In 2+1-dimensions (2+1D), a gapped quantum phase with no symmetry (_i.e._ a
topological order) can have a thermal Hall conductance
$\kappa_{xy}=c\frac{\pi^{2}k_{B}^{2}}{3h}T$, where the dimensionless $c$ is
called chiral central charge. If there is a $U_{1}$ symmetry, a gapped quantum
phase can also have a Hall conductance $\sigma_{xy}=\nu\frac{e^{2}}{h}$, where
the dimensionless $\nu$ is called filling fraction. In this paper, we derive
some quantization conditions of $c$ and $\nu$, via a cobordism approach to
define Chern–Simons topological invariants which are associated with $c$ and
$\nu$. In particular, we obtain quantization conditions that depend on the
ground state degeneracies on Riemannian surfaces, and quantization conditions
that depend on the type of spacetime manifolds where the topological partition
function is non-zero.

# Quantization of Chern-Simons topological invariants
for H-type and L-type quantum systems

Oscar Randal-Williams  DPMMS, Centre for Mathematical Sciences, Wilberforce
Road, Cambridge CB3 0WB, UK  Lokman Tsui  Department of Physics, Massachusetts
Institute of Technology, Cambridge, Massachusetts 02139, USA  Xiao-Gang Wen
Department of Physics, Massachusetts Institute of Technology, Cambridge,
Massachusetts 02139, USA

###### Contents

  1. I Introduction
  2. II The vector bundle on the moduli space for H-type bosonic systems with gapped liquid ground states
  3. III Chern–Simons invariants in 2+1D H-type bosonic $U_{1}$-SET and $U_{1}$-SPT orders
  4. IV Topological invariants in 2+1D H-type fermionic enriched topological orders with $U^{f}_{1}$ symmetry
  5. V The topological invariants for L-type topological orders
  6. VI The topological invariants for L-type topological orders with $U_{1}$ symmetry
  7. VII Summary
  8. A Characteristic numbers of 4-manifolds
  9. B Characteristic numbers of surface bundles over surfaces
  10. C Checking (29), (30), (31)
  11. D Group extension and trivialization

##  I Introduction

Different phases of matter are characterized by different orders in
them.Landau (1937a, b) Topological orderWen (1989); Wen and Niu (1990); Wen
(1990a) is a new kind of order beyond Landau symmetry breaking order.Landau
(1937a, b) It cannot be characterized by the local order parameters associated
with the symmetry breaking, but can be characterized by topological quantum
field theories.Witten (1989) Physically, we need to use new topological
quantum numbers to characterize topological orders. In 2+1D, the chiral
central charge $c$Wen (1991a, 1990b) (i.e. the thermal Hall conductance
$\kappa_{xy}=c\frac{\pi^{2}k_{B}^{2}}{3h}T$Kane and Fisher (1997)) of the edge
excitations is one such topological quantum number. When there is a $U_{1}$
symmetry, the Hall conductance $\sigma_{xy}=\nu\frac{e^{2}}{h}$ is another
such topological quantum number. In other words, the chiral central charge $c$
(partially) characterize 2+1D topological orders, and the dimensionless Hall
conductance $\nu$, together with $c$, (partially) characterize 2+1D
topological orders with $U_{1}$-symmetry.Chen  _et al._ (2010)

In this paper, we like to address the issue of the quantization of those
topological quantum numbers. At first sight, it seems that $c$ and $\nu$ are
not quantized since they do not have to be integers. On the other hand, $c$
and $\nu$ must be rational numbers, suggesting that they do satisfy certain
quantization conditions which can be complicated. A quantization condition on
$\nu$ for interacting systems was first obtained by Niu–Thouless–Wu in
LABEL:NTW8572. In this paper, we are going to generalize their result and
discuss those complicated quantization conditions that apply to all
topological orders with $U_{1}$-symmetry.

Before calculating those quantization conditions, we need to distinguish two
types of quantum systems: H-type and L-type.Kong and Wen (2014) The two types
of quantum systems have different quantization conditions.

The quantum systems in condensed matter physics are all H-type, i.e. are all
described by local Hamiltonians defined on smooth spatial manifolds. The
quantum systems in high energy theory and quantum field theory, such as the
topological quantum field theories, are L-type, i.e. are described by local
Lagrangian path integrals on smooth spacetime manifolds. Mathematically, the
L-type topological orders may correspond to unitary fully extended topological
quantum field theories. The H-type topological orders may correspond to
topless fully extended topological quantum field theories (or $d+\epsilon$
dimensional fully extended topological quantum field theories, see Walker ).

The gapped liquid statesZeng and Wen (2015); Swingle and McGreevy (2016) with
$U_{1}$ symmetry include $U_{1}$-symmetry enriched topological (SET)
ordersChen  _et al._ (2010); Lu and Vishwanath (2013); Mesaros and Ran (2013);
Hung and Wan (2013) and $U_{1}$-symmetry protected trivial (SPT) orders.Gu and
Wen (2009); Chen  _et al._ (2013); Gu and Wen (2014); Lu and Vishwanath
(2012); Liu and Wen (2013); Chen and Wen (2012); Senthil and Levin (2013) For
2+1D H-type gapped states with $U_{1}$ symmetry, the quantization of $c$ and
$\nu$ is determined from the ground state degeneracy on closed genus $g$
surfaces, and the main results are given by eqn. (III.1) and eqn. (IV.3).

In particular, for H-type invertible gapped states111By definition, invertible
gapped states have no fractionalized excitations. As a result, they have non-
degenerate ground state on closed smooth spatial manifolds. with $U_{1}$
symmetry, $c$ and $\nu$ satisfy the following quantization conditions:

bosonic systems: $\displaystyle c,\ \nu=0\text{ mod }2.$ fermionic systems:
$\displaystyle c,\ \nu,\ \frac{\nu-c}{2}=0\text{ mod }1.$ (1)

The even Hall conductance $\nu=0$ mod 2 for bosonic SPT phases was pointed out
in LABEL:LV1219,LW1224,CW1217,SL1204. We remark that the known H-type
invertible topological orders with $U_{1}$ symmetry do not saturate the above
quantization conditions. For example, we do not know any H-type bosonic
invertible topological orders with $c=2$. Also, we do not know any H-type
fermionic $U^{f}_{1}$ symmetric invertible topological orders with $\nu-c=2$.

For 2+1D L-type gapped states with $U_{1}$ symmetry, the quantization of $c$
and $\nu$ is determined by the non-vanishing topological partition functions
on certain types of spacetime manifolds, such as orientable or spinC. We find
that, for L-type invertible gapped phases with $U_{1}$ symmetry, $c$ and $\nu$
satisfy the following quantization conditions:

bosonic systems: $\displaystyle c=0\text{ mod }8,\ \ \nu=0\text{ mod }2.$
fermionic systems: $\displaystyle c,\ \nu,\ \frac{\nu-c}{8}=0\text{ mod }1.$
(2)

The above L-type gapped phases with $U_{1}$ symmetry have a framing anomaly,
i.e. the partition functions of the states dependent on the choices of the
framing of the spacetime manifold.Witten (1989)

If the microscopic Lagrangian path integral is manifestly independent of
choices of the framing (for example, only dependent on the diffeomorphism
equivalent classes of spacetime metrics), then the resulting partition must
also be independent of choices of the framing. The resulting phase is said to
be free of framing anomaly. In this case, $c$ must be multiple of 24, and the
above quantization conditions reduce to

bosonic systems: $\displaystyle c=0\text{ mod }24,\ \ \nu=0\text{ mod }2.$
fermionic systems: $\displaystyle c=0\text{ mod }24,\ \ \nu=0\text{ mod }8.$
(3)

It is interesting to note that the bosonic $E_{8}$ quantum Hall state with
$c=8$ (see eqn. (21) with $K$ given by the $E_{8}$ matrix (84)) is an H-type
topological order. But it cannot be realized by a topological quantum field
theory with no framing anomaly. In other words, it cannot be realized by a
Lagrangian path integral that manifestly only depends on the diffeomorphism
equivalent classes of spacetime metrics.

However, the $E_{8}$ state can be realized by a Lagrangian path integral with
framing anomaly (i.e. depends on the additional framing structure of spacetime
manifold).Witten (1989) One such realization is given by the dynamical
Chern–Simons theory eqn. (22) with $K=E_{8}$. Here we like to remark that it
is highly nontrivial to define the dynamical Chern–Simons theory eqn. (22) on
spacetime so that we can compute the partition function via a finite
calculation. For bosonic Chern–Simons theory eqn. (22) (where $K_{II}$ is
even), a non perturbative definition was recently given in LABEL:DW190608270,
by triangulating the spacetime and giving the triangulation a branching
structure. It appears that the branching structure plays a role of framing
structure.

On the other hand, the bosonic $E_{8}^{3}$ quantum Hall state (the stacking of
three bosonic $E_{8}$ quantum Hall state) can be described by a topological
quantum field theory with no framing anomaly, i.e. it can be realized by a
Lagrangian path integral that only depends on the diffeomorphism equivalent
classes of spacetime metrics (i.e. does not depend on the framing structure of
spacetime). An explicit construction of the Lagrangian path integral in terms
of $SO_{\infty}$ non-linear $\sigma$-model was given in LABEL:W1477.

###  I.1 Notations and conventions

We will abbreviate the cup product $a\smile b$ and the wedge product $a\wedge
b$ as $ab$. We will use $\overset{\scriptscriptstyle n}{=}$ to mean equal up
to a multiple of $n$, and use $\overset{\scriptscriptstyle\hskip
1.0pt\mathrm{d}}{=}$ to mean equal up to $\hskip 1.0pt\mathrm{d}f$ (i.e. up to
a coboundary). We will use $\lfloor x\rfloor$ to denote the greatest integer
less than or equal to $x$, and $\langle l,m\rangle$ for the greatest common
divisor of $l$ and $m$ (with $\langle 0,m\rangle\equiv m$).

We introduce a symbol $\leftthreetimes$ to construct fiber bundle
$E=F\leftthreetimes B$ from the fiber $F$ and the base space $B$:

$\displaystyle\text{pt}\to F\to E\to B\to\text{pt}.$ (4)

We will also use $\leftthreetimes$ to construct group extension of $G$ by $N$
Morandi (1997):

$\displaystyle 1\to N\to N\leftthreetimes_{e_{2},\alpha}G\to G\to 1.$ (5)

Here $e_{2}\in H^{2}[G;Z(N)]$ and $Z(N)$ is the center of $N$. Also $G$ may
have a non-trivial action on $Z(N)$ via $\alpha:G\to\text{Aut}(N)$. $e_{2}$
and $\alpha$ characterize different group extensions.

Also, we will use $Z_{n}=\\{1,\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi}{n}},\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt2\frac{2\pi}{n}},\cdots,\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt(n-1)\frac{2\pi}{n}}\\}$ to
denote an Abelian group, where the group multiplication is “$*$”. We use
$\mathbb{Z}_{n}=\\{0,1,\cdots,n-1\\}$ to denote an integer lifting of $Z_{n}$,
where “+” is done without mod-$n$. In this sense, $\mathbb{Z}_{n}$ is not a
group under “+”. But under a modified equality $\overset{\scriptscriptstyle
n}{=}$, $\mathbb{Z}_{n}$ is the $Z_{n}$ group under “+”. Similarly, we will
use ${\mathbb{R}/\mathbb{Z}}=[0,1)$ to denote an $\mathbb{R}$-lifting of
$U_{1}$ group. Under a modified equality $\overset{\scriptscriptstyle 1}{=}$,
${\mathbb{R}/\mathbb{Z}}$ is the $U_{1}$ group under “+”. In this paper, we
have expressions containing the addition “+” of $\mathbb{Z}_{n}$-valued or
${\mathbb{R}/\mathbb{Z}}$-valued, such as
$a^{\mathbb{Z}_{n}}_{1}+a^{\mathbb{Z}_{n}}_{2}$ where $a^{\mathbb{Z}_{n}}_{1}$
and $a^{\mathbb{Z}_{n}}_{2}$ are $\mathbb{Z}_{n}$-valued. Those additions “+”
are done without mod $n$ or mod 1. In this paper, we also have expressions
like $\frac{1}{n}a^{\mathbb{Z}_{n}}_{1}$. Such an expression converts a
$\mathbb{Z}_{n}$-valued $a^{\mathbb{Z}_{n}}_{1}$ to a
${\mathbb{R}/\mathbb{Z}}$-valued $\frac{1}{n}a^{\mathbb{Z}_{n}}_{1}$, by
viewing the $\mathbb{Z}_{n}$-value as a $\mathbb{Z}$-value. (In fact,
$\mathbb{Z}_{n}$ is a $\mathbb{Z}$ lifting of $Z_{n}$.)

##  II The vector bundle on the moduli space for H-type bosonic systems with
gapped liquid ground states

In this paper, we study topological invariants for gapped liquidZeng and Wen
(2015); Swingle and McGreevy (2016) ground states without excitations. In
order to characterize different classes of gapped liquid ground states, we
consider a moduli space – the space of Hamiltonians with on-site symmetry $G$
that have gapped liquid state as the ground state. Here we assume that the
Hamiltonians in the moduli space depend on the metrics $g_{ij}$ of the space
as well as the $G$-symmetry twist (i.e. the background $G$-gauge field $A_{i}$
from gauging the on-site $G$-symmetryWen (2013)). So for a closed
$n$-dimensional space $M^{n}$, with a certain topology as well as a
$G$-principle bundle for the $G$-symmetry twist on $M^{n}$, the moduli space
is the space of spatial metrics and $G$-connections, which is denoted as
${\cal M}_{G\leftthreetimes M^{n}}$. Here $G\leftthreetimes M^{n}$ describes
the $G$-principle bundle:

$\displaystyle\text{pt}\to G\to G\leftthreetimes M^{n}\to M^{n}\to\text{pt}.$
(6)

Also

we assume that the pairs $(g_{ij},A_{i})$ differ by gauge transformations and
diffeomorphisms are equivalent and represent the same point of the moduli
space ${\cal M}_{G\leftthreetimes M^{n}}$.

