# (3+1)D path integral state sums on curved U(1) bundles
and U(1) anomalies of (2+1)D topological phases

Ryohei Kobayashi  Maissam Barkeshli  Condensed Matter Theory Center and Joint
Quantum Institute, Department of Physics, University of Maryland, College
Park, Maryland 20472 USA

###### Abstract

Given the algebraic data characterizing any (2+1)D bosonic or fermionic
topological order with a global symmetry group $G=\mathrm{U}(1)\rtimes H$, we
construct a (3+1)D topologically invariant path integral in the presence of a
curved background $G$ gauge field, as an exact combinatorial state sum.
Specifically, the U(1) component of the $G$ gauge field can have a non-trivial
second Chern class, extending previous work that was restricted to flat $G$
bundles. Our construction expresses the U(1) gauge field in terms of a Villain
formulation on the triangulation, which includes a $1$-form $\mathbb{R}$ gauge
field and $2$-form $\mathbb{Z}$ gauge field. We develop a new graphical
calculus for anyons interacting with “Villain symmetry defects”, associated
with the $1$-form and $2$-form background gauge fields. This graphical
calculus is used to define the (3+1)D path integral, which can describe either
a bosonic or fermionic symmetry-protected topological (SPT) phase. For
example, we can construct the topological path integral on curved U(1) bundles
for the (3+1)D fermionic topological insulator in class AII and topological
superconductor in class AIII given appropriate (2+1)D fermionic symmetry
fractionalization data; these then give invariants of 4-manifolds with Spinc
or Pinc structures and their generalizations. The (3+1)D path integrals define
anomaly indicators for the (2+1)D topological orders; in the case of Abelian
(2+1)D topological orders, we derive by explicit computation all of the mixed
U(1) anomaly indicator formulas proposed by Lapa and Levin. We also propose a
Spinc generalization of the Gauss-Milgram sum, valid for super-modular
categories.

###### Contents

  1. I Introduction
    1. I.1 Summary of main results
      1. I.1.1 Construction of topological path integrals
      2. I.1.2 Abelian bosonic topological phases with $G=\mathrm{U}(1)$
      3. I.1.3 Abelian bosonic topological phases with $G=\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{\bf T}$
      4. I.1.4 Abelian bosonic topological phases with $G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{\bf T}$
      5. I.1.5 Abelian fermionic topological phases with $G_{f}=\mathrm{U}(1)^{f}$ and Spinc Gauss-Milgram formula
      6. I.1.6 Fermionic topological phases in class AII or AIII
    2. I.2 Organization of paper
  2. II U(1) gauge field on lattice: Villain formulation
    1. II.1 U(1) gauge field
    2. II.2 $\mathrm{U}(1)^{f}$ gauge field and Spinc structure
  3. III U(1) symmetry
    1. III.1 U(1) symmetry fractionalization
      1. III.1.1 Bosonic case: modular tensor category
      2. III.1.2 Fermionic case: super-modular tensor category
    2. III.2 New graphical calculus with Villain symmetry defects
  4. IV Bosonic state sum of (3+1)D TQFT with U(1) symmetry
  5. V Bosonic state sum of (3+1)D TQFT with $\mathrm{U}(1)\rtimes H$ symmetry
  6. VI Construction of fermionic topological phases: fermion condensation
    1. VI.1 ’t Hooft anomaly of a bosonic shadow theory $Z_{b}(M^{d},A_{b},f_{d-1})$
    2. VI.2 Anomaly cancellation and fermion condensation
    3. VI.3 $\mathcal{G}_{f}$ structure and the definition of $z_{c}$
    4. VI.4 Examples of $\mathcal{G}_{f}$ structures
      1. VI.4.1 class A: $\mathrm{Spin}^{c}$ structure
      2. VI.4.2 class AI: $\mathrm{Pin}^{\tilde{c}}_{-}$ structure
      3. VI.4.3 class AII: $\mathrm{Pin}^{\tilde{c}}_{+}$ structure
      4. VI.4.4 class AIII: $\mathrm{Pin}^{c}$ structure
  7. VII Bosonic shadow for (3+1)D fermionic SPT with curved U(1) bundle
  8. VIII Explicit evaluation of U(1) anomaly indicators for (2+1)D TQFT
    1. VIII.1 U(1) anomaly for (2+1)D Abelian TQFT: bosonic case
      1. VIII.1.1 General simplification of path integral for Abelian modular $\mathcal{C}$
      2. VIII.1.2 Partition function on $\mathbb{CP}^{2}$
      3. VIII.1.3 Partition function on $S^{2}\times S^{2}$
      4. VIII.1.4 $\mathrm{U}(1)\times\mathbb{Z}_{2}^{\mathbf{T}}$ symmetry: partition function on $S^{2}\times\mathbb{RP}^{2}$
    2. VIII.2 $\mathrm{U}(1)^{f}$ anomaly for (2+1)D Abelian Spinc TQFT: fermionic case
      1. VIII.2.1 General simplification of path integral for Abelian super-modular $\mathcal{C}$
      2. VIII.2.2 Partition function on $\mathbb{CP}^{2}$
      3. VIII.2.3 Partition function on $S^{2}\times S^{2}$
    3. VIII.3 Partition function on $\mathbb{RP}^{4}$: class DIII, AII, AIII
  9. IX Discussion
  10. X Acknowledgments
  11. A Review of (2+1)D anyon systems
    1. A.1 Notations of BTC
    2. A.2 Abelian BTC
    3. A.3  Modular and super-modular tensor categories
  12. B Review of symmetry fractionalization
    1. B.1  Topological symmetry and braided auto-equivalence
    2. B.2  Global symmetry and symmetry fractionalization: bosonic case
    3. B.3  Fermionic symmetries
    4. B.4 Locality-respecting natural isomorphisms in the fermionic case
  13. C Graphical calculus of Villain symmetry defects in the presence of $H$ symmetry defects
    1. C.1 Symmetry fractionalization data for $\mathrm{U}(1)\rtimes H$ symmetry
    2. C.2 Villain calculus for $\mathrm{U}(1)\rtimes H$ symmetry
  14. D Demonstration of Spinc Gauss-Milgram formula
    1. D.1 Moore-Read state: $(k,M)=(2,1)$
    2. D.2 Read-Rezayi state with $(k,M)=(3,1)$
    3. D.3 Read-Rezayi state with $(k,M)=(4,1)$
  15. E Twisted cohomology
  16. F Invariance of the partition function under the Pachner move
    1. F.1 Bosonic case
    2. F.2 Fermionic case

##  I Introduction

In the past several years, a comprehensive algebraic framework has emerged to
completely characterize symmetry fractionalization in both bosonic and
fermionic (2+1)D topological phases of matter in general [1, 2, 3]. In some
cases, a given pattern of symmetry fractionalization may be anomalous, in the
sense that it cannot occur in a purely (2+1)D system, but it can occur at the
(2+1)D surface of a (3+1)D symmetry-protected topological (SPT) state. In the
language of quantum field theory, the (2+1)D theory possesses a ’t Hooft
anomaly for the global symmetry group. A basic question then is to compute the
anomaly given the algebraic data that characterizes the (2+1)D topological
order; that is, to determine which (3+1)D SPT can host the given (2+1)D
topological phase at its surface. Since it has also been understood in recent
years that the Lieb-Schultz-Mattis theorem and its generalizations can be
rephrased in terms of mixed anomalies between translation symmetry and an on-
site symmetry [4, 5, 6, 7, 8], these questions have far-reaching implications
for placing constraints on the possible (2+1)D topological orders that can
emerge from a given microscopic model.

Recently it has been understood that one can in general compute anomalies of
(2+1)D topological orders as follows [9, 10, 11]. One can use the algebraic
data that defines the (2+1)D topological order and its possibly anomalous
symmetry fractionalization pattern for a symmetry group $G$ as input into the
construction of a (3+1)D topological path integral defined on a principal $G$
bundle. This (3+1)D path integral can be constructed in terms of an exact
combinatorial sum over a finite set of labelings of a triangulated 4-manifold
in the presence of a background $G$ gauge field. This construction defines an
invertible topological quantum field theory (TQFT) with $G$ symmetry, as
appropriate for describing (3+1)D SPT states, and can be viewed as a symmetry-
enriched generalization of the Crane-Yetter-Walker-Wang constructions [12,
13].

If the input is a fermionic topological order, i.e. a super-modular category,
with fermionic symmetry fractionalization, then the topological path integral
depends on a generalized spin structure as well, and defines an invertible
spin TQFT with a fermionic symmetry group $G_{f}$ [11].

The state sum path integral is quite powerful, as it can also be used to
derive a ground state wave function for the (3+1)D system and an exactly
solvable parent Hamiltonian. As such it provides a comprehensive, exact
description of the (3+1)D system although this perspective has only been
worked out in some cases [13, 14, 10].

A significant limitation of previously known constructions is that they only
give topological path integrals that can be defined on flat $G$-bundles. When
$G$ contains a continuous component, it is known that the path integral on
curved $G$ bundles is quite useful, and potentially necessary, to fully
characterize a (3+1)D SPT. For example, the famous (3+1)D topological
insulator in Class AII [15, 16] can be characterized by its U(1)
electromagnetic response, which contains a theta term at $\Theta=\pi$, and one
needs to introduce a non-trivial field strength for this term to be non-
trivial. In fact, all TQFT state sum constructions so far have only be defined
for flat $G$ bundles, and it is generally assumed that one must always
restrict to flat $G$ bundles.

In this paper, we show that when the symmetry $G$ contains a $\mathrm{U}(1)$
subgroup, one can generalize the previous constructions to define a (3+1)D
topological path integral on a curved bundle. More specifically,
$\mathrm{U}(1)$ bundles over (3+1)D manifolds can be completely characterized
by their second Chern number, in addition to the holonomies around non-
contractible cycles. Here we show that one can define a topological path
integral for any $\mathrm{U}(1)$ bundle, including any non-zero second Chern
number. In the case of fermionic topological phases, our construction allows
us to define topological path integrals on 4-manifolds with Spinc structures,
again with non-trivial Chern numbers.

Our construction proceeds by utilizing a Villain formulation for the
background $\mathrm{U}(1)$ gauge field. We view a $1$-form $\mathrm{U}(1)$
gauge field in terms of a $1$-form $\mathbb{R}$ gauge field and a $2$-form
$\mathbb{Z}$ gauge field. We then introduce a new kind of diagrammatic
formalism, which we refer to as a “Villain” graphical calculus, which
describes anyons interacting with symmetry defects associated with the
$1$-form $\mathbb{R}$ gauge field and $2$-form $\mathbb{Z}$ gauge field. We
show how this new graphical calculus can be used to consistently define the
(3+1)D topological path integral on curved $\mathrm{U}(1)$ bundles.

As an important application of our results, our constructions allow a
derivation of the anomaly indicator formulas presented in [17], which we
explicitly demonstrate in the case of Abelian topological phases. Our results
also provide a Spinc generalization of the Gauss-Milgram sum for super-modular
categories, which provides a simple explicit formula to determine the chiral
central charge $c_{-}\text{ mod }1$ of a super-modular category with
$\mathrm{U}(1)^{f}$ symmetry fractionalization just from the modular data and
fractional charge assignments.

###  I.1 Summary of main results

We consider both bosonic and fermionic (2+1)D topological orders with a global
symmetry that involves U(1). In the bosonic cases, the global symmetry group
has the form $G=\mathrm{U}(1)\rtimes H$. We allow $H$ to contain anti-unitary
symmetry group elements, and $H$ can act on U(1) by charge conjugation.

In the fermionic cases, the system has a fermionic global symmetry group
$G_{f}$. $G_{f}$ contains a fermion parity symmetry $\mathbb{Z}_{2}^{f}$, and
the group that acts on bosonic operators is expressed as
$G_{b}=G_{f}/\mathbb{Z}_{2}^{f}$. We take the bosonic symmetry group to be of
the form $G_{b}=\mathrm{U}(1)\rtimes H$. In general, $G_{f}$ is a nontrivial
extension of $G_{b}$ by $\mathbb{Z}_{2}^{f}$, whose extension is specified by
a 2-cocycle $\omega_{2}\in Z^{2}(BG_{b},\mathbb{Z}_{2})$. When $G_{f}$
involves a nontrivial extension of $\mathrm{U}(1)$ by $\mathbb{Z}_{2}^{f}$,
the fermion carries charge under the extended U(1) symmetry
$\mathbb{Z}_{2}^{f}\to\mathrm{U}(1)^{f}\to\mathrm{U}(1)$. More generally, the
existence of a non-trivial $\omega_{2}$ can be understood as the physical
fermion carrying fractional quantum numbers under $G_{b}$. In particular, the
above global symmetry group contains the Altland-Zirnbauer symmetry classes A,
AI, AII and AIII that involve U(1) symmetry. The specific 2-cocycle
$\omega_{2}(\mathbf{g},\mathbf{h})$ is determined by the action of $G_{f}$ on
local fermion operators in the microscopic theory [2, 3].

In this paper, we assume that we are given the data of a modular or super-
modular fusion category [18, 19, 20] $\mathcal{C}$ in the bosonic or fermionic
case respectively. In addition to $\mathcal{C}$, we are given the data that
specifies the symmetry fractionalization [1, 2, 3]. This data consists of a
group homomorphism $[\rho]:H\to\mathrm{Aut}(\mathcal{C})$, which specifies how
the symmetry $H$ permutes the anyons, and a set of U(1) phases
$\eta_{a}:G\times G\rightarrow\mathrm{U}(1)$ for each anyon $a$, which must
obey certain consistency equations and which are defined up to certain gauge
transformations. We review this data in Appendix B. The data $\\{\eta_{a}\\}$
in particular specify the $\mathrm{U}(1)$ fractional charges $\\{Q_{a}\\}$ of
each anyon $a$, that specifies how the $\mathrm{U}(1)$ symmetry is
fractionalized.

In this paper, given $\mathcal{C}$ and the symmetry fractionalization data, we
show how to construct a (3+1)D topologically invariant path integral coupled
to a background gauge field with non-trivial Chern class. In the fermionic
case, the path integral depends on both a $G_{b}$ background gauge field and a
twisted spin structure $\xi$.

The U(1) gauge field is expressed in a Villain form, as $({h},{c})$, which is
a pair of a 1-form $\mathbb{R}$ gauge field $h$ and a 2-form $\mathbb{Z}$
gauge field $c$. The integral of $c$ over a closed 2-cycle gives the first
Chern number of the $U(1)$ gauge field evaluated on that 2-cycle.

The state sum path integral is given by a symmetry-enriched generalization of
the Crane-Yetter-Walker-Wang construction, which utilizes the diagrammatic
calculus of (2+1)D topological order in the presence of symmetry defects [10,
11]. In order to define the path integral in the presence of the Villain
background gauge fields, we introduce a new graphical calculus that
characterizes the interaction between the anyons and the “Villain symmetry
defects,” which include codimension-1 and codimension-2 defects. This new
graphical calculus is regarded as encoding the symmetry action of 0-form
$\mathbb{R}$ and 1-form $\mathbb{Z}$ symmetry on the (2+1)D anyon system, and
allows us to define a (3+1)D topologically invariant theory coupled with the
generic Villain U(1) gauge field.

####  I.1.1 Construction of topological path integrals

Concretely, in the bosonic cases our (3+1)D path integral takes the form of
$Z(M^{4},(h,c),A_{H})$, where $M^{4}$ is a closed 4-manifold, $(h,c)$ is the
Villain gauge field subject to the action of $H$, and $A_{H}$ is a flat $H$
gauge field. In the fermionic cases, the path integral is given in the form of
$Z(M^{4},(h,c),A_{H},\xi_{\mathcal{G}})$ with the additional dependence on
twisted spin/pin structure $\xi_{\mathcal{G}}$ referred to as a
$\mathcal{G}_{f}$-structure in this paper. The $\mathcal{G}_{f}$-structure of
a manifold $M^{4}$ is specified by a choice of a 1-cochain $\xi_{\mathcal{G}}$
that satisfies

$\displaystyle\delta\xi_{\mathcal{G}}=w_{2}+w_{1}^{2}+2{c}+A_{H}^{*}\omega_{H},$
(1)

where $w_{k}\in H^{k}(M^{4},\mathbb{Z}_{2})$ is the $k^{\mathrm{th}}$ Stiefel-
Whitney class, and $\omega_{H}\in H^{2}(BH,\mathbb{Z}_{2})$ characterizes the
group extension by $\mathbb{Z}_{2}^{f}$ restricted to $H$. Here we view the
background $H$ gauge field as a map $A_{H}:M^{4}\rightarrow BH$, where $BH$ is
the classifying space of $H$, and $A_{H}^{*}$ denotes the pullback from
$H^{2}(BH,\mathbb{Z}_{2})$ to $H^{2}(M^{4},\mathbb{Z}_{2})$. For example, in
the case that $H$ is trivial, $\xi_{\mathcal{G}}$ is a Spinc structure on
oriented manifolds. When $H=\mathbb{Z}_{2}^{\mathbf{T}}$ with
$\mathbb{Z}_{2}^{\mathbf{T}}$ a time-reversal symmetry and $\omega_{H}$ is
trivial, $\xi_{\mathcal{G}}$ becomes a Pinc structure or
Pin${}^{\tilde{c}}_{-}$ structure depending on whether
$\mathbb{Z}_{2}^{\mathbf{T}}$ conjugates the U(1).

The fermionic path integral $Z(M^{4},(h,c),A_{H},\xi_{\mathcal{G}})$ is
constructed by first taking the super-modular category with $G_{b}$ symmetry,
and constructing a (3+1)D bosonic path integral using the new graphical
calculus involving Villain U(1) background gauge fields $(h,c)$. This gives a
(3+1)D path integral $Z_{b}(M^{4},(h,c),A_{H})$ for a bosonic topological
phase which contains a single non-trivial point-like excitation, which is a
fermion. We then condense this fermion, following the prescription of Refs.
[21, 11].

The fermion condensation is performed by first considering a $\mathbb{Z}_{2}$
2-form symmetry of the theory $Z_{b}(M^{4},(h,c),A_{H})$ generated by a Wilson
line of a fermion. We turn on the background gauge field $f_{3}\in
Z^{3}(M^{4},\mathbb{Z}_{2})$ of the 2-form symmetry, yielding a path integral
$Z_{b}(M^{4},(h,c),A_{H},f_{3})$. In the presence of the 3-form background
gauge field $f_{3}$, the path integral turns out to possess a ’t Hooft anomaly
of the $\mathbb{Z}_{2}$ 2-form symmetry characterized by a (4+1)D response
action

$\displaystyle
e^{iS_{5,b}}=(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{2}(f_{3})+(2{c}+A_{H}^{*}\omega_{H})\cup
f_{3}}$ (2)

where $\mathop{\mathrm{Sq}}\nolimits^{2}$ is a cohomology operation called
Steenrod square. Due to the ’t Hooft anomaly, $Z_{b}$ is no longer
topologically invariant in the presence of the 3-form background gauge field
$f_{3}$.

In order to obtain a topologically invariant path integral by condensing the
fermion, we need to appropriately compensate for these anomalies by including
an additional factor. The full fermionic path integral then takes the form:

$\displaystyle
Z(M^{4},(h,c),A_{H},\xi_{\mathcal{G}})=\frac{1}{\sqrt{|H^{2}(M^{4},\mathbb{Z}_{2})|}}\sum_{[f_{3}]\in
H^{3}(M^{4},\mathbb{Z}_{2})}Z_{b}(M^{4},(h,c),A_{H},f_{3})z_{c}(M^{4},f_{3},\xi_{\mathcal{G}}),$
(3)

where $z_{c}(M^{4},f_{3},\xi_{\mathcal{G}})$ is a fermionic theory coupled
with ${\mathcal{G}}_{f}$-structure $\xi_{\mathcal{G}}$, which has the same ’t
Hooft anomaly as that of $Z_{b}$ characterized by the response action Eq. (2).
By stacking the bosonic path integral $Z_{b}$ with $z_{c}$, we obtain a
fermionic theory free of ’t Hooft anomaly with respect to the diagonal 2-form
$\mathbb{Z}_{2}$ symmetry. We can gauge the diagonal 2-form symmetry by
summing over distinct $[f_{3}]$ classes, resulting in a fermionic theory $Z$.
This gauging process implements fermion condensation within the path integral
framework.

The definition of the fermionic theory $z_{c}(M^{4},f_{3},\xi_{\mathcal{G}})$
is based on the path integral over Grassmann variables that decorate the
triangulation of a manifold; for orientable manifolds this is given by the
construction of Gu and Wen [22, 21], while the generalization to non-
orientable manifolds was given in [23]. The full generalization for
$\xi_{\mathcal{G}}$ was obtained in [11].

