We study bosonic systems on a spacetime lattice (with imaginary time) defined
by path integrals of commuting fields. We introduce a concept of _branch-
independent bosonic_ (BIB) _systems_ , whose path integral is independent of
the branch structure of the spacetime simplicial complex, even for a spacetime
with boundaries. In contrast, a _generic lattice bosonic_ (GLB) _system_ ’s
path integral may depend on the branch structure.  We find the invertible
topological order characterized by the Stiefel-Whitney cocycle (such as 4+1d
w2w3) to be nontrivial for _branch-independent bosonic systems_ , but this
topological order and a trivial gapped tensor product state belong to the same
phase (via a smooth deformation without any phase transition) for _generic
lattice bosonic systems_. This implies that the invertible topological orders
in generic lattice bosonic systems on a spacetime lattice are not classified
by the oriented cobordism. The branch dependence on a lattice may be related
to the orthonormal frame of smooth manifolds and the framing anomaly of
continuum field theories. In general, the branch structure on a discretized
lattice may be related to a frame structure on a smooth manifold that
trivializes any Stiefel-Whitney classes. We construct branch-independent
bosonic systems to realize the w2w3 topological order, and its 3+1d gapped or
gapless boundaries. One of the gapped boundaries is a 3+1d $\mathbb{Z}_{2}$
gauge theory with (1) fermionic $\mathbb{Z}_{2}$ gauge charge particle which
trivializing w2 and (2) “fermionic” $\mathbb{Z}_{2}$ gauge flux line
trivializing w3. In particular, if the flux loop’s worldsheet is unorientable,
then the orientation-reversal 1d worldline must correspond to a fermion
worldline that _does not carry the $\mathbb{Z}_{2}$ gauge charge_. We also
explain why Spin and Spinc structures trivialize the w2w3 nonperturbative
global pure gravitational anomaly to zero (which helps to construct the
anomalous 3+1d gapped $\mathbb{Z}_{2}$ and gapless all-fermion U(1) gauge
theories), but the Spinh and Spin$\times_{\mathbb{Z}_{2}}$Spin$(n\geq 3)$
structures modify the w2w3 into a nonperturbative global mixed gauge-
gravitational anomaly, which helps to constrain Grand Unifications (e.g.,
$n=10,18$) or construct new models.

# 3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological
Order
for Branch-Independent Bosonic Systems

Zheyan Wan<EMAIL_ADDRESS>Yau Mathematical Sciences Center,
Tsinghua University, Beijing 100084, China  Juven Wang
<EMAIL_ADDRESS>Center of Mathematical Sciences and Applications,
Harvard University, MA 02138, USA  School of Natural Sciences, Institute for
Advanced Study, Einstein Drive, Princeton, NJ 08540, USA  Xiao-Gang Wen
<EMAIL_ADDRESS>Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139, USA

###### Contents

  1. I Introduction
  2. II Branch-Independent Bosonic System and Generic lattice Bosonic System
  3. III Boundaries of ${\rm w}_{2}{\rm w}_{3}$ invertible topological order
  4. IV 4d Boundary of SU(2) or Other Spin($n$) Internal Symmetric Theories
  5. V Acknowledgements
  6. A Spacetime Complex with Branch Structure, Cochains, Higher Cup Product
  7. B Comparison with Standard Mathematical Conventions and Stiefel-Whitney Class
  8. C Emergence of Half-Integer Spin and Fermi Statistics
  9. D Cobordism Group Data and Anomaly Classification
  10. E Oriented Bordism Groups and Manifold Generators
  11. F Combinatorial Formula for Stiefel-Whitney Classes
  12. G Compute ${\rm w}_{2}{\rm w}_{3}$ on Real Milnor, Wu, and Dold Manifolds
  13. H Generalized Wu Relation
  14. I Pullback Construction of Branch-Independent Bosonic Models
  15. J Background vs Dynamical Gauge Transformations
  16. K $\mathbb{Z}_{2}$ topological order with emergent fermion and higher dimensional bosonization

##  I Introduction

Gapped quantum states of matter (or more precisely, gapped quantum liquids
Zeng and Wen (2015); Swingle and McGreevy (2016)) with no symmetry can be
divided into two classes Wen (1990):
(1) States with no topological order. All those states belong to the trivial
phase represented by tensor product states, with no entanglement or short-
range entanglement.
(2) States with topological order Witten (1989); Wen and Niu (1990) (i.e.
gapped states with long-range entanglement Chen  _et al._ (2010)).
In the presence of global symmetry that is not spontaneously broken, the above
two classes can be further divided into some subclasses:
(1a) States with no topological order and no symmetry-protected topological
(SPT) order, or synonymously, symmetry-protected trivial (SPT) order, with no
entanglement or short-range entanglement.
(1b) States with no topological order but with nontrivial SPT order Gu and Wen
(2009); Kitaev (2011); Chen  _et al._ (2011, 2013); Lu and Vishwanath (2012);
Vishwanath and Senthil (2013); Gu and Wen (2014); Senthil (2015); Kapustin
_et al._ (2015); Freed and Hopkins (2016); Kapustin and Thorngren (2017); Wang
and Gu (2018); Barkeshli  _et al._ (2021), with short-range entanglement.
(2a) States with both topological order and symmetry. Those states are said to
have symmetry enriched topological (SET) orders Wen (2002, 2003); Kitaev
(2006); Wang and Vishwanath (2006); Kou  _et al._ (2008); Kou and Wen (2009);
Lu and Vishwanath (2016); Essin and Hermele (2013); Barkeshli  _et al._
(2019); Lan  _et al._ (2017); Bulmash and Barkeshli (2021); Aasen  _et al._
(2021), with long-range entanglement.
In this work, we aim to study those topological states of matter and their
boundaries for bosonic systems. We realize that even bosonic systems without
any symmetries can have many different types, such as bosonic systems on a
spacetime lattice with imaginary time, bosonic systems in continuum spacetime
with real or imaginary time, and bosonic systems defined via lattice
Hamiltonian with real continuous time. Those different bosonic theories
require different mathematical descriptions. In this work, we will only study
bosonic systems on a spacetime lattice with imaginary time, that satisfy the
reflection positivity. In fact, we will study a simpler problem – the so
called invertible topological states of matter in the bulk, and their
boundaries, with or without symmetry.

Stacking two topological states, $\mathscr{C}_{1}$ and $\mathscr{C}_{2}$, give
use a third topological state
$\mathscr{C}_{3}=\mathscr{C}_{1}\boxtimes\mathscr{C}_{2}$. The stacking
operation $\boxtimes$ makes the set of topological states into a monoid. (A
monoid is like a group except its elements may not have inverse.) If a
topological state $\mathscr{C}$ has an inverse under the stacking operation
$\boxtimes$, i.e. there exists a topological state $\mathscr{D}$ such that
$\mathscr{C}\boxtimes\mathscr{D}$ is a trivial product state, then
$\mathscr{C}$ will be called an invertible topological state. It turns out
that all SPT states are invertible. Only a small fraction of topologically
ordered states and SET states are invertible. For example, a fermionic integer
quantum Hall state is an invertible topologically ordered state (a fermionic
invertible SET with U(1) symmetry). An anti-ferromagnetic spin-1 Haldane chain
is a SPT state protected by spin rotation SO(3) or time reversal
$\mathbb{Z}_{2}^{T}$ symmetry, which is always invertible. (In contrast, an
anti-ferromagnetic spin-2 chain, another Haldane phase, has trivial SPT
order.)

Invertible topological states of bosonic systems are characterized by a simple
class of invertible topological invariants Wen (2014); Hung and Wen (2014);
Kapustin (2014); Kong and Wen (2014); Wang  _et al._ (2015); Witten (2016);
Freed and Hopkins (2016); Guo  _et al._ (2018). In this article, we will
derive the possible boundary theory from those topological invariants,
especially the 4+1d invertible topological order characterized by the Stiefel-
Whitney ${\rm w}_{2}{\rm w}_{3}$ topological invariant in 5d. There are some
earlier works in this direction Wang  _et al._ (2014); Kravec  _et al._
(2015); Kapustin (2014); Thorngren (2015, 2017); Wang  _et al._ (2018, 2019);
Wang and You (2021a, 2021); You and Wang (2022); Chen and Hsin (2021);
Fidkowski  _et al._ (2021) which construct boundary theories of the ${\rm
w}_{2}{\rm w}_{3}$ invertible topological order. In this work, we will present
a more complete and systematic derivation.

Branch-Independent Bosonic System vs Generic Lattice Bosonic System: Even
bosonic systems on a spacetime lattice can have different types. We find that
in order to discuss invertible topological order, we need to introduce a
concept of _branch-independent bosonic system_ (the L-type system studied in
Ref. Wen, 2015 happens to be a branch-independent bosonic system, L for
Lagrangian formulated on the spacetime lattice), whose partition function
computed from path integral is independent of the branch structures (or
branching structures) of spacetime lattice (i.e. spacetime simplicial
complex), even for spacetime with boundaries. Here a branch structure is a
local ordering of the vertices for each simplex (see Fig. 1 and Appendix A).
_Generic lattice bosonic systems_ do not have this requirement, and their path
integrals may depend on the branch structures of spacetime complex.

Figure 1: A branch structure of a simplicial complex is given by assigning
arrows to the links which do not form loops around every triangle. The branch
structure gives rise to a local order of vertices for every simplex. Stiefel-
Whitney cocycle a simplicial complex can be constructed after assigning a
branch structure (see Appendix F, Fig. 5 and Fig. 6).

Branch-independence is a property of lattice bosonic systems. It appears that
such a lattice property is related to frame-independence, a property of
continuum field theory. An orthonormal frame of a $d$-dimensional manifold is
a set of $d$ vector fields, which is orthonormal at every point respect to the
metrics tensor of the manifold. We will abbreviate orthonormal frame as frame
(which is also called the vielbein). After assigning a frame to a manifold, we
can define an SO$(d)$ connection to describe the curvature. Continuum field
theory explicitly depends on the SO$(d)$ connection and the frame. Thus the
partition function of the continuum field theory may depend on the frame of
the spacetime manifold. If the partition function does depend on the frame, we
say that the theory has a _framing anomaly_ Witten (1989). Otherwise, the
theory is _free of framing anomaly_.

In Ref. DeMarco and Wen, 2021, 2+1d generic lattice bosonic systems are
constructed to realize topological orders with any integral chiral central
charge and the corresponding gravitational Chern-Simons term. Since the
central charge is not 0 mod 24, those models contain framing anomaly and are
not frame-independent. This implies that the partition function of generic
lattice bosonic systems may depend on the frames of the spacetime manifold in
the continuum limit. This example and the above discussions suggest a close
relation between the branch structure on a lattice and the frame structure on
a continuum manifold.111 Branch structure on a discretized lattice vs Frame
structure on a smooth manifold that trivializes any Stiefel-Whitney classes: A
framing of a $d$-dimensional manifold of $M$ is a choice of trivialization of
its tangent bundle, hence a choice of a section of the corresponding frame
bundle. If there exists a framing on the tangent bundle $TM$, then there exist
$d$ linearly independent sections of the tangent bundle. If the $i$-th
Stiefel-Whitney class $w_{i}(TM)\neq 0$, then there can not exist $d-i+1$
linearly independent sections of the tangent bundle. So $w_{i}(TM)=0$ for
$1\leq i\leq d$. Thus,
(1) The tangential frame structure on a smooth manifold trivializes any
Stiefel-Whitney classes.
(2) We will later suggest that a branch structure on a discretized lattice or
on a simplicial complex also trivializes Stiefel-Whitney classes.
(3) Thus, our work suggests a possible relation: a branch structure on a
discretized lattice may be related to the frame structure on a smooth manifold
that trivializes any Stiefel-Whitney classes.