Since on every point of the moduli space, the corresponding Hamiltonian is in
a gapped liquid phase, the subspace formed by the degenerate ground states can
be viewed as the fiber at the point. This way, we obtain a vector bundle over
the moduli space ${\cal M}_{G\leftthreetimes M^{n}}$. It was conjectured in
LABEL:W9039 that

the topology of this vector bundle completely characterize the H-type
topological orders, i.e. gapped liquid phases with symmetry.

In particular, let us consider a closed subspace $B$ in ${\cal
M}_{G\leftthreetimes M^{n}}$. The vector bundle on ${\cal M}_{G\leftthreetimes
M^{n}}$ reduces to a vector bundle on $B$. In the next section, we concentrate
on this vector bundle on $B$, and try to relate the vector bundle to
topological term in the partition function of the quantum system under
consideration. We will follow closely the approach proposed in
LABEL:KW1458,KW200411904.

##  III Chern–Simons invariants in 2+1D H-type bosonic $U_{1}$-SET and
$U_{1}$-SPT orders

###  III.1 Bosonic gapped liquids with $U_{1}$ symmetry in 2-dimensional space

In this section, we consider gapped liquids with $U_{1}$ symmetry in
2-dimensional space for bosonic systems. Let $\Sigma_{g}$ be a $2$-dimensional
closed spatial manifold with a $U_{1}$ connection, that describes the twisted
global $U_{1}$ symmetry. The corresponding $U_{1}$-bundle is given by
$U_{1}\leftthreetimes\Sigma_{g}$. In this case, the moduli space ${\cal
M}_{U_{1}\leftthreetimes\Sigma_{g}}$ of the system is the space of metrics on
$\Sigma_{g}$ and the $U_{1}$-connections. The degenerate ground states on
$\Sigma_{g}$ give rise to a vector bundle over the moduli space ${\cal
M}_{U_{1}\leftthreetimes\Sigma_{g}}$, where the dimension of the vector is
given by the ground state degeneracy $D_{U_{1}\leftthreetimes\Sigma_{g}}$.

Let us consider a loop $S^{1}$ in the moduli space ${\cal
M}_{U_{1}\leftthreetimes\Sigma_{g}}$. The holonomy of the vector bundle around
the loop is given by a unitary matrix of $D_{U_{1}\leftthreetimes\Sigma_{g}}$
dimension: $W_{S^{1}}$. We note that the unitary matrix $W_{S^{1}}$ is the
non-Abelian geometric phaseWilczek and Zee (1984) of the degenerate ground
states under the adiabatic deformation around the loop $S^{1}$.

From the vector bundle, we can obtain a determinant bundle, whose holonomy
around the loop is given by the determinant: Det$(W_{S^{1}})$. This phase
factor is directly related to the topological term in the effective action
$S_{eff}=\int_{\Sigma_{g}\leftthreetimes S^{1}}\hskip
1.0pt\mathrm{d}^{3}x\;{\cal L}_{eff}$:

$\displaystyle\text{Det}(W_{S^{1}})=\Big{(}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\int_{\Sigma_{g}\leftthreetimes S^{1}}\hskip
1.0pt\mathrm{d}^{3}x\;{\cal
L}_{eff}}\Big{)}^{D_{U_{1}\leftthreetimes\Sigma_{g}}}$ (7)

The effective action may contain a gravitational Chern–Simons term and a
$U_{1}$ Chern–Simons term:Wen and Zee (1992a, b); Gromov  _et al._ (2015)

$\displaystyle{\cal L}_{eff}\hskip
1.0pt\mathrm{d}^{3}x=-2\pi\frac{c}{24}\omega_{3}+\frac{\nu}{4\pi}(A+sA^{SO_{2}})\hskip
1.0pt\mathrm{d}(A+sA^{SO_{2}})$
$\displaystyle=-2\pi\frac{c}{24}\omega_{3}+\frac{\nu}{4\pi}A_{eff}\hskip
1.0pt\mathrm{d}A_{eff}$ (8)

where $A$ is the $U_{1}$ connection 1-form describing the total
electromagnetic field, $s$ is the orbital spin carried by the charged
bosons,Wen and Zee (1992a, b) and $A^{SO_{2}}$ is the time-dependent
connection 1-form describing the $SO_{2}$ tangent bundle of the curved space.
$A_{eff}\equiv A+sA^{SO_{2}}$ is the effective $U_{1}$ connection 1-form. Also
$\omega_{3}$ is the gravitational Chern–Simons term that satisfy $\hskip
1.0pt\mathrm{d}\omega_{3}=p_{1}$, where $p_{1}$ is the first Pontryagin class
of the tangent bundle. In the above, $c$ is the chiral central charge of the
edge theory, and $\nu$ is proportional to the Hall conductance:

$\displaystyle\sigma_{xy}=\frac{\nu}{2\pi}=\nu\frac{e^{2}}{h}$ (9)

(in $e=\hbar=1$ unit).

We note that $\frac{\nu}{2\pi}\hskip 1.0pt\mathrm{d}A_{eff}=j$ is the
2-cocycle describing the conserved density and current of charged bosons in
the ground state. Thus $A_{eff}$ satisfy the quantization condition

$\displaystyle\int_{\Sigma_{g}}\frac{\nu}{2\pi}\hskip
1.0pt\mathrm{d}A_{eff}\in\mathbb{Z}$ (10)

for the closed 2-dimensional space $\Sigma_{g}$. In fact,
$\int_{\Sigma_{g}}\frac{\nu}{2\pi}\hskip 1.0pt\mathrm{d}A_{eff}$ is the number
of bosons in the ground state.

Now assume the $S^{1}$ to be the boundary of a 2-dimension subspace $B^{2}$ in
${\cal M}_{U_{1}\leftthreetimes\Sigma_{g}}$. Eqn. (7) can be rewritten
asZumino  _et al._ (1984)

$\displaystyle\text{Det}(W_{S^{1}})=\hskip
1.0pt\mathrm{e}^{D_{U_{1}\leftthreetimes\Sigma_{g}}\hskip
1.0pt\mathrm{i}\hskip 1.0pt2\pi\int_{\Sigma_{g}\leftthreetimes
B^{2}}-\frac{c}{24}p_{1}+\frac{\nu}{2}c_{1}^{2}},$ (11)

where $a=\frac{A_{eff}}{2\pi}$ and $c_{1}=\hskip 1.0pt\mathrm{d}a$ is the
first Chern class. If we shrink $S^{1}$ to a point, $B^{2}$ becomes a closed
2-dimensional subspace. In this case, $\text{Det}(W_{S^{1}})=1$ and

$\displaystyle
D_{U_{1}\leftthreetimes\Sigma_{g}}\int_{\Sigma_{g}\leftthreetimes
B^{2}}-\frac{c}{24}p_{1}+\frac{\nu}{2}c_{1}^{2}\in\mathbb{Z},$ (12) for any
orientable surface bundle $\Sigma_{g}\leftthreetimes B^{2}$ and for any
$U_{1}$ bundle whose first Chern class $c_{1}$ satisfies
$\nu\int_{\Sigma_{g}}c_{1}\in\mathbb{Z}$.

This result allows us to obtain quantization condition of chiral central
charge $c$ and the filling fraction $\nu$. In particular, every choice of a
surface bundle $\Sigma_{g}\leftthreetimes B^{2}$ and $U_{1}$ bundle will give
us a quantization condition of $c$ and $\nu$. The combination of all those
quantization conditions gives us the strictest constraint on the possible
values of $c$ and $\nu$. In the following, we will choose some special
combination of surface bundles $\Sigma_{g}\leftthreetimes B^{2}$ and $U_{1}$
bundles to obtain concrete quantization conditions of $c$ and $\nu$.

We note that $\int_{\Sigma_{g}\leftthreetimes B^{2}}p_{1}=0$ mod 12 for any
orientable surface bundles (also called $\Sigma_{g}$-bundles), by (87). If the
genus $g$ of the fiber $\Sigma_{g}$ is equal or less than 2, then
$\int_{\Sigma_{g}\leftthreetimes B^{2}}p_{1}=0$.Meyer (1973) If $g\geq 3$,
then we can always find a base manifold $B^{2}$ with a genus equal or less
than 111, such that there is a surface bundle $\Sigma_{g}\leftthreetimes
B^{2}$ with $\int_{\Sigma_{g}\leftthreetimes B^{2}}p_{1}=\pm 12$.Endo (1998)
If we choose the $U_{1}$ bundle to be trivial $c_{1}=0$, eqn. (12) gives us
the quantization condition on $c$ first proposed in LABEL:KW1458,KW200411904:

$\displaystyle\frac{c}{2}D_{U_{1}\leftthreetimes\Sigma_{g}}\in\mathbb{Z},\ \ \
\text{for }g\geq 3,$ (13)

Next, we consider a trivial surface bundle with $S^{2}=\Sigma_{0}^{2}$ as the
fiber: $\Sigma_{g}\leftthreetimes B^{2}=\Sigma_{0}\times B^{2}$. On such a
surface bundle, $c_{1}$ has a form $c_{1}=c_{1}^{B}+c_{1}^{\Sigma}$ where
$c_{1}^{\Sigma}$ lives on $\Sigma_{0}$. We have

$\displaystyle\frac{1}{2}\nu\int_{\Sigma_{0}\times
B^{2}}c_{1}^{2}=\frac{1}{2}\nu\int_{\Sigma_{0}\times
B^{2}}(c_{1}^{B}+c_{1}^{\Sigma})^{2}$
$\displaystyle=\nu\int_{\Sigma_{0}}c_{1}^{\Sigma}\times\int_{B^{2}}c_{1}^{B}$
(14)

Therefore, for $U_{1}$ bundles whose Chern class $c_{1}$ satisfying
$\nu\int_{\Sigma_{g}}c_{1}\in\mathbb{Z}$,
$\frac{1}{2}\nu\int_{\Sigma_{0}\times B^{2}}c_{1}^{2}$ is always an integer.
No constraint on $\nu$ is obtained. Also, $\int_{\Sigma_{0}\times
B^{2}}p_{1}=0$. Therefore, the trivial surface bundle
$\Sigma_{g}\leftthreetimes B^{2}=\Sigma_{0}\times B^{2}$ does not give us any
non-trivial quantization conditions for $c$ and $\nu$. So in the following, we
will assume $g>0$ for our space $\Sigma_{g}$.

Next, we consider the surface bundle $\Sigma_{g}\leftthreetimes
B^{2}=S^{1}_{x}\times S^{1}_{y}\times S^{1}_{a}\times S^{1}_{b}$, where
$S^{1}_{x}\times S^{1}_{y}=\Sigma_{1}$ is the space and $S^{1}_{a}\times
S^{1}_{b}=B^{2}$. We consider the $U_{1}$ bundle whose first Chern class has
the following form $c_{1}=c_{1}^{xa}+c_{1}^{yb}$, where $c_{1}^{xa}$ lives on
$S^{1}_{x}\times S^{1}_{a}$ and $c_{1}^{yb}$ lives on $S^{1}_{y}\times
S^{1}_{b}$. Since $\int_{\Sigma_{0}\leftthreetimes B^{2}}p_{1}=0$ and
$\int_{\Sigma_{1}}c_{1}^{xa}+c_{1}^{yb}=\int_{S^{1}_{x}\times
S^{1}_{y}}c_{1}^{xa}+c_{1}^{yb}=0$, the quantization condition eqn. (12)
becomes

$\displaystyle\ \ \ \
\frac{\nu}{2}D_{U_{1}\leftthreetimes\Sigma_{1}}\int_{S^{1}_{x}\times
S^{1}_{y}\times S^{1}_{a}\times S^{1}_{b}}(c_{1}^{xa}+c_{1}^{yb})^{2}$
$\displaystyle=\nu D_{U_{1}\leftthreetimes\Sigma_{1}}\int_{S^{1}_{x}\times
S^{1}_{a}}c_{1}^{xa}\int_{S^{1}_{y}\times S^{1}_{b}}c_{1}^{yb}.$ (15)

We can choose the $U_{1}$ bundles such that $\int_{S^{1}_{x}\times
S^{1}_{a}}c_{1}^{xa}=0,\pm 1$ and $\int_{S^{1}_{y}\times
S^{1}_{b}}c_{1}^{yb}=0,\pm 1$. We obtain the following quantization condition

$\displaystyle\nu D_{U_{1}\leftthreetimes\Sigma_{1}}\in\mathbb{Z}.$ (16)

The above result was first obtained by LABEL:NTW8572, using a similar
consideration. Because $\Sigma_{g}$ is the connected sum of $g$
$\Sigma_{1}$’s, the above reasoning can be generalized to the case
$\Sigma_{g}\leftthreetimes B^{2}=\Sigma_{g}\times S^{1}_{a}\times S^{1}_{b}$,
which allows us to obtain

$\displaystyle\nu D_{U_{1}\leftthreetimes\Sigma_{g}}\in\mathbb{Z},\ \ \ \
g\geq 1.$ (17)

This result generalizes that of LABEL:NTW8572.

To obtain an even stronger result, we consider a surface bundle over surface
$\Sigma_{g}\leftthreetimes\Sigma_{h}$. In Appendix B, we show that, for each
$g\geq 5$, there are many surface bundles
$\Sigma_{g}\leftthreetimes\Sigma_{h}$ such that each of them has a 2-cocycle
$c_{1}\in H^{2}(\Sigma_{g}\leftthreetimes\Sigma_{h};\mathbb{Z})$ satisfying
$\int_{\Sigma_{g}}c_{1}=0$. Let
$\int_{\Sigma_{g}\leftthreetimes\Sigma_{h}}c_{1}^{2}=\eta_{c}$ and
$\int_{\Sigma_{g}\leftthreetimes\Sigma_{h}}p_{1}=3\eta_{p}$ where $\eta_{c}$
and $\eta_{p}$ are integers. We find that the set of allowed
$(\eta_{p},\eta_{c})$ is given by (see Appendix B)

$\displaystyle\\{(\eta_{p},\eta_{c})\ |\ \eta_{p}\in 4\mathbb{Z},\
\eta_{c}\in\mathbb{Z}\\}.$ (18)

We obtain a stronger quantization condition on $\nu$ from eqn. (12):

$\displaystyle\frac{\nu}{2}D_{U_{1}\leftthreetimes\Sigma_{g}}\in\mathbb{Z},\ \
\ \ g\geq 5.$ (19)

To summarize

for a 2+1D bosonic gapped phase with $U_{1}$ symmetry, its chiral central
charge $c$ and filling fraction $\nu$ satisfy the following quantization
conditions $\displaystyle\frac{c}{2}D_{U_{1}\leftthreetimes\Sigma_{g}}$
$\displaystyle=0\text{ mod }1,\ \ \ \ g\geq 3,$ $\displaystyle\nu
D_{U_{1}\leftthreetimes\Sigma_{g}}$ $\displaystyle=0\text{ mod }1,\ \ \ \
g\geq 1,$ $\displaystyle\frac{\nu}{2}D_{U_{1}\leftthreetimes\Sigma_{g}}$
$\displaystyle=0\text{ mod }1,\ \ \ \ g\geq 5.$ (20) where
$D_{U_{1}\leftthreetimes\Sigma_{g}}$ is the ground state degeneracy on closed
space $\Sigma_{g}$.