In both bosonic and fermionic cases, the path integral $Z$ is invariant under
re-triangulations and also invariant under gauge transformations of background
gauge fields and $\mathcal{G}_{f}$-structure, and thus gives a topological
invariant of a manifold $M^{4}$ with a given $\mathcal{G}_{f}$-structure and
background gauge field with non-trivial Chern class.

We argue that the path integral $Z$ always defines an invertible topological
quantum field theory (TQFT), and therefore describes a (3+1)D SPT. This
implies that the path integral should give a $\mathrm{U}(1)$ phase which is a
smooth bordism invariant. More precisely, in the bosonic case, our path
integrals define elements of the Pontryagin dual of the smooth bordism group
of 4-manifolds equipped with (possibly curved) background $G$ gauge fields. In
the fermionic case, the bordism groups are equipped with (possibly curved)
background $G_{b}$ gauge fields and twisted spin structures (i.e.
$\mathcal{G}_{f}$ structures).

When the input (2+1)D topological order characterized by the (super)-modular
category $\mathcal{C}$ is Abelian, we can explicitly perform the computation
of the path integral $Z$ on various 4-manifolds, for both bosonic and
fermionic theories. These path integrals give anomaly indicator formulas for a
given topological order $\mathcal{C}$ with a global symmetry, that indicates
the presence or absence of ’t Hooft anomaly of the given (2+1)D topological
order.

Below we list the results of the computations for Abelian topological order
$\mathcal{C}$ as follows. For the Abelian topological order, the results shown
below produce all of the anomaly indicator formulas proposed by Lapa and Levin
in [17]. In the case with the time-reversal symmetry, the indicators were
derived in [24] based on a completely different method where the time-reversal
is replaced by a crystalline reflection symmetry.

####  I.1.2 Abelian bosonic topological phases with $G=\mathrm{U}(1)$

Let $\mathcal{C}$ be an Abelian unitary modular tensor category (UMTC) with
$G=\mathrm{U}(1)$ symmetry. We consider a path integral on $\mathbb{CP}^{2}$
with nontrivial Chern class $\int F/(2\pi)=C\in\mathbb{Z}$ evaluated on a
fundamental 2-cycle that generates
$H_{2}(\mathbb{CP}^{2},\mathbb{Z})=\mathbb{Z}$. The partition function is
given by 111The partition function is a number given by evaluating a path
integral on a closed manifold. In this paper, we sometimes use the
terminologies “partition function” and “path integral” interchangeably.

$\displaystyle
Z(\mathbb{CP}^{2},(h,c))=\frac{1}{\mathcal{D}}\sum_{a\in\mathcal{C}}e^{-2\pi
iQ_{a}\cdot C}\theta_{a},$ (4)

where $Q_{a}\in\mathbb{R}/\mathbb{Z}$ is a fractional U(1) charge of an anyon
$a$.

The above theory is conjectured to give the bosonic invertible TQFT with U(1)
symmetry. The invertible TQFT is classified up to isomorphisms as the
Pontryagin dual of the 4D oriented bordism group
$\Omega_{4}^{\mathrm{SO}}(B\mathrm{U}(1))=\mathbb{Z}\times\mathbb{Z}$, given
by
$\mathrm{Hom}(\Omega_{4}^{\mathrm{SO}}(B\mathrm{U}(1)),\mathrm{U}(1))=\mathrm{U}(1)\times\mathrm{U}(1)$.
The $\mathrm{U}(1)\times\mathrm{U}(1)$ corresponds to the two continuous theta
terms given by

$\displaystyle S=\frac{\Theta}{8\pi^{2}}\int F\wedge F+\frac{2\pi
c_{-}}{192\pi^{2}}\int\mathrm{Tr}(R\wedge R),$ (5)

The theta parameter $\Theta$ has $4\pi$ periodicity for bosonic phases, while
$c_{-}$ has periodicity 8. On a $\mathbb{CP}^{2}$ with the Chern number
$C\in\mathbb{Z}$, the above action is evaluated as
$e^{\frac{i\Theta}{2}C^{2}+\frac{2\pi i}{8}c_{-}}$. We can then read each
parameter from the indicator formula in Eq. (4) as

$\displaystyle\begin{split}e^{\frac{i\Theta}{2}}&=\frac{Z(\mathbb{CP}^{2},C=1)}{Z(\mathbb{CP}^{2},C=0)}=\frac{\sum_{a\in\mathcal{C}}e^{-2\pi
iQ_{a}}\theta_{a}}{\sum_{a\in\mathcal{C}}\theta_{a}}\\\ e^{\frac{2\pi
i}{8}c_{-}}&={Z(\mathbb{CP}^{2},C=0)}=\frac{1}{\mathcal{D}}{\sum_{a\in\mathcal{C}}\theta_{a}}\end{split}$
(6)

The latter formula for $e^{\frac{2\pi i}{8}c_{-}}$ reproduces the well-known
Gauss-Milgram formula for chiral central charge $c_{-}$ of a UMTC
$\mathcal{C}$, which corresponds to the framing anomaly characterized by the
4D response action in the second term of Eq. (5).

Note that here the $\mathrm{U}(1)\times\mathrm{U}(1)$ classification does not
classify invertible topological phases of matter, which correspond to
deformation classes of invertible TQFTs; therefore the formulas above do not
correspond to ’t Hooft anomalies of the (2+1)D phase, which are conceptually
distinct from the framing anomaly.

The $\Theta$ term also determines the Hall conductivity of the (2+1)D surface
to be $\sigma_{H}=\frac{1}{2\pi}\frac{\Theta}{2\pi}\text{ mod }2$, since the
boundary of the electromagnetic theta term (5) reduces locally to the response
action $\frac{\Theta}{8\pi^{2}}A\mathrm{d}A$, which gives the Hall response
$\sigma_{H}$ on the boundary. Thus, defining
$\overline{\sigma}_{H}=2\pi\sigma_{H}$, we have

$\displaystyle
e^{i\Theta/2}=e^{i\pi\overline{\sigma}_{H}}=\theta_{v}^{*}=\frac{\sum_{a\in\mathcal{C}}e^{-2\pi
iQ_{a}}\theta_{a}}{\sum_{a\in\mathcal{C}}\theta_{a}},$ (7)

where $\theta_{v}$ is the topological twist of the vison, which is the anyon
induced by $2\pi$ flux insertion (see Section III.1 for a more precise
definition of the vison).

We also consider the partition function on $S^{2}\times S^{2}$ with nontrivial
Chern numbers $C_{\alpha},C_{\beta}\in\mathbb{Z}$ evaluated on each
fundamental 2-cycle that generates $H_{2}(S^{2}\times
S^{2},\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}$. The partition function is given
by

$\displaystyle\begin{split}Z(S^{2}\times
S^{2},(h,c))=\frac{1}{\mathcal{D}}\sum_{a,b\in\mathcal{C}}e^{-2\pi iQ_{a}\cdot
C_{\beta}}e^{-2\pi iQ_{b}\cdot
C_{\alpha}}S^{*}_{{a},b}=(\theta_{v}^{*})^{2C_{\alpha}C_{\beta}}.\end{split}$
(8)

The topological action in Eq. (5) evaluates on this $S^{2}\times S^{2}$ as
$e^{i\Theta C_{\alpha}C_{\beta}}$. We hence have

$\displaystyle e^{i\Theta
C_{\alpha}C_{\beta}}=(\theta_{v}^{*})^{2C_{\alpha}C_{\beta}},$ (9)

which is consistent with Eq. (7).

####  I.1.3 Abelian bosonic topological phases with
$G=\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{\bf T}$

For the symmetry group $G=\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{\mathbf{T}}$,
the classification of (3+1)D invertible TQFT is given by
$(\mathbb{Z}_{2})^{3}$ [25]. These are partially accounted for by the four
choices of $(\Theta,c_{-})$ that are compatible with time-reversal invariance:
$(\Theta,c_{-})=(0,0),(2\pi,0),(0,4),(2\pi,4)$. The formulas given above for
$(\Theta,c_{-})$ therefore give anomaly indicator formulas to diagnose the ’t
Hooft anomalies of (2+1)D topological order that arise at the surface of these
bosonic topological insulators. The third factor of $\mathbb{Z}_{2}$ is
distinguished by the path integral on $\mathbb{RP}^{4}$ [9].

####  I.1.4 Abelian bosonic topological phases with
$G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{\bf T}$

Let $\mathcal{C}$ be an Abelian UMTC with
$G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{\mathbf{T}}$ symmetry. We consider a
partition function on $S^{2}\times\mathbb{RP}^{2}$ with the nontrivial Chern
number $\int_{S^{2}}F/(2\pi)=C\in\mathbb{Z}$ evaluated on $S^{2}$. We assume
that the time-reversal symmetry $\mathbb{Z}_{2}^{\mathbf{T}}$ does not permute
the label of anyons. The partition function is then given by

$\displaystyle\begin{split}Z(S^{2}\times\mathbb{RP}^{2},(h,c),A_{H})=\frac{1}{\mathcal{D}}\sum_{a,b\in\mathcal{C}}e^{-2\pi
iQ_{b}C}\eta_{a}^{\mathbf{T}}S_{a,b}\end{split}$ (10)

where we define $\eta_{a}^{\mathbf{T}}:=\eta_{a}(\mathbf{T},\mathbf{T})$,
which determines whether the anyon $a$ carries a local Kramers degeneracy.

(3+1)D invertible TQFTs with
$G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{\mathbf{T}}$ symmetry are classified by
$(\mathbb{Z}_{2})^{4}$ [25], and the partition function on
$S^{2}\times\mathbb{RP}^{2}$ diagnoses one of these four topological terms
given by

$\displaystyle e^{iS}=(-1)^{\int w_{1}^{2}\frac{F}{2\pi}}.$ (11)

The formula Eq. (10) with $C=1$ indicates whether a given symmetry
fractionalization class for the (2+1)D topological order has this mixed ’t
Hooft anomaly. The other three topological terms are

$\displaystyle(-1)^{\int
w_{1}^{4}},\quad(-1)^{\int\left(\frac{F}{2\pi}\right)^{2}},\quad(-1)^{\int
w_{2}^{2}},$ (12)

where the action $(-1)^{\int w_{1}^{4}}$ is detected by the partition function
on $\mathbb{RP}^{4}$ [9], and the rest are diagnosed by the indicators
$e^{\frac{i\Theta}{2}},e^{\frac{2\pi i}{8}c_{-}}$ obtained in Eq. (6).

####  I.1.5 Abelian fermionic topological phases with
$G_{f}=\mathrm{U}(1)^{f}$ and Spinc Gauss-Milgram formula

Let $\mathcal{C}$ be an Abelian super-modular tensor category with
$G_{b}=\mathrm{U}(1)$ symmetry. We consider a partition function $Z$ of a
fermionic theory on $\mathbb{CP}^{2}$ equipped with a Spinc structure
$\xi_{\mathcal{G}}$, with nontrivial Chern class $\int
F/(2\pi)=C\in\mathbb{Z}+1/2$ evaluated on a fundamental 2-cycle that generates
$H_{2}(\mathbb{CP}^{2},\mathbb{Z})=\mathbb{Z}$. The partition function is
given by

$\displaystyle
Z(\mathbb{CP}^{2},(h,c),\xi_{\mathcal{G}})=\frac{1}{\sqrt{2}\mathcal{D}}\sum_{a\in\mathcal{C}}e^{-2\pi
iQ_{a}\cdot C}\theta_{a},$ (13)

where $Q_{a}\in\mathbb{R}/(2\mathbb{Z})$ is a fractional U(1) charge of an
anyon $a$.

(3+1)D invertible Spinc TQFTs are classified by
$\mathrm{U}(1)\times\mathrm{U}(1)$ up to isomorphisms, which is the Pontryagin
dual of the bordism group
$\Omega_{4}^{\mathrm{Spin}^{c}}=\mathbb{Z}\times\mathbb{Z}$. The
$\mathrm{U}(1)\times\mathrm{U}(1)$ corresponds to the two continuous theta
terms in the topological effective action [26]

$\displaystyle S=\Theta_{1}I_{1}+\Theta_{2}I_{2},$ (14) $\displaystyle
I_{1}=\int\left(-\frac{1}{192\pi^{2}}\mathrm{Tr}(R\wedge
R)+\frac{1}{8\pi^{2}}F\wedge F\right),\quad I_{2}=\frac{1}{\pi^{2}}\int
F\wedge F,$ (15)

with the Dirac quantization condition $\int F/(2\pi)=\int w_{2}/2$ on closed
oriented 2-cycles. Both $I_{1}$ and $I_{2}$ are integers on Spinc manifolds,
so $\Theta_{1}$ and $\Theta_{2}$ have periodicity $2\pi$. On $\mathbb{CP}^{2}$
with the Chern number $C$, the above response action evaluates to
$\exp(i\Theta_{1}(-\frac{1}{8}+\frac{C^{2}}{2})+\Theta_{2}4C^{2})$. We can
then read off each theta parameter from the indicator formula

$\displaystyle e^{i\Theta_{1}}=e^{-2\pi
ic_{-}}=\frac{Z\left(\mathbb{CP}^{2},C=\frac{3}{2}\right)}{Z\left(\mathbb{CP}^{2},C=\frac{1}{2}\right)^{9}},\quad
e^{i\Theta_{2}}=Z\left(\mathbb{CP}^{2},C=\frac{1}{2}\right).$ (16)

Here, the chiral central charge $c_{-}$ of the Spinc topological order is
given by $c_{-}=-\Theta_{1}/(2\pi)$ mod 1.

The formula for $e^{i\Theta_{1}}$ can be regarded as a generalization of the
Gauss-Milgram formula Eq. (6) of $c_{-}$ to the Spinc case. In other words,
this provides a simple formula to determine the chiral central charge
$c_{-}\text{ mod }1$, given the data of the super-modular category with
$\mathrm{U}(1)^{f}$ fermionic symmetry fractionalization. For non-Abelian
topological orders, we conjecture that the partition function on
$\mathbb{CP}^{2}$ is given by

$\displaystyle
Z(\mathbb{CP}^{2},(h,c),\xi_{\mathcal{G}})=\frac{1}{\sqrt{2}\mathcal{D}}\sum_{a\in\mathcal{C}}d_{a}^{2}e^{-2\pi
iQ_{a}\cdot C}\theta_{a},$ (17)

and the formulas for the theta parameters are given by Eq. (16). In Appendix
D, we explicitly compute the indicator formula for $e^{i\Theta_{1}}$ for
several Read-Rezayi fractional quantum Hall states which are non-Abelian [27],
and demonstrate that it exactly gives $c_{-}$ mod 1.

Note that even though it is known that a super-modular category alone, without
any additional global symmetry, determines the chiral central charge
$c_{-}\text{ mod }1/2$, a simple explicit formula generalizing the Gauss-
Milgram sum is not known, although it is known that it should correspond to
the topological fermionic path integral evaluated on the K3 surface with a
choice of spin structure [28].

####  I.1.6 Fermionic topological phases in class AII or AIII

Here we consider general (2+1)D fermionic topological phases, which may be
non-Abelian, characterized by a general super-modular category $\mathcal{C}$,
in symmetry class AII or AIII. Symmetry class AII corresponds to
$G_{f}=[\mathrm{U}(1)^{f}\times\mathbb{Z}_{4}^{{\bf T},f}]/\mathbb{Z}_{2}$,
where $\mathbb{Z}_{4}^{{\bf T},f}$ is the non-trivial extension of
$\mathbb{Z}_{2}^{\bf T}$ by $\mathbb{Z}_{2}^{f}$, and the mod $\mathbb{Z}_{2}$
identifies the $Z_{2}^{f}$ from the two factors. Symmetry class AIII
corresponds to $G_{f}=\mathrm{U}(1)^{f}\times\mathbb{Z}_{2}^{\bf T}$. The
resulting TQFTs for class AII and AIII depend on a Pin${}^{\tilde{c}}_{+}$ or
Pinc structure, respectively.

In these cases we can compute the partition function on $\mathbb{RP}^{4}$ with
U(1) gauge field turned off, with the result

$\displaystyle Z(\mathbb{RP}^{4},\xi_{\mathcal{G}})$
$\displaystyle=\frac{1}{\sqrt{2}\mathcal{D}}\left(\sum_{x|x={\,{}^{\bf
T}}x}d_{x}\theta_{x}\eta^{\bf T}_{x}\pm i\sum_{x|x={\,{}^{\bf
T}}x\times\psi}d_{x}\theta_{x}\eta^{\bf T}_{x}\right)$ (18)

Here, $\xi_{\mathcal{G}}$ denotes the Pin${}^{\tilde{c}}_{+}$ or Pinc
structure, and the choice of sign in $\pm i$ depends on the possible two
choices of $\xi_{\mathcal{G}}$ structure. $\eta^{\bf T}_{x}$ is given by

$\eta_{a}^{\bf T}:=\begin{cases}\eta_{a}({\bf T},{\bf T})&\,{}^{\bf T}a=a\\\
\eta_{a}({\bf T},{\bf T})U_{\bf T}(a,\psi;a\psi)F^{a,\psi,\psi}&\,{}^{\bf
T}a=a\times\psi,\end{cases}$ (19)

and determines the Kramers degeneracy of the anyon $a$.

The computation of the $\mathbb{RP}^{4}$ partition function is essentially the
same as that done in [11] for Pin+ structure, which corresponds to class DIII
(where $G_{f}=\mathbb{Z}_{4}^{\mathbf{T},f}$). $\mathbb{RP}^{4}$ generates the
$\mathbb{Z}_{2}$ subclass of the Pin${}^{\tilde{c}}_{+}$ bordism group
$\Omega_{4}^{\mathrm{Pin}^{\tilde{c}}_{+}}=(\mathbb{Z}_{2})^{3}$, and
generates the $\mathbb{Z}_{8}$ subclass of the Pinc bordism group
$\Omega_{4}^{\mathrm{Pin}^{c}}=\mathbb{Z}_{8}\times\mathbb{Z}_{2}$ [29, 30].
These classes correspond to the topological insulator in class AII and
topological superconductor in class AIII respectively.

Therefore, given a (2+1)D fermionic topological order with a choice of
$\\{\eta_{a}^{\bf T}\\}$, if the theory is compatible with class AII, Eq. (18)
will evaluate to $\pm 1$, while if the theory is compatible with class AIII,
Eq. (18) will evaluate to an 8th root of unity.

One of the $\mathbb{Z}_{2}^{3}$ factors for the classification in class AII
corresponds to the conventional topological band insulator, with
electromagnetic response given by $\Theta_{1}=\pi$ in Eq. (15) on oriented
manifolds. The other two generators have topological actions given by [31]

$\displaystyle(-1)^{\int w_{1}^{4}},\quad(-1)^{\int w_{2}^{2}}.$ (20)

Since both the topological path integral for the conventional topological band
insulator and $(-1)^{\int w_{1}^{4}}$ evaluate to $-1$ on $\mathbb{RP}^{4}$
[32, 33], $Z(\mathbb{RP}^{4},\xi_{\mathcal{G}})$ cannot distinguish these two
phases. The indicator that can distinguish the topological insulator is given
by the $S^{2}\times S^{2}$ partition function with $\frac{1}{8\pi^{2}}\int
FF=1$. We conjecture that the $S^{2}\times S^{2}$ partition function has the
form

$\displaystyle Z(S^{2}\times
S^{2},(h,c))=\frac{1}{2\mathcal{D}}\sum_{a,b\in\mathcal{C}}d_{a}d_{b}e^{-2\pi
iQ_{a}\cdot C_{\beta}}e^{-2\pi iQ_{b}\cdot C_{\alpha}}S^{*}_{{a},b},$ (21)

which is regarded as a generalization of Eq. (8) to the non-Abelian and
fermionic case, proposed by [17].

The (3+1)D theory that generates the $\mathbb{Z}_{8}$ subclass of class AIII
is generated by a topological superconductor, which again has a
$\mathrm{U}(1)$ response $\Theta_{1}=\pi$ on oriented manifolds [32, 26]. The
other $\mathbb{Z}_{2}$ class is generated by the bosonic action $(-1)^{\int
w_{2}^{2}}$, which is diagnosed by $e^{i\Theta_{2}}$ in Eq. (16).

