p})}+d\lambda_{B}^{I({d-p-1})}+(-1)^{d}\lambda_{b}^{({d-p})}\right)\wedge
e^{I}\biggr{]}.$

The terms inside the large bracket in (101) is identical to (96). This
indicates that the mixed ’t Hooft anomaly is canceled by (97) with (98) and
(99).

Note that in the previous studies, such as [32, 33, 34], discussed anomaly
inflow when gauging subsystem symmetries in tensor gauge theories and exotic
field theories. Here we discuss the anomaly inflow mechanism for such
symmetries, described by foliated field theories. We leave elucidation of the
relation between the anomaly inflow discussed there and the present case for
future study.

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