We conjecture that the independence of the branch structure of spacetime
complex for lattice models implies the independence of frame structure of
spacetime manifold in the continuum limit. Under such a conjecture, the
branch-independent bosonic systems on lattice only give rise to continuum
effective field theories that are free of framing anomaly. As a result, a 2+1d
branch-independent bosonic system can only realize topological orders with
chiral central charge $c=0$ mod 24, where the 2+1d invertible topological
orders is generated by three copies of E8 quantum Hall states (say
E${}_{8}^{3}$ topological order). Indeed, using ${\rm SO}(\infty)$ non-linear
$\sigma$-model, Ref. Wen, 2015 constructed a branch-independent bosonic model
that realize the E${}_{8}^{3}$ topological order with $c=24$. For more
details, see Appendix I.

In this work, we find that the 4+1d invertible topological order characterized
by the Stiefel-Whitney class ${\rm w}_{2}{\rm w}_{3}$ is nontrivial for
branch-independent bosonic systems. On the other hand, The ${\rm w}_{2}{\rm
w}_{3}$ topological order and trivial tensor product states belong to the same
phase for generic lattice bosonic systems. This implies that we cannot use the
oriented cobordism  (i.e. manifolds with special orthogonal group SO
structures) Kapustin (2014); Kong and Wen (2014); Wang  _et al._ (2015); Freed
and Hopkins (2016) to classify invertible topological orders for the generic
lattice bosonic systems (where the lattice means the spacetime lattice).

Can we use oriented cobordism to classify invertible topological orders for
branch-independent bosonic systems? The oriented cobordism suggests a
$\mathbb{Z}$ class of 2+1d invertible bosonic topological orders, generated by
the E8 topological order.222The 2+1d Abelian bosonic topological orders are
classified (in a many-to-one fashion) by symmetric integral matrices with even
diagonalsWen and Zee (1992), which are called $K$-matrices. Those topological
orders are realized by $K$-matrix quantum Hall wavefunctions
$\Psi(z_{i}^{I})=\prod_{i<j;I,J}(z_{i}^{I}-z_{j}^{J})^{K_{IJ}}\hskip
1.0pt\mathrm{e}^{-\frac{1}{4}\sum|z_{i}^{I}|^{2}}$ and are described by
$K$-matrix U(1)-Chern-Simons theories $\frac{K_{IJ}}{4\pi}a_{I}\hskip
1.0pt\mathrm{d}a_{J}$. The $K$-matrices with $|\text{det}(K)|=1$ classify
invertible topological orders. Here the E8 topological order is an invertible
topological order described by a $K$-matrix given by the Cartan matrix of E8,
denoted as $K_{E_{8}}$. The 1+1d boundary carries the chiral central charge
$c=8$ Wen (1995). In contrast, the E${}_{8}^{3}$ topological order is
described by a $K$-matrix $K=K_{E_{8}}\oplus K_{E_{8}}\oplus K_{E_{8}}$ and
has its boundary carrying the chiral central charge $c=24$. It was suggested
that the $\mathbb{Z}$ class of the oriented cobordism is generated by the E8
topological order, see for example, Freed’s work Freed (2014) or Freed-Hopkins
Freed and Hopkins (2016). See Section 7 of Wan and Wang (2020) for an
elaborated interpretation of the related cobordism invariants. However,
currently, we do not have a realization of E8 topological order using branch-
independent bosonic systems. We only know a branch-dependent bosonic model
that realizes the E8 topological order DeMarco and Wen (2021), and a branch-
independent bosonic model that realizes the E${}_{8}^{3}$ topological order
Wen (2015). Thus it is not clear if oriented cobordism classifies invertible
topological orders for the branch-independent bosonic systems or not. In this
article, we will concentrate on branch-independent lattice bosonic systems.

The boundary theories of bulk invertible topological orders with no symmetry
belong to a special class: they are theories with gravitational anomalies. In
fact, the gravitational anomalies in field theories are classified by
invertible topological orders in one higher dimension Wen (2013); Kong and Wen
(2014). From this point of view, we study various anomalous theories with a
given gravitational anomaly on the 3+1d boundary of 4+1d ${\rm w}_{2}{\rm
w}_{3}$ topological order. We shall call this gravitational anomaly as the
${\rm w}_{2}{\rm w}_{3}$ anomaly, which is a _nonperturbative global
gravitational anomaly_.333Perturbative local anomalies are detectable via
infinitesimal gauge/diffeomorphism transformations continuously deformable
from the identity, captured by perturbative Feynman diagrams Alvarez-Gaume and
Witten (1984). Nonperturbative global anomalies are detectable via large
gauge/diffeomorphism transformations that cannot be continuously deformed from
the identity Witten (1985).

For example, it is known that the 4+1d invertible topological order has a
gapped 3+1d boundary described by a $\mathbb{Z}_{2}$ gauge theory with the
${\rm w}_{2}{\rm w}_{3}$ anomaly, where the $\mathbb{Z}_{2}$ gauge charge is
fermionic. However, the fermionic $\mathbb{Z}_{2}$ gauge charge does not fully
characterize the gravitational anomaly. In particular, there is also an
anomaly-free 3+1d $\mathbb{Z}_{2}$ gauge theory with a fermionic
$\mathbb{Z}_{2}$ gauge charge, i.e. there is a 3+1d lattice bosonic system
that can realize a $\mathbb{Z}_{2}$ gauge theory with a fermionic
$\mathbb{Z}_{2}$ gauge charge Levin and Wen (2003). In this work, we show that
the 2d worldsheet of the $\mathbb{Z}_{2}$ gauge flux in the spacetime must
carry a non-contractable 1d fermionic orientation-reversal worldline if the 2d
worldsheet is unorientable. This 1d fermionic orientation-reversal worldline
with neutral gauge charge is however distinct from the fermionic worldline of
$\mathbb{Z}_{2}$ gauge charge. This crucial property, together with the
fermionic $\mathbb{Z}_{2}$ gauge charge, characterizes the gravitational
anomaly.

The above result about the fermion worldline on unorientable worldsheet of
$\mathbb{Z}_{2}$ flux line was first obtained in Section 3.3 of Ref. Wang  _et
al._ , 2019. In this paper we give a different derivation of the result using
a path integral formulation on a spacetime lattice. This result was also
obtained recently in Ref. Chen and Hsin, 2021 using Hamiltonian formulation on
spatial lattice. The “fermionic” nature of the $\mathbb{Z}_{2}$ flux line can
also be characterized by the statistical hopping algebra for strings Fidkowski
_et al._ (2021), a generalization of the statistical hopping algebra for
particles Levin and Wen (2003).

###  I.1 Notations and conventions

We denote the ${\rm n^{\prime}}$d for the spacetime dimensions to be ${\rm
n^{\prime}}=({\rm n}+1)$ with an ${\rm n}$ dimensional space and a 1
dimensional time. Typically, in this article, the dimension always refer to
the spacetime dimension altogether. We may simply call the 0d $\mathbb{Z}_{2}$
gauge charge as $\mathbb{Z}_{2}$ charge, whose spacetime trajectory is a 1d
worldline. We may simply call the 1d $\mathbb{Z}_{2}$ gauge flux loop as
$\mathbb{Z}_{2}$ flux, which can be a 1d loop which bounds a 2d surface
enclosing gauge flux, whose spacetime trajectory is a 2d worldsheet.

In this work, we use a lot of formalisms of chain and cochain, as well as the
associated derivative cup product, Steenrod square, etc. A brief coverage of
those topic can found in Appendix A. The $\mathbb{Z}_{n}$ values are chosen to
be $\\{0,1,\cdots,n-1\\}$. In this article, we always use this set to extend
$\mathbb{Z}_{n}$ values to $\mathbb{Z}$ values, and treat
$\mathbb{Z}_{n}$-valued quantities as $\mathbb{Z}$-valued quantities. To help
to express $\mathbb{Z}_{n}$-valued relation using $\mathbb{Z}$-valued
quantities, we denote $\overset{\scriptscriptstyle n}{=}$ to mean equal up to
a multiple of $n$ (thus it is a mod $n$ relation: two sides of the equality
are equal mod $n$), and use $\overset{\scriptscriptstyle\hskip
1.0pt\mathrm{d}}{=}$ to mean equal up to a coboundary $\hskip
1.0pt\mathrm{d}f$ (i.e. with the coboundary operator $\hskip
1.0pt\mathrm{d}$).

We denote the Lorentz group as ${\rm SO}$ (for bosonic systems) and ${\rm
Spin}$ (for fermionic systems graded by the fermion parity
$\mathbb{Z}_{2}^{f}$). In ${\rm n}+1$d spacetime, the ${\rm SO}$ really means
the ${\rm SO}({\rm n}+1)$ for the Euclidean rotational symmetry group and the
${\rm SO}({\rm n},1)$ for the Lorentz rotational $+$ boost symmetry group; the
${\rm Spin}$ really means the ${\rm Spin}({\rm n}+1)$ for the Euclidean
rotational symmetry group and the ${\rm Spin}({\rm n},1)$ for the Lorentz
rotational $+$ boost symmetry group. We denote $\overset{\scriptscriptstyle
n,\hskip 1.0pt\mathrm{d}}{=}$ to mean equal up to a mod $n$ relation and also
equal up to a coboundary $\hskip 1.0pt\mathrm{d}f$. We will use the group
$N\rtimes_{e_{2},\alpha}G$ to describe the extension of a group $G$ by an
abelian group $N$ via

$1\to N\to N\rtimes_{e_{2},\alpha}G\to G\to 1,$

which is characterized by $e_{2}\in H^{2}(G;N)$ of the second cohomology
class, where $N$ is an abelian group with a $G$-action via
$\alpha:G\to\text{Aut}[N]$. We will also use
$G_{1}\times_{N}G_{2}\equiv\frac{G_{1}\times G_{2}}{N}$ to define as the
product group of $G_{1}$ and $G_{2}$ modding out their common normal subgroup
$N$. Other mathematical conventions and definitions (such as Stiefel-Whitney
class) can be found in Appendix B. We provide many appendices on the toolkits
of co/chain, co/cycle, cohomology, characteristic class, and cobordism.

_Dynamical gauge fields_ are associated with the gauge connections of gauge
bundles that are summed over in the path integral (or partition function).
_Background gauge fields_ are associated with the non-dynamical gauge
connections of gauge bundles that are fixed, not summed over in the path
integral. We will distinguish their gauge transformations: for dynamical
fields as _dynamical gauge transformations_ , for background fields as
_background gauge transformations_ , see Appendix J.

In this work, the _anomalous gauge theory_ merely means its partition function
alone is only _non-invariant_ under _background gauge transformations_ (but
still _invariant_ under _dynamical gauge transformations_) — namely, the
_anomalous gauge theory_ with only ’t Hooft anomaly of the global symmetry ’t
Hooft (1980) can still be _well-defined_ on the boundary of one-higher
dimensional invertible topological phase. The cancellation of _background
gauge transformations_ between the bulk and boundary theories are known as the
anomaly inflow Callan and Harvey (1985).

##  II Branch-Independent Bosonic System and Generic lattice Bosonic System

In this work, we are going to use cochains on a spacetime simplicial complex
as bosonic fields. In order to construct the action $S$ in the path integral,
using local Lagrangian term on each simplex, it is important to give the
vertices of each simplex a local order. A local scheme to order the vertices
is given by a branch structure Costantino (2005); Chen  _et al._ (2013, 2012).
A branch structure is a choice of orientation of each link in the complex so
that there is no oriented loop on any triangle (see Fig. 1). Relative to a
base branch structure, all other branch structure can be described by a
$\mathbb{Z}$-valued 1-cochain $s$ (see Appendix A.6). After assigning a branch
structure to the spacetime complex, we can define cup product
$\overset{\scriptscriptstyle s}{\underset{{\scriptscriptstyle}}{\smile}}$ of
cochains that depend on the branch structure $s$. For the base branch
structure $s=0$, we abbreviate $\overset{\scriptscriptstyle
s}{\underset{{\scriptscriptstyle}}{\smile}}$ by
$\overset{\scriptscriptstyle}{\underset{{\scriptscriptstyle}}{\smile}}$.