###  I.2 Organization of paper

This paper is organized as follows. In Sec. II, we introduce the Villain
formulation for the background U(1) gauge field utilized throughout this paper
to describe the curved U(1) bundle. In Sec. III, we first review the symmetry
fractionalization of U(1) symmetry. We then introduce a new graphical calculus
to describe the (2+1)D anyon systems coupled to the U(1) Villain gauge field,
which is beyond the scope of the existing symmetry fractionalization theory.
In Sec. IV, we provide the (3+1)D state sum path integral based on the Villain
graphical calculus in the simplest setup for a bosonic phase with U(1)
symmetry, which can describe the electromagnetic response of U(1) theta term.
In Sec. V, we generalize the bosonic state sum to the case with
$\mathrm{U}(1)\rtimes H$ symmetry, where $H$ is a symmetry group that can act
on $\mathrm{U}(1)$ by charge conjugation. In Sec. VI, we review the fermion
condensation utilized to construct a path integral of fermionic SPT phases
starting with a path integral of a bosonic theory, referred to as the bosonic
shadow. In Sec. VII, we construct a bosonic path integral that gives a (3+1)D
fermionic SPT phase with $\mathrm{U}(1)\rtimes H$ symmetry via fermion
condensation. For example, the resulting fermionic theory can describe Spinc
theta terms, a topological insulator in class AII and topological
superconductor in class AIII. In Sec. VIII, we compute the partition functions
of our path integrals for various manifolds, and prove the anomaly indicator
formulas proposed by Lapa and Levin for the case of Abelian topological phases
based on our path integral. We close with some discussion of open questions in
Sec. IX. Many technical discussions and detailed calculations are relegated to
appendices.

##  II U(1) gauge field on lattice: Villain formulation

###  II.1 U(1) gauge field

Let us consider a $\mathrm{U}(1)$ gauge theory on a $d$-dimensional manifold
$M$ equipped with a triangulation and a branching structure, following the
Villain-type formulation [34, 35]. That is, we start with a non-compact
$\mathbb{R}$ gauge field on each 1-simplex $h\in C^{1}(M,\mathbb{R})$. Then,
we gauge the $\mathbb{Z}$ 1-form symmetry of the theory by introducing a
2-form $\mathbb{Z}$ gauge field $c\in Z^{2}(M,\mathbb{Z})$ on each 2-simplex.
Here, $C^{k}(M,X)$ and $Z^{k}(M,X)$ denote a $k^{\mathrm{th}}$ cochain and
cocycle with coefficient $X$ respectively. We then postulate the gauge
invariance

$\displaystyle c\to c-\delta\lambda,\quad h\to h+\lambda$ (22)

with $\lambda\in C^{1}(M,\mathbb{Z})$. The resulting theory realizes a
$\mathbb{R}/\mathbb{Z}=\mathrm{U}(1)$ gauge theory whose field strength is
given by

$\displaystyle\frac{F}{2\pi}=c+\delta h,$ (23)

where the field strength $F$ is normalized so that it satisfies the Dirac
quantization condition $\int F=2\pi\mathbb{Z}$ on oriented 2-cycles. The usual
U(1) gauge transformation is realized by

$\displaystyle c\to c,\quad h\to h+\delta\chi$ (24)

with $\chi\in C^{0}(M,\mathbb{R})$.

Therefore, a background U(1) gauge field can be described by the pair of
background fields $(h,c)$, subject to the gauge transformation (22).

Note that one way to understand the above Villain formulation is as follows.
Usually to describe a $\mathrm{U}(1)$ gauge field, we define a
$\mathrm{U}(1)\simeq\mathbb{R}/\mathbb{Z}$ gauge field on links,
$a_{ij}\in\mathbb{R}/\mathbb{Z}$. The field strength in this case would be
$\delta a\in\mathbb{R}/\mathbb{Z}$; since this is a coboundary, the Chern
number will always vanish on closed manifolds. In order to describe a manifold
with non-trivial Chern number, we need to resolve a contribution to the flux
per plaquette that is not a coboundary. This can be done by defining a lift
$h$ of $a$ from $\mathbb{R}/\mathbb{Z}$ to $\mathbb{R}$, and introducing an
additional degree of freedom $c\in\mathbb{Z}$ on each $2$-simplex, which keeps
track of the integer part of the flux that is not a coboundary. The total flux
is then $F=c+\delta h$. Since we are really describing a $\mathrm{U}(1)$ gauge
field, the only gauge invariant quantities are $\mathbb{R}/\mathbb{Z}$-valued
holonomies through non-contractible cycles and the field strength $F$ through
each plaquette. Therefore we have the gauge transformation $h\rightarrow
h+\lambda$, $c\rightarrow c-\delta\lambda$ for $\lambda\in
C^{1}(M,\mathbb{Z})$, which keeps all physical quantities invariant.

###  II.2 $\mathrm{U}(1)^{f}$ gauge field and Spinc structure

We are also interested in fermionic theories, where the symmetry group
contains a $\mathrm{U}(1)^{f}$ factor. Here $\mathrm{U}(1)^{f}$ means that the
$\pi$ rotation in the $\mathrm{U}(1)$ is associated with fermion parity. In
other words, $\mathrm{U}(1)^{f}$ is a non-trivial central extension of
$\mathrm{U}(1)$ by fermion parity, $\mathbb{Z}_{2}^{f}$.

In this case, the fermionic path integral depends on a background
$\mathrm{U}(1)$ gauge field $A$ and a Spinc structure $\xi$, which is a
trivialization of $w_{2}+F/\pi\text{ mod 2}$:

$\displaystyle\delta\xi=w_{2}+\frac{F}{\pi}\text{ mod 2},$ (25)

where $w_{2}\in H^{2}(M,\mathbb{Z}_{2})$ is the 2nd Stiefel-Whitney class.

If we describe the $\mathrm{U}(1)$ gauge field in terms of $(h,c)$ as
discussed above, then $\int_{C}\frac{F}{2\pi}=\int_{C}c$ and so we require

$\displaystyle\int_{C}c=\frac{1}{2}\int_{C}w_{2}\quad\mod 1,$ (26)

for any oriented two-dimensional cycle $C$ in the spacetime. This is realized
by the gauge fields $(h,c)$ with the field strength $F/(2\pi)=c+\delta h$,
setting $h\in C^{1}(M,\mathbb{R})$, $2c\in\mathbb{Z}$ on each 2-simplex and
then $[2c]=[w_{2}]$ mod $2$ as an element of $H^{2}(M,\mathbb{Z}_{2})$. The
gauge transformation is again given by (22).

Note that in the above, the gauge field $A$ is the gauge field for the bosonic
symmetry $\mathrm{U}(1)^{f}/\mathbb{Z}_{2}^{f}\simeq\mathrm{U}(1)$, for which
the periodicity of the $\mathrm{U}(1)$ is $1/2$. One could choose a different
normalization for the gauge field so that the periodicity is $1$, which would
replace $F/\pi$ by $F/(2\pi)$ in the above equations. We will see that in the
normalization we chose above, the fermion $\psi$ has a charge
$Q_{\psi}=1\text{ mod 2}$ under the $\mathrm{U}(1)$; changing the
normalization would change the fermion charge to $1/2\text{ mod 1}$.

##  III U(1) symmetry

In this section, we discuss a braided tensor category (BTC) in the presence of
a global U(1) symmetry. See Appendix A for a general review of BTC. The global
symmetry of the BTC is described by the action of symmetry defects on its
anyon data. The symmetry defect is equivalent to the background gauge field of
the global symmetry inserted in its Poincaré dual. When a given symmetry
defect corresponds to the flat background gauge field, the description of
symmetry defects in the BTC is fully established within the framework of
symmetry fractionalization theory developed in [1, 2, 3] based on G-crossed
BTC, see Appendix B for a detailed review.

We want to construct a (3+1)D state sum path integral coupled with a curved
U(1) bundle. To do this, we extend the symmetry fractionalization theory to
deal with the non-flat background gauge field. This gives a new diagrammatic
calculus utilizing the Villain gauge fields, and enables us to construct a
(3+1)D state sum path integral coupled with Villain U(1) gauge field, starting
with a U(1) symmetry fractionalization data of the BTC. In Sec. III.1, we give
a brief explanation of U(1) symmetry fractionalization for flat U(1) gauge
field, and then extend the framework to the non-flat U(1) gauge field in Sec.
III.2. The symmetry fractionalization data and the calculus of Villain gauge
fields for $\mathrm{U}(1)\rtimes H$ symmetry is developed in Appendix C.

###  III.1 U(1) symmetry fractionalization

####  III.1.1 Bosonic case: modular tensor category

Here we briefly summarize the symmetry fractionalization data of unitary
modular tensor category (UMTC) with global $G=\mathrm{U}(1)$ symmetry
following [1, 4]. Most generally in BTC, the global symmetry acts on the
anyons and the topological state space through the action of a group
homomorphism

$\displaystyle[\rho]:G\rightarrow\text{Aut}(\mathcal{C}).$ (27)

The elements of $\text{Aut}(\mathcal{C})$ in general act by permuting the
anyons. Since $\text{Aut}(\mathcal{C})$ is a finite group, when
$G=\mathrm{U}(1)$ the map $[\rho_{\bf g}]$ must be trivial.

A given representative $\rho_{\bf g}$ defines $N_{ab}^{c}\times N_{ab}^{c}$
matrices $U_{\bf g}(a,b;c)$, which characterize the action of $\rho_{\bf g}$
on the fusion and splitting spaces in a given basis. Once a representative
$\rho$ is specified, the symmetry fractionalization pattern is characterized
by the phases $\eta_{a}({\bf g},{\bf h})$, defined diagrammatically in Fig. 1.
These $\eta$ and $U$’s are subject to the consistency equations

$\displaystyle\begin{split}{U}_{\mathbf{g}}({a},{b};{c}){U}_{\mathbf{h}}({a},{b};{c})&={U}_{\mathbf{gh}}({a},{b};{c})\frac{\eta_{c}({\bf
g},{\bf h})}{\eta_{a}({\bf g},{\bf h})\eta_{b}({\bf g},{\bf h})},\\\
{\eta}_{a}({\bf h},{\bf k}){\eta}_{a}({\bf gh},{\bf k})&={\eta}_{a}({\bf
g},{\bf h}){\eta}_{a}({\bf g},{\bf hk}).\end{split}$ (28)

Figure 1: Anyon lines (black) passing through branch sheets (orange) and
graphical definitions of the $U$ and $\eta$ symbols.

For a given set of $\\{\eta,U\\}$, one can define fractional charges of anyons
that characterizes the U(1) symmetry fractionalization as follows. For a fixed
anyon $a$, let $n$ be the smallest integer such that $a^{n}$ contains the
identity as a fusion product. Choose a sequence of anyons $a,a^{2},\dots
a^{n}=1$ such that $a\times a^{k}$ contains $a^{k+1}$ as a fusion product.
Then define a fractional charge $Q_{a}\in\mathbb{R}/\mathbb{Z}$ as [2]

$\displaystyle e^{2\pi
iQ_{a}}:=\prod_{m=1}^{n-1}\eta_{a}\left(\frac{1}{n},\frac{m}{n}\right)U_{\frac{1}{n}}(a,a^{m};a^{m+1}),$
(29)

where the elements of $\mathrm{U}(1)=\mathbb{R}/\mathbb{Z}$ is labeled by the
numbers in $[0,1)$. One can check that the quantity $e^{2\pi iQ_{a}}$ is
gauge-invariant. Since the map $[\rho]$ is trivial, we can fix a gauge where
$U=1$, so that $\eta$ satisfies ${\eta_{c}({\bf g},{\bf h})}=\eta_{a}({\bf
g},{\bf h})\eta_{b}({\bf g},{\bf h})$ when $N^{c}_{ab}>0$. One can then see
that

$\displaystyle e^{2\pi iQ_{a}}e^{2\pi iQ_{b}}=e^{2\pi iQ_{c}}\quad\text{when
$N_{a,b}^{c}\neq 0$.}$ (30)

In the gauge where $U=1$, we can write the phases $\eta_{a}({\bf g},{\bf h})$
as

$\displaystyle\eta_{a}({\bf g},{\bf h})=M_{a,\mathfrak{t}({\bf g},{\bf h})},$
(31)

for $\mathfrak{t}\in Z^{2}(B\mathrm{U}(1),\mathcal{A})$. A representative
2-cocycle $\mathfrak{t}$ is given by

$\displaystyle\mathfrak{t}({\bf g},{\bf h})=v^{{\bf g}+{\bf h}-[{\bf g}+{\bf
h}]},$ (32)

where $v\in\mathcal{A}$ is referred to as the vison,
$\mathbf{g},\mathbf{h}\in\mathbb{R}/\mathbb{Z}$ takes the values in $[0,1)$,
and $[\mathbf{g}+\mathbf{h}]$ means the sum mod 1.

The fractional charge can be written as [4]

$\displaystyle e^{2\pi iQ_{a}}=M_{a,v},$ (33)

in terms of which we then have

$\displaystyle\eta_{a}(\mathbf{g},\mathbf{h})=e^{2\pi
iQ_{a}(\mathbf{g}+\mathbf{h}-[\mathbf{g}+\mathbf{h}])}.$ (34)

####  III.1.2 Fermionic case: super-modular tensor category

In fermionic topological phases, we consider the fermionic
$G_{f}=\mathrm{U}(1)^{f}=\mathbb{R}/\mathbb{Z}$ global symmetry given by the
symmetry extension

$\displaystyle\mathbb{Z}_{2}^{f}\to G_{f}\to G_{b},$ (35)

where $G_{b}=\mathrm{U}(1)$ is the bosonic symmetry group. Note that
$G_{b}=\mathbb{R}/(\mathbb{Z}/2)$ has periodicity 1/2, since we want the
fermionic group $G_{f}=\mathbb{R}/\mathbb{Z}$ to have periodicity 1.

In this case, the symmetry fractionalization for the anyons is formulated by
considering the symmetry action of $G_{b}$ on the super-modular category
$\mathcal{C}$.

We demand, as reviewed in Appendix B, that [2, 3]

$\displaystyle U_{\bf g}(\psi,\psi;1)$ $\displaystyle=1$ (36)
$\displaystyle\eta_{\psi}({\bf g},{\bf h})$ $\displaystyle=\omega_{2}({\bf
g},{\bf h})$ (37)

where $[\omega_{2}]\in H^{2}(BG_{b},\mathbb{Z}_{2})$ is the cohomology class
specifying $G_{f}$ as a group extension of $G_{b}$ by $\mathbb{Z}_{2}$.

For fermionic symmetry fractionalization defined for super-modular categories,
the first step is to pick a map

$\displaystyle[\rho]:G_{b}\rightarrow\text{Aut}_{LR}(\mathcal{C}),$ (38)

where $\text{Aut}_{LR}(\mathcal{C})$ is the group of locality-respecting auto-
equivalences of $\mathcal{C}$ [2, 3].

Given a choice of $[\omega_{2}]$ and $[\rho]$, there can be obstructions to
defining any consistent pattern of symmetry fractionalization. In the case of
$G_{f}=\mathrm{U}(1)^{f}$, one can define symmetry fractionalization only for
those super-modular categories $\mathcal{C}$ for which the map
$\Upsilon_{\psi}$, is locality-respecting [2]. (see Appendix B for a
definition of $\Upsilon_{\psi}$ and the locality respecting condition). For
the case of $\mathrm{U}(1)^{f}$, one can also show that the map $[\rho]$ must
be the identity map for those super-modular categories that admit a consistent
definition of symmetry fractionalization [2].

For super-modular categories where there is no obstruction to defining
$\mathrm{U}(1)^{f}$ symmetry fractionalization, we can define fractional
charges of anyons analogously to the bosonic case. For a fixed anyon $a$, let
$n$ be the smallest integer such that $a^{n}$ contains the identity as a
fusion product. Choose a sequence of anyons $a,a^{2},\dots a^{n}=1$ such that
$a\times a^{k}$ contains $a^{k+1}$ as a fusion product. Then define a
fractional charge $Q_{a}\in\mathbb{R}/(2\mathbb{Z})$ as [2]

$\displaystyle e^{\pi
iQ_{a}}:=\prod_{m=1}^{n-1}\eta_{a}\left(\frac{1}{2n},\frac{m}{2n}\right)U_{\frac{1}{2n}}(a,a^{m};a^{m+1}),$
(39)

where the elements of $G_{b}$ is labeled by numbers in $[0,1/2)$.222Note that
our normalizations are different by a factor of $2$ compared with [2].

Since $[\rho]$ is the identity map, there is a gauge in which we can set
$U=1$, so that $\eta$ satisfies ${\eta_{c}({\bf g},{\bf h})}=\eta_{a}({\bf
g},{\bf h})\eta_{b}({\bf g},{\bf h})$ when $N^{c}_{ab}>0$. Then one can see
that $e^{\pi iQ_{a}}$ satisfies

$\displaystyle e^{\pi iQ_{a}}e^{\pi iQ_{b}}=e^{\pi iQ_{c}}\quad\text{when
$N_{ab}^{c}\neq 0$}.$ (40)

Since $\omega_{2}({\bf g},{\bf h})=e^{2\pi
i(\mathbf{g}+\mathbf{h}-[\mathbf{g}+\mathbf{h}]_{1/2})}$, the fractional
charge of a transparent fermion $\psi$ satisfies

$\displaystyle Q_{\psi}=1\mod 2.$ (41)

For fixed fractional charges $e^{\pi iQ_{a}}$, an example of
$\eta_{a}(\mathbf{g},\mathbf{h})$ realizing the fractional charges $e^{2\pi
iQ_{a}}$ is given by

$\displaystyle\eta_{a}(\mathbf{g},\mathbf{h})=e^{2\pi
iQ_{a}(\mathbf{g}+\mathbf{h}-[\mathbf{g}+\mathbf{h}]_{1/2})},$ (42)

where $\mathbf{g},\mathbf{h}\in\mathbb{R}/(\mathbb{Z}/2)$ takes the values in
$[0,1/2)$, and $[\mathbf{g}+\mathbf{h}]_{1/2}$ means the sum mod 1/2.

###  III.2 New graphical calculus with Villain symmetry defects

Here, we provide a new diagrammatic calculus to describe the action of U(1)
symmetry defects on anyons of the BTC, by utilizing the Villain formulation of
the curved U(1) background gauge field. This is done by extending the
diagrammatics of U(1) symmetry fractionalization to the case with non-flat
gauge fields described by the Villain gauge field $(h,c)$. The new
diagrammatics takes the symmetry fractionalization data of the BTC
characterized by the fractional charge $Q_{a}\in\mathbb{R}/\mathbb{Z}$ as an
input. This is utilized to define a (3+1)D state sum path integral coupled
with curved U(1) gauge fields, based on a given symmetry fractionalization
data for a BTC.

Let us consider the U(1) Villain gauge field $(h,c)\in
C^{1}(M,\mathbb{R})\times Z^{2}(M,\mathbb{Z})$. Since the codimension-1
symmetry defect sheet (Poincaré dual of $h$) is now valued in $\mathbb{R}$,
the fractional charge of anyons should also take value in $\mathbb{R}$. Let us
denote the charge $q_{a}\in\mathbb{R}$ of an anyon $a$, which is defined by
taking the $\mathbb{R}$ lift of $Q_{a}\in\mathbb{R}/\mathbb{Z}$. We describe
the $\mathbb{R}$ symmetry action on anyons in the diagrammatics as shown in
Fig. 2, which projects the 3D configuration of defects onto a plane. The
symmetry actions are then encoded in the diagrammatic calculus shown in Fig.
3. We note that the calculus shown in Fig. 3 depends on a choice of the
$\mathbb{R}$ lift $q_{a}\in\mathbb{R}$ of the fractional charge
$Q_{a}\in\mathbb{R}/\mathbb{Z}$, but we will later see in Sec. IV that the
resulting (3+1)D state sum topological invariant does not depend on the
$\mathbb{R}$ lift of fractional charges. Our (3+1)D state sum path integral
hence gives a well-defined theory based on a given data of $\\{Q_{a}\\}$.

Figure 2: The $\mathbb{R}$ symmetry defect is realized as a codimension-1
plane in (2+1)D, and we describe the symmetry action on the anyons by the
diagrammatics given by projecting the 3D picture onto a plane, where the
$\mathbb{R}$ defects are expressed as orange lines. The orange bold dot at the
junction of $\mathbb{R}$ defects indicates that $h$ is not closed in general
and $\mathbb{R}$ defects can end at the junction. Figure 3: The graphical
calculus for the $\mathbb{R}$ symmetry defects. (a) This figure reflects that
the anyon $a$ is charged by $e^{2\pi iq_{a}h}$ under the $\mathbb{R}$ symmetry
action. (b) The difference of the $\mathbb{R}$ symmetry action under fusion of
anyons is expressed via the difference of $\mathbb{R}$ fractional charges
$\delta q_{a}:=q(a_{ij})+q(a_{jk})-q(a_{ik})$.