###  III.2 Examples

####  III.2.1 Bosonic $K$-matrix Abelian topological orders

We check $\frac{c}{2}D_{U_{1}\leftthreetimes\Sigma_{g}}\in\mathbb{Z}$ for 2+1D
bosonic Abelian topological orders described by the following multi-layer wave
function characterized by symmetric integral matrix $K$ with even diagonal:

$\displaystyle\Psi(z_{i}^{I})=\prod_{I,i<j}(z_{i}^{I}-z_{j}^{I})^{K_{II}}\hskip-5.0pt\prod_{I<J,i,j}(z_{i}^{I}-z_{j}^{J})^{K_{IJ}}\hskip
1.0pt\mathrm{e}^{-\sum_{I,i}\frac{|z_{i}^{I}|^{2}}{4}},$ (21)

where $z_{i}^{I}$ the complex coordinates of the $i^{\text{th}}$ boson in
$I^{\text{th}}$-layer. The effective field theory of the above state is
described by $K$-matrix Abelian Chern–Simons theory:

$\displaystyle S_{K}=\int\frac{K_{IJ}}{4\pi}a_{I}\hskip
1.0pt\mathrm{d}a_{J}+\frac{q_{I}}{2\pi}A\hskip 1.0pt\mathrm{d}a_{I}.$ (22)

where the charge-vector $q_{I}$ are integers, describing the $U_{1}$ charge of
the bosons in $I^{\text{th}}$-layer. It is believedWen and Zee (1992c) that
the $K$-matrix Abelian Chern–Simons theory can realize all the 2+1D Abelian
topological ordors.

After integrating out the matter field $a_{I}$, we obtain the effective theory
eqn. (III.1) with

$\displaystyle c=\text{sgn}(K)=n_{+}-n_{-},\ \ \ \
\nu=\boldsymbol{q}^{\top}K^{-1}\boldsymbol{q}.$

where $n_{\pm}$ are the number of eigenvalues of $K$ with sign $\pm$. The
ground state degeneracy is given by

$\displaystyle D_{U_{1}\leftthreetimes\Sigma_{g}}=|{\rm Det}K|^{g}.$

So eqn. (12) states that the product $\text{sgn}(K)|{\rm Det}K|^{g}$ is an
even integer for $g>2$. To check this, note that $\text{sgn}(K)=n\text{ mod
}2$, where $n$ is the dimension of $K$. So it suffices to check that

$\displaystyle{\rm Det}K\in 2\mathbb{Z}~{}~{}~{}~{}~{}~{}~{}\text{for odd
$n$}.$ (23)

To see this, write

$\displaystyle{\rm
Det}K=\sum_{\sigma}(-1)^{\sigma}\prod_{I=1}^{n}K_{I\sigma(I)}$ (24)

which is summed over the permutations $\sigma$ of $\\{1,\dots,n\\}$ and
$(-1)^{\sigma}$ is its sign. Note that any $\sigma$ which involves a diagonal
element of $K$ contributes an even term to the sum, so we only need to
consider the permutations where $\sigma(I)\neq I$ for all $I$. Every
permutation $\sigma$ can be expressed as a number of disjoint cycles. The sum
of all the lengths of the cycles is $n$, which is odd by assumption. So there
exists a cycle whose length is odd. Such cycle cannot have length 1, since we
assumed $\sigma(I)\neq I$. So it has length $\geq 3$. The existence of such a
cycle implies $\sigma\neq\sigma^{-1}$, since reversing the cycle gives a
different permutation. Moreover $\sigma$ and $\sigma^{-1}$ contributes equally
to the sum (24), since

$\displaystyle\prod_{I=1}^{n}K_{I\sigma^{-1}(I)}=\prod_{I=1}^{n}K_{\sigma^{-1}(I)I}=\prod_{I=1}^{n}K_{I\sigma(I)}.$

So the two terms together is even. So (24) is even.

The conditions on $\nu$, eqn. (17) and eqn. (19), now become

$\displaystyle{\rm
Det}(K)\;\boldsymbol{q}^{\top}K^{-1}\boldsymbol{q}\in\mathbb{Z},\ \ |{\rm
Det}(K)|^{5}\frac{\boldsymbol{q}^{\top}K^{-1}\boldsymbol{q}}{2}\in\mathbb{Z},$
(25)

for all integral vectors $\boldsymbol{q}$. The first expression is satisfied
because ${\rm Det}(K)K^{-1}$ is an integer matrix. For the second expression,
we only need to consider the case when ${\rm Det}(K)$ is odd. By (23) this
implies the $n$, the dimension of $K$, is even. So the submatrix of $K$ by
deleting the $i$th row and $i$th column has odd dimension for all $i$.
Applying (23) to this submatrix, we see that it has even determinant. Hence
the diagonals of the cofactor matrix of $K$ are even, which means ${\rm
Det}(K)K^{-1}$ has even diagonals. From this, the second expression reads

$\displaystyle|{\rm Det}(K)|^{5}\frac{q_{I}K^{-1}_{IJ}q_{J}}{2}$
$\displaystyle=\frac{1}{2}|{\rm
Det}(K)|^{5}\big{(}\sum_{I}q^{2}_{I}K^{-1}_{II}+\sum_{I\neq
J}q_{I}K^{-1}_{IJ}q_{J}\big{)}$ $\displaystyle=|{\rm
Det}(K)|^{5}\big{(}\frac{\sum_{I}q^{2}_{I}K^{-1}_{II}}{2}+\sum_{I<J}q_{I}K^{-1}_{IJ}q_{J}\big{)}\in\mathbb{Z}.$

From the above result, we find that, for Abelian bosonic topological orders
with $U_{1}$ symmetry, the filling fraction is quantized as

$\displaystyle\nu$
$\displaystyle=\frac{2\times\text{integer}}{\text{gcd}(2D_{1},D_{1}^{5})}$
$\displaystyle=\begin{cases}\frac{1}{D_{1}}\times\text{integer},&\text{ if
}D_{1}=\text{even}\\\ \frac{2}{D_{1}}\times\text{integer},&\text{ if
}D_{1}=\text{odd}\\\ \end{cases}$ (26)

where $D_{1}=|{\rm Det}(K)|$ is the ground state degeneracy on torus. In
particular, for bosonic $U_{1}$-SPT order, $D_{1}=|{\rm Det}(K)|=1$ and
$\nu=0$ mod 2. We note that the bosonic $\nu=1/m$ Laughlin state has
$D_{1}=m=$ even, and saturates the above quantization condition.

####  III.2.2 Bosonic non-Abelian topological orders described by $SU(2)_{k}$
Chern–Simons Theory

The following wave function

$\displaystyle\Psi(z_{i})=\big{(}\chi_{k}(z_{i})\big{)}^{2}$ (27)

describes a bosonic non-Abelian topological orders of charge-1 bosons, where
$\chi_{k}(z_{i})$ is the fermion wave function of $k$-filled Landau levels.Wen
(1991b) The low energy effective theory is the $SU(2)_{k}$ Chern–Simons
theory. The edge is described by a $U_{1}^{2k}/SU(2)_{k}$ WZW theory, whose
chiral central charge and filling fraction areWen (1991b)

$\displaystyle c=2k-\frac{3k}{k+2},\ \ \ \ \nu=\frac{k}{2}.$ (28)

Using the $SU(2)_{k}$ Chern–Simons theory, we find that there are $k+1$ anyons
in the bulk labeled by $s=m/2$ where $m=0,1,\dots,k$. The ground state
degeneracy is given by

$\displaystyle
D_{U_{1}\leftthreetimes\Sigma_{g}}=\sum_{m}S_{0m}^{-2(g-1)}\in\mathbb{Z}$

where

$\displaystyle
S_{mn}=\sqrt{\frac{2}{k+2}}\sin\big{(}\frac{(m+1)(n+1)\pi}{k+2}\big{)}$

So eqn. (III.1) reads

$\displaystyle\frac{c}{2}D_{U_{1}\leftthreetimes\Sigma_{g}}=\frac{-3k}{2(k+2)}\big{(}\frac{k+2}{2}\big{)}^{g-1}\sum_{m=1}^{k+1}\csc^{2(g-1)}\frac{m\pi}{k+2}\in\mathbb{Z}$
(29)

which is verified in Appendix C for all integers $k\geq 0$ and $g\geq 3$. We
also have

$\displaystyle\nu
D_{U_{1}\leftthreetimes\Sigma_{g}}=\frac{k}{2}\big{(}\frac{k+2}{2}\big{)}^{g-1}\sum_{m=1}^{k+1}\csc^{2(g-1)}\frac{m\pi}{k+2}\in\mathbb{Z}$
(30)

for $g\geq 1$, and

$\displaystyle\frac{\nu}{2}D_{U_{1}\leftthreetimes\Sigma_{g}}=\frac{k}{4}\big{(}\frac{k+2}{2}\big{)}^{g-1}\sum_{m=1}^{k+1}\csc^{2(g-1)}\frac{m\pi}{k+2}\in\mathbb{Z}$
(31)

for $g\geq 5$. We also verify the above two relations in Appendix C.

###  III.3 A general point of view

In general, we may consider a boson system in $d$-dimensional space with an
internal symmetry described by group $G_{b}$. The bosons may carry non-zero
integer spin and may transform non-trivially under the spatial rotation
$SO_{d}$. We embed $SO_{d}$ into $SO_{\infty}\equiv SO$. Thus, bosons
transform under the full symmetry group $G_{bSO}=G_{b}\times SO$. On curved
spacetime with $G_{b}$-symmetry twist, we have a $G_{bSO}$ connection
$a^{G_{bSO}}$ on the spacetime. The connection $a^{G_{bSO}}$ is a special one.
When we project $G_{bSO}$ to $SO$: $G_{bSO}\xrightarrow{\pi}SO$, the $G_{bSO}$
connection is projected into an $SO$ connection:
$a^{G_{bSO}}\xrightarrow{\pi}a^{SO}$, and such an $SO$ connection $a^{SO}$
must be the connection for the tangent bundle of the spacetime. The low energy
effective Lagrangian $L_{eff}$ (III.1) is a Chern–Simons term for group
$G_{bSO}$. This point of view will help us to understand the effective
Lagrangian for fermion system. Since $G_{bSO}=G_{b}\times SO$, the low energy
effective Lagrangian $L_{eff}$ (III.1) is a sum of the gravitational
Chern–Simons term and a Chern–Simons term for group $G_{b}$.

##  IV Topological invariants in 2+1D H-type fermionic enriched topological
orders with $U^{f}_{1}$ symmetry

###  IV.1 Symmetry twist of fermion system

We have seen that to probe the topological properties of a bosonic gapped
liquid phase with symmetry $G_{b}$, we can use the symmetry twist described by
a $G_{b}$-connection $a^{SO}$, plus the curved spacetime described by the
$SO$-connection $a^{SO}$ of the tangent bundle. In other words, we can use the
$G_{bSO}$-connection $a^{G_{bSO}}$ and effective action $S_{eff}(a^{G_{bSO}})$
to probe the topological properties. Here, we will discuss the symmetry twist
for fermion systems, which turns out to be mixed with spacetime curvature.

For a fermion system in $d$-dimensional space with an internal symmetry, its
symmetry group $G_{f}$ has a central $\mathbb{Z}_{2}^{f}$ subgroup:

$\displaystyle\text{id}\to\mathbb{Z}_{2}^{f}\to G_{f}\to G_{b}\to\text{id}.$
(32)

We also write $G_{f}=\mathbb{Z}_{2}^{f}\leftthreetimes_{e_{2}}G_{b}$, where
$e_{2}$ is a group 2-cocycle

$\displaystyle e_{2}\in H^{2}(G_{b};\mathbb{Z}_{2}^{f})$ (33)

describing the group extension (32) (see Appendix D). The fermion carry half-
integer spins and transform non-trivially under the spatial rotation $SO_{d}$,
which is enlarged to $SO$. Thus, fermions transform under the full symmetry
group $G_{fSO}=G_{f}\leftthreetimes SO$. On curved spacetime with
$G_{f}$-symmetry twist, we have a $G_{fSO}$ connection $a^{G_{fSO}}$ on the
spacetime. The connection $a^{G_{fSO}}$ is a special one. When we project
$G_{fSO}$ to $SO$: $G_{fSO}\xrightarrow{\pi}SO$, the $G_{fSO}$ connection is
projected into an $SO$ connection: $a^{G_{fSO}}\xrightarrow{\pi}a^{SO}$, and
such an $SO$ connection $a^{SO}$ must be the connection for the tangent bundle
of the spacetime. The low energy effective Lagrangian $L_{eff}$ is a
Chern–Simons term for group $G_{fSO}$. Because $G_{fSO}=G_{f}\leftthreetimes
SO\neq G_{f}\times SO$, the low energy effective Lagrangian $L_{eff}$ (III.1)
in general cannot be written as a sum of the gravitational Chern–Simons term
for group $SO$ and a Chern–Simons term for group $G_{f}$.