We find that for two cocycles, $f$ and $g$, $f\overset{\scriptscriptstyle
s}{\underset{{\scriptscriptstyle}}{\smile}}g-f\overset{\scriptscriptstyle}{\underset{{\scriptscriptstyle}}{\smile}}g$
is coboundary, that depends on $s,f,g$. Let us use $\hskip
1.0pt\mathrm{d}\nu(s,f,g)$ to denote such a coboundary (for details, see
Appendix A.6):

$\displaystyle f\overset{\scriptscriptstyle
s}{\underset{{\scriptscriptstyle}}{\smile}}g+\hskip
1.0pt\mathrm{d}\nu(s,f,g)=f\overset{\scriptscriptstyle}{\underset{{\scriptscriptstyle}}{\smile}}g.$
(1)

Using derivative and cup product of the cochains, we can construct a local
action $S$. So, in general, the action amplitude, $\hskip
1.0pt\mathrm{e}^{-S}$, may depend on the choices of the branch structures.

###  II.1 Branch-Independent Bosonic (BIB) System

Now we are ready to define the _branch-independent bosonic system_ : A branch-
independent bosonic system is defined by a path integral on a branch spacetime
simplicial complex, such that the value of the path integral is independent of
the choices of branch structures, even when the spacetime has boundaries.

Let us give an example of branch-independent bosonic system. The bosonic
system has two fields: a $\mathbb{Z}_{2}$-valued 1-cochain field
$a^{\mathbb{Z}_{2}}$ and a $\mathbb{Z}_{2}$-valued 2-cochain field
$b^{\mathbb{Z}_{2}}$, which give rise to the following partition function

$\displaystyle Z=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}({\rm w}_{2}^{s}+\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})\overset{\scriptscriptstyle
s}{\underset{{\scriptscriptstyle}}{\smile}}({\rm w}_{3}^{s}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}$ $\displaystyle\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{\partial
M^{5}}\nu(s,{\rm w}_{2}^{s}+\hskip 1.0pt\mathrm{d}a^{\mathbb{Z}_{2}},{\rm
w}_{3}^{s}+\hskip 1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})},$ (2)

where $\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}$ sums over all the cochain
fields. The summation $\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}$ in the
path integral is known as a summation of degrees of freedom. Here, the ${\rm
w}_{n}^{s}$ is the $n^{\text{th}}$ Stiefel-Whitney cocycle computed from a
simplicial complex $M^{5}$ with a branch structure $s$, as described in Ref.
Goldstein and Turner, 1976 and in Appendix F. The cocycle ${\rm w}_{n}^{s}$ is
a representation of the Stiefel-Whitney class $w_{n}(TM)$ of the tangent
bundle $(TM)$ of the spacetime manifold $M$.444Here we treat ${\rm w}_{n}$ as
the Stiefel-Whitney cocycle. In contrast, the mathematical definition of
Stiefel-Whitney class as cohomology class and characteristic class $w_{n}$ is
given in Appendix B. Because the cohomology class is also a cocycle, so we may
also abuse the notation to write Stiefel-Whitney class $w_{n}$ in terms of
cocycle ${\rm w}_{n}$.  A cohomology class is an equivalence class that has
many representatives; any elements in the equivalence class are
representatives. The $w_{n}$ is Stiefel-Whitney class, also it can be referred
to as Stiefel-Whitney cocycle when we choose a representative ${\rm w}_{n}$;
the $w_{n}$ is a Stiefel-Whitney number only when $n$ is the top spacetime
dimension.
The Stiefel-Whitney product cocycle ${\rm w}_{2}{\rm w}_{3}$ is a
representative of the cup product of two Stiefel-Whitney classes ($w_{2}$ and
$w_{3}$). When the ${\rm w}_{2}{\rm w}_{3}$ paired with the fundamental class,
as $\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}$, is called the Stiefel-Whitney number.
We omit the normalization factor here in the partition function (II.1).

Let ${\rm w}_{n}$ be the Stiefel-Whitney cocycle for the base branch
structure: ${\rm w}_{n}\equiv{\rm w}_{n}^{s=0}$. In general, ${\rm w}_{n}^{s}$
depends on the branch structure $s$ on $M^{5}$. However, ${\rm w}_{n}^{s}-{\rm
w}_{n}$ is a coboundary:

$\displaystyle{\rm w}_{n}^{s}={\rm w}_{n}+\hskip 1.0pt\mathrm{d}v_{n-1}(s).$
(3)

We can show that (II.1) is independent of branch structure $s$. First, from
the definition of $\nu$, with a bulk ${M^{5}}$ but without any boundary
${\partial M^{5}}$, we have

$\displaystyle Z$
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}({\rm
w}_{2}^{s}+\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})\overset{\scriptscriptstyle}{\underset{{\scriptscriptstyle}}{\smile}}({\rm
w}_{3}^{s}+\hskip 1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}$
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}({\rm
w}_{2}^{s}+\hskip 1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})({\rm w}_{3}^{s}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}.$ (4)

In this article, we abbreviate the cup product for the base branch structure,
$f\smile g$, as $fg$. From the relation between ${\rm w}_{n}^{s}$ and ${\rm
w}_{n}$, we have

$\displaystyle Z$
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}({\rm
w}_{2}+\hskip 1.0pt\mathrm{d}v_{1}(s)+\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})({\rm w}_{3}+\hskip
1.0pt\mathrm{d}v_{2}(s)+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}}$
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}({\rm
w}_{2}+\hskip 1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})({\rm w}_{3}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}.$ (5)

We see that the partition function (II.1) is indeed independent of branch
structure.

The partition function for a generic bosonic system on a spacetime complex $M$
in a quantum liquid phase has a form

$\displaystyle Z(M)$ $\displaystyle=\hskip
1.0pt\mathrm{e}^{-S^{\text{eff}}}Z^{\text{top}}(M),$ (6)

in a thermodynamic limit, where $S^{\text{eff}}$ $=\int_{M}$ energy-density is
the non-universal volume part, and $Z^{\text{top}}(M)$ is the universal
topological partition function (i.e. the topological invariant) that
characterizes the topological order. (The topological partition function
$Z^{\text{top}}$ and its isolation are discussed in much more details in Ref.
Kong and Wen, 2014; Wen, 2017; Wen and Wang, 2020.) For a branch-independent
bosonic system, the topological invariant $Z^{\text{top}}(M)$ does not depend
on the branch structures on $M$, and possibly, nor the framing of the
spacetime manifold in continuum limit. This leads to the conjecture that the
topological invariant $Z^{\text{top}}(M)$ is a cobordism invariant Kapustin
(2014); Kong and Wen (2014); Freed and Hopkins (2016) for invertible
topological orders in the branch-independent bosonic systems.

The above example (II.1) is exactly solvable when $M^{5}$ has no boundary,
since the partition function can be calculated exactly

$\displaystyle Z=2^{N_{l}+N_{f}}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}$ (7)

where $N_{l}$ is the number of links (namely 1-simplices that can be paired
with $a^{\mathbb{Z}_{2}}$) and $N_{f}$ is the number of faces (namely
2-simplices that can be paired with $b^{\mathbb{Z}_{2}}$) in $M^{5}$. Thus the
summation $\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}$ gives a
$2^{N_{l}+N_{f}}$ factor. After dropping the non-universal volume term $\hskip
1.0pt\mathrm{e}^{-S^{\text{eff}}}=2^{N_{l}+N_{f}}$, we see that the
topological partition function is given by

$\displaystyle Z^{\text{top}}(M^{5})=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}.$ (8)

The nontrivial topological invariant $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}$ suggests
that the bosonic model (II.1) realizes a nontrivial topological order. The
topological order is invertible since the topological invariant is a phase
factor for any closed spacetime $M^{5}$. We will refer such an invertible
topological order as the ${\rm w}_{2}{\rm w}_{3}$-topological order.

###  II.2 Generic Lattice Bosonic (GLB) System

We also like to define a concept of _generic lattice bosonic system_ , as a
bosonic system on a spacetime simplicial complex, whose path integral may or
may not depend on the branch structures. Indeed, when we define an generic
lattice bosonic system, the local branch structure of spacetime complex is
given, and the definition may depend on the choices of the local branch
structure.

Let us give an example of a generic lattice bosonic system, that happens to
have “no degrees of freedom.” Here “no degrees of freedom” means that there is
only one term in the path integral, and also means that there is only one
state $|0\rangle_{j}$ per single site $j$. By definition, the ground state of
such a system is a _trivial tensor product state_ $\otimes_{j}|0\rangle_{j}$
(or, simply, a _product state_). Since the generic lattice bosonic system may
have an action that depend on the branch structure on the spacetime complex
$M^{5}$, we can choose the action amplitude to be

$\displaystyle\hskip 1.0pt\mathrm{e}^{-S}$ $\displaystyle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}^{s}\overset{\scriptscriptstyle
s}{\underset{{\scriptscriptstyle}}{\smile}}{\rm w}_{3}^{s}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{\partial
M^{5}}\nu(s,{\rm w}_{2}^{s},{\rm w}_{3}^{s})}$ $\displaystyle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}^{s}\overset{\scriptscriptstyle}{\underset{{\scriptscriptstyle}}{\smile}}{\rm
w}_{3}^{s}}=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}^{s}{\rm w}_{3}^{s}}.$ (9)

This is possible since a branch simplicial complex fully determines the cup
product and the cocycles ${\rm w}_{n}^{s}$ that represents Stiefel-Whitney
cohomology classes Gaiotto and Kapustin (2016); Wang and Gu (2018). The ${\rm
w}_{n}^{s}$ depends on the branch structure and different choices of branch
structures $s$ can only change ${\rm w}_{n}^{s}$ by a coboundary. Thus, when
$M^{5}$ has boundaries, the action amplitude $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}^{s}{\rm w}_{3}^{s}}$
will depend on the branch structure. So such an action amplitude is only
allowed by generic lattice bosonic systems.

Since there is no degrees of freedom, the path integral is only a summation of
one term and the topological partition function is given by the action
amplitude

$\displaystyle Z^{\text{top}}(M^{5})=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}^{s}{\rm w}_{3}^{s}}.$
(10)

This implies that

In generic lattice bosonic systems (GLB), a cobordism invariant $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}$ can be realized by a system with no degrees of freedom,
and corresponds to a trivial tensor product state with no SPT nor topological
order. Here “no degrees of freedom” means that there is only one term in the
path integral, and also means that there is only one state $|0\rangle_{j}$ per
single site $j$. So the Hilbert space only contains a single tensor product
state $\otimes_{j}|0\rangle_{j}$ for all sites.

In contrast,

In branch-independent bosonic systems (BIB), a cobordism invariant $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}$ cannot be realized by a system with no degrees of freedom,
and thus corresponds to a nontrivial invertible topological order.

In other words, we can smoothly deform the topological order characterized by
$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}$ into a trivial product state in the parameter space (or
landscape) of generic lattice bosonic systems, but such a smooth deformation
path does not exist in the parameter space (or landscape) of branch-
independent bosonic systems. The smooth deformation is defined as the
continuous and differentiable tuning of any coupling strength of
Lagrangian/Hamiltonian terms in the parameter space, that do not cause any
phase transitions.