The 2-form field $c\in Z^{2}(M,\mathbb{Z})$ is regarded as a defect line for a
$\mathbb{Z}$ 1-form symmetry, which acts on an anyon by the mutual braiding,
as shown in Fig. 4. This corresponds to the Aharanov-Bohm phase between the
unit magnetic flux carried by the defect line and the charged anyon.

Figure 4: The action of the $\mathbb{Z}$ 1-form symmetry on anyons. The $c$
line acts on anyons as a ’t Hooft line which emits Aharanov-Bohm phase by
braiding with an anyon.

We have to check that the symmetry actions of the U(1) symmetry defect $(h,c)$
are independent of the gauge transformation of the $\mathbb{Z}$ 1-form
symmetry $(h,c)\to(h+\lambda,c-\delta\lambda)$ with $\lambda\in
C^{1}(M,\mathbb{Z})$, which is regarded as redundancy to express the same
configuration of U(1) background gauge field. This puts a constraint on the
fractional charges $\\{q_{a}\\}$ carried by anyons. To see this, consider a
shift of the symmetry defect $h\to h+\delta\Lambda$ by $\Lambda\in
C^{0}(M,\mathbb{Z})$ using the gauge transformation $\lambda=\delta\Lambda$.
Since $c$ is invariant under this transformation, the process does not
generate the defect line for $c$. When the $h$ defect shifts across the fusion
vertex of anyons as shown in Fig. 3 (b) by this transformation, it emits a
phase $e^{-2\pi i(\delta q)\delta\Lambda}$, which should always be 1. We hence
require that

$\displaystyle e^{2\pi iq_{a}}e^{2\pi iq_{b}}=e^{2\pi iq_{c}}\quad\text{when
$N_{a,b}^{c}\neq 0$,}$ (43)

which is satisfied by Eq. (30), since $q_{a}$ is $\mathbb{R}$ lift of $Q_{a}$.
We note that the symmetry actions of the defects $(h,c)$ on anyons illustrated
in Figs. 2, 4 are consistent with the gauge invariance under
$(h,c)\to(h+\lambda,c-\delta\lambda)$ with $\lambda\in C^{1}(M,\mathbb{Z})$.
To see this, consider shifting the gauge field locally by
$(h,c)=(\lambda,-\delta\lambda)$, which generates a small disc of a symmetry
defect $h=\lambda$ in the spacetime, bounded by the $c=-\delta\lambda$ defect
line. One can see that this object acts trivially on anyons, since $h$ acts as
$e^{2\pi i\lambda}$, while $c$ acts as $e^{-2\pi i\lambda}$ by braiding, and
these contributions cancel out. See Fig. 5.

Figure 5: The local $\mathbb{Z}$ gauge transformation generates a bubble of
symmetry defects $(h,c)=(\lambda,-\delta\lambda)$. This bubble acts trivially
on anyons since the 0-form symmetry defect acts as $e^{2\pi i\lambda}$ by
crossing with the end of the defect, while $c$ acts as $e^{-2\pi i\lambda}$ by
braiding, and these contributions cancel out. This makes our diagrammatics
consistent with the invariance under the $\mathbb{Z}$ gauge transformation.

The background gauge field $(h,c)\in C^{1}(M,\mathbb{R})\times
Z^{2}(M,\mathbb{Z})$ is regarded as the symmetry defect of a $\mathbb{R}$
0-form symmetry and $\mathbb{Z}$ 1-form symmetry respectively. Our calculus is
then understood as encoding the action of $\mathbb{Z}$ 1-form symmetry in
addition to the 0-form $\mathbb{R}$ symmetry action, in a way respecting the
gauge invariance Eq. (22). We will see that this diagrammatic calculus makes
it possible to define a (3+1)D state-sum topological invariant correctly
coupled with the Villain U(1) gauge field.

The diagrammatic calculus developed in this context is therefore beyond the
usual one used to describe symmetry fractionalization [1, 2, 3], which only
considers flat background gauge fields for the 0-form symmetries. Note that
the full formalism of G-crossed BTCs does incorporate non-flat gauge fields,
corresponding to cases in which the codimension-1 symmetry defects end at a
codimension-2 line in space-time; however, the precise relationship between
the formalism we have developed here and the usual one for G-crossed BTCs is
unclear.

#### Villain symmetry defects for $\mathrm{U}(1)^{f}$ symmetry of super-
modular tensor categories

In the case of super-modular categories with $\mathrm{U}(1)^{f}$ symmetry, the
local fermion $\psi$ carries charge $1$ under the $\mathrm{U}(1)^{f}$
symmetry. Under the bosonic
$\mathrm{U}(1)=\mathrm{U}(1)^{f}/\mathbb{Z}_{2}^{f}$ symmetry, we choose the
convention that $\psi$ continues to carry charge $1$, so that the bosonic
$\mathrm{U}(1)=\mathbb{R}/(\mathbb{Z}/2)$, which is half the quantization of
the $\mathrm{U}(1)^{f}$.

In this case, we use the Villain gauge fields $(h,c)$ to describe the
background $\mathrm{U}(1)$ gauge field. The $\mathbb{R}$ symmetry action of
the $h$ defects on an anyon $a\in\mathcal{C}$ are again described in terms of
the fractional charges $q_{a}\in\mathbb{R}$, which in this case is defined by
lifting the U(1) fractional charges $Q_{a}\in\mathbb{R}/(2\mathbb{Z})$ to
$\mathbb{R}$.

The symmetry defects act on anyons according to the diagrammatics shown in
Fig. 2, 4. The fractional charges are subject to the constraint

$\displaystyle e^{\pi iq_{a}}e^{\pi iq_{b}}=e^{\pi iq_{c}}\quad\text{when
$N_{a,b}^{c}\neq 0$,}$ (44)

reflecting Eq. (40).

Due to Eq. (41), the fractional charge of the transparent fermion $\psi$
satisfies

$\displaystyle q_{\psi}=1\quad\mod 2.$ (45)

##  IV Bosonic state sum of (3+1)D TQFT with U(1) symmetry

In this section, we provide a path integral state sum for a (3+1)D bosonic
topological phase coupled with U(1) Villain gauge field. The input data to our
construction of a bosonic state sum is a unitary modular tensor category
(UMTC) $\mathcal{C}$, U(1) gauge field $(h,c)$, and U(1) charge
$Q_{a}\in\mathbb{R}/\mathbb{Z}$ assigned for each anyon $a\in\mathcal{C}$.
Given a 4-manifold $M^{4}$, we pick a triangulation and a branching structure
(i.e. a local ordering of vertices). Each simplex can then be either a $+$
simplex or a $-$ simplex, depending on whether the ordering agrees with the
orientation or not. We associate to each simplex the following data:

  * •

$1$-simplex $ij$: a link variable $h_{ij}\in\mathbb{R}$

  * •

$2$-simplex $ijk$: a face variable $c_{ijk}\in\mathbb{Z}$, an anyon
$a_{ijk}\in\mathcal{C}$

  * •

$3$-simplex $ijkl$: an anyon $b_{ijkl}\in\mathcal{C}$ obeying certain rules

The 3-simplex data assumes $N_{ij}^{k}\leq 1$, and is determined as follows.
Consider a particular $3$-simplex $\langle 0123\rangle$, with link variables
$h_{ij}\in\mathbb{R}$ on its 1-simplices, anyons $a_{ijk}$ and face variables
$c_{ijk}\in\mathbb{Z}$ on its 2-simplices. This is shown in Fig. 6. Then we
demand that the anyon $b_{0123}$ placed on this 3-simplex obeys

$N_{a_{023},a_{012}}^{b_{0123}}\neq 0\text{ and
}N_{a_{013},a_{123}}^{b_{0123}}\neq 0$ (46)

where $N_{a,b}^{c}$ are fusion coefficients.

In the language of category theory, the 2-simplex data is a simple object of
$\mathcal{C}$, and the data on the 3-simplex is an element of the space
Hom$\left(a_{013}\otimes a_{123},a_{013}\otimes a_{012}\right)$, i.e. an
element of $\bigoplus_{b}V^{b}_{013,123}\otimes V_{b}^{023,{012}}$. In other
words, if we had allowed $N_{ab}^{c}>1$, we would also need to associate
elements of a fusion space and a splitting space to the 3-simplices; this
generalization is straightforward. The picture for each 3-simplex in Fig. 6 is
projected onto a 2d plane, and then we get a graphical calculus for each
3-simplex as described in Fig. 7.

Figure 6: Anyon lines and gauge fields on a 3-simplex. $h_{ij}$ is described
as a yellow surface, $c_{ijk}$ is a blue line, and anyons $a_{ijk}$ are black
lines Poincaré dual of $2$-simplices. $a_{ijk}$ lines are dislocated from
$c_{ijk}$ lines, and intersect with a $h_{23}$ surface at a red point. The
directions of $c$ lines are assigned so that $c+\delta h$ becomes a gauge-
invariant field strength. That is, the direction of $c$ is the same as that of
$\delta h$ which is given by the “right-handed screw” rule on each 2-simplex,
e.g., directed outwards on $\braket{123}$. Figure 7: Graphical calculus for a
3-simplex with $+$ or $-$ orientation. The orange line denotes the symmetry
defect of the 0-form $\mathbb{R}$ symmetry, while the blue line denotes the
symmetry defect of the 1-form $\mathbb{Z}$ symmetry.

We now define our path integral. Given a labeling of all the simplices, we
will compute an anyon diagram for each 4-simplex based on the labeling. We
multiply the results, and then sum over all labelings. Let $\Delta^{4}$ be a
4-simplex, and let $\epsilon(\Delta^{4})=\pm$ be its orientation. Then define
complex numbers $Z^{\epsilon(\Delta^{4})}(\Delta^{4})$ in the following
diagrammatic way, shown in Fig. 8. For each 3-simplex in $\Delta^{4}$, take
the diagram obtained in Fig. 7 given the induced orientation on that
3-simplex. Lay out these five diagrams in a plane such that anyon lines shared
between two 3-simplices are near each other. Then connect up all of the lines
and symmetry defects, sliding symmetry defects along anyon lines and bending
symmetry defects far from anyon lines as necessary to obtain a closed diagram.
These anyon diagrams arise from projecting the boundary 3-simplices, which
form a triangulation of $S^{3}$, into the plane. There are an arbitrary choice
in this definition of the 15j symbol; in Fig. 7, we could have resolved the
four-fold fusion in a different channel, related by an $F$-move to our current
choice. In [10], it was shown that the partition function does not depend on
the choice in the definition. In addition, there are choices for resolving the
blue $c$ lines in Fig. 7; we could have resolved the blue lines on the left of
the black anyon lines. One can easily find that the choice does not affect on
the value of the 15j symbol. Also, one can notice that the angles of orange
$h$ lines are freely changed in going from Fig. 7 to Fig. 8. These changes of
angles do not affect on the value of the 15j symbol either, and one can freely
bend the $h$ lines to obtain Fig. 8.

Figure 8: 15j symbol for a $+$ simplex. The blue lines represent the dual of
the $c$ field. The blue lines are behind the black lines of anyons, except for
the the case that $c_{234}$ line crosses over the $a_{234}$ line. The 15j
symbol of a $-$ simplex gives the complex conjugate for that of a $+$ simplex.

The evaluation of the $15$j symbol involves the symmetry action of Villain
gauge fields on anyons defined in Sec. III.2. This requires us to define a
$\mathbb{R}$ lift $q_{a}\in\mathbb{R}$ of the fractional charge $Q_{a}$ that
satisfies $q_{a}=Q_{a}$ mod 1. We can regard the fractional charge of an anyon
$a_{ijk}$, $q_{ijk}:\langle ijk\rangle\to\mathbb{R}$ as a 2-cochain $q\in
C^{2}(M^{4},\mathbb{R})$, satisfying $(\delta
q)_{0123}=q_{123}-q_{023}+q_{013}-q_{012}=0$ mod 1 on each 3-simplex due to
the charge conservation (43).

The $15$j symbol is now computed as

$\displaystyle\begin{split}Z^{+}(01234)&=e^{-2\pi iq_{012}(c+\delta
h)_{234}-2\pi i(\delta q)_{0123}h_{34}}Z_{0}^{+}(01234),\\\
Z^{-}(01234)&=e^{2\pi iq_{012}(c+\delta h)_{234}+2\pi i(\delta
q)_{0123}h_{34}}Z_{0}^{-}(01234),\end{split}$ (47)

We note that the factor $e^{-2\pi iq_{012}(c+\delta h)_{234}-2\pi i(\delta
q)_{0123}h_{34}}$ can be expressed as $e^{-2\pi i(q\cup(c+\delta h)-2\pi
i(\delta q)\cup h)_{01234}}$ using the cup product. $Z_{0}^{\pm}(01234)$ is
the $15$j symbol without any symmetry defects [12],

$\displaystyle Z^{+}_{0}(01234)=\mathcal{N}_{01234}\sum_{d,a\in\mathcal{C}}$
$\displaystyle
F^{024,234,012}_{d,0234,a}R_{a}^{012,234}(F^{024,012,234}_{d})^{-1}_{a,0124}F^{014,124,234}_{d,0124,1234}\times$
$\displaystyle\times(F^{014,134,123}_{d})^{-1}_{1234,0134}F^{034,013,123}_{d,0134,0123}\times(F^{034,023,012}_{d})^{-1}_{0123,0234}$
(48) $\displaystyle
Z^{-}_{0}(01234)=\mathcal{N}_{01234}\sum_{d,a\in\mathcal{C}}$
$\displaystyle(F^{024,234,012}_{d})^{-1}_{a,0234}(R_{a}^{012,234})^{-1}F^{024,012,234}_{d,0124,a}\left(F^{014,124,234}_{d}\right)^{-1}_{1234,0124}\times$
$\displaystyle\times
F^{014,134,123}_{d,0134,1234}\left(F^{034,013,123}_{d}\right)^{-1}_{0123,0134}\times
F^{034,023,012}_{d,0234,0123}$ (49)

where $\\{F,R\\}$ are $F$ and $R$ symbols of BTC given in Appendix A. The
normalization factor $\mathcal{N}_{01234}$ is given by

$\mathcal{N}_{01234}=\sqrt{\frac{\prod_{\Delta^{3}\in
3\text{-simplices}}d_{b_{\Delta^{3}}}}{\prod_{\Delta^{2}\in
2\text{-simplices}}d_{a_{\Delta^{2}}}}},$ (50)

where the products are over $3$-simplices $\Delta^{3}$ and $2$-simplices
$\Delta^{2}$ of the $4$-simplex $01234$, and $d_{a}$ is the quantum dimension
of anyon $a$. We may now define the path integral. On a closed 4-manifold
$M^{4}$, we define

$\begin{split}Z(M^{4},(h,c))&=\sum_{\\{a,b\\}}{\mathcal{D}^{2(N_{0}-N_{1})-\chi}}\frac{\prod_{\Delta^{2}\in\mathcal{T}^{2}}d_{a_{\Delta^{2}}}\prod_{\Delta^{4}\in\mathcal{T}^{4}}Z^{\epsilon(\Delta^{4})}(\Delta^{4})}{\prod_{\Delta^{3}\in\mathcal{T}^{3}}d_{b_{\Delta^{3}}}}\\\
&=\sum_{\\{a,b\\}}{\mathcal{D}^{2(N_{0}-N_{1})-\chi}}\frac{\prod_{\Delta^{2}\in\mathcal{T}^{2}}d_{a_{\Delta^{2}}}\prod_{\Delta^{4}\in\mathcal{T}^{4}}Z_{0}^{\epsilon(\Delta^{4})}(\Delta^{4})}{\prod_{\Delta^{3}\in\mathcal{T}^{3}}d_{b_{\Delta^{3}}}}\prod_{\Delta^{4}\in\mathcal{T}^{4}}e^{-2\pi
i\epsilon(\Delta^{4})(q\cup(c+\delta h)+\delta q\cup h)}.\end{split}$ (51)

Here $\mathcal{T}^{k}$ denotes the set of $k$-simplices and
$N_{k}=|\mathcal{T}^{k}|$ is the number of $k$-simplices in the triangulation.
$\chi$ is the Euler characteristic of $M^{4}$. The sum is over all possible
labelings of the anyons $\\{a\\}$, $\\{b\\}$ on the $2$\- and $3$-simplices.

In Appendix F, we show that the above partition function is invariant under
the Pachner move (i.e., re-triangulations), which guarantees the topological
invariance of $Z(M^{4},(h,c))$.

We note that the partition function $Z(M^{4},(h,c))$ on a closed manifold is
independent of the choice of $\mathbb{R}$ lift $q_{a}$ of the fractional
charges, since the change of the lift $q\to q+\Lambda$ with $\Lambda\in
C^{2}(M^{4},\mathbb{Z})$ shifts the partition function by

$\displaystyle e^{-2\pi i\int(\Lambda\cup(c+\delta h)+\delta\Lambda\cup
h)}=e^{-2\pi i\int\delta(\Lambda\cup h)}=1.$ (52)

The path integral thus gives a well-defined (3+1)D invariant based on the data
of U(1) fractional charge $Q_{a}\in\mathbb{R}/\mathbb{Z}$.

Since the input category $\mathcal{C}$ is a modular tensor category, we expect
that the resulting (3+1)D path integral defines an invertible TQFT, i.e.
$|Z(M^{4},(h,c))|=1$. When $(h,c)=0$, the above path integral reduces to the
Crane-Yetter path integral [12], which is known to define an invertible TQFT
when $\mathcal{C}$ is modular. It is generally expected that if
$|Z(M^{4},A=0)|=1$, for any background gauge field $A$, then $|Z(M^{4},A)|=1$
for $A\neq 0$ as well, although we are not aware of a rigorous proof. This is
indeed borne out in a variety of examples studied in Sec. VIII and in closely
related constructions studied in previous works [10, 11].

Since we expect that $Z$ defines an invertible TQFT, it should be a bordism
invariant, and so we expect that in general the path integral defined above
evaluates to:

$\displaystyle Z(M^{4},(h,c))=e^{\frac{2\pi i}{8}c_{-}\sigma(M^{4})+i\Theta
C_{2}},$ (53)

where $\sigma(M^{4})$ is the signature of $M^{4}$, $c_{-}$ is the chiral
central charge, $C_{2}$ is the total second Chern number of the
$\mathrm{U}(1)$ bundle determined from $c$. $\Theta$ defines the
electromagnetic theta term, which, as discussed in Section I.1, is determined
by the topological twist of the vison, $e^{i\Theta/2}=\theta_{v}^{*}$.

Finally, we note that our construction above assumed that $\mathcal{C}$ is
modular, however this was not really an essential ingredient. As long as we
have a set of charges $Q_{a}$ for each anyon $a$ that is compatible with the
fusion rules, we can see that the above path integral will be well-defined and
topologically invariant. What is less clear, if $\mathcal{C}$ is not modular,
is when such a set of charges can be defined, and whether it provides an
exhaustive characterization of $\mathrm{U}(1)$ symmetry fractionalization.

##  V Bosonic state sum of (3+1)D TQFT with $\mathrm{U}(1)\rtimes H$ symmetry

We can consider bosonic topological phases with $\mathrm{U}(1)\rtimes H$
global symmetry, by incorporating into the state sum the flat background gauge
field of the symmetry group $H$ following [10].

The $H$ symmetry action is classified by a $\mathbb{Z}_{2}$ grading
corresponding to whether $\mathbf{g}\in H$ has a unitary or anti-unitary
action on the category:

$\displaystyle s(\mathbf{g})=\left\\{\begin{array}[]{ll}+1&\text{if
$\mathbf{g}$ is unitary}\\\ -1&\text{if $\mathbf{g}$ is anti-unitary}\\\
\end{array}\right.$ (56)

The symmetry group $\mathrm{U}(1)\rtimes H$ is characterized by the $H$-action
on U(1), corresponding to whether the group elements of $H$ complex conjugate
the U(1) elements. This is given by the map

$\displaystyle\sigma_{\mathbf{g}}:(h,c)\to\sigma(\mathbf{g})\cdot(h,c),$ (57)

with $\sigma(\mathbf{g})\in\\{\pm 1\\}$. Then, the $\mathrm{U}(1)\rtimes H$
gauge field is represented as the U(1) gauge field, defined by the pair $h\in
C^{1}(M,\mathbb{R}),c\in Z_{\sigma}^{2}(M,\mathbb{Z})$, together with a flat
$H$ gauge field $A_{H}$. Here, $Z_{\sigma}^{d}(M,\mathbb{Z})$ means the
twisted cocycle by the $H$ action, see Appendix E for an explanation of
twisted cohomology. Accordingly, the U(1) gauge transformation in the presence
of an $H$ gauge field also gets twisted by the $H$ action as

$\displaystyle c\to c-\delta_{\sigma}\lambda,\quad h\to
h+\lambda+\delta_{\sigma}\chi$ (58)

with $\lambda\in C^{1}(M,\mathbb{Z})$, $\chi\in C^{0}(M,\mathbb{R})$. Since
$H$ acts on the U(1) by conjugation, in the diagrammatics the $H$ defect acts
on the labels of anyons together with the U(1) gauge field $(h,c)$ when the
$H$ defect crosses over the $h$ and $c$ lines.