There is another way to describe full fermionic symmetry group
$G_{fSO}=G_{f}\leftthreetimes SO$:

$\displaystyle G_{fSO}=(G_{f}\times Spin)/\mathbb{Z}_{2}^{f}$ (34)

We note that the spin group $Spin\equiv
Spin_{\infty}=\mathbb{Z}_{2}^{f}\leftthreetimes_{\mathrm{w}_{2}}SO$ contains a
central subgroup $\mathbb{Z}_{2}^{f}$ such that $Spin/\mathbb{Z}_{2}^{f}=SO$,
where $\mathrm{w}_{2}\in H^{2}(SO;\mathbb{Z}_{2}^{f})$ is the group 2-cocycle
describing the group extension

$\displaystyle\text{id}\to\mathbb{Z}_{2}^{f}\to Spin\to SO\to\text{id}.$ (35)

$G_{f}=\mathbb{Z}_{2}^{f}\leftthreetimes_{e_{2}}G_{b}$ also contains a central
subgroup $\mathbb{Z}_{2}^{f}$ such that $G_{f}/\mathbb{Z}_{2}^{f}=G_{b}$.
$G_{fSO}$ is obtained by identify these two $\mathbb{Z}_{2}^{f}$ subgroups in
$G_{f}\times Spin$. Let us described the $G_{fSO}$-connection using this point
of view.

We first triangulate the spacetime, where the vertices are labeled by
$i,j,k,\cdots$, the links labelled by $ij,jk,\cdots$, etc. A
$G_{f}$-connection on the triangulation is described by the group elements of
$G_{f}$ on the links: $a^{G_{f}}_{ij}\in G_{f}$. We can view
$G_{f}=\mathbb{Z}_{2}^{f}\leftthreetimes_{e_{2}}G_{b}$ and use the pair
$(a^{G_{b}}_{ij},a^{\mathbb{Z}^{f}_{2}}_{ij})$ to label the group elements of
$G_{f}$, where $a^{G_{b}}_{ij}\in G_{b}$ and
$a^{\mathbb{Z}^{f}_{2}}_{ij}\in\mathbb{Z}_{2}^{f}$. Similarly, a
$Spin$-connection on the triangulation is described by the group elements of
$Spin$ on the links: $a^{Spin}_{ij}\in Spin$. We can view
$Spin=\mathbb{Z}_{2}^{f}\leftthreetimes_{\mathrm{w}_{2}}SO$ and use the pair
$(a^{SO}_{ij},\widetilde{a}^{\mathbb{Z}^{f}_{2}}_{ij})$ to label the group
elements of $Spin$, where $a^{SO}_{ij}\in SO$ and
$\widetilde{a}^{\mathbb{Z}^{f}_{2}}_{ij}\in\mathbb{Z}_{2}^{f}$. Now, the
$G_{fSO}$-connection is obtained by requiring
$\widetilde{a}^{\mathbb{Z}^{f}_{2}}_{ij}=a^{\mathbb{Z}^{f}_{2}}_{ij}$. Also
the $SO$-connection $a^{SO}_{ij}$ describes the tangent bundle of the
spacetime.

Now, let us describe a nearly flat $G_{fSO}$-connection that describe the
curved spacetime and twisted symmetry. The nearly flat condition is given by

$\displaystyle a^{G_{fSO}}_{ij}a^{G_{fSO}}_{ij}\approx a^{G_{fSO}}_{ik}$ (36)

for all the triangles $ijk$. It implies that the $G_{f}$-connection is nearly
flat

$\displaystyle(a^{G_{b}}_{ij},a^{\mathbb{Z}^{f}_{2}}_{ij})\circ(a^{G_{b}}_{jk},a^{\mathbb{Z}^{f}_{2}}_{jk})\approx(a^{G_{b}}_{ik},a^{\mathbb{Z}^{f}_{2}}_{ik}),$
(37)

and the $Spin$-connection is nearly flat

$\displaystyle(a^{SO}_{ij},a^{\mathbb{Z}^{f}_{2}}_{ij})\circ(a^{SO}_{jk},a^{\mathbb{Z}^{f}_{2}}_{jk})\approx(a^{SO}_{ik},a^{\mathbb{Z}^{f}_{2}}_{ik}).$
(38)

In other words (see Appendix D)

$\displaystyle a^{G_{b}}_{ij}a^{G_{b}}_{jk}$ $\displaystyle\approx
a^{G_{b}}_{ik},$ $\displaystyle a^{SO}_{ij}a^{SO}_{jk}$ $\displaystyle\approx
a^{SO}_{ik},$ (39)

and

$\displaystyle
a^{\mathbb{Z}^{f}_{2}}_{ij}+a^{\mathbb{Z}^{f}_{2}}_{jk}-a^{\mathbb{Z}_{2}^{f}}_{ik}$
$\displaystyle\overset{\scriptscriptstyle
2}{=}e_{2}(a^{G_{b}}_{ij},a^{G_{b}}_{jk})=(e_{2})_{ijk},$ $\displaystyle
a^{\mathbb{Z}^{f}_{2}}_{ij}+a^{\mathbb{Z}^{f}_{2}}_{jk}-a^{\mathbb{Z}_{2}^{f}}_{ik}$
$\displaystyle\overset{\scriptscriptstyle
2}{=}\mathrm{w}_{2}(a^{SO}_{ij},a^{SO}_{jk})=(\mathrm{w}_{2})_{ijk}.$ (40)

The above can be rewritten as

$\displaystyle\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}^{f}_{2}}\overset{\scriptscriptstyle
2}{=}e_{2}(a^{G_{b}}),\ \ \ \ \hskip
1.0pt\mathrm{d}a^{\mathbb{Z}^{f}_{2}}\overset{\scriptscriptstyle
2}{=}\mathrm{w}_{2}(a^{SO}).$ (41)

Since $a^{SO}$ is the connection of the tangent bundle of the spacetime,
$\mathrm{w}_{2}(a^{SO})=\mathrm{w}_{2}$ is the second Stiefel–Whitney class of
the tangent bundle of the spacetime. We see that the $G_{b}$-connection and
the $SO$-connection on spacetime are not arbitrary. They must satisfy the
constraint

$\displaystyle e_{2}(a^{G_{b}}_{ij},a^{G_{b}}_{jk})\overset{\scriptscriptstyle
2}{=}\mathrm{w}_{2}(a^{SO}_{ij},a^{SO}_{jk})=\mathrm{w}_{2}.$ (42)

In other words, the nearly flat $G_{b}$ bundle on the spacetime (describing
the symmetry twist) is not arbitrary. Its nearly flat connection $a^{G_{b}}$
must satisfy

$\displaystyle e_{2}(a^{G_{b}})\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}\mathrm{w}_{2}.$ (43)

###  IV.2 $U^{f}_{1}$ symmetry and spinC structure

Let us consider an example of the above result. A fermion system with fermion
number conservation has a $U^{f}_{1}$ symmetry. Assume all the odd charges of
$U^{f}_{1}$ are fermionic, then $U^{f}_{1}$ has a $\mathbb{Z}_{2}^{f}$
subgroup generated by the $\pi$-rotation, which corresponds to the fermion-
number-parity. It is more convenient to view $U^{f}_{1}$ as a group extension
of $U_{1}$ by $\mathbb{Z}_{2}^{f}$, i.e.
$U^{f}_{1}=\mathbb{Z}_{2}^{f}\leftthreetimes_{e_{2}}U_{1}$ where $e_{2}$ is
the $\mathbb{Z}_{2}$ valued group 2-cocycle that generates
$H^{2}(U_{1},\mathbb{Z}_{2})$ (see Appendix D).

Using ${\mathbb{R}/\mathbb{Z}}$-valued $a^{({\mathbb{R}/\mathbb{Z}})^{f}}$ to
label group elements in $U^{f}_{1}$, ${\mathbb{R}/\mathbb{Z}}$-valued
$a^{{\mathbb{R}/\mathbb{Z}}}$ to label group elements in $U_{1}$, and
$\mathbb{Z}_{2}$-valued $a^{\mathbb{Z}_{2}^{f}}$ to label group elements in
$\mathbb{Z}_{2}^{f}$, we find that

$\displaystyle
a^{({\mathbb{R}/\mathbb{Z}})^{f}}=\frac{1}{2}a^{{\mathbb{R}/\mathbb{Z}}}+\frac{1}{2}a^{\mathbb{Z}_{2}^{f}},$
(44)

Here ${\mathbb{R}/\mathbb{Z}}=[0,1)$ and
$\mathbb{Z}_{n}=\\{0,1,\cdots,n-1\\}$. Now the group multiplication in
$U^{f}_{1}$ can be rewritten as

$\displaystyle a^{({\mathbb{R}/\mathbb{Z}})^{f}}_{1}$
$\displaystyle+a^{({\mathbb{R}/\mathbb{Z}})^{f}}_{2}=\frac{1}{2}a^{{\mathbb{R}/\mathbb{Z}}}_{1}+\frac{1}{2}a^{{\mathbb{R}/\mathbb{Z}}}_{2}+\frac{1}{2}a^{\mathbb{Z}_{2}^{f}}_{1}+\frac{1}{2}a^{\mathbb{Z}_{2}^{f}}_{2}$
$\displaystyle=\frac{1}{2}(a^{{\mathbb{R}/\mathbb{Z}}}_{1}+a^{{\mathbb{R}/\mathbb{Z}}}_{2}-\lfloor
a^{{\mathbb{R}/\mathbb{Z}}}_{1}+a^{{\mathbb{R}/\mathbb{Z}}}_{2}\rfloor)$
$\displaystyle\ \ \ \ \
+(\frac{1}{2}a^{\mathbb{Z}_{2}^{f}}_{1}+\frac{1}{2}a^{\mathbb{Z}_{2}^{f}}_{2}+\lfloor
a^{{\mathbb{R}/\mathbb{Z}}}_{1}+a^{{\mathbb{R}/\mathbb{Z}}}_{2}\rfloor)$
$\displaystyle=\frac{1}{2}(a^{{\mathbb{R}/\mathbb{Z}}}_{1}+a^{{\mathbb{R}/\mathbb{Z}}}_{2}-\lfloor
a^{{\mathbb{R}/\mathbb{Z}}}_{1}+a^{{\mathbb{R}/\mathbb{Z}}}_{2}\rfloor)$
$\displaystyle\ \ \ \ \
+(\frac{1}{2}a^{\mathbb{Z}_{2}^{f}}_{1}+\frac{1}{2}a^{\mathbb{Z}_{2}^{f}}_{2}+e_{2}(a^{{\mathbb{R}/\mathbb{Z}}}_{1},a^{{\mathbb{R}/\mathbb{Z}}}_{2})),$
(45)

where $\lfloor x\rfloor$ denotes the largest integer smaller than or equal to
$x$, and

$\displaystyle
e_{2}(a^{{\mathbb{R}/\mathbb{Z}}}_{1},a^{{\mathbb{R}/\mathbb{Z}}}_{2})=\lfloor
a^{{\mathbb{R}/\mathbb{Z}}}_{1}+a^{{\mathbb{R}/\mathbb{Z}}}_{2}\rfloor$ (46)

which is a $\mathbb{Z}_{2}$-valued group 2-cocycle $e_{2}\in
H^{2}(U_{1};\mathbb{Z}_{2})$ that characterizes the group extension of $U_{1}$
by $\mathbb{Z}_{2}^{f}$:

$\displaystyle U_{1}^{f}=\mathbb{Z}_{2}^{f}\leftthreetimes_{e_{2}}U_{1}.$ (47)

Note that
$e_{2}(a^{{\mathbb{R}/\mathbb{Z}}}_{1},a^{{\mathbb{R}/\mathbb{Z}}}_{2})$ is a
smooth function of
$a^{{\mathbb{R}/\mathbb{Z}}}_{1},a^{{\mathbb{R}/\mathbb{Z}}}_{2}$ when
$0<a^{{\mathbb{R}/\mathbb{Z}}}_{1}<\frac{1}{2}$ and
$0<a^{{\mathbb{R}/\mathbb{Z}}}_{2}<\frac{1}{2}$. But it has discontinuities in
other places. It turns out that

$\displaystyle e_{2}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}c_{1}^{U_{1}}$ (48)

where $c_{1}^{U_{1}}\in H^{2}(U_{1},\mathbb{Z})$ is the first Chern class of
$U_{1}$ viewed as group 2-cocycle.

The “Chern Class” for $U^{f}_{1}$ and the Chern Class for $U_{1}$ are related
(see eqn. (44))

$\displaystyle c_{1}^{U_{1}^{f}}=\frac{1}{2}c_{1}^{U_{1}},$ (49)

where $c_{1}^{U_{1}}=\hskip 1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}}$ and
$c_{1}^{U_{1}^{f}}=\hskip 1.0pt\mathrm{d}a^{({\mathbb{R}/\mathbb{Z}})^{f}}$.
We like to point out that the $U^{f}_{1}$-connection
$a^{({\mathbb{R}/\mathbb{Z}})^{f}}$ corresponds to the electromagnetic gauge
potential plus the connection of the curved space:

$\displaystyle A_{eff}=2\pi a^{({\mathbb{R}/\mathbb{Z}})^{f}},$ (50)

and the Chern class $c_{1}^{U_{1}^{f}}$ corresponds to the field strength of
the effective $U^{f}_{1}$-connection

$\displaystyle F_{eff}=\hskip 1.0pt\mathrm{d}A_{eff}=2\pi c_{1}^{U_{1}^{f}}.$
(51)

We find that for fermion system with $U^{f}_{1}$ symmetry, the Chern number on
a closed 2-dimension space $C^{2}$ satisfies

$\displaystyle\int_{C^{2}}c_{1}^{U_{1}^{f}}=\frac{1}{2\pi}\int_{C^{2}}F_{eff}\overset{\scriptscriptstyle
1}{=}\frac{1}{2}\int_{C^{2}}\mathrm{w}_{2}.$ (52)

In other words, on a spin manifold $\mathrm{w}_{2}\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}0$, the Chern number
$\frac{1}{2\pi}\int_{C^{2}}F_{eff}$ is quantized as integers. On non-spin
manifold, the fermion system with $U^{f}_{1}$ symmetry can still be defined,
provided that the Chern number $\frac{1}{2\pi}\int_{C^{2}}F_{eff}$ is
quantized as half-integers when
$\int_{C^{2}}\mathrm{w}_{2}\overset{\scriptscriptstyle 2}{=}1$.