To construct the above mentioned smooth deformation in the parameter space of
generic lattice bosonic systems, let us consider the following generic lattice
bosonic system

$\displaystyle Z=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}({\rm w}_{2}+\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})({\rm w}_{3}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}\hskip
1.0pt\mathrm{e}^{-U\int_{M^{5}}|a^{\mathbb{Z}_{2}}|^{2}+|b^{\mathbb{Z}_{2}}|^{2}},$
(11)

where

$\displaystyle\int_{M^{5}}|a^{\mathbb{Z}_{2}}|^{2}\equiv\sum_{(ij)}|a^{\mathbb{Z}_{2}}_{ij}|^{2},\
\ \
\int_{M^{5}}|b^{\mathbb{Z}_{2}}|^{2}\equiv\sum_{(ijk)}|b^{\mathbb{Z}_{2}}_{ijk}|^{2}.$
(12)

Changing $U$ leads to the smooth deformation. When $U=0$, the path integral
(11) is (II.1). In the limit $U\to\infty$, the path integral (11) becomes
(10). The path integral (11) is still exactly solvable when $M^{5}$ has no
boundaries

$\displaystyle Z$
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}({\rm
w}_{2}+\hskip 1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})({\rm w}_{3}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}\hskip
1.0pt\mathrm{e}^{-U\int_{M^{5}}|a^{\mathbb{Z}_{2}}|^{2}+|b^{\mathbb{Z}_{2}}|^{2}}$
$\displaystyle=(1+\hskip 1.0pt\mathrm{e}^{-U})^{N_{l}+N_{f}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}.$ (13)

The system is gapped, namely, with a short-range correlation for any values of
$U$. So the deformation from $U=0$ to $U=+\infty$ is a smooth deformation that
does _not_ cause any phase transition. This is why a ${\rm w}_{2}{\rm w}_{3}$
topological order and a product state belong to the same phase for generic
lattice bosonic systems.

From the above discussion, it is clear that invertible topological orders have
different classifications for branch-independent bosonic systems and for
generic lattice bosonic systems. However, the two kinds of invertible
topological orders are still related. Let TP denotes a classification of
“Topological Phases” following the notation of Freed-Hopkins Freed and Hopkins
(2016). Let $\mathrm{TP}_{d}(\text{BIB})$ be the Abelian group that classifies
the topological phases invertible topological orders for branch-independent
bosonic (BIB) systems. Let $\mathrm{TP}_{d}(\text{GLB})$ be the Abelian group
that classifies invertible topological orders for generic lattice bosonic
(GLB) systems. Since invertible topological orders for branch-independent
bosonic systems can also be viewed as invertible topological orders for
generic lattice bosonic systems, we have a group homomorphism

$\displaystyle\mathrm{TP}_{d}(\text{BIB})\xrightarrow{p_{d}}\mathrm{TP}_{d}(\text{GLB}).$
(14)

Since invertible topological order characterized by Stiefel-Whitney classes
all become trivial for generic lattice bosonic systems, the map $p_{d}$ sends
all the Stiefel-Whitney class to zero in the topological invariant. The map
$p_{d}$ is not injective. The map $p_{d}$ may also not be surjective. So it
may only tell us a subset of invertible topological orders for generic lattice
bosonic systems. In the rest of this article, we will mainly concentrate on
branch-independent bosonic systems and its ${\rm w}_{2}{\rm w}_{3}$ invertible
topological order.

###  II.3 Lattice Bosonic Systems with time reversal symmetry

The above discussion can be easily generalized to bosonic systems with time
reversal symmetry, defined via path integrals on a spacetime lattice with a
real action amplitude $\hskip 1.0pt\mathrm{e}^{-S}$. In order words, we
restrict ourselves to consider only the real action amplitudes $\hskip
1.0pt\mathrm{e}^{-S}$, as a way to implement time reversal symmetry,

The simplest bosonic invertible topological order with time reversal symmetry
(or time-reversal SPT order) appears in 2d and is characterized by the
topological invariant $(-)^{\int_{M^{2}}{\rm w}_{1}^{2}}$ on a closed
spacetime $M^{2}$. A branch-independent bosonic model that realizes the SPT
order is given by

$\displaystyle Z=\sum_{g^{\mathbb{Z}_{2}}}(-)^{\int_{M^{2}}({\rm w}_{1}+\hskip
1.0pt\mathrm{d}g^{\mathbb{Z}_{2}})^{2}}$ (15)

where $\sum_{g^{\mathbb{Z}_{2}}}$ sums over all $\mathbb{Z}_{2}$-valued
0-cochains $g^{\mathbb{Z}_{2}}$ and the spacetime $M^{2}$ can have boundaries.
The branch-independence is ensured by the invariance of the action amplitude
$(-)^{\int_{M^{2}}({\rm w}_{1}+\hskip 1.0pt\mathrm{d}g^{\mathbb{Z}_{2}})^{2}}$
under the following transformation

$\displaystyle{\rm w}_{1}\to{\rm w}_{1}+\hskip 1.0pt\mathrm{d}v_{0},\ \ \ \
g^{\mathbb{Z}_{2}}\to g^{\mathbb{Z}_{2}}+v_{0},$ (16)

even when $M^{2}$ has boundaries.

We also have a generic lattice bosonic model given by

$\displaystyle Z=\sum_{g^{\mathbb{Z}_{2}}}(-)^{\int_{M^{2}}({\rm w}_{1}+\hskip
1.0pt\mathrm{d}g^{\mathbb{Z}_{2}})^{2}}\hskip
1.0pt\mathrm{e}^{-U\int_{M^{2}}|g^{\mathbb{Z}_{2}}|^{2}}.$ (17)

The model has the time-reversal symmetry since the action amplitude is real.
The model is exactly solvable when $M^{2}$ has no boundaries, and has the
following partition function.

$\displaystyle Z=(1+\hskip 1.0pt\mathrm{e}^{-U})^{N_{v}}(-)^{\int_{M^{2}}{\rm
w}_{1}^{2}}$ (18)

where $N_{v}$ is the number of the vertices in the spacetime complex $M^{2}$.
Since the partition function only depends on the area of spacetime $M^{2}$,
but not depends on the shape of spacetime $M^{2}$, the model (17) is gapped
(i.e. has short range correlations) for any values of $U$. There is no phase
transtion as we change $U$.

When $U=0$, the model (17) becomes (15) and realize the time-reversal SPT
order characterized by topological invariant $(-)^{\int_{M^{2}}{\rm
w}_{1}^{2}}$. When $U=\infty$, the model (17) becomes a model with no degrees
of freedom which must correspond to a trivial tensor product state. We see
that, for generic lattice bosonic models with time reversal symmetry, the
time-reversal SPT order characterized by topological invariant
$(-)^{\int_{M^{2}}{\rm w}_{1}^{2}}$ and the trivial tensor product state
belong to the same phase. This result suggests that the bosonic invertible
topological orders with time-reversal symmetry (i.e. with real action
amplitudes) for generic lattice bosonic systems are not classified by
unoriented cobordism (i.e. manifolds with orthogonal group O structures).

##  III Boundaries of ${\rm w}_{2}{\rm w}_{3}$ invertible topological order

###  III.1 The branch independence and background gauge invariance

In the last section, we studied a 5d branch-independent bosonic model on
spacetime complex $M^{5}$ defined by the path integral (II.1). That path
integral can be simplified by using the base branch structure to define the
cup product:

$\displaystyle Z=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}({\rm
w}_{2}^{s}+\hskip 1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})({\rm w}_{3}^{s}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}.$ (19)

When $M^{5}$ is a closed manifold with no boundary, the above path integral
gives rise to a topological invariant (also known as a cobordism invariant)
$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}$. In this section, we consider the case when $M^{5}$ has a
boundary. We shall obtain the possible boundary theories for the ${\rm
w}_{2}{\rm w}_{3}$-topological order.

One might guess that, when $M^{5}$ has a boundary, the partition function is
still given by $Z(M^{5})=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}^{s}{\rm w}_{3}^{s}}$. If this was true, we
could conclude that the boundary is gapped, since the partition function
$Z(M^{5})=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}^{s}{\rm w}_{3}^{s}}$ does not depend on the
metrics on the boundary $B^{4}\equiv\partial M^{5}$. (A gapless system must
have a partition function that depends on the metrics (i.e. the shape and
size) of the spacetime.)

Such a gapped boundary is possible for generic lattice bosonic systems, but it
is impossible for the branch-independent bosonic model (19). This is because
the partition function in (19) is independent of the branch structure on
$M^{5}$ even when $M^{5}$ has boundaries, while our guess $Z(M^{5})=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}^{s}{\rm w}_{3}^{s}}$ depends on the branch structure, since

$\displaystyle{\rm w}_{2}^{s}={\rm w}_{2}+\hskip 1.0pt\mathrm{d}v_{1}(s),\ \ \
\ {\rm w}_{3}^{s}={\rm w}_{3}+\hskip 1.0pt\mathrm{d}v_{2}(s).$ (20)

Thus $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}^{s}{\rm w}_{3}^{s}}$ cannot be the partition
function of (19) which must be independent of the branch structure.

Due to (20), we see that we can encode the branch structure independence, via
the invariance under the following transformation

$\displaystyle{\rm w}_{2}\to{\rm w}_{2}+\hskip 1.0pt\mathrm{d}v_{1},\ \ \ \
{\rm w}_{3}\to{\rm w}_{3}+\hskip 1.0pt\mathrm{d}v_{2}.$ (21)

Thus the branch independence of (19) can be rephrased as the invariance of the
following path integral

$\displaystyle Z=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}({\rm
w}_{2}+\hskip 1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})({\rm w}_{3}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}.$ (22)

under the above transformation (21), even for spacetime $M^{5}$ with
boundaries. We refer to (21) as a “background gauge transformation,” which is
a change in the parameters in the Lagrangian rather than a change in the
dynamical fields in the Lagrangian. See the comparison between “background
gauge transformation” and “dynamical gauge transformation” in Appendix J.

In the following, we will use the background gauge invariance under (21) to
ensure the independence of branch structures. For the branch-independent
bosonic model (22), the boundary must be non-trivial. We may assume the
boundary to be described by

$\displaystyle Z(M^{5})=\sum_{\phi}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm
w}_{3}-\int_{\partial M^{5}}{\cal L}_{\text{bndry}}(\phi,{\rm w}_{2},{\rm
w}_{3})}.$ (23)

The boundary Lagrangian ${\cal L}_{\text{bndry}}$ is not invariant under the
background gauge transformation (21), which cancels the non-invariance of
${\rm w}_{2}{\rm w}_{3}$ in $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}$ when
$M^{5}$ has boundaries. This cancellation of non-invariances of the bulk and
boundary theories is actually the idea of anomaly inflow Callan and Harvey
(1985).

###  III.2 4d $\mathbb{Z}_{2}$-gauge boundary of the ${\rm w}_{2}{\rm w}_{3}$
topological order

In this section, we are going to explore the possibility that the boundary
Lagrangian ${\cal L}_{\text{bndry}}$ describes a $\mathbb{Z}_{2}$ gauge
theory, more precisely the dynamical Spin structure summed over in the path
integral.

####  III.2.1 Effective boundary theory

A 4d $\mathbb{Z}_{2}$ gauge theory can be described by $\mathbb{Z}_{2}$-valued
1-cochain $a^{\mathbb{Z}_{2}}$ and 2-cochain $b^{\mathbb{Z}_{2}}$ fields (for
example, see Banks and Seiberg (2011); Wen (2017); Putrov  _et al._ (2017))
with the following path integral

$\displaystyle Z$
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\overset{\scriptscriptstyle
s}{\underset{{\scriptscriptstyle}}{\smile}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{\partial
B^{4}}\nu(s,a^{\mathbb{Z}_{2}},\hskip 1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}$
(24) $\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\overset{\scriptscriptstyle}{\underset{{\scriptscriptstyle}}{\smile}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}}=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}},$

where $\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}$ is a summation over
$\mathbb{Z}_{2}$-valued 1-cochains $a^{\mathbb{Z}_{2}}$ and 2-cochains
$b^{\mathbb{Z}_{2}}$ on $B^{4}$. But such a theory is invariant under the
background gauge transformation (21), and cannot cancel the non-invariant of
$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}$.