As explained in Appendix C, the graphical calculus of Villain gauge fields in
the presence of $H$ defects is defined based on the $\mathrm{U}(1)\rtimes H$
symmetry fractionalization data. This is constructed by the U(1) fractional
charge $Q_{a}$ satisfying the transformation law

$\displaystyle Q_{{}^{\mathbf{g}}a}=s(\mathbf{g})\sigma(\mathbf{g})Q_{a}\mod
1,$ (59)

and the symmetry fractionalization data $U,\eta$ for $H$ symmetry. Hence, our
construction takes as input a modular tensor category $\mathcal{C}$ describing
the bosonic topological order, symmetry fractionalization data $U$ and $\eta$
for $H$ on $\mathcal{C}$, and the fractional charge of anyons
$Q_{a}\in\mathbb{R}/\mathbb{Z}$ for U(1) symmetry satisfying Eq. (59). See
Appendix C for the construction of $\mathrm{U}(1)\rtimes H$ symmetry
fractionalization data and the Villain calculus in the presence of $H$
symmetry defects.

The 15j symbol on a 4-simplex is given by connecting the diagrams on each
boundary 3-simplex shown in Fig. 9. The resulting 15j symbol is given by Fig.
10. On a $+$ or $-$ 4-simplex, the 15j symbol evaluates as

$\displaystyle\begin{split}Z^{+}(01234)&=e^{-2\pi
i^{42}q_{012}(c+\delta_{\sigma}h)_{234}-2\pi
i^{43}(\delta_{s\sigma}q)_{0123}h_{34}}\cdot\frac{U_{34}(013,123,0123)^{s(34)}}{U_{34}(023,^{32}012,0123)^{s(34)}\eta_{012}(23,34)^{s(24)}}\cdot
Z_{0}^{+}(01234),\\\ Z^{-}(01234)&=e^{2\pi
i^{42}q_{012}(c+\delta_{\sigma}h)_{234}+2\pi
i^{43}(\delta_{s\sigma}q)_{0123}h_{34}}\cdot\frac{U_{34}(023,^{32}012,0123)^{s(34)}\eta_{012}(23,34)^{s(24)}}{U_{34}(013,123,0123)^{s(34)}}\cdot
Z_{0}^{-}(01234),\end{split}$ (60)

where ${}^{lm}q_{ijk}$ means the $\mathbf{g}_{lm}$ action on the U(1) charge,
${}^{lm}q_{ijk}:=q(^{lm}a_{ijk})$. Here, $q_{a}$ satisfies

${}^{ji}q_{a}=s(\mathbf{g}_{ji})\sigma(\mathbf{g}_{ji})q_{a}.$ (61)

which follows from Eq. (59). Then, the $\mathbb{R}$-lifted fractional charge
$q_{ijk}:\langle ijk\rangle\to\mathbb{R}$ of anyons can be regarded as a
2-cochain $q\in C^{2}(M^{4},\mathbb{R})$ that satisfies $\delta_{s\sigma}q=0$
mod 1, where $\delta_{s\sigma}$ is a twisted coboundary defined by the $H$
action on $q_{a}$ in Eq. (61). We note that the first term is neatly expressed
as twisted cup product (see Appendix E)

$\displaystyle e^{\mp 2\pi
i(q\cup_{s\sigma}(c+\delta_{\sigma}h)+\delta_{s\sigma}q\cup_{s\sigma}h)}.$
(62)

Then

$\displaystyle Z^{+}_{0}(01234)=\mathcal{N}_{01234}\sum_{d,a\in\mathcal{C}}$
$\displaystyle
F^{024,234,^{42}012}_{d,0234,a}R_{a}^{{}^{42}012,234}(F^{024,^{42}012,234}_{d})^{-1}_{a,0124}F^{014,124,234}_{d,0124,1234}\times$
$\displaystyle\times(F^{014,134,^{43}123}_{d})^{-1}_{1234,0134}F^{034,^{43}013,^{43}123}_{d,0134,^{43}0123}\times(F^{034,^{43}023,^{42}012}_{d})^{-1}_{{}^{43}0123,0234}$
(63) $\displaystyle
Z^{-}_{0}(01234)=\mathcal{N}_{01234}\sum_{d,a\in\mathcal{C}}$
$\displaystyle(F^{024,234,^{42}012}_{d})^{-1}_{a,0234}(R_{a}^{{}^{42}012,234})^{-1}F^{024,^{42}012,234}_{d,0124,a}\left(F^{014,124,234}_{d}\right)^{-1}_{1234,0124}\times$
$\displaystyle\times
F^{014,134,^{43}123}_{d,0134,1234}\left(F^{034,^{43}013,^{43}123}_{d}\right)^{-1}_{{}^{43}0123,0134}\times
F^{034,^{43}023,^{42}012}_{d,0234,^{43}0123}$ (64)

Figure 9: Graphical calculus for a 3-simplex with $+$ or $-$ orientation. The
red line labeled by $\mathbf{g}_{23}\in H$ denotes the $H$-symmetry defect.
Figure 10: 15j symbol for a $+$ simplex. The blue lines represent the dual of
the $c$ field. The blue lines are behind the black lines of anyons, except for
the the case that $c_{234}$ line crosses over the $a_{234}$ line. Here id
means the identity in $H$, and the blue dot denotes the background domain of
$H$ symmetry, see Appendix B, C for the graphical calculus.

In the presence of the defects for anti-unitary symmetry $H$, the space-time
manifold $M^{4}$ in general becomes unoriented. In that case, the anti-unitary
symmetry defect is realized as the Poincaré dual of the first Stiefel-Whitney
class by $A_{H}^{*}s=w_{1}$, reflecting that the symmetry defect reverses the
orientation of the space-time manifold. See [10, 11] for detailed description.

The partition function is then given by

$\begin{split}Z(M^{4},(h,c),A_{H})&=\sum_{\\{a,b\\}}{\mathcal{D}^{2(N_{0}-N_{1})-\chi}}\frac{\prod_{\Delta^{2}\in\mathcal{T}^{2}}d_{a_{\Delta^{2}}}\prod_{\Delta^{4}\in\mathcal{T}^{4}}Z^{\epsilon(\Delta^{4})}(\Delta^{4})}{\prod_{\Delta^{3}\in\mathcal{T}^{3}}d_{b_{\Delta^{3}}}}.\end{split}$
(65)

We show the invariance of the path integral under the Pachner move in Appendix
F, which shows that our path integral defines a topological invariant of
$M^{4}$.

To check that the partition function is independent of the $\mathbb{R}$ lift
of the fractional charge $Q_{a}$, let us study the effect of a local
redefinition of the charge $q_{a}\to q_{a}+\Lambda$, where $\Lambda\in
C^{2}(M^{4},\mathbb{Z})$ is supported on a single 2-simplex. Since the
partition function is invariant under $H$ gauge transformation, one can move
the $H$ symmetry defect away from $\Lambda$. Then, the shift of the partition
function is given by

$\displaystyle\prod_{\Delta^{4}}e^{-2\pi
i\epsilon(\Delta^{4})\delta(\Lambda\cup h)}=1.$ (66)

This shows that the path integral gives a well-defined (3+1)D invariant based
on the data of U(1) fractional charge $Q_{a}\in\mathbb{R}/\mathbb{Z}$.

For the same reason discussed in the preceding section, the (3+1)D path
integral $Z(M^{4},(h,c),A_{H})$ is expected to define an invertible TQFT with
$G=\mathrm{U}(1)\rtimes H$ symmetry. Therefore $Z(M^{4},(h,c),A_{H})$ is
expected to be a smooth bordism invariant, and can be viewed as defining an
element of the Pontyragin dual of the smooth bordism group for 4-manifolds
equipped with background $G$ gauge fields.

##  VI Construction of fermionic topological phases: fermion condensation

Our constructions described above can also be extended to describe fermionic
topological phases with $\mathrm{U}(1)$ symmetry, where the fermion carries
charge under the $\mathrm{U}(1)$, following the general procedure of [11, 21].
In this case, the input into the construction is the data of a (2+1)D
fermionic topological order, characterized by a super-modular category, and
fermionic symmetry fractionalization [2, 3].

Since we are interested in fermionic topological phases with
$\mathrm{U}(1)^{f}$ symmetry, our resulting TQFT will depend on a
$\mathrm{Spin}^{c}=(\mathrm{Spin}\times\mathrm{U}(1))/\mathbb{Z}_{2}$
structure on our space-time 4-manifold $M^{4}$. More generally, the full
fermionic symmetry group, including the space-time symmetry, is a group
extension $\mathbb{Z}_{2}^{f}\to\mathcal{G}_{f}\to\mathcal{G}_{b}$,
corresponding to a lift of the bosonic symmetry group
$\mathcal{G}_{b}=\mathrm{SO}(d)\rtimes G_{b}$. To define such a lift, the
fermionic theory in general requires a variant of the Spinc structure, twisted
by the elements of $G_{b}$, which we refer to as a
$\mathcal{G}_{f}$-structure, $\xi_{\mathcal{G}}$ (see Eq. (73)). A general
review of $\mathcal{G}_{f}$-structures is provided in Appendix of [11]. The
internal bosonic symmetry group $G_{b}=\mathcal{G}_{b}/\mathrm{SO}(d)$, and
recall that the internal fermionic symmetry group $G_{f}$ is a
$\mathbb{Z}_{2}^{f}$ extension of $G_{b}$, $\mathbb{Z}_{2}^{f}\to G_{f}\to
G_{b}$, characterized by $[\omega_{2}]\in H^{2}(BG_{b},\mathbb{Z}_{2})$.

The construction proceeds by starting with a path integral
$Z_{b}(M^{4},A_{b})$ for a bosonic topological phase, which is referred to as
the bosonic shadow. $Z_{b}$ is defined by using the constructions in the
preceding sections, with a super-modular category and symmetry
fractionalization data as input. Here $A_{b}$ is a background gauge field for
the bosonic symmetry group $G_{b}$. Since the super-modular category has an
invisible particle, the fermion $\psi$, the resulting (3+1)D TQFT is non-
invertible, and corresponds to a dynamical (3+1)D $\mathbb{Z}_{2}$ gauge
theory with a single topologically non-trivial point-like particle, which is a
fermion (see [11] for an extended discussion in the context of flat $G_{b}$
bundles). To obtain an invertible fermionic topological phase, i.e a fermionic
SPT, we must condense the fermion to Higgs the $\mathbb{Z}_{2}$ gauge field.
The fermion condensation can be done by defining the bosonic path integral
$Z_{b}$ in the presence of fermion world-lines, and then summing over all
possible configurations of wordlines. Mathematically, this is achieved by
coupling the theory to a background $3$-form $\mathbb{Z}_{2}$ gauge field
$f_{3}\in Z^{3}(M^{4},\mathbb{Z}_{2})$, whose Poincare dual defines the
fermion worldlines, to obtain $Z_{b}(M^{4},A_{b},f_{3})$.

Due to the Fermi statistics of the fermion, the path integral
$Z_{b}(M^{4},A_{b},f_{3})$ is not actually gauge invariant, and thus possesses
a number of ’t Hooft anomalies involving the 2-form $\mathbb{Z}_{2}$ symmetry
and $0$-form $G_{b}$ symmetry. The fermion condensation procedure requires
carefully canceling these anomalies by combining the theory with another,
intrinsically fermionic theory, whose path integral is denoted
$z_{c}(M^{4},f_{3},\xi_{\mathcal{G}})$. Physically this corresponds to the
statement that for a topological phase with an emergent fermion, the only way
to “condense” the fermion $\psi$ is to stack another trivial topological phase
with a physical (not emergent) fermion, $c$, and condense the composite $\psi
c$.

Below, we briefly review the fermion condensation procedure for general
$d$-dimensional theories, starting with a bosonic path integral
$Z_{b}(M^{d},A_{b},f_{d-1})$, and we summarize the ’t Hooft anomalies carried
by the bosonic theory $Z_{b}(M^{d},A_{b},f_{d-1})$. We will see how the
anomalies of the bosonic theory $Z_{b}$ is canceled out by stacking with the
trivial fermionic theory $z_{c}$, and how the anomalous bosonic theory is
converted to a topologically invariant fermionic theory free of anomalies by
the fermion condensation procedure.

###  VI.1 ’t Hooft anomaly of a bosonic shadow theory
$Z_{b}(M^{d},A_{b},f_{d-1})$

The global symmetry of the bosonic shadow theory is a 0-form symmetry $G_{b}$
with the background gauge field $A_{b}$ and a $\mathbb{Z}_{2}$ $(d-1)$-form
symmetry generated by the Wilson line of the fermionic particle $\psi$ with
the background gauge field $f_{d-1}\in Z^{d-1}(M^{d},\mathbb{Z}_{2})$.

In this paper, the bosonic symmetry group $G_{b}$ has the form of
$G_{b}=\mathrm{U}(1)\rtimes H$, where $H$ is a symmetry group that can act on
$\mathrm{U}(1)$ by charge conjugation
$\sigma:H\to\mathrm{Aut}(\mathrm{U}(1))$. The $U(1)$ part of the $G_{b}$ gauge
field $A_{b}$, is characterized by the Villain fields $(h,c)$, and we use
$A_{H}$ to denote the background $H$ gauge field.

The extension class $[\omega_{2}]$ that characterizes the group extension
$\mathbb{Z}_{2}^{f}\to G_{f}\to G_{b}$ satisfies

$\displaystyle A_{b}^{*}\omega_{2}=2c+A_{H}^{*}\omega_{H}\quad\mod 2,$ (67)

where $\omega_{H}\in H^{2}(BH,\mathbb{Z}_{2})$ characterizes the group
extension restricted to $H$, and $c$ is the Chern class $2c\in
H^{2}_{\sigma}(M^{d},\mathbb{Z})$ twisted by the $H$ action on
$\mathrm{U}(1)$. Here we slightly abuse notation and view the gauge fields
$A_{b}$ and $A_{H}$ as maps $A_{b}:M\rightarrow BG_{b}$,
$A_{H}:M^{d}\rightarrow BH$, and $A_{b}^{*}$, $A_{H}^{*}$ denote their
pullbacks.

In the presence of background $H$ gauge fields, the U(1) Villain gauge field
is realized by the gauge fields $(h,c)$ with the field strength $c+\delta h$,
setting $h\in C^{1}_{\sigma}(M^{d},\mathbb{R})$, $2c\in
Z^{2}_{\sigma}(M^{d},\mathbb{Z})$ on each 2-simplex. The gauge transformation
is given by

$\displaystyle c\to c-\delta_{\sigma}{\lambda},\quad h\to
h+\lambda+\delta_{\sigma}\chi,$ (68)

with $\lambda\in C^{1}(M^{d},\mathbb{Z}),\chi\in C^{0}(M^{d},\mathbb{R})$.

Now let us explain the ’t Hooft anomaly of the bosonic shadow theory
$Z_{b}(M^{d},A_{b},f_{d-1})$ constructed in this paper. The ’t Hooft anomaly
of $Z_{b}(M^{d},A_{b},f_{d-1})$ is characterized by the $(d+1)$-dimensional
response action

$\displaystyle
e^{iS_{d+1,b}}=(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{2}(f_{d-1})+A_{b}^{*}\omega_{2}\cup
f_{d-1}}=(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{2}(f_{d-1})+(2c+A_{H}^{*}\omega_{H})\cup
f_{d-1}}.$ (69)

The ’t Hooft anomaly means the non-invariance of the partition function
$Z_{b}(M^{d},A_{b},f_{d-1})$ under the gauge transformations and re-
triangulations. That is, given two triangulations $T$ and $T^{\prime}$ of
$M^{d}$, we consider a triangulation of $M^{d}\times[0,1]$ which reduces to
$T$ and $T^{\prime}$ on the two boundaries. We also consider extending the
definition of $A_{b}$ and $f_{d-1}$ to $M^{d}\times[0,1]$, which reduce to
their values on the initial and final triangulations. Then the change in phase
of $Z_{b}$ is given by

$\displaystyle
Z_{b}((M^{d},T),A_{b},f_{d-1})=Z_{b}((M^{d},T^{\prime}),A_{b}^{\prime},f_{d-1}^{\prime})(-1)^{\int_{M\times[0,1]}\mathop{\mathrm{Sq}}\nolimits^{2}(f_{d-1})+A_{b}^{*}\omega_{2}\cup
f_{d-1}}.$ (70)

Here we have explicitly written the dependence on the triangulation $T$ in the
path integral. Note that we could also consider a different manifold
$(M^{d})^{\prime}$ on the right-hand side instead, in which case we consider a
bordism from $M^{d}$ to $(M^{d})^{\prime}$. The ’t Hooft anomaly reflects that
the excitation $\psi$ behaves as a fermionic particle that carries the
nontrivial symmetry fractionalization characterized by $\omega_{2}$ under the
$G_{b}$ symmetry. See [11] for the detailed description.

###  VI.2 Anomaly cancellation and fermion condensation

We obtain a re-triangulation invariant, anomaly-free path integral, by
“stacking” the bosonic theory $Z_{b}$ with another theory which transforms in
exactly the same way as to cancel these anomalies. Since we want a fermionic
theory after stacking, this other theory should be an intrinsically fermionic
theory. An important requirement of such a fermionic theory is that it should
depend on a choice of a $\mathcal{G}_{f}$-structure $\xi_{\mathcal{G}}$, which
is required to specify how to couple the theory with $G_{b}$ internal symmetry
to fermions. Thus, we consider the product:

$\displaystyle
Z_{b}((M^{d},T),A_{b},f_{d-1})z_{c}((M^{d},T),f_{d-1},\xi_{\mathcal{G}}).$
(71)

Here $z_{c}$ is the path integral of a fermionic theory, i.e. which contains
physical fermions, and will be defined in the next subsection. Note that
$z_{c}$ depends on the $G_{b}$-connection $A_{b}$ implicitly through the
$\mathcal{G}_{f}$ structure $\xi_{\mathcal{G}}$ which we will also define in
the next subsection. When $z_{c}$ is defined appropriately, the product above
will be re-triangulation invariant, anomaly-free, and independent of a change
of $\xi_{\mathcal{G}}$ by a 1-coboundary.

Finally, the topologically invariant path integral for our fermionic theory
can then be obtained by condensing the fermions of the bosonic shadow by
summing over all inequivalent choices of $f_{d-1}$:

$\displaystyle Z(M^{d},A_{b},\xi_{\mathcal{G}})\propto\sum_{[f_{d-1}]\in
H^{d-1}(M^{d},\mathbb{Z}_{2})}Z_{b}((M^{d},T),A_{b},f_{d-1})z_{c}((M^{d},T),f_{d-1},\xi_{\mathcal{G}}).$
(72)

For the bosonic shadow theory that we construct in Sec. VII, the resulting
theory after fermion condensation $Z(M^{d},A_{b},\xi_{\mathcal{G}})$ defines
an invertible fermionic TQFT with fermionic $G_{f}$ symmetry. As such, it is a
smooth bordism invariant for 4-manifolds equipped with $G_{b}$ background
gauge fields and $\xi_{\mathcal{G}}$ structures.

###  VI.3 $\mathcal{G}_{f}$ structure and the definition of $z_{c}$

The definition of $z_{c}$ depends on a $\mathcal{G}_{f}$ structure, which is a
$1$-cochain with the property that 333In fact, the precise definition of
$z_{c}$ requires $\xi_{\mathcal{G}}$ to be a $(d-1)$-chain
$\xi_{\mathcal{G}}\in C_{d-1}(M^{d},\mathbb{Z}_{2})$, while
$A_{b}^{*}\omega_{2}$ in the RHS of Eq. (73) is defined as a cocycle. The
precise formulation of $\xi_{\mathcal{G}}$ hence needs a map
$f_{\infty}:Z^{k}(M^{d},\mathbb{Z}_{2})\to Z_{d-k}(M^{d},\mathbb{Z}_{2})$ that
turns a cochain to a chain, and define
$\delta\xi_{\mathcal{G}}=w_{2}+w_{1}^{2}+f_{\infty}(A_{b}^{*}\omega_{2})$,
with $w_{k}$ the chain representative of $k^{\mathrm{th}}$ Stiefel-Whitney
class. See [11] for the detailed description.