We see that for fermion systems, there is a constraint on the $U^{f}_{1}$
connection $a^{U^{f}_{1}}=\frac{1}{2\pi}A_{eff}$ and the curved spacetime.
When $A_{eff}=0$, this constraint requires that
$\mathrm{w}_{2}\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}0$,
i.e. the spacetime to be spin. This leads to the general impression that the
appearance of fermions requires the spacetime manifold to be spin. Here we see
that the appearance of $U^{f}_{1}$ fermions requires the spacetime manifold to
be spin only when there is no background effective $U^{f}_{1}$ gauge field. In
the presence of effective background $U^{f}_{1}$ gauge field, the spacetime
may not be spin. This result can be summarized more precisely by the following
statement: the appearance of $U^{f}_{1}$ fermions requires the spacetime
manifold to be spinC, and the $U^{f}_{1}$ connection
$a^{({\mathbb{R}/\mathbb{Z}})^{f}}$ is a spinC structure.

We know that an orientable manifold is spin if and only if its
$\mathrm{w}_{2}\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}0$.
Similarly, an orientable manifold is spinC if and only if
$\beta_{2}\mathrm{w}_{2}\overset{\scriptscriptstyle\hskip
1.0pt\mathrm{d}}{=}0$. Here $\beta_{2}=\frac{1}{2}\hskip 1.0pt\mathrm{d}$ is
the the Bockstein homomorphism. In order words, an orientable manifold $M$ is
spinC if and only if its $\mathrm{w}_{2}$ is the mod 2 reduction of a
$\mathbb{Z}$-valued 2-cocycle in $H^{2}(M;\mathbb{Z})$. Also a complex vector
bundle of rank $n$ can be viewed as a real vector bundle of rank $2n$. The
$n^{\text{th}}$ Chern class $c_{n}^{U}$ of the complex vector bundle and the
$2n^{\text{th}}$ Stiefel–Whitney class are related

$\displaystyle\mathrm{w}_{2n}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}c_{n}^{U}.$ (53)

This implies that a complex manifold is always spinC.

###  IV.3 Fermionic gapped liquids with $U^{f}_{1}$ symmetry in 2-dimensional
space

Now, we consider a gapped liquids with $U^{f}_{1}$ symmetry in 2-dimensional
space for fermionic systems. The effective action may contain a gravitational
Chern–Simons term and a $U^{f}_{1}$ Chern–Simons term:

$\displaystyle{\cal L}_{eff}\hskip
1.0pt\mathrm{d}^{3}x=-2\pi\frac{c}{24}\omega_{3}+\frac{\nu}{4\pi}A_{eff}\hskip
1.0pt\mathrm{d}A_{eff}$ (54)

where $A_{eff}$ is the $U^{f}_{1}$ connection 1-form describing the total
electromagnetic field, plus the connection from the curved space due to the
orbital spin of the fermions. Follow the same discussion for the bosonic case,
we find that the quantization conditions of $c$ and $\nu$ for fermionic case
can be obtained from (see eqn. (12))

$\displaystyle
D_{U_{1}\leftthreetimes\Sigma_{g}}\int_{\Sigma_{g}\leftthreetimes
B^{2}}-\frac{c}{24}p_{1}+\frac{\nu}{8}(c_{1}^{U_{1}})^{2}\in\mathbb{Z},$ (55)
for any orientable surface bundle $\Sigma_{g}\leftthreetimes B^{2}$ and for
any $U_{1}$ bundle whose first Chern class $c_{1}^{U_{1}}$ satisfies
$\frac{1}{2}\nu\int_{\Sigma_{g}}c_{1}^{U_{1}}\in\mathbb{Z}$ and
$c_{1}^{U_{1}}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}\mathrm{w}_{2}$.

Here $c_{1}^{U_{1}}=\hskip 1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}}$,
$a^{{\mathbb{R}/\mathbb{Z}}}=2\frac{A_{eff}}{2\pi}$ is the $U_{1}$-connection
for charge-$2$ bosons (fermion pairs), and $\mathrm{w}_{2}$ is the second
Stiefel–Whitney class of the tangent bundle of $\Sigma_{g}\leftthreetimes
B^{2}$.

Next, we choose some special surface bundles $\Sigma_{g}\leftthreetimes B^{2}$
and the allowed $U_{1}$ bundles to obtain quantization conditions of $\nu$ and
$c$. Let us choose the surface bundle to be $\Sigma_{g}\leftthreetimes
B^{2}=S^{1}_{x}\times S^{1}_{y}\times S^{1}\times S^{1}$, where
$S^{1}_{x}\times S^{1}_{y}$ is the space and $S^{1}\times S^{1}=B^{2}$. The
second Stiefel–Whitney class of such a surface bundle is trivial
$\mathrm{w}_{2}\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}0$.
Thus the allowed $U_{1}$ bundles satisfy $\int_{C^{2}}c_{1}^{U_{1}}=$ even
integers. We consider the following form pf
$c_{1}^{U_{1}}=c_{1}^{x}+c_{1}^{y}$, where $c_{1}^{x}$ lives on
$S^{1}_{x}\times S^{1}$ and $c_{1}^{y}$ lives on $S^{1}_{y}\times S^{1}$.
Since $\int_{\Sigma_{0}\leftthreetimes B^{2}}p_{1}=0$ and
$\int_{S^{1}_{x}\times S^{1}_{y}}c_{1}^{x}+c_{1}^{y}=0$, the quantization
condition eqn. (55) becomes

$\displaystyle\ \ \ \
\frac{\nu}{8}D_{U_{1}\leftthreetimes\Sigma_{1}}\int_{S^{1}_{x}\times
S^{1}_{y}\times S^{1}\times S^{1}}(c_{1}^{x}+c_{1}^{y})^{2}$
$\displaystyle=\frac{\nu}{4}D_{U_{1}\leftthreetimes\Sigma_{1}}\int_{S^{1}_{x}\times
S^{1}}c_{1}^{x}\int_{S^{1}_{y}\times S^{1}}c_{1}^{y}.$ (56)

We can choose the $U_{1}$ bundle such that $\int_{S^{1}_{x}\times
S^{1}}c_{1}^{x}=0,\pm 2$ and $\int_{S^{1}_{y}\times S^{1}}c_{1}^{y}=0,\pm 2$.
We obtain the following quantization condition

$\displaystyle\nu D_{U_{1}\leftthreetimes\Sigma_{1}}\in\mathbb{Z}.$ (57)

The above result was first obtained by LABEL:NTW8572. Again, because
$\Sigma_{g}$, $g\geq 1$, is the connected sum of $g$ $\Sigma_{1}$’s, the above
reasoning can be generalized to the case $\Sigma_{g}\leftthreetimes
B^{2}=\Sigma_{g}\times S^{1}\times S^{1}$ whose second Stiefel–Whitney class
is still trivial $\mathrm{w}_{2}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}0$, which allow us to obtain

$\displaystyle\nu D_{U_{1}\leftthreetimes\Sigma_{g}}\in\mathbb{Z},\ \ \ \
g\geq 1.$ (58)

If $\Sigma_{g}\leftthreetimes B^{2}$ is spin, then
$\int_{\Sigma_{g}\leftthreetimes B^{2}}p_{1}=0$ mod 48 by (88). We explain in
Appendix B that as long as $g\geq 9$ there is such a spin surface bundle with
$\int_{\Sigma_{g}\leftthreetimes B^{2}}p_{1}=\pm 48$, in which case we may
choose the $U_{1}$-bundle to be trivial. This gives us a quantization of $c$
(see LABEL:KW200411904):

$\displaystyle 2cD_{U_{1}\leftthreetimes\Sigma_{g}}\in\mathbb{Z},\ \ \
\text{for }g\geq 9.$ (59)

To obtain additional quantization condition, we consider a more general
surface bundle over surface $\Sigma_{g}\leftthreetimes B^{2}$. In Appendix B,
we show that, for each $g\geq 5$, there are many surface bundles
$\Sigma_{g}\leftthreetimes B^{2}$ equipped with a 2-cocycle $c_{1}^{U_{1}}\in
H^{2}(\Sigma_{g}\leftthreetimes B^{2};\mathbb{Z})$ satisfying (1)
$\int_{\Sigma_{g}}c_{1}^{U_{1}}=0$ and (2)
$c_{1}^{U_{1}}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}\mathrm{w}_{2}$. Let $\int_{\Sigma_{g}\leftthreetimes
B^{2}}(c_{1}^{U_{1}})^{2}=\eta_{c}$ and $\int_{\Sigma_{g}\leftthreetimes
B^{2}}p_{1}=3\eta_{p}$ where $\eta_{c}$ and $\eta_{p}$ are integers. Eqn. (55)
becomes

$\displaystyle
D_{U_{1}\leftthreetimes\Sigma_{g}}(-\eta_{p}\frac{c}{8}+\eta_{c}\frac{\nu}{8})\in\mathbb{Z},$
(60)

In Appendix B, we find that the set of allowed $(\eta_{p},\eta_{c})$ is
exactly given by the integer pairs that satisfy

$\displaystyle\eta_{c}\overset{\scriptscriptstyle
8}{=}\eta_{p}\overset{\scriptscriptstyle 4}{=}0.$ (61)

Now eqn. (60) implies that $\nu
D_{U_{1}\leftthreetimes\Sigma_{g}}\in\mathbb{Z}$ and two new quantization
conditions

$\displaystyle cD_{U_{1}\leftthreetimes\Sigma_{g}}$
$\displaystyle\in\mathbb{Z},\ \ \ \ g\geq 5,$
$\displaystyle(-\frac{c}{2}+\frac{\nu}{2})D_{U_{1}\leftthreetimes\Sigma_{g}}$
$\displaystyle\in\mathbb{Z},\ \ \ \ g\geq 5.$ (62)

To summarize

for a 2+1D fermionic gapped phase with $U^{f}_{1}$ symmetry, its chiral
central charge $c$ and filling fraction $\nu$ satisfy the following
quantization conditions $\displaystyle cD_{U_{1}\leftthreetimes\Sigma_{g}}$
$\displaystyle=0\text{ mod }1,\ \ \ \ g\geq 5,$ $\displaystyle\nu
D_{U_{1}\leftthreetimes\Sigma_{g}}$ $\displaystyle=0\text{ mod }1,\ \ \ \
g\geq 1,$ $\displaystyle(-c+\nu)D_{U_{1}\leftthreetimes\Sigma_{g}}$
$\displaystyle=0\text{ mod }2,\ \ \ \ g\geq 5,$ (63) where
$D_{U_{1}\leftthreetimes\Sigma_{g}}$ is the ground state degeneracy on closed
space $\Sigma_{g}$.

###  IV.4 H-type invertible fermionic $U^{f}_{1}$-enriched topological orders

Let us discuss a simple example. The invertible fermionic $U^{f}_{1}$-enriched
topological order are classified by $K$-matrix (see eqn. (22))

$\displaystyle K=(1)^{\oplus m}\oplus(-1)^{\oplus n}$ (64)

and charge vector $\boldsymbol{q}$ with odd-integer components. Its $c$ and
$\nu$ are given by

$\displaystyle c={\rm Tr}(K),\ \ \ \
\nu=\sum_{i=1}^{m}q_{i}^{2}-\sum_{i=m+1}^{m+n}q_{i}^{2}.$ (65)

The ground state degeneracy is always
$D_{U_{1}\leftthreetimes\Sigma_{g}}=|{\rm Det}(K)|=1$. We see that $-c+\nu=0$
mod 8, which satisfies (but does not saturate) eqn. (IV.3).

##  V The topological invariants for L-type topological orders

###  V.1 Topological partition function for $L$-type topological order

Consider a bosonic system in gapped liquid phase described by a path integral
on $n$D spacetime $M^{n}$. After integrating out the all the dynamical degrees
of freedom, we will obtain a partition function that has the following form

$\displaystyle Z(M^{n})=\hskip 1.0pt\mathrm{e}^{-\int_{M^{n}}\hskip
1.0pt\mathrm{d}^{n}x\,\varepsilon(x)}Z^{\text{top}}(M^{n})$ (66)

where $\varepsilon(x)$ is the energy density of the ground state. The term
$Z^{\text{top}}(M^{n})$ is called the topological partition function and is a
topological invariants of the 2+1D topological order. To be more precise,
$Z^{\text{top}}(M^{n})$ is a complex function on ${\cal M}_{M^{n}}$, which is
the moduli space of the spacetime $M^{n}$. i.e. ${\cal M}_{M^{n}}$ is a space
of metrics $g_{\mu\nu}$ of a closed manifold. The $g_{\mu\nu}$’s differ by
diffeomorphisms are equivalent and represent the same point of the moduli
space ${\cal M}_{M^{n}}$.

We like to point out that for certain spacetime topologies, the spacetime
$M^{n}$ must contain world-line of point-like excitations, and/or world-sheet
of string-like excitations, etc. For those certain spacetime topologies
$Z^{\text{top}}(M^{n})=0$ on the corresponding moduli space ${\cal
M}_{M^{n}}$. For other spacetime topologies, the spacetime $M^{n}$ does not
contain any excitations. In this case $Z^{\text{top}}(M^{n})$ is always non-
zero on the corresponding moduli space ${\cal M}_{M^{n}}$. If
$Z^{\text{top}}(M^{n})$ was zero at an isolated point (or isolated lower
dimensional subspace of ${\cal M}_{M^{n}}$), then a small perturbation will
make $Z^{\text{top}}(M^{n})$ non-zero. This causes the diverging change in the
effective action $S_{eff}=\ln(Z^{\text{top}}(M^{n}))$, which indicates that
the system to be gapless. ThusKong and Wen (2014)

on the moduli space of gapped systems, $Z^{\text{top}}(M^{n})$ is either
always zero (when the spacetime must contain excitations), or always non-zero
(when the spacetime is filled by gapped ground state).

We also like to conjecture that

on the moduli space of gapped systems, $|Z^{\text{top}}(M^{n})|$ is a
constant.

Physically, since the system is gapped, the $Z^{\text{top}}(M^{n})$ should not
depend on the size (i.e. the metrics) of the spacetime, we naively expect
$Z^{\text{top}}(M^{n})$ to be constant. However, the such a naively
expectation is not totally correct. We will see that the phase of
$Z^{\text{top}}(M^{n})$ can depend on the metrics. So we conjecture that
$|Z^{\text{top}}(M^{n})|$ is a constant.