We can add coupling to ${\rm w}_{2}$ and ${\rm w}_{3}$ to fix this problem and
obtain the following boundary theory (with the bulk topological invariant
included)

$\displaystyle\ \ \ \ Z(M^{5},B^{4}=\partial M^{5})$ (25)
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}\text{ on
}B^{4}}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm w}_{3}+{\rm
w}_{2}b^{\mathbb{Z}_{2}}}.$

Indeed, such a partition function is independent of the branch structure,
since it is invariant under the background gauge transformation (21). To see
this point, we note that the change in ${\rm w}_{2}$ and ${\rm w}_{3}$ can be
absorbed by $a^{\mathbb{Z}_{2}}$ and $b^{\mathbb{Z}_{2}}$. In other words, the
action amplitude $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{\partial M^{5}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm w}_{3}+{\rm
w}_{2}b^{\mathbb{Z}_{2}}}$ is invariant under the following generalized
transformation

$\displaystyle a^{\mathbb{Z}_{2}}$ $\displaystyle\to
a^{\mathbb{Z}_{2}}+v_{1},$ $\displaystyle b^{\mathbb{Z}_{2}}$
$\displaystyle\to b^{\mathbb{Z}_{2}}+v_{2}$ $\displaystyle{\rm w}_{2}$
$\displaystyle\to{\rm w}_{2}+\hskip 1.0pt\mathrm{d}v_{1},$ $\displaystyle{\rm
w}_{3}$ $\displaystyle\to{\rm w}_{3}+\hskip 1.0pt\mathrm{d}v_{2}.$ (26)

So (25) is a boundary theory of the branch-independent bosonic theory (22).

In fact we can obtain (25) directly from (22) by assuming $M^{5}$ to have
boundaries and not adding anything on the boundaries:

$\displaystyle Z$
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}\text{ on
}M^{5}}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}({\rm w}_{2}+\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})({\rm w}_{3}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}})}$ (27)
$\displaystyle=2^{N_{l}^{b}+N_{f}^{b}}\hskip-17.07164pt\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}\text{
on }B^{4}}\hskip-17.07164pt\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm w}_{3}+{\rm
w}_{2}b^{\mathbb{Z}_{2}}}$

where $N_{2}^{b}$ and $N_{f}^{b}$ are the number of links and faces in $M^{5}$
that are not on the boundary $\partial M^{5}$.

Both (24) and (25) describe some kinds of $\mathbb{Z}_{2}$ gauge theories.
However, the two $\mathbb{Z}_{2}$ gauge theory is very different. Eqn. (24) is
anomaly-free and can be realized by a branch-independent bosonic model in 4d.
In fact (24) itself is a branch-independent bosonic model in 4d that realize
the anomaly-free $\mathbb{Z}_{2}$ gauge theory. On the other hand, (25) has an
invertible gravitational anomaly. It can only be realized as a boundary of 5d
invertible topological order. In our example, (22) is a branch-independent
bosonic model that realizes the 5d invertible topological order, and (25) is a
boundary theory of of the 5d model (22).

Due to different gravitational anomalies, the two $\mathbb{Z}_{2}$ gauge
theories (24) and (25) have different properties. In the $\mathbb{Z}_{2}$
gauge theory (24), the $\mathbb{Z}_{2}$ gauge charge is a bosonic particle and
the $\mathbb{Z}_{2}$ gauge flux line behave like a bosonic string. On the
other hand, in the $\mathbb{Z}_{2}$ gauge theory (25), the $\mathbb{Z}_{2}$
gauge charge is a fermionic particle and the $\mathbb{Z}_{2}$ gauge flux line
has certain fermionic nature.

To see the above result, we include the worldline of $\mathbb{Z}_{2}$ gauge
charge and worldsheet of $\mathbb{Z}_{2}$ gauge flux into the boundary theory
(25):

$\displaystyle\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm w}_{3}+{\rm
w}_{2}b^{\mathbb{Z}_{2}}}\cdot$ $\displaystyle\ \ \ \ \ \ \ \ \ \hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}l_{3}a^{\mathbb{Z}_{2}}+s_{2}b^{\mathbb{Z}_{2}}}$ (28)

where $l_{3}$ and $s_{2}$ are $\mathbb{Z}_{2}$-valued 3- and 2-cocycles which
correspond to the Poincaré dual of the worldline and the worldsheet. But the
above action with the worldlines and worldsheets is not invariant under the
transformation (III.2.1) , and thus is not a correct boundary theory for
branch-independent bosonic systems. We may fix this problem by considering the
following modified boundary theory

$\displaystyle\sum_{{a^{\mathbb{Z}_{2}}},{b^{\mathbb{Z}_{2}}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm
w}_{3}+b^{\mathbb{Z}_{2}}{\rm w}_{2}}$ $\displaystyle\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}l_{3}a^{\mathbb{Z}_{2}}+s_{2}b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}l_{3}{\rm
w}_{2}+s_{2}{\rm w}_{3}}.$ (29)

In the above, we have assumed that $l_{3}$ and $s_{2}$ on the boundary $B^{4}$
can be extended to the bulk $M^{5}$ as cocycles. The invariance of the above
path integral under transformation (III.2.1) can be checked directly.

But the above expression also has a problem: it depends on how we extend
$l_{3}$ and $s_{2}$ on the boundary $B^{4}$ to the bulk $M^{5}$. To fix this
problem, we propose the path integral

$\displaystyle Z=\sum_{{a^{\mathbb{Z}_{2}}},{b^{\mathbb{Z}_{2}}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm w}_{3}+{\rm
w}_{2}b^{\mathbb{Z}_{2}}}$ $\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}l_{3}a^{\mathbb{Z}_{2}}+s_{2}b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}+l_{3}{\rm
w}_{2}+\text{Sq}^{2}{\beta_{2}}s_{2}+s_{2}{\rm w}_{3}},$ (30)

that is fully invariant under the transformations in (III.2.1). Here Sq is the
Steenrod square (90) and $\beta_{2}$ is the generalized Bockstein homomorphism
(79) that act on cocycles (see Appendix A for details).

Let us first show that the term $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}+l_{3}{\rm
w}_{2}+\text{Sq}^{2}\beta_{2}s_{2}+s_{2}{\rm w}_{3}}$ only depends on the
fields on the boundary $B^{4}=\partial M^{5}$, so that the above path integral
is well-defined. To do so, we note that, according to Wu relation for a
$\mathbb{Z}_{2}$-valued cocycle $x_{d-n}$ in the top $d$-dimension on the
complex $M^{d}$:

$\displaystyle\text{Sq}^{n}x_{d-n}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}u_{n}x_{d-n},$ (31)

where $u_{n}$ is Wu class:

$\displaystyle u_{0}$ $\displaystyle\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}1,\ \ \ \ \ u_{1}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}{\rm w}_{1},\ \ \ \ \ u_{2}\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}{\rm w}_{1}^{2}+{\rm w}_{2},$ $\displaystyle
u_{3}$ $\displaystyle\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}{\rm w}_{1}{\rm w}_{2},\ \ \ \ \
u_{4}\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}{\rm
w}_{1}^{4}+{\rm w}_{2}^{2}+{\rm w}_{1}{\rm w}_{3}+{\rm w}_{4},$ (32)
$\displaystyle u_{5}$ $\displaystyle\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}{\rm w}_{1}^{3}{\rm w}_{2}+{\rm w}_{1}{\rm w}_{2}^{2}+{\rm
w}_{1}^{2}{\rm w}_{3}+{\rm w}_{1}{\rm w}_{4},$ $\displaystyle u_{6}$
$\displaystyle\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}{\rm
w}_{1}^{2}{\rm w}_{2}^{2}+{\rm w}_{1}^{3}{\rm w}_{3}+{\rm w}_{1}{\rm
w}_{2}{\rm w}_{3}+{\rm w}_{3}^{2}+{\rm w}_{1}^{2}{\rm w}_{4}+{\rm w}_{2}{\rm
w}_{4},$ $\displaystyle u_{7}$ $\displaystyle\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}{\rm w}_{1}^{2}{\rm w}_{2}{\rm w}_{3}+{\rm
w}_{1}{\rm w}_{3}^{2}+{\rm w}_{1}{\rm w}_{2}{\rm w}_{4},$ $\displaystyle
u_{8}$ $\displaystyle\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}{\rm w}_{1}^{8}+{\rm w}_{2}^{4}+{\rm w}_{1}^{2}{\rm
w}_{3}^{2}+{\rm w}_{1}^{2}{\rm w}_{2}{\rm w}_{4}+{\rm w}_{1}{\rm w}_{3}{\rm
w}_{4}+{\rm w}_{4}^{2}$ $\displaystyle\ \ \ \ +{\rm w}_{1}^{3}{\rm w}_{5}+{\rm
w}_{3}{\rm w}_{5}+{\rm w}_{1}^{2}{\rm w}_{6}+{\rm w}_{2}{\rm w}_{6}+{\rm
w}_{1}{\rm w}_{7}+{\rm w}_{8}.$

From (31) and (32), we can show that $\text{Sq}^{2}l_{3}+l_{3}{\rm w}_{2}$ is
a coboundary on oriented $M^{5}$ with ${\rm w}_{1}\overset{\scriptscriptstyle
2}{=}0$. Thus $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}+l_{3}{\rm w}_{2}}$ only depends on
$l_{3}$ on the boundary $B^{4}=\partial M^{5}$.

The term $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}+l_{3}{\rm w}_{2}}$ makes $l_{3}$ on the
boundary to be a fermion worldline via a higher dimensional bosonization
Bhardwaj  _et al._ (2017); Wen (2017); Kapustin and Thorngren (2017); Lan  _et
al._ (2019); Chen (2020).555 _Higher dimensional bosonization_ here especially
in Lan  _et al._ (2019) means to use the purely bosonic commuting fields (i.e.
cochains) with only Steenrod algebras (but without using Grassmann variables)
to represent the fermionic systems with the fermionic parity symmetry
$\mathbb{Z}_{2}^{f}$. The fermionic system here may be regarded as a system
with emergent fermions living on the boundary of bosonic topological order.
See Appendix K for a summary. (For details, see Appendix C.) In other words,
the anomalous $\mathbb{Z}_{2}$ gauge theory on the boundary has a special
property that the $\mathbb{Z}_{2}$ gauge charge is a fermion.