$\displaystyle\delta\xi_{\mathcal{G}}=w_{2}+w_{1}^{2}+A_{b}^{*}\omega_{2}.$
(73)

As mentioned above, the $\mathcal{G}_{f}$-structure $\xi_{\mathcal{G}}$ allows
us to specify a lift from the full bosonic symmetry (including space-time
symmetries) of the field theory to the fermionic theory. A detailed
description of $\mathcal{G}_{f}$-structure can be found in Appendix of [11].
The fermionic path integral $z_{c}$ then has the form of

$\displaystyle
z_{c}((M^{d},T),f_{d-1},\xi_{\mathcal{G}})=\sigma((M^{d},T),f_{d-1})(-1)^{\int_{M^{d}}\xi_{\mathcal{G}}\cup
f_{d-1}}.$ (74)

Here, $\sigma((M^{d},T),f_{d-1})$ is referred to as the Gu-Wen Grassmann
integral constructed in [22, 21, 23, 36]. The Grassmann integral
$\sigma((M^{d},T),f_{d-1})$ is not invariant under the gauge transformation
and re-triangulation, and the shift is characterized by the
$(d+1)$-dimensional bulk response action

$\displaystyle
e^{iS_{d+1,\sigma}}=(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{2}(f_{d-1})+(w_{2}+w_{1}^{2})\cup
f_{d-1}}.$ (75)

It is known that the combination
$\mathop{\mathrm{Sq}}\nolimits^{2}(f_{d-1})+(w_{2}+w_{1}^{2})\cup f_{d-1}$ is
trivial in cohomology according to the Wu formula [37]. This means that
$S_{d+1,\sigma}$ should be regarded as an action of a trivial
$(d+1)$-dimensional invertible topological phase. One can regard the Grassmann
integral $\sigma((M^{d},T),f_{d-1})$ as giving the trivial boundary of the
$(d+1)$-dimensional trivial invertible theory determined by $S_{d+1,\sigma}$.

Meanwhile, the fermionic action $(-1)^{\int_{M^{d}}\xi_{\mathcal{G}}\cup
f_{d-1}}$ has an ’t Hooft anomaly given by

$\displaystyle
e^{iS_{d+1,\xi}}=(-1)^{\int(w_{2}+w_{1}^{2}+A_{b}^{*}\omega_{2})\cup
f_{d-1}}.$ (76)

Altogether, we see that the ’t Hooft anomaly of the combined action
$Z_{b}((M^{d},T),A_{b},f_{d-1})z_{c}((M^{d},T),f_{d-1},\xi_{\mathcal{G}})$
exactly cancels as

$\displaystyle e^{iS_{d+1,b}}e^{iS_{d+1,\sigma}}e^{iS_{d+1,\xi}}=1.$ (77)

Thus one can gauge the $(d-2)$-form symmetry $f_{d-1}$ of the combined theory
to perform the fermion condensation as described in Eq. 72.

###  VI.4 Examples of $\mathcal{G}_{f}$ structures

Here we show the examples of $\mathcal{G}_{f}$ structures that correspond to
the Altland-Zirnbauer class A, AI, AII, AIII. See Table 1 for a summary of the
groups $G_{f}$, $G_{b}$, $\mathcal{G}_{f}$, and $\mathcal{G}_{b}$ for these
symmetry classes.

####  VI.4.1 class A: $\mathrm{Spin}^{c}$ structure

Spinc structure can be obtained by setting $G_{b}=\mathrm{U}(1)$, and
$A_{b}^{*}\omega_{2}=2c$ mod 2. There is no time-reversal symmetry, so we take
$M^{d}$ as an oriented manifold. The $\mathcal{G}_{f}$ structure is then given
by

$\displaystyle\delta\xi_{\mathcal{G}}=w_{2}+2c\quad\mod 2,$ (78)

which implies that $[w_{2}]=[2c]$ as an element of
$H^{2}(M^{d},\mathbb{Z}_{2})$. This results in the Dirac quantization of the
Spinc gauge field in Eq. (26).

####  VI.4.2 class AI: $\mathrm{Pin}^{\tilde{c}}_{-}$ structure

$\mathrm{Pin}^{\tilde{c}}_{-}$ structure can be obtained by setting
$G_{b}=\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{\mathbf{T}}$ with
$\mathbb{Z}_{2}^{\mathbf{T}}$ that conjugates the U(1) charge. The symmetry
extension class is given by $A_{b}^{*}\omega_{2}=2c$ mod 2, implying that
$\mathbf{T}^{2}=1$. $M^{d}$ is in general unoriented, and the
$\mathcal{G}_{f}$ structure is then given by

$\displaystyle\delta\xi_{\mathcal{G}}=w_{2}+w_{1}^{2}+2c\quad\mod 2,$ (79)

which implies that $[w_{2}+w_{1}^{2}]=[2c]$ as an element of
$H^{2}(M^{d},\mathbb{Z}_{2})$. The Dirac quantization of the Villain gauge
field is given by

$\displaystyle\int_{C}c=\frac{1}{2}\int_{C}(w_{2}+w_{1}^{2})\quad\mod 1,$ (80)

for any oriented two-dimensional cycle $C$ in the spacetime.

####  VI.4.3 class AII: $\mathrm{Pin}^{\tilde{c}}_{+}$ structure

$\mathrm{Pin}^{\tilde{c}}_{+}$ structure can be obtained by setting
$G_{b}=\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{\mathbf{T}}$ with
$\mathbb{Z}_{2}^{\mathbf{T}}$ that conjugates the U(1) charge. The symmetry
extension class is given by $A_{b}^{*}\omega_{2}=2c+w_{1}^{2}$ mod 2, implying
that $\mathbf{T}^{2}=(-1)^{f}$. $M^{d}$ is in general unoriented, and the
$\mathcal{G}_{f}$ structure is then given by

$\displaystyle\delta\xi_{\mathcal{G}}=w_{2}+2c\quad\mod 2,$ (81)

which implies that $[w_{2}]=[2c]$ as an element of
$H^{2}(M^{d},\mathbb{Z}_{2})$. The Dirac quantization of the Villain gauge
field is given by

$\displaystyle\int_{C}c=\frac{1}{2}\int_{C}w_{2}\quad\mod 1,$ (82)

for any oriented two-dimensional cycle $C$ in the spacetime.

####  VI.4.4 class AIII: $\mathrm{Pin}^{c}$ structure

$\mathrm{Pin}^{c}$ structure can be obtained by setting
$G_{b}=\mathrm{U}(1)\times\mathbb{Z}_{2}^{\mathbf{T}}$ with
$\mathbb{Z}_{2}^{\mathbf{T}}$ that does not conjugate the U(1) charge. The
symmetry extension class is given by $A_{b}^{*}\omega_{2}=2c$ mod 2. $M^{d}$
is in general unoriented, and the $\mathcal{G}_{f}$ structure is then given by

$\displaystyle\delta\xi_{\mathcal{G}}=w_{2}+2c\quad\mod 2,$ (83)

which implies that $[w_{2}]=[2c]$ as an element of
$H^{2}(M^{d},\mathbb{Z}_{2})$. The Dirac quantization of the Villain gauge
field is given by

$\displaystyle\int_{C}c=\frac{1}{2}\int_{C}w_{2}\quad\mod 1,$ (84)

for any oriented two-dimensional cycle $C$ in the spacetime.

Cartan | $G_{b}$ | $G_{f}$ | $\mathcal{G}_{b}$ | $\mathcal{G}_{f}$
---|---|---|---|---
A | $\mathrm{U}(1)$ | $\mathrm{U}(1)^{f}$ | $\mathrm{SO}\times\mathrm{U}(1)$ | $\mathrm{Spin}^{c}$
AI | $\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{\bf T}$ | $\mathrm{U}(1)^{f}\rtimes\mathbb{Z}_{2}^{\bf T}$ | $\mathrm{O}\ltimes\mathrm{U}(1)$ | $[\mathrm{Pin}^{-}\ltimes\mathrm{U}(1)]/\mathbb{Z}_{2}$
AII | $\mathrm{U}(1)\rtimes\mathbb{Z}_{2}^{\bf T}$ | $[\mathrm{U}(1)^{f}\rtimes\mathbb{Z}_{4}^{{\bf T},f}]/\mathbb{Z}_{2}$ | $\mathrm{O}\ltimes\mathrm{U}(1)$ | $[\mathrm{Pin}^{+}\ltimes\mathrm{U}(1)]/\mathbb{Z}_{2}$
AIII | $\mathrm{U}(1)\times\mathbb{Z}_{2}^{\bf T}$ | $\mathrm{U}(1)^{f}\times\mathbb{Z}_{2}^{\bf T}$ | $\mathrm{O}\times\mathrm{U}(1)$ | $\mathrm{Pin}^{c}$
Table 1: Altland-Zirnbauer class involving U(1), and corresponding global
symmetries.

##  VII Bosonic shadow for (3+1)D fermionic SPT with curved U(1) bundle

In this section, we construct a path integral $Z_{b}((M^{4},T),A_{b},f_{3})$
that corresponds to the bosonic shadow discussed in the previous section.
Compared to Sec. IV, V, here we take as input a super-modular category
$\mathcal{C}$ with fermionic symmetry fractionalization data, and we couple
the theory to a non-zero background flat $3$-form $\mathbb{Z}_{2}$ gauge field
$f_{3}$.

Since the $G_{b}=\mathrm{U}(1)\rtimes H$ gauge field $A_{b}$ is specified by a
pair of flat $H$ gauge field $A_{H}$ and the Villain gauge field $(h,c)$, we
sometimes write the bosonic shadow as $Z_{b}((M^{4},T),A_{H},(h,c),f_{3})$.

Our construction takes as input a super-modular category $\mathcal{C}$
describing the fermionic topological order, the bosonic symmetry group $G_{b}$
and the class $[\omega_{2}]\in H^{2}(BG_{b},\mathbb{Z}_{2})$ specifying
$G_{f}$, symmetry fractionalization data $U$ and $\eta$ for $H$ on
$\mathcal{C}$, and the fractional charges of anyons
$Q_{a}\in\mathbb{R}/(2\mathbb{Z})$ for U(1) symmetry reviewed in Sec. III.1.2,
which satisfy

$\displaystyle Q_{{}^{\mathbf{g}}a}=s(\mathbf{g})\sigma(\mathbf{g})Q_{a}\mod
2.$ (85)

In Appendix C, we see that these data define the symmetry fractionalization
data for $\mathcal{C}$.

As reviewed in Appendix B, we require

$\displaystyle U_{\bf g}(\psi,\psi;1)$ $\displaystyle=1$ (86)
$\displaystyle\eta_{\psi}({\bf g},{\bf h})$ $\displaystyle=\omega_{2}({\bf
g},{\bf h})$ (87)

Subject to the first constraint, the consistency condition Eq. (177) between
$\eta$ and $U$ implies that $\eta_{\psi}({\bf g,h})=\pm 1$, which turns the
consistency condition for $\eta_{\psi}$ Eq. (178) into an (untwisted)
$\mathbb{Z}_{2}$-valued cocycle condition, so the second condition makes
sense.

Given a triangulation and branching structure $T$ of $M^{4}$, we assign the
following fixed background data:

  * •

A cocycle $f\in Z^{3}(M^{4},\mathbb{Z}_{2})$ representing the background
fermion line, which consists of an assignment $f_{ijkl}$ on each 3-simplex
satifying $f_{0123}+f_{0124}+f_{0134}+f_{0234}+f_{1234}=0\text{ (mod 2)}$ on
each 4-simplex $\braket{01234}$ (using the additive $\mathbb{Z}_{2}$
notation).

  * •

A flat gauge field $A_{H}$ which consists of group elements ${\bf g}_{ij}\in
H$ on each edge $\braket{ij}$ satifying ${\bf g}_{ij}{\bf g}_{jk}={\bf
g}_{ik}$

  * •

A 1-form Villain gauge field $h\in C^{1}(M^{4},\mathbb{R})$ which consists of
group elements $h_{ij}\in\mathbb{R}$ on each edge $\braket{ij}$

  * •

A 2-form Villain gauge field $c$ with $2c\in Z^{2}_{\sigma}(M^{4},\mathbb{Z})$
which consists of group elements $c_{ijk}\in\mathbb{Z}/2$ on each face
$\braket{ijk}$.

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

  * •

To each 2-simplex $\braket{ijk}$, 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 vector space $V^{\psi^{f_{ijkl}}\times
b}_{ijl,jkl}\otimes V_{b}^{ikl,{}^{{\bf lk}}ijk}$ .

Where it does not lead to ambiguity, we will often ignore the distinction
between a simplex and the anyon or group element data assigned to it, i.e.
simply write ${\bf g}_{ij}$ as $ij$, $a_{ijk}$ as $ijk$, or $b_{ijkl}$ as
$ijkl$. We will always refer to the 3-form gauge field as $f_{ijkl}$.

The evaluation of the $15$j symbol in Fig.11 involves the symmetry action of
Villain gauge fields on anyons defined in Sec. III.2. This definition uses a
$\mathbb{R}$ lift $q_{a}\in\mathbb{R}$ of the fractional charge
$Q_{a}\in\mathbb{R}/(2\mathbb{Z})$ that satisfies $q_{a}=Q_{a}$ mod 2.

We will also need to assign to each $4$-simplex $\Delta_{4}$ of $M$ an
orientation $\epsilon(\Delta_{4})=\pm$. If $M$ is orientable, this simply
amounts to choosing a global orientation of $M$. We then assign an
orientation-dependent amplitude $Z_{b}^{\epsilon(\Delta_{4})}(\Delta_{4})$ to
each 4-simplex $\Delta_{4}$ of $M$. This amplitude is given diagrammatically
in Fig. 11, with a normalization factor

$\mathcal{N}_{01234}=\sqrt{\frac{\prod_{\Delta_{3}\in\text{3-simplices}}d_{b_{\Delta_{3}}}}{\prod_{\Delta_{2}\in\text{2-simplices}}d_{a_{\Delta_{2}}}}}.$
(88)

Now we construct a (3+1)D bosonic shadow theory $Z_{b}(M^{4},A_{H},(h,c),f)$
for the fermionic SPT phase based on the $\mathcal{G}_{f}$ structure. With the
gauge $F^{\psi,\psi,a}=F^{a,\psi,\psi}=1$ for all $a\in\mathcal{C}$, we can
evaluate the diagram in Fig. 11 as

$\displaystyle\begin{split}Z^{+}_{b}(01234)&=e^{-2\pi
i(^{42}q_{012}(c+\delta_{\sigma}h)_{234}+q_{\psi^{f_{0123}}}c_{034}+q_{\psi^{f_{1234}}}c_{014}+^{43}(\delta_{s\sigma}q)_{0123}h_{34})}Z_{0,b}^{+}(01234),\\\
Z^{-}_{b}(01234)&=e^{2\pi
i(^{42}q_{012}(c+\delta_{\sigma}h)_{234}+q_{\psi^{f_{0123}}}c_{034}+q_{\psi^{f_{1234}}}c_{014}+^{43}(\delta_{s\sigma}q)_{0123}h_{34})}Z_{0,b}^{-}(01234),\end{split}$
(89)

where ${}^{lm}q_{ijk}$ means the $\mathbf{g}_{lm}\in H$ action on the U(1)
charge $q_{ijk}$. One can regard $q_{ijk}:\braket{ijk}\to\mathbb{R}$ as a
2-cochain $q\in C^{2}(M^{4},\mathbb{R})$ which satisfies $\delta_{s\sigma}q=f$
mod 2. This mod 2 relation is a consequence of Eq. (44) and Eq. (45) on each
3-simplex.

$Z_{0,b}^{\pm}$ is the 15j symbol without U(1) gauge field $(h,c)$ expressed
as

$\begin{split}Z_{0,b}^{+}(01234)=&\frac{1}{\eta_{012}(23,34)^{s(24)}}\Big{(}\frac{U_{34}(013,123;0123\times
f_{0123})}{U_{34}(023,{{}^{32}}012;0123)U_{34}(f_{0123},0123;0123\times
f_{0123})}\Big{)}^{s(34)}\\\
&\cdot\Big{(}F^{f_{0123}\,,\,{{}^{43}}013\,,\,123}_{0123\times
f_{0123}\,,\,{{}^{43}}013\times
f_{0123}\,,\,0123}\,\,\,\big{(}F^{034\,,\,f_{0123}\,,\,{{}^{43}}013\times
f_{0123}}_{0134\times f_{0134}\,,\,034\times
f_{0123}\,,\,{{}^{43}}013}\big{)}^{*}\,\,\,R^{f_{0123}\,,\,034}_{034\times
f_{0123}}\Big{)}\\\
&\cdot\left(F^{f_{0234}\,,\,034\,,\,{{}^{43}}023}_{0234\,,\,034\times
f_{0234}\,,\,0234\times f_{0234}}\right)\\\
&\cdot\Big{(}F^{f_{0124}\,,\,024\,,\,{{}^{42}}012}_{0124\,,\,024\times
f_{0124}\,,\,0124\times f_{0124}}\,\,\,F^{f_{0124}\,,\,024\times
f_{0124}\,,\,234}_{0234\,,\,024\,,\,0234\times
f_{0124}}\,\,\,F^{f_{0124}\,,\,034\times
f_{0234}\,,\,{{}^{43}}023}_{0234\times f_{0124}\,,\,034\times f_{0234}\times
f_{0124}\,,\,0234}\Big{)}\\\ &\cdot\Big{(}\big{(}F^{f_{0134}\,,\,034\times
f_{0123}\,,\,{{}^{43}}013\times f_{0123}}_{0134\,,\,034\times f_{0123}\times
f_{0134}\,,\,0134\times f_{0134}}\big{)}^{*}\Big{)}\\\
&\cdot\Big{(}\big{(}F^{f_{1234}\,,\,134\,,\,{{}^{43}}123}_{1234\,,\,134\times
f_{1234}\,,\,1234\times
f_{1234}}\big{)}^{*}\,\,\,F^{014\,,\,f_{1234}\,,\,134\times
f_{1234}}_{0134\,,\,014\times
f_{1234}\,,\,134}\,\,\,\big{(}R^{f_{1234}\,,\,014}_{014\times
f_{1234}}\big{)}^{*}\\\ &\quad\quad\big{(}F^{f_{1234}\,,\,014\,,\,134\times
f_{1234}}_{0134\,,\,014\times f_{1234}\,,\,0134\times
f_{1234}}\big{)}^{*}\,\,\,\big{(}F^{f_{1234}\,,\,034\times f_{0123}\times
f_{0134}\,,\,{{}^{43}}013\times f_{0123}}_{0134\times f_{1234}\,,\,034\times
f_{0123}\times f_{0134}\times f_{1234}\,,\,0134}\big{)}^{*}\,\Big{)}\\\
&\cdot\Bigg{(}\sum_{d,a\in\mathcal{C}}F^{024\times
f_{0124}\,,\,234\,,\,{{}^{42}}012}_{d\,,\,0234\times
f_{0124}\,,\,a}\,\,R^{{{}^{42}}012\,,\,234}_{a}\,\,\big{(}F^{024\times
f_{0124}\,,\,012\,,\,234}_{d\,,\,0124\,,\,a}\big{)}^{*}\,\,F^{014\,,\,124\,,\,234}_{d\,,\,0124\,,\,1234}\\\
&\quad\quad\quad\quad\quad\big{(}F^{014\,,\,134\times
f_{1234}\,,\,{{}^{43}}123}_{d\,,\,0134\times
f_{1234}\,,\,1234}\big{)}^{*}\,\,F^{034\times f_{0234}\times
f_{0124}\,,\,{{}^{43}}013\times f_{0123}\,,\,{{}^{43}}123}_{d\,,\,0134\times
f_{1234}\,,\,{{}^{43}}0123\times f_{0123}}\,\,\big{(}F^{034\times
f_{0234}\times f_{0124}\,,\,{{}^{43}}023\,,\,{{}^{42}}012}_{d\,,\,0234\times
f_{0124}\,,\,{{}^{43}}0123\times
f_{0123}}\big{)}^{*}\,\,d_{d}\Bigg{)}\end{split}$ (90)

where we mean $f_{ijkl}=(\psi)^{f(ijkl)}$, $s(ij)=s({\bf g}_{ij})$,
${{}^{lk}}ijk={{}^{{\bf g}_{lk}}a_{ijk}}$, etc. On a $-$ simplex, the weight
is simply given by the complex conjugate,

$Z_{b}^{-}(01234)=(Z_{b}^{+}(01234))^{*}.$ (91)

The partition function is then given by

$\begin{split}Z_{b}((M^{4},T),A_{H},(h,c),f_{3})&=\sum_{\\{a,b\\}}{\mathcal{D}^{2(N_{0}-N_{1})-\chi}}\frac{\prod_{\Delta^{2}\in\mathcal{T}^{2}}d_{a_{\Delta^{2}}}\prod_{\Delta^{4}\in\mathcal{T}^{4}}Z_{b}^{\epsilon(\Delta^{4})}(\Delta^{4})}{\prod_{\Delta^{3}\in\mathcal{T}^{3}}d_{b_{\Delta^{3}}}}\\\
&=\sum_{\\{a,b\\}}{\mathcal{D}^{2(N_{0}-N_{1})-\chi}}\frac{\prod_{\Delta^{2}\in\mathcal{T}^{2}}d_{a_{\Delta^{2}}}\prod_{\Delta^{4}\in\mathcal{T}^{4}}Z_{0,b}^{\epsilon(\Delta^{4})}(\Delta^{4})}{\prod_{\Delta^{3}\in\mathcal{T}^{3}}d_{b_{\Delta^{3}}}}\prod_{\Delta^{4}\in\mathcal{T}^{4}}e^{-2\pi
i\epsilon(\Delta^{4})(q\cup_{s\sigma}(c+\delta_{\sigma}h)+c\cup_{1}f+\delta_{s\sigma}q\cup_{s\sigma}h)}\end{split}$
(92)

Figure 11: 15j symbol of $Z_{b}$ for a $+$ simplex. The blue lines represent
the dual of the $c$ field. The blue lines cross over the pink lines of $f$
fields, and cross behind the black lines of anyons, except for the the case
that $c_{234}$ line crosses over the $a_{012}$ line. Here id means the
identity in $H$, and the blue dot denotes the background domain of $H$
symmetry, see Appendix B, C for the graphical calculus.