The metrics dependence of the phase of $Z^{\text{top}}(M^{n})$ has a
topological reason: they come from the gravitational Chern–Simons terms
$\omega_{n}(a^{SO})$ that depend on the $SO$ connection $a^{SO}$ of the
tangent bundle

$\displaystyle Z^{\text{top}}(M^{n})=|Z^{\text{top}}(M^{n})|\hskip
1.0pt\mathrm{e}^{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int_{M^{n}}\kappa\omega_{n}(a^{SO})}.$ (67)

The gravitational Chern–Simons terms exist only in spacetime dimensions
$n\overset{\scriptscriptstyle 4}{=}3$.

When the tangent bundle of the spacetime is non-trivial, the $SO$ connection
$a^{SO}$ of the tangent bundle is not globally defined, and as the result the
gravitational Chern–Simons action $\int_{M^{n}}\kappa\omega_{n}(a^{SO})$ is
not well defined. In this paper, we use a cobordism approach trying to define
the difference of two gravitational Chern–Simons actions as

$\displaystyle\ \ \ \
\int_{M^{n}_{1}}\kappa\omega_{n}(a^{SO})-\int_{M^{n}_{2}}\kappa\omega_{n}(a^{SO})$
$\displaystyle\equiv\int_{N^{n+1}}\kappa\hskip
1.0pt\mathrm{d}\omega_{n}(a^{SO})=\int_{N^{n+1}}\kappa p(a^{SO})$ (68)

where $M^{1}_{1}$ and $M^{n}_{2}$ are the boundary of $N^{n+1}$: $\partial
N^{n+1}=M^{n}_{1}\sqcup-M^{n}_{2}$, and $p(a^{SO})=\hskip
1.0pt\mathrm{d}\omega_{n}(a^{SO})$ is a product of Pontryagin classes. Note
that $a^{SO}$ in $p(a^{SO})$ is the $SO$ -connection of the tangent bundle for
$N^{n+1}$. Since $N^{n+1}$ satisfying $\partial
N^{n+1}=M^{n}_{1}\sqcup-M^{n}_{2}$ is not unique, the different choices of
$N^{n+1}$ may give different
$\int_{M^{n}_{1}}\kappa\omega_{n}(a^{SO})-\int_{M^{n}_{2}}\kappa\omega_{n}(a^{SO})$.
However, if the ambiguity is an integer, then $\frac{\hskip
1.0pt\mathrm{e}^{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int_{M^{n}_{1}}\kappa\omega_{n}(a^{SO})}}{\hskip
1.0pt\mathrm{e}^{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int_{M^{n}_{2}}\kappa\omega_{n}(a^{SO})}}$ is well defined. This leads
to a quantization of $\kappa$:

$\displaystyle\int_{N^{n+1}}\kappa p(a^{SO})\in\mathbb{Z},\ \ \ \partial
N^{n+1}=\emptyset.$ (69)

So according to the cobordism approach, the phase of the partition function
$\hskip 1.0pt\mathrm{e}^{2\pi\hskip 1.0pt\mathrm{i}\hskip
1.0pt\int_{M^{n}}\kappa\omega_{n}(a^{SO})}$ can be well defined only when
$\kappa$ satisfy a quantization condition eqn. (69).

###  V.2 The quantization of $c$ for 2+1D bosonic topological orders with non-
zero partition function

Let us apply the above result for 2+1D L-type bosonic topological orders. Here
we make a crucial assumption that the partition function is non-zero for all
closed _orientable_ spacetime manifolds

$\displaystyle Z^{\text{top}}(M^{3})=|Z^{\text{top}}(M^{3})|\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
c}{24}\int_{M^{3}}\omega_{3}}\neq 0.$ (70)

Not all 2+1D L-type bosonic topological orders satisfy this condition. But at
least, the 2+1D L-type bosonic invertible topological ordersKong and Wen
(2014); Freed (2014); Kapustin (2014) are believed to satisfy this condition.

For 2+1D L-type bosonic topological orders satisfying this condition, their
chiral central charges satisfy

$\displaystyle\frac{c}{24}\int_{N^{4}}p_{1}\in\mathbb{Z},\ \ \ \forall\ \
\partial N^{4}=\emptyset.$ (71)

Since the partition function is non-zero on all closed _orientable_
3-manifold, here we choose $N^{4}$ to an arbitrary closed _orientable_
4-manifold. This is a key assumption in our cobordism approach.

We know that for all different closed _orientable_ 4-manifolds $N^{4}$, the
set $\\{\int_{N^{4}}p_{1}\\}$ is given by $3\mathbb{Z}$. We find that for
those L-type bosonic topological orders, their chiral central charges are
quantized as

$\displaystyle\frac{c}{8}\in\mathbb{Z}.$ (72)

We conclude that, at least for the 2+1D L-type bosonic invertible topological
orders, their chiral central charges satisfy $c=0$ mod 8.

###  V.3 The quantization of $c$ for 2+1D bosonic topological orders with
vanishing partition functions only on non-spin manifolds and for 2+1D
fermionic invertible topological orders

It was pointed out that a bosonic topological order with emergent fermions
must have vanishing topological partition functions on non-spin manifolds.
Here we consider a subclass of 2+1D bosonic topological order with emergent
fermions, such that topological partition functions vanishes _only_ on non-
spin manifolds. The topological partition functions are non-zero on all closed
_orientable spin_ manifolds. We believe that the partition functions for
fermionic invertible topological order satisfy this condition. Also the
bosonic topological order obtained by gauging the $Z_{2}^{f}$ symmetry in
fermionic invertible topological order satisfy this condition. In this case,
the quantization condition becomes

$\displaystyle\frac{c}{24}\int_{N^{4}}p_{1}\in\mathbb{Z},\ \ \ \forall\ \
\partial N^{4}=\emptyset,\ \ \mathrm{w}_{2}=0.$ (73)

i.e. we also require $N^{4}$ to be _orientable_ and _spin_. As explained in
Appendix A, the set of integers which may be realized as $\int_{N}p_{1}$ for
closed spin 4-manifolds $N^{4}$ is $48\mathbb{Z}$. We find that for those
simple L-type bosonic topological orders with emergent fermions, their chiral
central charges are quantized as

$\displaystyle 2c\in\mathbb{Z}.$ (74)

We conclude that, at least for the 2+1D L-type fermionic invertible
topological orders, their chiral central charges satisfy $c=0$ mod
$\frac{1}{2}$.

###  V.4 Framing anomaly

We have mentioned that the partition function is a function on the moduli
space ${\cal M}_{M^{3}}$ of different closed spacetime manifolds $M^{3}$. In
fact, in the presence of gravitational Chern–Simons term, such a statement is
incorrect. The partition function is actually a function on the moduli space
${\cal M}_{M^{3}}^{F}$ of different closed spacetime manifolds $M^{3}$ _with
framing structure_. In other words, the partition functions for the same
manifold but with different framing structures may have different values.

So, what is a framing structure? Give a $n$-manifold $M^{n}$ and its tangent
bundle $TM$, its stabilized tangent bundle is given by
$TM\oplus\mathbb{R}^{\infty}$. The manifold $M^{n}$ is _framable_ if the
stabilized tangent bundle is trivial. A trivialization of the stabilized
tangent bundle (i.e. a choice of a global basis of the stabilized tangent
bundle) is a framing structure.

It turns out that all orientable 3-manifold $M^{3}$ is framable. This gives us
a way to define the gravitational Chern–Simons term. If the tangent bundle
$TM$ of $M^{3}$ is non-trivial, the $SO_{3}$ connection $a^{SO_{3}}$ for the
tangent bundle $TM$ cannot be globally defined on $M^{3}$, which makes the
gravitational Chern–Simons term $\int_{M^{3}}\omega_{3}(a^{SO_{3}})$ not well
defined. But if we embed $SO_{3}$ into $SO_{\infty}\equiv SO$, then the
corresponding $a^{SO}$ connection can be globally defined on $M^{3}$ (since
the stabilized tangent bundle is trivial), which makes the gravitational
Chern–Simons term $\int_{M^{3}}\omega_{3}(a^{SO})$ well defined. But there are
different ways to turn a $a^{SO_{3}}$-connection into a globally defined
$SO$-connection $a^{SO}$, which corresponds to different trivializations (i.e.
different choices of framing structures). Such different choices of framing
structures can change the gravitational Chern–Simons term
$\int_{M^{3}}\omega_{3}(a^{SO})$ by an arbitrary integer. We see that if
$c\neq 0$ mod 24, then the phase of partition function $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\frac{2\pi
c}{24}\int_{M^{3}}\omega_{3}(a^{SO})}$ will depend on the framing structures
and the partition function is not a function on the moduli space ${\cal
M}_{M^{3}}$ of 3-manifolds. This phenomenon is called framing anomaly. In this
case, the partition function is a function on the moduli space ${\cal
M}_{M^{3}}^{F}$ of 3-manifolds with framing structures.

In general, H-type topological orders have $c\neq 0$ mod 24, and thus
correspond to L-type topological orders with framing anomaly. The L-type
topological orders without framing anomaly must have
$c\overset{\scriptscriptstyle 24}{=}0$. In this paper, we will mainly discuss
L-type topological orders with framing anomaly. Their partition function is a
function on the moduli space ${\cal M}_{M^{3}}^{F}$ of 3-manifolds with
framing structures.

##  VI The topological invariants for L-type topological orders with $U_{1}$
symmetry

###  VI.1 Quantization of $c$ and $\nu$ for bosonic $U_{1}$-SET orders with
non-zero partition functions

A L-type bosonic $U_{1}$-enriched topological orders may contain two
Chern–Simons terms given by eqn. (III.1). Here we assume that the partition
function is non-zero for all closed _orientable_ spacetime with any
$U_{1}$-bundle on it. Repeating the arguments in the last section, but for
spacetime with a $U_{1}$ connection
$a^{{\mathbb{R}/\mathbb{Z}}}=\frac{A_{eff}}{2\pi}$, we obtain the following
quantization for $c$ and $\nu$, for bosonic $U_{1}$-enriched topological
orders whose topological partition functions are non-zero on any closed
orientable spacetime:

$\displaystyle\int_{N^{4}}-\frac{c}{24}p_{1}+\frac{\nu}{2}c_{1}^{2}\in\mathbb{Z},\
\ \ \forall\ \partial N^{4}=\emptyset\text{ and $\forall\ U_{1}$ bundles}.$
(75)

Here $N^{4}$ is an arbitrary _orientable_ 4-manifold with an arbitrary
$U_{1}$-bundle on it. Let $3\eta_{p}=\int_{N^{4}}p_{1}$ and
$\eta_{c}=\int_{N^{4}}c_{1}^{2}$, where $\eta_{p}$ and $\eta_{c}$ are
integers. In Appendix A we show that the pairs $(\eta_{p},\eta_{c})$ which may
be realized is given by $(\mathbb{Z},\mathbb{Z})$. This leads to the
quantization of $c$ and $\nu$:

$\displaystyle c=0\text{ mod }8,\ \ \ \ \ \nu=0\text{ mod }2.$ (76)

The above result, at least, applies to L-type invertible bosonic
$U_{1}$-enriched topological orders and L-type bosonic $U_{1}$-SPT orders .

###  VI.2 Quantization of $c$ and $\nu$ for fermionic $U_{1}^{f}$-enriched
topological orders with non-zero partition functions

A L-type fermionic $U_{1}^{f}$-enriched topological orders also contain two
Chern–Simons terms given by eqn. (III.1). For fermionic $U_{1}$-SET orders
whose topological partition functions are non-zero on any closed smooth
_orientable spin C_ spacetime manifolds, the quantization for $c$ and $\nu$ is
given by

$\displaystyle\int_{N^{4}}-\frac{c}{24}p_{1}+\frac{\nu}{8}(c_{1}^{U_{1}})^{2}\in\mathbb{Z},\
\ \ \forall\ \partial N^{4}=\emptyset\text{ and }\forall\
c_{1}^{U_{1}}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}\mathrm{w}_{2}.$ (77)

Here $N^{4}$ is an arbitrary closed smooth _orientable spin C_ 4-manifold, as
implied by the condition $c_{1}^{U_{1}}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}\mathrm{w}_{2}$. Let $3\eta_{p}=\int_{N^{4}}p_{1}$ and
$\eta_{c}=\int_{N^{4}}(c_{1}^{U_{1}})^{2}$, where $\eta_{p}$ and $\eta_{c}$
are integers. As explained in Appendix A, we may find such spinC 4-manifolds
realizing any integers $\eta_{p}$ and $\eta_{c}$ as long as they satisfy

$\displaystyle\eta_{p}\overset{\scriptscriptstyle 8}{=}\eta_{c}.$ (78)

This leads to the quantization of $c$ and $\nu$:

$\displaystyle c$ $\displaystyle=0\text{ mod }1,\ \ \ \ \ \nu=0\text{ mod }1,$
$\displaystyle-c+\nu$ $\displaystyle=0\text{ mod }8.$ (79)

The above result, at least, applies to L-type invertible fermionic
$U_{1}^{f}$-enriched topological orders and L-type fermionic $U_{1}^{f}$-SPT
orders.

For L-type fermionic $U_{1}^{f}$-SPT orders, the central charge vanishes $c=0$
and $\nu=0$ mod 8. This agrees with the result in LABEL:W1447. We also note
that the fermionic invertible $U_{1}^{f}$-SET states discussed in Sec. IV.4
satisfy the above quantization condition, and actually saturate the
quantization condition.

We remark that the combination of invertible fermionic $U_{1}^{f}$-enriched
topological orders and fermionic $U_{1}^{f}$-SPT orders (i.e. the invertible
fermionic gapped liquid phases with $U^{f}_{1}$ symmetry) is classified via
spinC cobordism in LABEL:FH160406527, which is given by $\mathbb{Z}^{2}$ in
2+1D. This agrees with our result that those state are labeled by two integers
$\\{(\eta_{p},\eta_{c})\ |\ \eta_{p}\overset{\scriptscriptstyle
8}{=}\eta_{c}\\}$. However, the values of $c$ and $\nu$ are not discussed in
LABEL:FH160406527.