There is another way to understand why the $\mathbb{Z}_{2}$ gauge charge is a
fermion. The ${\rm w}_{2}{\rm w}_{3}$-topological order, as a bosonic state,
can live on any 5-dimensional orientable manifold with a ${\rm SO}(5)$
connection for the tangent bundle. One way to obtain a gapped boundary of
${\rm w}_{2}{\rm w}_{3}$-topological order is to trivialize ${\rm w}_{2}$ for
the ${\rm SO}(5)$ connection on the boundary. Such a trivialization can be
viewed as a group extension ${\rm Spin}(5)=\mathbb{Z}_{2}\rtimes_{{\rm
w}_{2}}{\rm SO}(5)$ via

$\displaystyle 1\to\mathbb{Z}_{2}\to{\rm Spin}(5)\to{\rm SO}(5)\to 1.$ (33)

This is consistent with the fact that the Spin manifold is necessary and
sufficient condition for its second Stiefel-Whitney class ${\rm w}_{2}=0$ for
its tangent bundle. Trivialize ${\rm w}_{2}$ on the boundary can be thought as
promoting the ${\rm SO}(5)$ connection in the bulk to a ${\rm Spin}(5)$
connection on the boundary, where ${\rm Spin}(5)$ is a $\mathbb{Z}_{2}$
central extension of ${\rm SO}(5)$. This implies that the boundary is
described by a twisted $\mathbb{Z}_{2}$ gauge theory, where the
$\mathbb{Z}_{2}$ connection 1-cochain $a^{\mathbb{Z}_{2}}$ satisfies

$\displaystyle\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}_{2}}\overset{\scriptscriptstyle 2}{=}{\rm
w}_{2},$ (34)

instead of $\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}_{2}}\overset{\scriptscriptstyle 2}{=}0$. The
above relation can be obtained from (25) by integrating out
$b^{\mathbb{Z}_{2}}$ first. In this case, the $\mathbb{Z}_{2}$ charge couples
to the ${\rm Spin}(5)$ connection on the boundary, instead of
$\mathbb{Z}_{2}\times{\rm SO}(5)$ connection. So the $\mathbb{Z}_{2}$ charge
carries a half-integer spin representation of the spacetime ${\rm Spin}$ group
if we interpret the extended normal $\mathbb{Z}_{2}$ as the fermion parity
$\mathbb{Z}_{2}^{f}$ in (33) as

$\displaystyle 1\to\mathbb{Z}_{2}^{f}\to{\rm Spin}(5)\to{\rm SO}(5)\to 1.$
(35)

The odd $\mathbb{Z}_{2}$ gauge charge in (33) is also the half-integer spin
representation of ${\rm Spin}$ group, which is also odd under the fermion
parity $\mathbb{Z}_{2}^{f}$ in (35). Then according to the usual lattice
belief in terms of the spacetime-spin statistics relation, the half-integer
spin representation of this $\mathbb{Z}_{2}$ gauge charge is also a fermion.
The Spin structure, which contains the emergent fermion parity
$\mathbb{Z}_{2}^{f}$ on the boundary, is also called the emergent dynamical
Spin structure Wang  _et al._ (2019).

However, a 4d $\mathbb{Z}_{2}$ gauge theory with a fermionic $\mathbb{Z}_{2}$
gauge charge may not have gravitational anomaly, since such theory can be
realized by a 4d bosonic model

$\displaystyle Z$
$\displaystyle=\sum_{a^{\mathbb{Z}_{2}},b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}b^{\mathbb{Z}_{2}}({\rm w}_{2}+\hskip
1.0pt\mathrm{d}a^{\mathbb{Z}_{2}})}.$ (36)

The action amplitude is invariant under the following transformation, even
when $B^{4}$ has boundaries

$\displaystyle a^{\mathbb{Z}_{2}}\to a^{\mathbb{Z}_{2}}+v_{1},\ \ \
b^{\mathbb{Z}_{2}}\to b^{\mathbb{Z}_{2}},\ \ \ {\rm w}_{2}\to{\rm
w}_{2}+\hskip 1.0pt\mathrm{d}v_{1}.$ (37)

After including the worldline of $\mathbb{Z}_{2}$ charge and the worldsheet of
$\mathbb{Z}_{2}$ flux, the above becomes

$\displaystyle Z=\sum_{{a^{\mathbb{Z}_{2}}},{b^{\mathbb{Z}_{2}}}\text{ on
}B^{4}}$ $\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+b^{\mathbb{Z}_{2}}{\rm w}_{2}}$ (38)
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}l_{3}a^{\mathbb{Z}_{2}}+s_{2}b^{\mathbb{Z}_{2}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}+l_{3}{\rm w}_{2}},$

which is the path integral description of the anomaly-free $\mathbb{Z}_{2}$
gauge theory with fermionic $\mathbb{Z}_{2}$ charge Levin and Wen (2003)

Therefore, the $\mathbb{Z}_{2}$-flux line must also have some special
properties as a realization of the anomaly in the $\mathbb{Z}_{2}$ gauge
theory (25). Let us first show that $\text{Sq}^{2}\beta_{2}s_{2}+s_{2}{\rm
w}_{3}$ is also a coboundary on $M^{5}$. Using

$\displaystyle\text{Sq}^{1}x$ $\displaystyle\overset{\scriptscriptstyle
2}{=}\beta_{2}x,$ $\displaystyle\text{Sq}^{1}({\rm w}_{j})$
$\displaystyle\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}{\rm
w}_{1}{\rm w}_{j}+(j-1){\rm w}_{j+1},$ (39)

we find that

$\displaystyle\beta_{2}{\rm w}_{2}\overset{\scriptscriptstyle
2}{=}\text{Sq}^{1}{\rm w}_{2}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}{\rm w}_{1}{\rm w}_{2}+{\rm w}_{3}.$ (40)

Noticing ${\rm w}_{1}\overset{\scriptscriptstyle 2}{=}0$, we find that

$\displaystyle s_{2}{\rm w}_{3}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}s_{2}\beta_{2}{\rm w}_{2}\overset{\scriptscriptstyle
2}{=}{\rm w}_{2}\beta_{2}s_{2}+{\beta_{2}({\rm
w}_{2}s_{2})}\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}{\rm
w}_{2}\beta_{2}s_{2}.$ (41)

The first equality uses (40), and the second equality uses the chain rule. The
third equality uses $\beta_{2}({\rm w}_{2}s_{2})\overset{\scriptscriptstyle
2}{=}\text{Sq}^{1}({\rm w}_{2}s_{2})\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}{\rm w}_{1}{\rm w}_{2}s_{2}\overset{\scriptscriptstyle
2}{=}0$, where we have also used the Wu relation (31) and ${\rm
w}_{1}\overset{\scriptscriptstyle 2}{=}0$. Now, according to (41), we have
${\rm w}_{2}\beta_{2}s_{2}+s_{2}{\rm w}_{3}\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}0$. Then using (31) and ${\rm
w}_{1}\overset{\scriptscriptstyle 2}{=}0$, we have
$\text{Sq}^{2}\beta_{2}s_{2}+s_{2}{\rm w}_{3}\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}0$, and $\text{Sq}^{2}\beta_{2}s_{2}+{\rm
w}_{2}\beta_{2}s_{2}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}0$. Thus $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\beta_{2}s_{2}+s_{2}{\rm w}_{3}}$ and $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\beta_{2}s_{2}+{\rm w}_{2}\beta_{2}s_{2}}$
only differ by a surface term on $B^{4}=\partial M^{5}$. Also, $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\beta_{2}s_{2}+{\rm w}_{2}\beta_{2}s_{2}}$
itself is a surface term on $B^{4}=\partial M^{5}$.

The term $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\beta_{2}s_{2}+{\rm w}_{2}\beta_{2}s_{2}}$
makes $\beta_{2}s_{2}$ on the boundary to be the Poincaré dual of a fermion
worldline via a higher dimensional bosonization Wen (2017); Kapustin and
Thorngren (2017); Lan  _et al._ (2019). We note that if the 2d worldsheet for
the $\mathbb{Z}_{2}$ flux loop is _orientable_ , then its Poincaré dual
$s_{2}$ is a $\mathbb{Z}$-valued 2-cocycle thus $s_{2}\in
Z^{2}(M^{5};\mathbb{Z})$. In this case $\beta_{2}s_{2}=\frac{1}{2}\hskip
1.0pt\mathrm{d}s_{2}=0$ because the cocycle condition imposes $\hskip
1.0pt\mathrm{d}s_{2}=0$.

Therefore a nontrivial $\beta_{2}s_{2}$ comes from an _unorientable_ 2d
worldsheet.666Although we require the _oriented_ and _orientable_ special
orthogonal SO symmetry for this 4d boundary and 5d bulk theory, we do allow
_unorientable_ worldsheets on 2d submanifolds. Earlier we wrote for _oriented_
worldsheet $s_{2}\in Z^{2}(M^{5};\mathbb{Z})$ with the topological term:
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\beta_{2}s_{2}+{\rm
w}_{2}\beta_{2}s_{2}}=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\beta_{2}s_{2}+{\rm w}_{3}s_{2}}.$ (42)
However, for _unorientable_ worldsheets $s_{2}\in
Z^{2}(M^{5};\mathbb{Z}_{2})$, we can use Steenrod square $\text{Sq}^{1}$ to
rewrite (42) as  $\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\text{Sq}^{1}s_{2}+{\rm
w}_{2}\text{Sq}^{1}s_{2}}=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\text{Sq}^{1}s_{2}+{\rm w}_{3}s_{2}}.$ (43)
In Appendix H, we prove a generalized Wu relation
$\displaystyle\text{Sq}^{2}\text{Sq}^{1}x_{d-3}=({\rm w}_{1}^{3}+{\rm
w}_{3})x_{d-3}$ (44) on a manifold $M^{d}$ where ${\rm w}_{i}$ is the Stiefel-
Whitney class of $M^{d}$.  On an unorientable worldsheet, we have a worldline
that marks the reversal of the orientation, whose Poincaré dual is
$\beta_{2}s_{2}$. So such an orientation-reversal worldline corresponds to a
fermion worldline.

In other words, the anomalous $\mathbb{Z}_{2}$ gauge theory (25) on the
boundary has a special property that an unorientable worldsheet of the
$\mathbb{Z}_{2}$-flux carries a non-contractable fermionic worldline. Such a
fermionic worldline corresponds to the orientation reversal loop on the
unorientable worldsheet.

####  III.2.2 Trivialization picture

We have used the trivialization picture to understand the half-integer spin
and the Fermi statistic of the $\mathbb{Z}_{2}$ charge. Can we use the similar
trivialization picture to understand the above “fermionic” properties of
$\mathbb{Z}_{2}$-flux lines?

In the above we have associated the Fermi statistic of the $\mathbb{Z}_{2}$
charge (described by $l_{3}$ in spacetime $B^{4}$) with the topological
invariant

$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}l_{3}a^{\mathbb{Z}_{2}}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}+l_{3}{\rm w}_{2}},\ \ \ B^{4}=\partial
M^{5},$ (45)

expressed in one higher dimension $M^{5}$. Similarly, we can associate the
“Fermi statistic” of the $\mathbb{Z}_{2}$ flux line (described by $s_{2}$ in
spacetime $B^{4}$) with the topological invariant

$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}s_{2}b^{\mathbb{Z}_{2}}+\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}\text{Sq}^{1}s_{2}+s_{2}{\rm w}_{3}},\ \ \
B^{4}=\partial M^{5}.$ (46)

To gain a better understanding of the “Fermi” statistics of $\mathbb{Z}_{2}$
charged particle and $\mathbb{Z}_{2}$ flux line, We like to express
topological invariants in the same dimension rather than one-higher dimension.