In Appendix F, we will see that the bosonic shadow theory has the expected ’t
Hooft anomaly characterized by the response action

$\displaystyle
S_{5,b}=(-1)^{\int\mathop{\mathrm{Sq}}\nolimits^{2}(f)+(2c+A_{H}^{*}\omega_{H})\cup
f}$ (93)

To check that the partition function is independent of the $\mathbb{R}$ lift
of the fractional charge $Q_{a}$, let us study the effect of a local
redefinition of the charge $q_{a}\to q_{a}+2\Lambda$, where $\Lambda\in
C^{2}(M^{4},\mathbb{Z})$ is supported on a single 2-simplex. Firstly, we move
the $H$ symmetry defect away from $\Lambda$, which has the effect of shifting
the partition function by some phase $e^{i\phi}$ according to the ’t Hooft
anomaly. Then, we redefine the fractional charge $q_{a}\to q_{a}+2\Lambda$.
the shift of the partition function under is given by

$\displaystyle\prod_{\Delta^{4}}e^{-2\pi
i\epsilon(\Delta^{4})\delta(2\Lambda\cup h)}=1.$ (94)

We then perform the inverse gauge transformation for the $H$ symmetry defect
to the initial configuration. This shifts the partition function by
$e^{-i\phi}$, since the response action $S_{5,b}$ of the ’t Hooft anomaly is
independent of the fractional charge $q_{a}$. After all, we can see that the
partition function is not shifted by $q_{a}\to q_{a}+\Lambda$. This shows that
the path integral gives a well-defined (3+1)D invariant based on the data of
U(1) fractional charge $Q_{a}\in\mathbb{R}/(2\mathbb{Z})$.

##  VIII Explicit evaluation of U(1) anomaly indicators for (2+1)D TQFT

The (3+1)D invertible TQFTs that are defined by the (2+1)D data can be viewed
as defining the ’t Hooft anomaly of the (2+1)D theory. One can uniquely
identify the (3+1)D invertible TQFTs by computing the partition function on
the generating manifolds for the appropriate bordism groups, which define
anomaly indicator formulas [9, 17, 10, 11].

Below, we will show that the path integrals defined in the preceding sections
can be given a relatively simple formula in the case where the input category
$\mathcal{C}$ is Abelian. This allows us to explicitly compute anomaly
indicators for a variety of physically relevant symmetry groups.

###  VIII.1 U(1) anomaly for (2+1)D Abelian TQFT: bosonic case

####  VIII.1.1 General simplification of path integral for Abelian modular
$\mathcal{C}$

For (2+1)D Abelian modular category, one can conveniently compute the path
integral $Z(M^{4},(h,c))$ by using the particular parameterization of $(F,R)$
given in Eq. (149) and Eq. (150), with
$\mathcal{A}=\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}\times\dots\times\mathbb{Z}_{n_{k}}$.

First, we note that for Abelian theories, the 15j symbol $Z_{0}^{\pm}(01234)$
given in Eq. (47) simplifies to

$\displaystyle Z^{+}_{0}(01234)=$ $\displaystyle
R^{012,234}F^{014,124,234}(F^{014,134,123})^{-1}F^{034,013,123}(F^{034,023,012})^{-1}$
(95) $\displaystyle Z^{-}_{0}(01234)=$
$\displaystyle(R^{012,234})^{-1}\left(F^{014,124,234}\right)^{-1}F^{014,134,123}\left(F^{034,013,123}\right)^{-1}F^{034,023,012}$
(96)

One can express the 15j symbol in terms of a quadratic function of
$\\{a_{i}\\}$. To do this, we utilize the hexagon equation for BTC

$\displaystyle\begin{split}R^{a,c}F^{a,c,b}R^{b,c}&=F^{c,a,b}R^{a\times
b,c}F^{a,b,c},\end{split}$ (97)

In the gauge that we are using for $F$ (Eq. (149)) we have
$F^{a,b,c}=F^{a,c,b}$, so that the above simplifies as

$\displaystyle\begin{split}F^{a,b,c}=R^{b,a}R^{c,a}(R^{b\times
c,a})^{-1}.\end{split}$ (98)

One can then see that

$\displaystyle\begin{split}Z_{0}^{+}(01234)&=R^{012,234}(R^{234,014}R^{124,014}(R^{1234,014})^{-1})((R^{134,014})^{-1}(R^{123,014})^{-1}R^{1234,014})\\\
&\times(R^{123,034}R^{013,034}(R^{0123,034})^{-1})((R^{023,034})^{-1}(R^{012,034})^{-1}R^{0123,034})\\\
&=R^{012,234}\times(R^{234,014}R^{124,014}(R^{134,014})^{-1}(R^{123,014})^{-1})\\\
&\times(R^{123,034}R^{013,034}(R^{023,034})^{-1}(R^{012,034})^{-1})\\\
&=\prod_{i}(\theta_{p_{i}})^{\hat{a}_{i}\cup\hat{a}_{i}+\hat{a}_{i}\cup_{1}\delta\hat{a}_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{\hat{a}_{i}\cup\hat{a}_{j}+\hat{a}_{j}\cup_{1}\delta\hat{a}_{i}}.\end{split}$
(99)

Here, we write the anyon $a\in\mathcal{A}$ as $a=\prod_{i}p_{i}^{a_{i}}$ with
generators $\\{p_{i}\\}$ for $1\leq i\leq k$, and the assignment of anyons on
2-simplices defines a 2-cocycle $a_{i}\in Z^{2}(M^{4},\mathbb{Z}_{n_{i}})$ for
each $i$. $\hat{a}_{i}$ is the integral lift of $a_{i}\in
Z^{2}(M^{4},\mathbb{Z}_{n_{i}})$ that maps $x\in\mathbb{Z}_{n_{i}}$ to
$x\in\mathbb{Z}$. $(M_{p_{i},p_{j}})^{n_{j}}=M_{p_{i},1}=1$, so
$M_{p_{i},p_{j}}$ is an $n_{j}$-th root of unity (since $M_{ab}$ is symmetric,
$M_{p_{i},p_{j}}$ is also an $n_{i}$-th root of unity). Since
$\delta\hat{a}_{i}$ is the integer multiple of $n_{i}$, the 15j symbol further
simplifies as

$\displaystyle\begin{split}Z_{0}^{+}(01234)&=\prod_{i}(\theta_{p_{i}})^{\hat{a}_{i}\cup\hat{a}_{i}+\hat{a}_{i}\cup_{1}\delta\hat{a}_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{{a}_{i}\cup{a}_{j}}.\end{split}$
(100)

The partition function is then expressed as

$\displaystyle\begin{split}Z(M^{4},(h,c))=\sum_{\\{a_{j}\\}}&\mathcal{D}^{2(N_{0}-N_{1})-\chi}\prod_{i}e^{-2\pi
i\int(q(p_{i}^{a_{i}})\cup(c+\delta h)+\delta q(p_{i}^{a_{i}})\cup
h)}(\theta_{p_{i}})^{\int(\hat{a}_{i}\cup\hat{a}_{i}+\hat{a}_{i}\cup_{1}\delta\hat{a}_{i})}\prod_{i<j}(M_{p_{i},p_{j}})^{\int{a}_{i}\cup{a}_{j}},\end{split}$
(101)

where $e^{\int S}$ means the path integral
$\prod_{\Delta^{4}\in\mathcal{T}^{4}}e^{\epsilon(\Delta^{4})S(\Delta^{4})}$.
Note that here we use the notation $q(a)$ instead of $q_{a}$.

In general, the topological twist $\theta_{p}$ for an Abelian BTC is a $2n$-th
root of unity when the anyon $p$ generates a $\mathbb{Z}_{n}$ group under
fusion for even $n$, to further satisfy
$(\theta_{p})^{2n}=M_{p,p}^{n}=M_{p,p^{n}}=1$. Meanwhile, $\theta_{p}$ further
becomes an $n$-th root of unity for odd $n$, due to
$(\theta_{p})^{n^{2}}=\theta_{p^{n}}=1$. Here, for a given $a\in
Z^{2}(M,\mathbb{Z}_{n})$ let us introduce an operation $\mathcal{P}(a)$ known
as the Pontryagin square,

$\displaystyle\mathcal{P}(a):=\begin{cases}&\hat{a}\cup\hat{a}+\hat{a}\cup_{1}\delta\hat{a}\in
Z^{4}(M^{4},\mathbb{Z}_{2n})\quad\text{for even $n$}\\\ &a\cup a\in
Z^{4}(M^{4},\mathbb{Z}_{n})\quad\text{for odd $n$},\end{cases}$ (102)

where $\hat{a}$ here denotes the $\mathbb{Z}_{2n}$ lift of $a$.
$\mathcal{P}(a_{i})$ is known to define a cohomology operation, which gives a
map $\mathcal{P}:H^{2}(M^{4},\mathbb{Z}_{n})\to H^{4}(M^{4},\mathbb{Z}_{2n})$
for even $n$, and $\mathcal{P}:H^{2}(M^{4},\mathbb{Z}_{n})\to
H^{4}(M^{4},\mathbb{Z}_{n})$ for odd $n$ [38, 39, 40]. Using the Pontryagin
square, the above expression can be further simplified as

$\displaystyle\begin{split}Z(M^{4},(h,c))=&\sum_{\\{[a_{j}]\in
H^{2}(M^{4},\mathbb{Z}_{n_{j}})\\}}\mathcal{D}^{2(N_{0}-N_{1})-\chi}\prod_{i}{|B^{2}(M^{4},\mathbb{Z}_{n_{i}})|}\prod_{i}e^{-2\pi
i\int(q(p_{i}^{a_{i}})\cup(c+\delta h)+\delta q(p_{i}^{a_{i}})\cup
h)}(\theta_{p_{i}})^{\mathcal{P}(a_{i})}\prod_{i<j}(M_{p_{i},p_{j}})^{\int{a}_{i}\cup{a}_{j}}\\\
=&\prod_{i}\sqrt{n_{i}}^{-\chi}\frac{|H^{0}(M^{4},\mathbb{Z}_{n_{i}})|}{|H^{1}(M^{4},\mathbb{Z}_{n_{i}})|}\cdot\sum_{\\{[a_{j}]\in
H^{2}(M^{4},\mathbb{Z}_{n_{j}})\\}}\prod_{i}e^{-2\pi
i\int(q(p_{i}^{a_{i}})\cup(c+\delta h)+\delta q(p_{i}^{a_{i}})\cup
h)}(\theta_{p_{i}})^{\mathcal{P}(a_{i})}\prod_{i<j}(M_{p_{i},p_{j}})^{\int{a}_{i}\cup{a}_{j}}\end{split}$
(103)

Here, we have written the normalization factor using
$\mathcal{D}^{2}=\prod_{i}n_{i}$ and the formula

$\displaystyle
n^{N_{0}-N_{1}}|B^{2}(M,\mathbb{Z}_{n})|=\frac{|H^{0}(M,\mathbb{Z}_{n})|}{|H^{1}(M,\mathbb{Z}_{n})|}$
(104)

The proof is by considering the chain complex
$1\xrightarrow{\delta_{0}}(\mathbb{Z}_{n})^{N_{0}}\xrightarrow{\delta_{1}}(\mathbb{Z}_{n})^{N_{1}}\xrightarrow{\delta_{2}}(\mathbb{Z}_{n})^{N_{2}}\xrightarrow{\delta}\to\cdots$.
Note that $|B^{2}(M,\mathbb{Z}_{n})|=|\text{im}(\delta_{2})|$. And by
definition,
$|\text{ker}(\delta_{1})|/|\text{im}(\delta_{0})|=|H^{0}(M,\mathbb{Z}_{n})|$
and
$|\text{ker}(\delta_{2})|/|\text{im}(\delta_{1})|=|H^{1}(M,\mathbb{Z}_{n})|$.
Further we have $|\text{ker}(\delta_{1})||\text{im}(\delta_{1})|=n^{N_{0}}$
and $|\text{ker}(\delta_{2})||\text{im}(\delta_{2})|=n^{N_{1}}$. Putting these
all together gives the above claim.

In the following subsections, we explicitly compute the partition functions on
spacetime manifolds of our interest.

####  VIII.1.2 Partition function on $\mathbb{CP}^{2}$

Firstly, let us evaluate the partition function of our theory on
$\mathbb{CP}^{2}$ with U(1) gauge field. We consider the U(1) gauge field with
the nontrivial 1st Chern class $[c]\in H^{2}(\mathbb{CP}^{2},\mathbb{Z})$.
Since the field strength is entirely given by $c$, $\delta h=0$, and so
without loss of generality, we can take the $\mathbb{R}$ field $h=0$ by a
gauge transformation.

Since the generator of $H^{2}(\mathbb{CP}^{2},\mathbb{Z}_{n_{i}})$ is given by
a $\mathbb{Z}_{n_{i}}$ reduction of that of
$H^{2}(\mathbb{CP}^{2},\mathbb{Z})$, one can pick a representative $a_{i}$ of
$[a_{i}]\in H^{2}(\mathbb{CP}^{2},\mathbb{Z}_{n_{i}})$ that lifts to an
integral cocycle, where we can write
$\mathcal{P}(a_{i})=\hat{a}_{i}\cup\hat{a}_{i}$. Since $\mathcal{P}(a_{i})$ is
a cohomology operation, any representative of a fixed cohomology class
$[a_{i}]$ gives the same result. Hence, we just write
$\mathcal{P}(a_{i})=\hat{a}_{i}\cup\hat{a}_{i}$ for $M^{4}=\mathbb{CP}^{2}$ in
the following discussion.

The partition function is then expressed as

$\displaystyle\begin{split}Z(\mathbb{CP}^{2},(h,c))=\prod_{i}\sqrt{n_{i}}^{-\chi}\frac{|H^{0}(\mathbb{CP}^{2},\mathbb{Z}_{n_{i}})|}{|H^{1}(\mathbb{CP}^{2},\mathbb{Z}_{n_{i}})|}\cdot\sum_{\\{[a_{j}]\in
H^{2}(\mathbb{CP}^{2},\mathbb{Z}_{n_{j}})\\}}&\prod_{i}e^{-2\pi i\int
q(p_{i}^{a_{i}})\cup
c}(\theta_{p_{i}})^{\int\hat{a}_{i}\cup\hat{a}_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{\int{a}_{i}\cup{a}_{j}}.\end{split}$
(105)

Now, with some abuse of notation, we identify the 2-form field $[a_{i}]\in
H^{2}(\mathbb{CP}^{2},\mathbb{Z}_{n_{i}})$ as a $\mathbb{Z}_{n_{i}}$-valued
number $a_{i}\in\mathbb{Z}_{n_{i}}$, by rewriting the cohomology as
$a_{i}[x]\in H^{2}(\mathbb{CP}^{2},\mathbb{Z}_{n_{i}})$ with a generator
$[x]$. Using $\chi(\mathbb{CP}^{2})=3$,
$|H^{0}(\mathbb{CP}^{2},\mathbb{Z}_{n})|=n$,
$|H^{1}(\mathbb{CP}^{2},\mathbb{Z}_{n})|=1$, the normalization factor is
simply given by $1/\mathcal{D}$. The partition function is then expressed as

$\displaystyle\begin{split}Z(\mathbb{CP}^{2},(h,c))=\frac{1}{\mathcal{D}}\sum_{\\{a_{j}\in\mathbb{Z}_{n_{j}}\\}}&e^{-2\pi
i\int q(p_{i}^{a_{i}})\cup
c}(\theta_{p_{i}})^{\hat{a}_{i}^{2}}\prod_{i<j}(M_{p_{i},p_{j}})^{{a}_{i}{a}_{j}}.\end{split}$
(106)

To further simplify the expression, we introduce an anyon
$p=\prod_{i}p_{i}^{a_{i}}$. We then have

$\displaystyle\theta_{p}=\prod_{i}\theta_{p_{i}^{a_{i}}}\prod_{i<j}M_{p_{i}^{a_{i}},p_{j}^{a_{j}}}=\prod_{i}(\theta_{p_{i}})^{\hat{a}_{i}^{2}}\prod_{i<j}(M_{p_{i},p_{j}})^{{a}_{i}{a}_{j}}.$
(107)

Hence, we get

$\displaystyle\begin{split}Z(\mathbb{CP}^{2},(h,c))=\frac{1}{\mathcal{D}}\sum_{p\in\mathcal{C}}e^{-2\pi
iq(p)\cdot C}\theta_{p},\end{split}$ (108)

where $C\in\mathbb{Z}$ denotes the Chern number given by integrating $[c]\in
H^{2}(\mathbb{CP}^{2},\mathbb{Z})$ on the fundamental 2-cycle. Using
$q_{p}=Q_{p}$ mod 1, we get the final expression

$\displaystyle\begin{split}Z(\mathbb{CP}^{2},(h,c))=\frac{1}{\mathcal{D}}\sum_{p\in\mathcal{C}}e^{-2\pi
iQ_{p}\cdot C}\theta_{p}.\end{split}$ (109)

One can immediately see that this partition function becomes a phase, as also
discussed in [17]. Using a vison $v\in\mathcal{C}$, we can rewrite the
fractional charge as

$\displaystyle e^{-2\pi iQ_{p}\cdot C}=M_{p,v^{-C}}=\frac{\theta_{p\times
v^{-C}}}{\theta_{p}\theta_{v^{-C}}}.$ (110)

The partition function is then given by a phase

$\displaystyle
Z(\mathbb{CP}^{2},(h,c))=(\theta_{\overline{v}})^{-C^{2}}\cdot\frac{1}{\mathcal{D}}\sum_{p\in\mathcal{C}}\theta_{p}=(\theta_{{v}})^{-C^{2}}\cdot
e^{\frac{2\pi i}{8}c_{-}}.$ (111)

####  VIII.1.3 Partition function on $S^{2}\times S^{2}$

Next we consider the manifold $S^{2}\times S^{2}$. By simplifying the
normalization factor, we have

$\displaystyle\begin{split}Z(S^{2}\times
S^{2},(h,c))=\frac{1}{\mathcal{D}^{2}}\sum_{\\{[a_{j}]\in H^{2}(S^{2}\times
S^{2},\mathbb{Z}_{n_{j}})\\}}&\prod_{i}e^{-2\pi i\int q(p_{i}^{a_{i}})\cup
c}(\theta_{p_{i}})^{\int\hat{a}_{i}\cup\hat{a}_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{\int{a}_{i}\cup{a}_{j}}.\end{split}$
(112)

Note that $[a_{i}]$ is again lifted to the integral cohomology
$[\hat{a}_{i}]$, and then we used $\delta\hat{a}_{i}=0$. We express
$[a_{j}]\in H^{2}(S^{2}\times S^{2},\mathbb{Z}_{n_{j}})$ using two
$\mathbb{Z}_{n_{j}}$-valued numbers $[a_{j}]=\alpha_{j}[x]+\beta_{j}[y]$,
where $x$ and $y$ are fundamental 2-cycles. Then