##  VII Summary

For 2+1D gapped liquid phases with $U_{1}$ symmetry, the central charge $c$
and the dimensionless Hall conductance $\nu$ are described by the Chern–Simons
terms in the partition function. However, for arbitrary smooth spacetime
3-manifolds and for arbitrary $U_{1}$-bundle over spacetime 3-manifolds, it is
highly non-trivial to define the Chern–Simons terms. In this paper, we use a
cobordism approach to define the Chern–Simons terms. We find that for H-type
quantum systems, the 4-manifolds used in the cobordism approach must be a
surface bundle. For L-type quantum systems, the 4-manifolds must have the same
type as that of the spacetime 3-manifolds, such that the partition functions
are non-zero. This leads to different quantization conditions for $c$ and
$\nu$. In particular, for the H-type quantum systems, the quantization of $c$
and $\nu$ depends on the ground state degeneracies on Riemannian surfaces.
While for the L-type quantum systems, the quantization of $c$ and $\nu$
depends on the type of spacetime manifolds where the topological partition
function is non-zero.

ORW is supported by the ERC under the European Union’s Horizon 2020 research
and innovation programme (grant agreement No. 756444), and by a Philip
Leverhulme Prize from the Leverhulme Trust. LT is supported by the Croucher
Fellowship for Postdoctoral Research. XGW is partially supported by NSF
DMS-1664412 and by the Simons Collaboration on Ultra-Quantum Matter, which is
a grant from the Simons Foundation (651440).

##  Appendix A Characteristic numbers of 4-manifolds

The most fundamental invariant of a closed oriented 4-manifold $M$ is its
intersection form

$\displaystyle Q_{M}:$ $\displaystyle\,\,H^{2}(M^{4};\mathbb{Z})\times
H^{2}(M^{4};\mathbb{Z})\to\mathbb{Z}$ $\displaystyle(a,b)\mapsto\langle
a\smile b,[M^{4}]\rangle=\int_{M}ab.$

Under connected sum the intersection form satisfies

$\displaystyle Q_{M\\#N}=Q_{M}\oplus Q_{N}.$

By Poincaré duality it is unimodular, and by commutativity of the cup-product
it is symmetric and therefore has a signature, denoted $\sigma(M)$. By a
theorem of Thom, $M$ is the boundary of a 5-manifold if and only if its
signature is 0.

Hirzebruch’s signature theorem relates the signature to Pontrjagin classes,
and for a 4-manifold gives

$\displaystyle\int_{M}p_{1}(TM)=3\cdot\sigma(M).$ (80)

By definition of the Wu class $u_{2}$ for any $\mathbb{Z}_{2}$-cohomology
class $x$ we have $\int_{M}x^{2}=\int_{M}x\cdot u_{2}$. As $M$ is oriented, so
its first Stiefel–Whitney class vanishes, and
$u_{2}\equiv\mathrm{w}_{2}+\mathrm{w}_{1}^{2}=\mathrm{w}_{2}$, so

$\displaystyle\int_{M}x^{2}=\int_{M}x\cdot\mathrm{w}_{2}.$ (81)

Thus if $M$ is spin, i.e. $\mathrm{w}_{2}=0$, then the form $Q_{M}$ is even.
If $M$ is spin then by Rochlin’s theorem its signature is divisible by 16, and
so

$\displaystyle\int_{M}p_{1}(TM)\overset{\scriptscriptstyle 48}{=}0.$ (82)

Finally consider spinC 4-manifolds $M$, with $c_{1}\in H^{2}(M;\mathbb{Z})$
the associated Chern class. Then $c_{1}$ reduces modulo 2 to $\mathrm{w}_{2}$,
so is a characteristic element of $Q_{M}$. In this case we may also form the
characteristic number $\int_{M}c_{1}^{2}$. By an elementary property of
symmetric forms over $\mathbb{Z}$ (see II.5.2 of Milnor and Husemoller (1973))
we have

$\displaystyle\int_{M}c_{1}^{2}\overset{\scriptscriptstyle 8}{=}\sigma(M).$
(83)

We now describe to what extent these characteristic numbers may be realized.

Oriented 4-manifolds. The manifold $\mathbb{C}P^{2}$ has $Q_{\mathbb{C}P^{2}}$
given by the 1-by-1 matrix $(1)$, so has signature $1$; similarly
$\overline{\mathbb{C}P}^{2}$ has signature $-1$. By taking connected sums we
may therefore realize every element of $\mathbb{Z}$ as the signature of an
oriented 4-manifold, so may realize every element of $3\mathbb{Z}$ as
$\int_{M}p_{1}(TM)$.

Oriented 4-manifolds with $U_{1}$-bundle. The manifold
${\mathbb{C}P}^{2}\\#\overline{\mathbb{C}P}^{2}$ has signature 0, and
cohomology ring
$H^{*}({\mathbb{C}P}^{2}\\#\overline{\mathbb{C}P}^{2};\mathbb{Z})=\mathbb{Z}[x,y]/(x^{2}+y^{2},x^{3},y^{3})$.
Taking the $U_{1}$-bundle with $c_{1}=x$ it has
$\int_{{\mathbb{C}P}^{2}\\#\overline{\mathbb{C}P}^{2}}c_{1}^{2}=1$, and taking
instead the $U_{1}$-bundle with $c_{1}=y$ it has
$\int_{{\mathbb{C}P}^{2}\\#\overline{\mathbb{C}P}^{2}}c_{1}^{2}=-1$. By
forming connected sums of $\mathbb{C}P^{2}$ and $\overline{\mathbb{C}P}^{2}$
with trivial $U_{1}$-bundle, and
${\mathbb{C}P}^{2}\\#\overline{\mathbb{C}P}^{2}$ with the $U_{1}$-bundles just
described, we can realize any element of $\mathbb{Z}^{2}$ as
$(\sigma(N),\int_{N}c_{1}^{2})$ for an oriented 4-manifold $N$ with a
$U_{1}$-bundle over it.

Spin 4-manifolds The 4-manifold $K3$ is spin, and has intersection form

$\displaystyle Q_{K3}=-E_{8}^{\oplus 2}\oplus H^{\oplus 3},$

where

$\displaystyle H=\begin{pmatrix}0&1\\\ 1&0\\\ \end{pmatrix},\ \ \ \
E_{8}=\begin{pmatrix}2&1&0&0&0&0&0&0\\\ 1&2&1&0&0&0&0&0\\\ 0&1&2&1&0&0&0&0\\\
0&0&1&2&1&0&0&0\\\ 0&0&0&1&2&1&0&1\\\ 0&0&0&0&1&2&1&0\\\ 0&0&0&0&0&1&2&0\\\
0&0&0&0&1&0&0&2\\\ \end{pmatrix}$ (84)

so has signature $-16$. By taking connected sums of $K3$ and its orientation-
reversed analogue $\overline{K3}$ we may realize every element of
$16\mathbb{Z}$ as the signature of a spin 4-manifold, so may realize every
element of $48\mathbb{Z}$ as $\int_{M}p_{1}(TM)$.

SpinC 4-manifolds. The manifold $\mathbb{C}P^{2}$ has cohomology ring
$H^{*}(\mathbb{C}P^{2};\mathbb{Z})=\mathbb{Z}[x]/(x^{3})$. Its usual complex
structure induces a spinC-structure having $c_{1}=3x$, and so having

$\displaystyle\sigma(\mathbb{C}P^{2})$ $\displaystyle=1$
$\displaystyle\frac{1}{8}\left(\sigma(\mathbb{C}P^{2})-\int_{\mathbb{C}P^{2}}c_{1}^{2}\right)$
$\displaystyle=-1.$

The manifold $\mathbb{C}P^{1}\times\mathbb{C}P^{1}$ has cohomology ring
$H^{*}(\mathbb{C}P^{1}\times\mathbb{C}P^{1};\mathbb{Z})=\mathbb{Z}[x,y]/(x^{2},y^{2})$.
Its usual complex structure induces a spinC-structure having $c_{1}=2x+2y$,
and so having

$\displaystyle\sigma(\mathbb{C}P^{1}\times\mathbb{C}P^{1})$ $\displaystyle=0$
$\displaystyle\frac{1}{8}\left(\sigma(\mathbb{C}P^{1}\times\mathbb{C}P^{1})-\int_{\mathbb{C}P^{1}\times\mathbb{C}P^{1}}c_{1}^{2}\right)$
$\displaystyle=-1.$

Therefore by taking connected sums, and changing orientations, there are spinC
4-manifolds $M$ realising any values of $\sigma(M)$ and $\int_{M}c_{1}^{2}$
satisfying (83).

##  Appendix B Characteristic numbers of surface bundles over surfaces

Let $\pi:E^{4}\to B^{2}$ be a smooth fiber bundle, with $E$ and $B$ closed
oriented manifolds and fiber the oriented genus $g$ surface $\Sigma_{g}$.
Suppose we are given a $U_{1}$-bundle on $E$, or equivalently a class
$c_{1}\in H^{2}(E;\mathbb{Z})$. This appendix describes the various
characteristic classes and numbers which arise in this situation, and then
explains the extent to which these may be realized.

###  B.1 Definitions and relations

Let $T_{\pi}E=\mathrm{Ker}(D\pi:TE\to\pi^{*}TB)$ denote the subbundle of the
tangent bundle of $E$ consisting of tangent vectors parallel to the fibers of
$\pi$, giving a decomposition $TE=T_{\pi}E\oplus\pi^{*}TB$. The tangent bundle
of an orientable surface, such as $B$, becomes trivial upon the addition of a
trivial line bundle, and hence its Stiefel–Whitey and Pontryagin classes
vanish. We therefore have

$\displaystyle p_{1}(TE)=p_{1}(T_{\pi}E)$ $\displaystyle=:p_{1}\in
H^{4}(E;\mathbb{Z})$
$\displaystyle\mathrm{w}_{2}(TE)=\mathrm{w}_{2}(T_{\pi}E)$
$\displaystyle=:\mathrm{w}_{2}\in H^{2}(E;\mathbb{Z}_{2}).$

In addition, the orientations of $TE$ and $TB$ induce an orientation of the
2-dimensional bundle $T_{\pi}E$, and we have its Euler class

$e(T_{\pi}E)=:e\in H^{2}(E;\mathbb{Z}).$

As for any oriented 2-dimensional bundle, we have the identities

$\displaystyle p_{1}$ $\displaystyle=e^{2}\in H^{4}(E;\mathbb{Z})$ (85)
$\displaystyle r_{2}(e)$ $\displaystyle=\mathrm{w}_{2}\in
H^{2}(E;\mathbb{Z}_{2}),$ (86)

where $r_{2}:H^{*}(-;\mathbb{Z})\to H^{*}(-;\mathbb{Z}_{2})$ denotes reduction
modulo 2.

Using these classes we may form $\int_{E}p_{1}$, which by (85) agrees with
$\int_{E}e^{2}$, known as the first Miller–Morita–Mumford class. By (80) we
have $\int_{E}p_{1}=3\cdot\sigma(E)$ so $\int_{E}p_{1}$ is always divisible by
3, but in fact more is true: as a consequence of the Atiyah–Singer index
theorem applied to the fiberwise signature operator we have

$\displaystyle\int_{E}p_{1}\overset{\scriptscriptstyle 12}{=}0.$ (87)

This may be obtained from p. 555 of LABEL:MoritaChar, using that
$B_{2}=\tfrac{1}{6}$. If in addition $\mathrm{w}_{2}=0$, then (82) gives

$\displaystyle\int_{E}p_{1}\overset{\scriptscriptstyle 48}{=}0.$ (88)

Now let us suppose that we are further given a $c_{1}\in H^{2}(E;\mathbb{Z})$.
Using this we may form $\int_{E}c_{1}^{2}$ and $\int_{E}e\cdot c_{1}$. As $e$
reduces modulo 2 to $\mathrm{w}_{2}$ by (86), (81) gives

$\displaystyle\int_{E}c_{1}^{2}\overset{\scriptscriptstyle 2}{=}\int_{E}e\cdot
c_{1}.$ (89)

Let us now suppose in addition that $r_{2}(c_{1})=\mathrm{w}_{2}$, i.e. that
$c_{1}$ provides a fiberwise spinC-structure, and investigate its
consequences. As the Euler class $e$ also reduces to $\mathrm{w}_{2}$ modulo
2, it follows from the Bockstein sequence that $c_{1}=e+2\bar{c}_{1}$ for some
$\bar{c}_{1}\in H^{2}(E;\mathbb{Z})$. Then we have
$\int_{E}c_{1}^{2}=\int_{E}e^{2}+4\int_{E}(e\bar{c}_{1}+\bar{c}_{1}^{2})$
which by the divisibility results above we can write in terms of integers as
$\int_{E}c_{1}^{2}=12\cdot\left(\frac{\int_{E}p_{1}}{12}\right)+8\cdot\left(\frac{\int_{E}(e\bar{c}_{1}+\bar{c}_{1}^{2})}{2}\right)$.
This implies

$\displaystyle\int_{E}c_{1}^{2}\overset{\scriptscriptstyle
8}{=}\frac{\int_{E}p_{1}}{3},$ (90)

recovering (83). Similarly, $\int_{E}e\cdot
c_{1}=12\cdot\left(\frac{\int_{E}p_{1}}{12}\right)+2\cdot\int_{E}e\cdot\bar{c}_{1}$
so we have

$\displaystyle\int_{E}e\cdot c_{1}\overset{\scriptscriptstyle 2}{=}0.$ (91)

###  B.2 Realising characteristic numbers

The Madsen–Weiss theorem Madsen and Weiss (2007) and its variants Cohen and
Madsen (2009); Galatius (2006) can be used to establish the existence of
surface bundles with given geometric structure and characteristic numbers in a
highly indirect way, assuming that the genus $g$ is sufficiently large in
comparison with the dimension of the base. For the structure considered here,
of a surface bundle equipped with a $U_{1}$-bundle (equivalently a second
integral cohomology class) on the total space and 2-dimensional base, this has
been carried out in Ebert and Randal-Williams (2012). The conclusion is as
follows.

Firstly, for any $g\geq 5$ there are oriented $\Sigma_{g}$-bundles
$\pi:E^{4}\to B^{2}$ and classes $c_{1}\in H^{2}(E;\mathbb{Z})$ which realize
any values of

$\displaystyle\int_{\Sigma_{g}}c_{1},\ \int_{E}p_{1},\ \int_{E}c_{1}^{2},\
\int_{E}e\cdot c_{1}$

as long as they satisfy (87) and (89). This is obtained by combining Theorem A
(iv) and Theorem C of LABEL:ERW and using that every second homology class may
be represented by a map from a surface $B^{2}$.