We start with the path integral (25) describing the $\mathbb{Z}_{2}$ boundary
of the ${\rm w}_{2}{\rm w}_{3}$ invertible bosonic topological order. Then we
add the worldline for $\mathbb{Z}_{2}$ charge and worksheet for
$\mathbb{Z}_{2}$ flux line. But here we assume the worldline and worldsheet
are boundaries. Thus their Poincaré dual’s, $l_{3}$ and $s_{2}$, are
coboundaries

$\displaystyle l_{3}=\hskip 1.0pt\mathrm{d}\tilde{l}_{2},\ \ \ \ s_{2}=\hskip
1.0pt\mathrm{d}\tilde{s}_{1}.$ (47)

Adding the worldline for $\mathbb{Z}_{2}$ charge and worksheet for
$\mathbb{Z}_{2}$ flux line to the boundary spacetime $B^{4}$, we obtain the
following path integral

$\displaystyle Z=\sum_{{a^{\mathbb{Z}_{2}}},{b^{\mathbb{Z}_{2}}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm
w}_{3}+b^{\mathbb{Z}_{2}}{\rm w}_{2}}$ $\displaystyle\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{B^{4}}(\hskip
1.0pt\mathrm{d}\tilde{l}_{2})a^{\mathbb{Z}_{2}}+(\hskip
1.0pt\mathrm{d}\tilde{s}_{1})b^{\mathbb{Z}_{2}}}.$ (48)

But the new term break the invariance under (III.2.1), which to ensure the
branch independence. To fix this, we consider

$\displaystyle Z=\sum_{{a^{\mathbb{Z}_{2}}},{b^{\mathbb{Z}_{2}}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}{\rm w}_{3}}\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm
w}_{3}+b^{\mathbb{Z}_{2}}{\rm w}_{2}}$ $\displaystyle\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{B^{4}}(\hskip
1.0pt\mathrm{d}\tilde{l}_{2})a^{\mathbb{Z}_{2}}+\tilde{l}_{2}{\rm
w}_{2}+(\hskip
1.0pt\mathrm{d}\tilde{s}_{1})b^{\mathbb{Z}_{2}}+\tilde{s}_{1}{\rm w}_{3}}.$
(49)

However, the fix causes another problem: $\tilde{l}_{2}$ and
$\tilde{l}_{2}+\bar{l}_{2}$ described the same worldline if $\bar{l}_{2}$ is a
$\mathbb{Z}_{2}$-valued cocycle; $\tilde{s}_{1}$ and
$\tilde{s}_{1}+\bar{s}_{1}$ described the same worldsheet if $\bar{s}_{1}$ is
a $\mathbb{Z}_{2}$-valued cocycle. Therefore, the path integral must be
invariant under the following transformations

$\displaystyle\tilde{l}_{2}$ $\displaystyle\to\tilde{l}_{2}+\bar{l}_{2},$
$\displaystyle\hskip 1.0pt\mathrm{d}\bar{l}_{2}$
$\displaystyle\overset{\scriptscriptstyle 2}{=}0;$
$\displaystyle\tilde{s}_{1}$ $\displaystyle\to\tilde{s}_{1}+\bar{s}_{1},$
$\displaystyle\hskip 1.0pt\mathrm{d}\bar{s}_{1}$
$\displaystyle\overset{\scriptscriptstyle 2}{=}0.$ (50)

To have such an invariance, we consider

$\displaystyle Z$
$\displaystyle=\hskip-5.0pt\sum_{{a^{\mathbb{Z}_{2}}},{b^{\mathbb{Z}_{2}}}}\hskip-5.0pt\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}{\rm w}_{3}}\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}a^{\mathbb{Z}_{2}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}+a^{\mathbb{Z}_{2}}{\rm
w}_{3}+b^{\mathbb{Z}_{2}}{\rm w}_{2}}$ (51) $\displaystyle\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{B^{4}}(\hskip
1.0pt\mathrm{d}\tilde{l}_{2})a^{\mathbb{Z}_{2}}+\tilde{l}_{2}{\rm
w}_{2}+\mathbb{Sq}^{2}\tilde{l}_{2}+(\hskip
1.0pt\mathrm{d}\tilde{s}_{1})b^{\mathbb{Z}_{2}}+\tilde{s}_{1}{\rm
w}_{3}+\mathbb{Sq}^{2}\mathbb{Sq}^{1}\tilde{s}_{1}},$

where $\mathbb{Sq}^{n}$ is the generalized Steenrod square introduced in Ref.
Lan  _et al._ , 2019, which acts on cochains and is defined in (91).
$\mathbb{Sq}^{n}$ is equal to Pontryagin square mod 2 when acting on
$n$-cochains. Thus the generalized Steenrod square $\mathbb{Sq}^{n}$ is in
general _not_ the same as the convention Pontryagin square studied in
Pontrjagin (1942); Whitehead (1949).

Using (99), we find

$\displaystyle(\tilde{l}_{2}+\bar{l}_{2}){\rm
w}_{2}+\mathbb{Sq}^{2}(\tilde{l}_{2}+\bar{l}_{2})\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}(\tilde{l}_{2}+\bar{l}_{2}){\rm
w}_{2}+\mathbb{Sq}^{2}\tilde{l}_{2}+\mathbb{Sq}^{2}\bar{l}_{2}$
$\displaystyle\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}(\tilde{l}_{2}+\bar{l}_{2}){\rm
w}_{2}+\mathbb{Sq}^{2}\tilde{l}_{2}+{\rm
w}_{2}\bar{l}_{2}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}\tilde{l}_{2}{\rm w}_{2}+\mathbb{Sq}^{2}\tilde{l}_{2},$
(52)

where we have used $\mathbb{Sq}^{2}\bar{l}_{2}\overset{\scriptscriptstyle
2}{=}\text{Sq}^{2}\bar{l}_{2}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}({\rm w}_{2}+{\rm w}_{1}^{2})\bar{l}_{2}$ and ${\rm
w}_{1}\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}0$ for $B^{4}$.
This allows us to show the invariance of the path integral (51) under
$\tilde{l}_{2}\to\tilde{l}_{2}+\bar{l}_{2}$.

Similarly, using (99) and (93), we find

$\displaystyle\ \ \ \ (\tilde{s}_{1}+\bar{s}_{1}){\rm
w}_{3}+\mathbb{Sq}^{2}\mathbb{Sq}^{1}(\tilde{s}_{1}+\bar{s}_{1})$
$\displaystyle\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}(\tilde{s}_{1}+\bar{s}_{1}){\rm
w}_{3}+\mathbb{Sq}^{2}\mathbb{Sq}^{1}\tilde{s}_{1}+\mathbb{Sq}^{2}\mathbb{Sq}^{1}\bar{s}_{1}$
$\displaystyle\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}\tilde{s}_{1}{\rm w}_{3}+\bar{s}_{1}\mathbb{Sq}^{1}{\rm
w}_{2}+\mathbb{Sq}^{2}\mathbb{Sq}^{1}\tilde{s}_{1}+{\rm
w}_{2}\mathbb{Sq}^{1}\bar{s}_{1}$ (53)

where we have used $\mathbb{Sq}^{1}{\rm w}_{2}\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}{\rm w}_{3}$ when ${\rm
w}_{1}\overset{\scriptscriptstyle 2,\hskip 1.0pt\mathrm{d}}{=}0$. Using (103)
and (79), we have

$\displaystyle\ \ \ \ \bar{s}_{1}\mathbb{Sq}^{1}{\rm w}_{2}+{\rm
w}_{2}\mathbb{Sq}^{1}\bar{s}_{1}\overset{\scriptscriptstyle
2}{=}\bar{s}_{1}\beta_{2}{\rm w}_{2}+{\rm w}_{2}\beta_{2}\bar{s}_{1}$
$\displaystyle\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}(\beta_{2}{\rm w}_{2})\bar{s}_{1}+{\rm
w}_{2}\beta_{2}\bar{s}_{1}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}\beta_{2}({\rm
w}_{2}\bar{s}_{1})\overset{\scriptscriptstyle\hskip 1.0pt\mathrm{d}}{=}0$ (54)

Therefore

$\displaystyle(\tilde{s}_{1}+\bar{s}_{1}){\rm
w}_{3}+\mathbb{Sq}^{2}\mathbb{Sq}^{1}(\tilde{s}_{1}+\bar{s}_{1})\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}\tilde{s}_{1}{\rm
w}_{3}+\mathbb{Sq}^{2}\mathbb{Sq}^{1}\tilde{s}_{1}.$ (55)

This allows us to show the invariance of the path integral (51) under
$\tilde{s}_{1}\to\tilde{s}_{1}+\bar{s}_{1}$. Thus (51) is a correct boundary
theory for ${\rm w}_{2}{\rm w}_{3}$ topological order.

Let us examine the $\tilde{l}_{2}$ terms in the theory. The term $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}\mathbb{Sq}^{2}\tilde{l}_{2}}$, quadratic in
$\tilde{l}_{2}$, gives the $\mathbb{Z}_{2}$-charge (described by $l_{3}=\hskip
1.0pt\mathrm{d}\tilde{l}_{2}$) a Fermi statistics. The accompanying linear
term $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}{\rm w}_{2}\tilde{l}_{2}}$ gives the
$\mathbb{Z}_{2}$-charge a half integer spin, which is associated with the
statement that fermion is related to the trivialization of ${\rm w}_{2}$ (for
details, see Appendix C).

Now let us examine the $\tilde{s}_{1}$ terms. Due to the very similar
structure, we can say that the term $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}\mathbb{Sq}^{2}\mathbb{Sq}^{1}\tilde{s}_{1}}$ gives the
$\mathbb{Z}_{2}$-flux line (described by $s_{2}=\hskip
1.0pt\mathrm{d}\tilde{s}_{1}$) an unusual statistics. We may also say that the
accompanying linear term $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}{\rm w}_{3}\tilde{s}_{1}}$ is associated with the
trivialization of ${\rm w}_{3}$. So the unusual statistics of the
$\mathbb{Z}_{2}$-flux line is associated with the trivialization of ${\rm
w}_{3}$, just like the Fermi statistics of a particle is associated with the
trivialization of ${\rm w}_{2}$.

One way to characterize the unusual statistics of the $\mathbb{Z}_{2}$-flux
line is to notice that $\hskip
1.0pt\mathrm{d}(\mathbb{Sq}^{1}\tilde{s}_{1})=\mathbb{Sq}^{1}\hskip
1.0pt\mathrm{d}\tilde{s}_{1}=\mathbb{Sq}^{1}s_{2}\overset{\scriptscriptstyle
2}{=}\beta_{2}s_{2}$ describes the Poincaré dual of the orientation reversal
line of the worldsheet of the $\mathbb{Z}_{2}$-flux line. So the orientation
reversal line of the worldsheet behaves like a fermion worldline due to the
term $\mathbb{Sq}^{2}(\mathbb{Sq}^{1}\tilde{s}_{1})$.

From the relation $\beta_{2}{\rm w}_{2}\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}{\rm w}_{3}$, we see that the trivialization of ${\rm
w}_{2}$ on the boundary, also implies a trivialization of ${\rm w}_{3}$. The
worldsheet $s_{2}$ of the $\mathbb{Z}_{2}$-flux line couples to a 2-cochain
$b^{\mathbb{Z}_{2}}$ via $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{B^{4}}s_{2}b^{\mathbb{Z}_{2}}}$ (see (III.2.1)). For our
anomalous $\mathbb{Z}_{2}$ gauge theory, the ${\rm w}_{3}$ is trivialized as a
coboundary via the split into a lower-dimensional cochain
$b^{\mathbb{Z}_{2}}$, namely

$\displaystyle\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}_{2}}\overset{\scriptscriptstyle 2}{=}{\rm
w}_{3}.$ (56)

Such a relation can be obtained from (25) by integrating out
$a^{\mathbb{Z}_{2}}$ first. But (56) is not independent from (34), because
${\rm w}_{3}\overset{\scriptscriptstyle\hskip
1.0pt\mathrm{d},2}{=}\text{Sq}^{1}{\rm w}_{2}$.

To summarize, the 4+1D invertible topological order has a boundary described
by
(1) a dynamical $\mathbb{Z}_{2}$ gauge theory with gravitational anomaly.
(2) In the continuum limit, the gauge charge transforms as
$\mathbb{Z}_{2}\rtimes_{{\rm w}_{2}}{\rm SO}(\infty)={\rm Spin}(\infty)$ under
the $\mathbb{Z}_{2}$ gauge transformation and spacetime rotation.
(3) Such a $\mathbb{Z}_{2}$ gauge charge is a fermion in the spin-statistics.
(4) The _orientation-reversal_ worldline on the _unorientable_ worldsheet of
$\mathbb{Z}_{2}$-flux loop is a fermion worldline. But such a fermion
worldline does not carry the $\mathbb{Z}_{2}$ gauge charge.