$\displaystyle\begin{split}Z(S^{2}\times
S^{2},(h,c))=\frac{1}{\mathcal{D}^{2}}\sum_{\\{\alpha_{j},\beta_{j}\\}}&\prod_{i}e^{-2\pi
iq(p_{i})(\alpha_{i}C_{\beta}+\beta_{i}C_{\alpha})}(\theta_{p_{i}})^{2\alpha_{i}\beta_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{\alpha_{i}\beta_{j}+\beta_{i}\alpha_{j}}.\end{split}$
(113)

where we label the 1st Chern class $[c]=C_{\alpha}[x]+C_{\beta}[y]$ with
$C_{\alpha},C_{\beta}\in\mathbb{Z}$. Let
$p_{\alpha}=\prod_{i}p_{i}^{\alpha_{i}}$,
$p_{\beta}=\prod_{i}p_{i}^{\beta_{i}}$. Then the braiding between
$p_{\alpha}$, $p_{\beta}$ is given by

$\displaystyle
M_{p_{\alpha},p_{\beta}}=\prod_{i}(\theta_{p_{i}})^{2\alpha_{i}\beta_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{\alpha_{i}\beta_{j}+\beta_{i}\alpha_{j}}.$
(114)

We then get the final expression

$\displaystyle\begin{split}Z(S^{2}\times
S^{2},(h,c))=\frac{1}{\mathcal{D}}\sum_{{p}_{\alpha},p_{\beta}\in\mathcal{C}}e^{-2\pi
iQ(p_{\alpha})C_{\beta}}e^{-2\pi
iQ(p_{\beta})C_{\alpha}}S^{*}_{{p}_{\alpha},p_{\beta}}.\end{split}$ (115)

To see that the partition function becomes a phase, we again rewrite it using
Eq. (110) as

$\displaystyle\begin{split}Z(S^{2}\times
S^{2},(h,c))&=\sum_{p_{\alpha},p_{\beta}\in\mathcal{C}}{S}^{*}_{\overline{p}_{\alpha},v^{C_{\beta}}}M_{{p}_{\beta},v^{-C_{\alpha}}}S_{\overline{p}_{\alpha},p_{\beta}}\\\
&=\sum_{p_{\beta}\in\mathcal{C}}M_{{p}_{\beta},v^{C_{-\alpha}}}\delta_{v^{C_{\beta}},{p}_{\beta}}\\\
&=M_{v^{C_{\beta}},v^{-C_{\alpha}}}=(\theta_{v})^{-2C_{\alpha}C_{\beta}}.\end{split}$
(116)

####  VIII.1.4 $\mathrm{U}(1)\times\mathbb{Z}_{2}^{\mathbf{T}}$ symmetry:
partition function on $S^{2}\times\mathbb{RP}^{2}$

Here we compute the partition function with symmetry group
$G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{\mathbf{T}}$. We consider the case where
time-reversal symmetry on unitary modular tensor category (UMTC) does not
permute the anyons. In that case, the consistency equations involving the
$H=\mathbb{Z}_{2}^{\mathbf{T}}$ symmetry data $\eta$ greatly simplifies:

$\displaystyle{U_{\mathbf{T}}(a,b;c)}^{*}U_{\mathbf{T}}(a,b;c)=\frac{\eta_{c}(\mathbf{T},\mathbf{T})}{\eta_{a}(\mathbf{T},\mathbf{T})\eta_{b}(\mathbf{T},\mathbf{T})},\quad\eta_{a}(\mathbf{T},\mathbf{T})={\eta_{a}(\mathbf{T},\mathbf{T})}^{*}.$
(117)

This means that $\eta_{a}(\mathbf{T},\mathbf{T})=\pm 1$ and satisfies
$\eta_{a}(\mathbf{T},\mathbf{T})\eta_{b}(\mathbf{T},\mathbf{T})=\eta_{a\times
b}(\mathbf{T},\mathbf{T})$ for Abelian UMTC. The rest of the $\eta$’s are set
to be trivial by suitable gauge transformations,

$\displaystyle\eta_{1}(\mathbf{g},\mathbf{h})=\eta_{a}(1,\mathbf{g})=\eta_{a}(\mathbf{g},1)=1.$
(118)

We will simply write $\eta_{a}^{\mathbf{T}}:=\eta_{a}(\mathbf{T},\mathbf{T})$.

Further, we have $\theta_{a}=\pm 1$ and $M_{a,b}=\pm 1$ for any pair of
anyons, since the $\mathbf{T}$ symmetry does not permute anyons. This means
that $a^{2}$ for some $a\in\mathcal{A}$ is transparent, since
$M_{a^{2},b}=(M_{a,b})^{2}=1$ for any $b\in\mathcal{A}$. $a^{2}$ hence becomes
a trivial anyon in UMTC, so the group of Abelian anyons has the form of
$\mathcal{A}=(\mathbb{Z}_{2})^{k}$.

When the symmetry does not permute anyons in the Abelian anyon system for
$G=\mathrm{U}(1)\times\mathbb{Z}_{2}^{\mathbf{T}}$, one can set
$U_{\mathbf{g}}(a,b;c)=1$ by a suitable natural isomorphism. 444In that case,
the consistency condition Eq. (175), Eq. (176) imply that
$U_{\mathbf{g}}(a,b)$ defines an element of
$Z^{2}(B\mathcal{A},\mathrm{U}(1))$ for fixed $\mathbf{g}\in H$, which is
symmetric under the exchange of $a,b$. For an Abelian discrete group
$\mathcal{A}$, one can easily show that any symmetric function
$U_{\mathbf{g}}(a,b)$ is trivial in cohomology
$H^{2}(B\mathcal{A},\mathrm{U}(1))$. Hence it can be set as 1 by a suitable
natural isomorphism, which is regarded as a shift of $U$ by coboundary
$\delta\gamma(a,b):=(\gamma(a)\gamma(b))/\gamma(a\times b)$. We hence consider
the case where $U_{\mathbf{g}}(a,b;c)=1$, and ignore the contribution of $U$
symbols.

We consider the case with trivial U(1) holonomy $h=0$, and the nontrivial
Chern number on $S^{2}$ given by

$\displaystyle\int_{S^{2}}c=C\in\mathbb{Z}.$ (119)

By simplifying the normalization factor using $\chi=2$ and
$|H^{0}(S^{2}\times\mathbb{RP}^{2},\mathbb{Z}_{2})|=|H^{1}(S^{2}\times\mathbb{RP}^{2},\mathbb{Z}_{2})|=2$,
the partition function is then obtained as

$\displaystyle\begin{split}Z(S^{2}\times\mathbb{RP}^{2},(h,c))=\frac{1}{\mathcal{D}^{2}}&\sum_{\\{[a_{j}]\in
H^{2}(S^{2}\times\mathbb{RP}^{2},\mathbb{Z}_{2})\\}}\prod_{i}e^{-2\pi i\int
q({p_{i}})a_{i}\cup
c}(\theta_{p_{i}})^{\int{a}_{i}\cup{a}_{i}}(\eta^{\mathbf{T}}_{p_{i}})^{\int
a_{i}\cup
w_{1}^{2}}\prod_{i<j}(M_{p_{i},p_{j}})^{\int{a}_{i}\cup{a}_{j}}.\end{split}$
(120)

Here we have used that $\eta_{p_{012}}(23,34)$ on a single 4-simplex with an
anyon $p_{012}=\prod_{i}p_{i}^{a_{i}(012)}$ can be expressed as
$\prod_{i}(\eta^{\mathbf{T}}_{p_{i}})^{a_{i}\cup w_{1}^{2}}$, where $w_{1}$ is
the 1st Stiefel-Whitney class. This can be seen by
$\eta_{p_{012}}(23,34)=\prod_{i}(\eta_{p_{i}}(23,34))^{a_{i}(012)}=\prod_{i}(\eta_{p_{i}}^{\mathbf{T}})^{a_{i}(012)s(\mathbf{g}_{23})s(\mathbf{g}_{34})}$,
and $s(\mathbf{g}_{ij})=w_{1}(ij)$. We express $[a_{j}]\in
H^{2}(S^{2}\times\mathbb{RP}^{2},\mathbb{Z}_{2})$ using a
$\mathbb{Z}_{2}$-valued number $\alpha_{j}$, $\beta_{j}$ in terms of
$[a_{j}]=\alpha_{j}[x]+\beta_{j}[y]$, where $x$ and $y$ are fundamental
2-cocycles of $S^{2}$ and $\mathbb{RP}^{2}$ respectively. Then

$\displaystyle\begin{split}Z(S^{2}\times\mathbb{RP}^{2},(h,c))=\frac{1}{\mathcal{D}^{2}}\sum_{\\{\alpha_{j},\beta_{j}\\}}\prod_{i}e^{-2\pi
iq(p_{i})\beta_{i}C}(\theta_{p_{i}})^{2\alpha_{i}\beta_{i}}(\eta^{\mathbf{T}}_{p_{i}})^{\alpha_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{\alpha_{i}\beta_{j}+\alpha_{j}\beta_{i}}.\end{split}$
(121)

Let $p_{\alpha}=\prod_{i}p_{i}^{\alpha_{i}}$,
$p_{\beta}=\prod_{i}p_{i}^{\beta_{i}}$. The braiding between $p_{\alpha}$,
$p_{\beta}$ is then given by

$\displaystyle M_{p_{\alpha},p_{\beta}}=\mathcal{D}\cdot
S_{p_{\alpha},p_{\beta}}=\prod_{i}(\theta_{p_{i}})^{2\alpha_{i}\beta_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{\alpha_{i}\beta_{j}+\beta_{i}\alpha_{j}}.$
(122)

We then get the final expression

$\displaystyle\begin{split}Z(S^{2}\times\mathbb{RP}^{2},(h,c))&=\frac{1}{\mathcal{D}}\sum_{\\{p_{\alpha},p_{\beta}\in\mathcal{C}\\}}e^{-2\pi
iq(p_{\beta})C}\eta^{\mathbf{T}}_{p_{\alpha}}S_{p_{\alpha},p_{\beta}}.\end{split}$
(123)

To see that the partition function becomes a phase, we again rewrite it using
Eq. (110), and that $\eta^{\mathbf{T}}_{p}$ can be expressed as $M_{p,w}$ with
some anyon $w\in\mathcal{C}$. Then

$\displaystyle\begin{split}Z(S^{2}\times\mathbb{RP}^{2},(h,c))&=\mathcal{D}\sum_{p_{\alpha},p_{\beta}\in\mathcal{C}}S_{p_{\alpha},w}S_{p_{\beta},v^{-C}}S_{p_{\alpha},p_{\beta}}\\\
&=\mathcal{D}\sum_{p_{\alpha},p_{\beta}\in\mathcal{C}}S_{p_{\alpha},w}\overline{S}_{p_{\beta},v^{-C}}S_{p_{\alpha},p_{\beta}}=\mathcal{D}\sum_{p_{\alpha}\in\mathcal{C}}S_{p_{\alpha},w}\delta_{p_{\alpha},v^{C}}\\\
&=M_{v^{C},w}.\end{split}$ (124)

###  VIII.2 $\mathrm{U}(1)^{f}$ anomaly for (2+1)D Abelian Spinc TQFT:
fermionic case

####  VIII.2.1 General simplification of path integral for Abelian super-
modular $\mathcal{C}$

We evaluate the the bosonic shadow theory of the fermionic state sum (92) in
the case that $\mathcal{C}$ is Abelian. In that case, the anyons form a group
$\mathcal{A}=\mathbb{Z}_{2}^{\psi}\times\mathbb{Z}_{n_{1}}\times\mathbb{Z}_{n_{2}}\times\dots\times\mathbb{Z}_{n_{k}}$,
where $\mathbb{Z}_{2}^{\psi}$ is generated by a transparent fermion $\psi$. We
fix generators $p_{i}$ of $\mathbb{Z}_{n_{i}}$ for $0\leq i\leq k$ with
$p_{0}=\psi$ and $n_{0}=2$. An element $a\in\mathcal{A}$ is then expressed as
$a=\prod_{i}p_{i}^{a_{i}}$. $(F,R)$ symbols are again given given in Eq. (149)
and Eq. (150).

We immediately see that $F^{\psi,b,c}=1,R^{\psi,b}=R^{b,\psi}=(-1)^{b_{0}}$
for any $b,c\in\mathcal{A}$.

Based on these $(F,R)$ symbols, the 15j symbol $Z_{0,b}^{\pm}(01234)$ is
expressed in a simpler form by using Eq. (98) as

$\displaystyle\begin{split}Z_{0,b}^{+}(01234)=&\Big{(}\big{(}F^{034\,,\,f_{0123}\,,\,013\times
f_{0123}}_{0134\times f_{0134}\,,\,034\times
f_{0123}\,,\,013}\big{)}^{*}\,\,\,R^{f_{0123}\,,\,034}_{034\times
f_{0123}}\cdot F^{014\,,\,f_{1234}\,,\,134\times f_{1234}}_{0134\,,\,014\times
f_{1234}\,,\,134}\,\,\,\big{(}R^{f_{1234}\,,\,014}_{014\times
f_{1234}}\big{)}^{*}\Big{)}\\\ &\cdot\Bigg{(}F^{024\times
f_{0124}\,,\,234\,,\,012}_{d\,,\,0234\times
f_{0124}\,,\,a}\,\,R^{012\,,\,234}_{a}\,\,\big{(}F^{024\times
f_{0124}\,,\,012\,,\,234}_{d\,,\,0124\,,\,a}\big{)}^{*}\,\,F^{014\,,\,124\,,\,234}_{d\,,\,0124\,,\,1234}\big{(}F^{014\,,\,134\times
f_{1234}\,,\,123}_{d\,,\,0134\times f_{1234}\,,\,1234}\big{)}^{*}\,\,\\\
&\quad\quad\quad\quad\quad F^{034\times f_{0234}\times f_{0124}\,,\,013\times
f_{0123}\,,\,123}_{d\,,\,0134\times f_{1234}\,,\,0123\times
f_{0123}}\,\,\big{(}F^{034\times f_{0234}\times
f_{0124}\,,\,023\,,\,012}_{d\,,\,0234\times f_{0124}\,,\,0123\times
f_{0123}}\big{)}^{*}\Bigg{)}\\\ =&(R^{013\times
f_{0123},034})^{-1}R^{013,034}R^{134\times f_{1234},014}(R^{134,014})^{-1}\\\
&\cdot R^{012,234}\cdot R^{124,014}R^{234,014}\cdot(R^{134\times
f_{1234},014}R^{123,014})^{-1}\\\ &\cdot R^{013\times f_{0123},034\times
f_{0234}\times f_{0124}}R^{123,034\times f_{0234}\times
f_{0124}}\cdot(R^{023,034\times f_{0234}\times f_{0124}}R^{012,034\times
f_{0234}\times f_{0124}})^{-1}\\\
=&(-1)^{f_{0234}(a_{0})_{013}+f_{0124}(a_{0})_{013}+f_{0234}(a_{0})_{123}+f_{0124}(a_{0})_{123}+f_{0234}(a_{0})_{023}+f_{0124}(a_{0})_{023}+f_{0234}(a_{0})_{012}+f_{0124}(a_{0})_{012}}\\\
&\cdot(-1)^{f_{0123}(f_{0234}+f_{0124})}\\\
&R^{012,234}\cdot(R^{234,014}R^{124,014}(R^{134,014})^{-1}(R^{123,014})^{-1})\\\
&\cdot(R^{123,034}R^{013,034}(R^{023,034})^{-1}(R^{012,034})^{-1})\\\
=&R^{012,234}\cdot(R^{234,014}R^{124,014}(R^{134,014})^{-1}(R^{123,014})^{-1})\\\
&\cdot(R^{123,034}R^{013,034}(R^{023,034})^{-1}(R^{012,034})^{-1})\\\
=&\prod_{i}(\theta_{p_{i}})^{\hat{a}_{i}\cup\hat{a}_{i}+\hat{a}_{i}\cup_{1}\delta\hat{a}_{i}}\prod_{i<j}(M_{p_{i},p_{j}})^{\hat{a}_{i}\cup\hat{a}_{j}+\hat{a}_{j}\cup_{1}\delta\hat{a}_{i}},\end{split}$
(125)

where we used $\delta(a_{0})=f$ in the fourth equation. The partition function
is then given by

$\displaystyle\begin{split}Z_{b}(M^{4},(h,c),f)=\sum_{\\{a_{j}\\}}&\mathcal{D}^{2(N_{0}-N_{1})-\chi}e^{-2\pi
i\int c\cup_{1}f}\prod_{i}e^{-2\pi i\int(q(p_{i}^{a_{i}})\cup(c+\delta
h)+\delta q(p_{i}^{a_{i}})\cup
h)}(\theta_{p_{i}})^{\int(\hat{a}_{i}\cup\hat{a}_{i}+\hat{a}_{i}\cup_{1}\delta\hat{a}_{i})}\prod_{i<j}(M_{p_{i},p_{j}})^{\int{a}_{i}\cup{a}_{j}}\end{split}$
(126)

####  VIII.2.2 Partition function on $\mathbb{CP}^{2}$

We evaluate the partition function of our theory on $\mathbb{CP}^{2}$ with
Spinc structure. We consider the gauge fields $h=0$ by suitable gauge
transformations, and we have $\int c=C\in\mathbb{Z}+1/2$ with the integral
taken over a fundamental 2-cycle of $\mathbb{CP}^{2}$. By using the similar
logic to Sec. VIII.1.2, we have

$\displaystyle\begin{split}Z_{b}(\mathbb{CP}^{2},(h,c),f)=&\mathcal{D}^{2(N_{0}-N_{1})-\chi}\prod_{0<i}{|B^{2}(\mathbb{CP}^{2},\mathbb{Z}_{n_{i}})|}\sum_{\\{a_{j}\in\mathbb{Z}_{n_{j}}|0<j\\},}\prod_{0<i}e^{-2\pi
i\int q(p_{i}^{a_{i}})\cup
c}(\theta_{p_{i}})^{\hat{a}_{i}^{2}}\prod_{0<i<j}(M_{p_{i},p_{j}})^{{a}_{i}{a}_{j}}\\\
&\cdot\sum_{\begin{subarray}{c}a_{0}\in
C^{2}(\mathbb{CP}^{2},\mathbb{Z}_{2}),\\\
\delta(a_{0})=f\end{subarray}}(-1)^{\int 2c\cup_{1}f}(-1)^{\int
q(\psi^{a_{0}})\cup 2c}(-1)^{\int a_{0}\cup a_{0}+a_{0}\cup_{1}f}\end{split}$
(127)

When $f=0$, the path integral is computed as

$\displaystyle\begin{split}Z_{b}(\mathbb{CP}^{2},(h,c),f=0)=\frac{1}{\mathcal{D}}\sum_{p\in\mathcal{C}}e^{-2\pi
iq(p)\cdot C}\theta_{p}.\end{split}$ (128)

We then get the partition function of the invertible fermionic theory coupled
to the Spinc structure $\xi_{\mathcal{G}}$ by performing the fermion
condensation:

$Z(M,A_{b},\xi_{\mathcal{G}})=\frac{1}{\mathcal{N}}\sum_{f_{3}\in
H^{3}(M,\mathbb{Z}_{2})}Z_{b}(M,A_{b},f_{3})z_{c}(M,\xi_{\mathcal{G}},f_{3}).$

In [11], the normalization constant is shown to be
$\mathcal{N}=\sqrt{|H^{2}(M^{4},\mathbb{Z}_{2})|}$. With this normalization,
it is argued in [11] that the partition function of the path integral
$Z(M,A_{b},\xi_{\mathcal{G}})$ becomes a phase for general configuration of
$A_{b}$.

Since $H^{3}(\mathbb{CP}^{2},\mathbb{Z}_{2})$ is trivial, we only need to
consider the case where $f=0$. Using $z_{c}(M,\xi_{\mathcal{G}},f_{3}=0)=1$,
we obtain

$\displaystyle
Z(\mathbb{CP}^{2},(h,c),\xi_{\mathcal{G}})=\frac{1}{\sqrt{2}}Z_{b}(\mathbb{CP}^{2},(h,c),f_{3}=0)=\frac{1}{\sqrt{2}\mathcal{D}}\sum_{p\in\mathcal{C}}e^{-2\pi
iQ_{p}\cdot C}\theta_{p},$ (129)

where we used $q_{p}=Q_{p}$ mod 2.

####  VIII.2.3 Partition function on $S^{2}\times S^{2}$