Secondly, for any $g\geq 5$ there are oriented $\Sigma_{g}$-bundles
$\pi:E^{4}\to B^{2}$ and classes $c_{1}\in H^{2}(E;\mathbb{Z})$ satisfying
$r_{2}(c_{1})=\mathrm{w}_{2}$ which realize any values of

$\displaystyle\int_{\Sigma_{g}}c_{1},\ \int_{E}p_{1},\ \int_{E}c_{1}^{2},\
\int_{E}e\cdot c_{1}$

as long as they satisfy (87), (90), and (91). This is obtained by taking
$c_{1}=e+2\bar{c}_{1}$ and translating the conditions on
$\int_{\Sigma_{g}}\bar{c}_{1},\int_{E}p_{1},\int_{E}\bar{c}_{1}^{2},\int_{E}e\cdot\bar{c}_{1}$
from the previous paragraph to conditions on
$\int_{\Sigma_{g}}c_{1},\int_{E}p_{1}\int_{E}c_{1}^{2},\int_{E}e\cdot c_{1}$.

Thirdly, the analogous analysis for spin surface bundles though not equipped
with a additional line bundle, has been carried out in Randal-Williams (2012).
It follows from Example 1.11 of loc. cit. that for any $g\geq 9$ there are
Spin $\Sigma_{g}$-bundles $\pi:E^{4}\to B^{2}$ realising any value of
$\int_{E}p_{1}$ satisfying (88).

##  Appendix C Checking (29), (30), (31)

###  C.1 Spread polynomials $S_{n}(x)$

The spread polynomials $S_{n}(x)$ are defined by
$S_{n}(\sin^{2}\theta)=\sin^{2}(n\theta)$. They have the explicit
formWildberger (2005):

$\displaystyle S_{n}(x)=x\sum_{m=0}^{n-1}\frac{n}{n-m}{2n-1-m\choose
m}(-4x)^{n-m-1}$

which has roots $\\{\sin^{2}(\frac{m\pi}{n}):m=0,\dots,n-1\\}$. Putting
$n=k+2$ and replacing $x\rightarrow 1/x$, the polynomial

$\displaystyle x^{k+2}S_{k+2}(\frac{1}{x})=\sum_{m=0}^{k+1}\alpha_{m}x^{m}$
$\displaystyle\alpha_{m}:=\frac{k+2}{k+2-m}{2k+3-m\choose m}(-4)^{k-m+1}$

has roots $\\{\csc^{2}\frac{m\pi}{k+2}:m=1,\dots,k+1\\}$.

###  C.2 Symmetric polynomials $e_{l}$ and $p_{l}$

Let $\xi_{m}:=\csc^{2}\frac{m\pi}{k+2}$. Their elementary symmetric
polynomials are given by

$\displaystyle e_{l}:$ $\displaystyle=\sum_{1\leq m_{1}<\dots<m_{l}\leq
k+1}\xi_{m_{1}}\dots\xi_{m_{l}}=\frac{(-1)^{l}\alpha_{k-l+1}}{\alpha_{k+1}}$
$\displaystyle=\frac{4^{l}}{(k+2)(l+1)}{k+2+l\choose 2l+1}$

for $l\leq k+1$. For $l>k+1$, $e_{l}:=0$. Define

$\displaystyle E_{l}:$
$\displaystyle=\frac{(k+2)^{l}}{2^{l}}e_{l}=2^{l}(k+2)^{l-2}\frac{k+2}{l+1}{k+2+l\choose
2l+1}$ $\displaystyle=2^{l}(k+2)^{l-2}\big{[}{k+l+3\choose 2l+2}+{k+l+2\choose
2l+2}\big{]}.$

Note that $E_{1}={k+3\choose 3}\in\mathbb{Z}$, which is odd iff $k=0\text{ mod
}4$, so $kE_{l}\in 4\mathbb{Z}$. For $l\geq 2$, $E_{l}\in 4\mathbb{Z}$.
Moreover $\frac{3E_{2}}{k+2}\in 2\mathbb{Z}$ and $\frac{E_{l}}{k+2}\in
2\mathbb{Z}$ for $l\geq 3$.

Define the $l$-th power sums $p_{l}:=\sum_{m=1}^{k+1}\xi_{m}^{l}$. They are
related to $e_{l}$’s via Newton–Girard FormulasSéroul (2000):

$\displaystyle p_{l}=(-1)^{l+1}le_{l}+\sum_{j=1}^{l-1}(-1)^{j+1}e_{j}p_{l-j}.$
(92)

###  C.3 Checking (29)

Define $n_{l}=\frac{3k}{k+2}\frac{(k+2)^{l}}{2^{l}}p_{l}$. The expression eqn.
(29) to be checked is

$\displaystyle n_{g-1}\in 2\mathbb{Z}.$

We will show this by induction for $g\geq 3$. From (92) we have

$\displaystyle
n_{l}=(-1)^{l+1}lk\big{(}\frac{3E_{l}}{k+2}\big{)}+\sum_{j=1}^{l-1}(-1)^{j+1}E_{j}n_{l-j}.$
(93)

Note that $n_{1}=\frac{(k+3)(k+1)k}{2}\in\mathbb{Z}$, which is odd iff
$k=2\text{ mod }4$. For $l=2$, we have

$\displaystyle n_{2}=-2k\big{(}\frac{3E_{2}}{k+2}\big{)}+E_{1}n_{1}.$

where the first term is even since the factor in bracket is even as discussed
before. The second term is even since $E_{1}$ and $n_{1}$ cannot be odd at the
same time. For $l\geq 3$, it can be seen that $n_{l}\in 2\mathbb{Z}$
inductively since both terms of (93) are even integers.

###  C.4 Checking (30)

Define $m_{l}=k\frac{(k+2)^{l}}{2^{l}}p_{l}$. The expression eqn. (30) to be
checked is

$\displaystyle m_{g-1}\in 2\mathbb{Z}.$

We will show this by induction for $g\geq 1$. From (92) we have

$\displaystyle
m_{l}=(-1)^{l+1}lkE_{l}+\sum_{j=1}^{l-1}(-1)^{j+1}E_{j}m_{l-j}.$ (94)

Note that $m_{0}=k(k+1)\in 2\mathbb{Z}$. For $l\geq 1$, the first term in (94)
is even since $kE_{l}$ is even. Hence $m_{l}\in 2\mathbb{Z}$ for all $l\geq 0$
by induction.

###  C.5 Checking (31)

The expression eqn. (31) to be checked is

$\displaystyle\frac{m_{g-1}}{2}\in 2\mathbb{Z}.$

for $g\geq 6$. We will show a stronger result that this holds for all $g\geq
2$ by induction. (94) can be rewritten as

$\displaystyle\frac{m_{l}}{2}$
$\displaystyle=(-1)^{l+1}l\big{(}\frac{kE_{l}}{2}\big{)}+\sum_{j=1}^{l-1}(-1)^{j+1}E_{j}\big{(}\frac{m_{l-j}}{2}\big{)}$
$\displaystyle\frac{m_{l}}{2}$ $\displaystyle\overset{\scriptscriptstyle
2}{=}\sum_{j=1}^{l-1}E_{j}\big{(}\frac{m_{l-j}}{2}\big{)}\overset{\scriptscriptstyle
2}{=}\begin{cases}E_{1}\big{(}\frac{m_{l-1}}{2}\big{)}&\text{for }l\geq 2\\\
0&\text{for }l=1.\end{cases}$ (95)

where we used the fact that $\frac{kE_{l}}{2}\in 2\mathbb{Z}$ for $l\geq 1$
and $E_{j}\in 4\mathbb{Z}$ for $j\geq 2$. Note that
$\frac{m_{0}}{2}=\frac{k(k+1)}{2}$ is odd iff $k=1,2$ mod 4. Also recall that
$E_{1}\in\mathbb{Z}$. It is straightforward to read off that
$\frac{m_{1}}{2}\in 2\mathbb{Z}$ and hence $\frac{m_{l}}{2}\in 2\mathbb{Z}$
for all $l\geq 1$ which follows by induction.

##  Appendix D Group extension and trivialization

Consider an extension of a group $H$

$\displaystyle A\to G\to H$ (96)

where $A$ is an Abelian group with group multiplication given by $x+y\in A$
for $x,y\in A$. Such a group extension is denoted by $G=A\leftthreetimes H$.
It is convenient to label the elements in $G$ as $(h,x)$, where $h\in H$ and
$x\in A$. The group multiplication of $G$ is given by

$\displaystyle(h_{1},x_{1})(h_{2},x_{2})=(h_{1}h_{2},x_{1}+\alpha(h_{1})\circ
x_{2}+e_{2}(h_{1},h_{2})).$ (97)

where $e_{2}$ is a function

$\displaystyle e_{2}:H\times H\to A,$ (98)

and $\alpha$ is a function

$\displaystyle\alpha:H\to\text{Aut}(A).$ (99)

We see that group extension is defined via $e_{2}$ and $\alpha$. The
associativity

$\displaystyle[(h_{1},x_{1})(h_{2},x_{2})](h_{3},x_{3})=(h_{1},x_{1})[(h_{2},x_{2})](h_{3},x_{3})]$
(100)

requires that

$\displaystyle x_{1}+\alpha(h_{1})\circ
x_{2}+e_{2}(h_{1},h_{2})+\alpha(h_{1}h_{2})\circ
x_{3}+e_{2}(h_{1}h_{2},h_{3})$ $\displaystyle=\alpha(h_{1})\circ
x_{2}+\alpha(h_{1})\alpha(h_{2})\circ x_{3}+\alpha(h_{1})\circ
e_{2}(h_{2},h_{3})]$ $\displaystyle\ \ \ \ +x_{1}+e_{2}(h_{1},h_{2}h_{3})$
(101)

or

$\displaystyle\alpha(h_{1})\alpha(h_{2})=\alpha(h_{1}h_{2})$ (102)

and

$\displaystyle
e_{2}(h_{1},h_{2})-e_{2}(h_{1},h_{2}h_{3})+e_{2}(h_{1}h_{2},h_{3})$
$\displaystyle-\alpha(h_{1})\circ e_{2}(h_{2},h_{3})=0.$ (103)

Such a $e_{2}$ is a group 2-cocycle $e_{2}\in H^{2}({\cal B}H;A_{\alpha})$,
where $H$ has a non-trivial action on the coefficient $A$ as described by
$\alpha$. Also, $\alpha$ is a group homomorphism $\alpha:H\to\text{Aut}(A)$.
We see that the $A$ extension from $H$ to $G$ is described by a group
2-cocycle $e_{2}$ and a homomorphism $\alpha$. Thus we can more precisely
denote the group extension by $G=A\leftthreetimes_{e_{2},\alpha}H$.

Note that the homomorphism $\alpha:H\to\text{Aut}(A)$ is in fact the action by
conjugation in $G$,

$\displaystyle(h,0)(\mathbf{1},x)=(h,\alpha(h)\circ x)$
$\displaystyle=(\mathbf{1},\alpha(h)\circ x)(h,0),$
$\displaystyle\Rightarrow(h,0)(\mathbf{1},x)(h,0)^{-1}$
$\displaystyle=(\mathbf{1},\alpha(h)\circ x).$ (104)

Thus, $\alpha$ is trivial if and only if $A$ lies in the center of $G$. This
case is called a central extension, where the action $\alpha$ will be omitted.

Our way to label group elements in $G$:

$\displaystyle g=(h,x)\in G$ (105)

defines two projections of $G$:

$\displaystyle\pi:G\to H,\ \ \ \pi(g)=h,$ $\displaystyle\sigma:G\to A,\ \ \
\sigma(g)=x.$ (106)

$\pi$ is a group homomorphism while $\sigma$ is a generic function. Using the
two projections, $g_{1}g_{2}=g_{3}$ can be written as

$\displaystyle\ \ \ \ (\pi(g_{1}),\sigma(g_{1}))(\pi(g_{2}),\sigma(g_{2}))$
$\displaystyle=[\pi(g_{1})\pi(g_{2}),\sigma(g_{1})+\alpha(\pi(g_{1}))\circ\sigma(g_{2})+e_{2}(\pi(g_{1}),\pi(g_{2}))]$
$\displaystyle=(\pi(g_{3}),\sigma(g_{3}))=(\pi(g_{1}g_{2}),\sigma(g_{1}g_{2}))$
(107)

We see that the group cocycle $e_{2}(h_{1},h_{2})$ in $H^{2}({\cal B}H;A)$ can
be pullback to give a group cocycle $e_{2}(\pi(g_{1}),\pi(g_{2}))$ in
$H^{2}({\cal B}G;A)$, and such a pullback is a coboundary

$\displaystyle
e_{2}(\pi(g_{1}),\pi(g_{2}))=-\sigma(g_{1})+\sigma(g_{1}g_{2})-\alpha(\pi(g_{1}))\circ\sigma(g_{2}),$
(108)

i.e. an element in $B^{2}({\cal B}G;A_{\alpha})$, where $G$ has a non-trivial
action on the coefficient $A$ as described by $\alpha$.

The above result can be put in another form. Consider the homomorphism

$\displaystyle\varphi:{\cal B}G\to{\cal B}H$ (109)

where $G=A\leftthreetimes_{e_{2}}H$, and $e_{2}$ is a $A$-valued 2-cocycle on
${\cal B}H$. The homomorphism $\varphi$ sends an link of ${\cal B}G$ labeled
by $a^{G}_{ij}\in G$ to an link of ${\cal B}H$ labeled by
$a^{H}_{ij}=\pi(a^{G}_{ij})\in H$. The pullback of $e_{2}$ by $\varphi$,
$\varphi^{*}e_{2}$, is always a coboundary on ${\cal B}G$

The above discussion also works for continuous group, if we only consider a
neighborhood near the group identity $\mathbf{1}$. In this case,
$e_{2}(h_{1},h_{2})$ and $\alpha(h)$ are continuous functions on such a
neighborhood. But globally, $e_{2}(h_{1},h_{2})$ and $\alpha(h)$ may not be
continuous functions.

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