####  III.2.3 A physical consequence of the fermion-carrying
$\mathbb{Z}_{2}$-flux worldsheet

In a usual anomaly-free $\mathbb{Z}_{2}$-gauge theory, if we proliferate the
$\mathbb{Z}_{2}$-flux worldsheets in spacetime, we will get new gapped state,
which corresponds to a confined phase $\mathbb{Z}_{2}$-gauge theory with no
topological order. Why proliferating the $\mathbb{Z}_{2}$-flux worldsheet can
give rise to a gapped state? This is because the path integral amplitude for
the $\mathbb{Z}_{2}$-flux worldsheets is positive. If the surface tension of
the worldsheet is zero, the equal weight superposition of the worldsheet leads
to a short-range correlated state.

For the anomalous $\mathbb{Z}_{2}$-gauge theory on the boundary of ${\rm
w}_{2}{\rm w}_{3}$ topological order, the path integral amplitude for the
$\mathbb{Z}_{2}$-flux worldsheets is no longer positive. It contains a $\pm$
sign from the braiding of the fermionic world lines carried by the
unorientable worldsheet. In this case, we are not sure that the proliferation
the $\mathbb{Z}_{2}$-flux worldsheets can give rise to a gapped
$\mathbb{Z}_{2}$ confined state. This result is consistent with our
expectation that the anomalous $\mathbb{Z}_{2}$ gauge theory at the boundary
of the ${\rm w}_{2}{\rm w}_{3}$ topological order cannot have a trivial
confined phase.

If we only allow orientable worldsheet of the $\mathbb{Z}_{2}$-flux lines,
then the path integral amplitude for those orientable worldsheet can be all
positive. We can have a phase where the orientable worldsheet proliferate. But
the proliferation of orientable worldsheet give rise to a $\mathrm{U}(1)$
gauge theory, instead of $\mathbb{Z}_{2}$ confined phase. Such a
$\mathrm{U}(1)^{f}$-gauge boundary of the ${\rm w}_{2}{\rm w}_{3}$ topological
order will be discussed in the next subsection.

###  III.3 4d $\mathrm{U}(1)^{f}$-gauge boundary of the ${\rm w}_{2}{\rm
w}_{3}$ topological order

The anomalous $\mathbb{Z}_{2}$-gauge theory is only one possible boundaries of
the ${\rm w}_{2}{\rm w}_{3}$ topological order. In this section, we will
discuss another boundary — a $\mathrm{U}(1)$-gauge theory with gravitational
anomaly. To obtain such an anomalous boundary $\mathrm{U}(1)$-gauge theory, we
first rewrite the topological invariant ${\rm w}_{2}{\rm
w}_{3}\overset{\scriptscriptstyle 2}{=}{\rm w}_{2}\text{Sq}^{1}{\rm w}_{2}$ as

$\displaystyle Z^{\text{top}}(M^{5})$ $\displaystyle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm
w}_{2}\text{Sq}^{1}{\rm w}_{2}}$ $\displaystyle=\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\pi\int_{M^{5}}{\rm w}_{2}\beta_{2}{\rm
w}_{2}}=\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}\frac{1}{2}\hskip 1.0pt\mathrm{d}{\rm w}_{2}},$
(57)

where we have used $\text{Sq}^{1}{\rm w}_{2}\overset{\scriptscriptstyle
2,\hskip 1.0pt\mathrm{d}}{=}\beta_{2}{\rm w}_{2}$ (see (103)) and
$\beta_{2}{\rm w}_{2}=\frac{1}{2}\hskip 1.0pt\mathrm{d}{\rm w}_{2}$ (see
(79)). In (III.3), both $({\rm w}_{2})(\beta_{2}{\rm w}_{2})$ and $({\rm
w}_{2})(\frac{1}{2}\hskip 1.0pt\mathrm{d}{\rm w}_{2})$ pair between the
$\mathbb{Z}_{2}$-valued ${\rm w}_{2}$ and the $\mathbb{Z}$-valued
$(\beta_{2}{\rm w}_{2})$ or $(\frac{1}{2}\hskip 1.0pt\mathrm{d}{\rm w}_{2})$,
which altogether can be well-defined in the $\mathbb{Z}_{2}$ value.

We find that if $M^{5}$ has a ${\rm Spin}^{c}$ structure, then
$\text{Sq}^{1}{\rm
w}_{2}=\text{Sq}^{1}(c_{1}\;\text{mod}\;2)\overset{\scriptscriptstyle 2,\hskip
1.0pt\mathrm{d}}{=}0$, thus $\beta_{2}{\rm
w}_{2}\overset{\scriptscriptstyle\hskip 1.0pt\mathrm{d}}{=}0$ and
$Z^{\text{top}}(M^{5})=1$ (See a proof in Appendix D.1’s Remark 3). Thus, the
${\rm w}_{2}{\rm w}_{3}$ is _not_ a cobordism invariant for the ${\rm
Spin}^{c}$ structure. In this case, the ${\rm SO}(\infty)$ connection on
$M^{5}$ can be lifted into a $\mathrm{U}(1)^{f}\rtimes_{\frac{1}{2}{\rm
w}_{2}}{\rm SO}(\infty)$ connection. The $\mathrm{U}(1)^{f}$ implies that

$\mathrm{U}(1)^{f}\supset\mathbb{Z}_{2}^{f}$

contains the fermion parity as a normal subgroup. Here
$\mathrm{U}(1)^{f}\rtimes_{\frac{1}{2}{\rm w}_{2}}{\rm SO}(\infty)$ is the
$\mathrm{U}(1)^{f}$ extension of ${\rm SO}(\infty)$ characterized by
$\frac{1}{2}{\rm w}_{2}\in H^{2}(\mathrm{B}{\rm
SO}(\infty);\mathbb{R}/\mathbb{Z})$.

Such a group extension to a total group ${\rm Spin}^{c}\equiv{\rm
Spin}\times_{\mathbb{Z}_{2}}\mathrm{U}(1)^{f}\equiv\frac{{\rm
Spin}\times\mathrm{U}(1)^{f}}{{\mathbb{Z}_{2}^{f}}}$ via the short exact
sequence

$\displaystyle 1\to\mathrm{U}(1)^{f}\to{\rm Spin}^{c}(5)\to{\rm SO}(5)\to 1$
(58)

implies the ${\rm w}_{2}{\rm w}_{3}$ in SO is trivialized in ${\rm Spin}^{c}$.
The ${\rm Spin}^{c}$ structure, which contains the emergent
$\mathrm{U}(1)^{f}\supset\mathbb{Z}_{2}^{f}$ on the boundary, is also called
the emergent dynamical ${\rm Spin}^{c}$ structure Wang  _et al._ (2019).

The above discussions also implies that the ${\rm w}_{2}{\rm w}_{3}$
topological order has another boundary described by a $\mathrm{U}(1)^{f}$
gauge theory with gravitational anomaly. To write down such a
$\mathrm{U}(1)^{f}$ gauge theory, we start with a 5d branch-independent
bosonic model that realize the ${\rm w}_{2}{\rm w}_{3}$ invertible topological
order:

$\displaystyle Z=\sum_{a^{\mathbb{R}/\mathbb{Z}^{f}},b^{\mathbb{Z}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}({\rm w}_{2}+2\hskip
1.0pt\mathrm{d}a^{\mathbb{R}/\mathbb{Z}^{f}})(\beta_{2}{\rm w}_{2}+\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}})}.$ (59)

The brunching-independence is ensured by the invariance of the above path
integral under the following dynamical gauge transformations ($\alpha_{0}$)
and background gauge transformations ($v_{1}$ and $v_{2}$):

$\displaystyle a^{{\mathbb{R}/\mathbb{Z}}^{f}}$ $\displaystyle\to
a^{{\mathbb{R}/\mathbb{Z}}^{f}}+\hskip
1.0pt\mathrm{d}\alpha_{0}+\frac{1}{2}v_{1},$ $\displaystyle b^{\mathbb{Z}}$
$\displaystyle\to b^{\mathbb{Z}}+v_{2},$ (60) $\displaystyle{\rm w}_{2}$
$\displaystyle\to{\rm w}_{2}-\hskip 1.0pt\mathrm{d}v_{1}-2v_{2},$
$\displaystyle\beta_{2}{\rm w}_{2}$ $\displaystyle\to\beta_{2}{\rm
w}_{2}-{\beta_{2}\hskip 1.0pt\mathrm{d}v_{1}}-\hskip
1.0pt\mathrm{d}v_{2}=\beta_{2}{\rm w}_{2}-\hskip 1.0pt\mathrm{d}v_{2}.$

The $v_{1}\in C^{1}(M^{5};\mathbb{Z})$ and $v_{2}\in C^{2}(M^{5};\mathbb{Z})$
are $\mathbb{Z}$-valued 1- and 2-cochains and $\alpha_{0}\in
C^{0}(B^{4};{\mathbb{R}/\mathbb{Z}})$ is a ${\mathbb{R}/\mathbb{Z}}$-valued
function (i.e. 0-cochain). Note that $\beta_{2}\hskip 1.0pt\mathrm{d}v_{1}=0$
because $\beta_{2}=\frac{1}{2}\hskip 1.0pt\mathrm{d}$, while
$\frac{1}{2}v_{1}\in C^{1}(M^{5},{\mathbb{R}/\mathbb{Z}})$, and $\hskip
1.0pt\mathrm{d}\hskip 1.0pt\mathrm{d}=\hskip 1.0pt\mathrm{d}^{2}=0$ on
$C^{1}(M^{5},{\mathbb{R}/\mathbb{Z}})$. Here the gauge transformation of
$a^{{\mathbb{R}/\mathbb{Z}}^{f}}$ is related to that of
$\frac{1}{2}a^{\mathbb{Z}_{2}}$ where the gauge transformation of
$a^{\mathbb{Z}_{2}}$ is in (III.2.1).

When $M^{5}$ has boundaries and after integrating out
$a^{\mathbb{R}/\mathbb{Z}^{f}},b^{\mathbb{Z}}$ in the bulk, we obtain the
following boundary theory

$\displaystyle
Z=\hskip-8.99994pt\sum_{a^{\mathbb{R}/\mathbb{Z}^{f}},b^{\mathbb{Z}}}\hskip-6.00006pt$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}\beta_{2}{\rm w}_{2}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}a^{{\mathbb{R}/\mathbb{Z}}^{f}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}}+a^{{\mathbb{R}/\mathbb{Z}}^{f}}\beta_{2}{\rm
w}_{2}+\frac{1}{2}{\rm w}_{2}b^{\mathbb{Z}}}$ $\displaystyle\hskip
1.0pt\mathrm{e}^{-\frac{1}{2}g\int_{B^{4}}|b+\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}+\frac{1}{2}{\rm w}_{2}|^{2}}$
(61)

where $a^{{\mathbb{R}/\mathbb{Z}}^{f}}\in
C^{1}(B^{4};{\mathbb{R}/\mathbb{Z}})$ is a ${\mathbb{R}/\mathbb{Z}}$-valued
1-cochain777Note that the isomorphism ${\mathbb{R}/\mathbb{Z}}=\mathrm{U}(1)$,
however we use ${\mathbb{R}/\mathbb{Z}}$ to emphasize the group operation is
addition as in ${\mathbb{R}/\mathbb{Z}}$, instead of multiplication in
$\mathrm{U}(1)$. Also we denote
${\mathbb{R}/\mathbb{Z}}^{f}=\mathrm{U}(1)^{f}\supset\mathbb{Z}_{2}^{f}$ to
include the fermion parity normal subgroup. and $b^{\mathbb{Z}}\in
C^{2}(B^{4};\mathbb{Z})$ is a $\mathbb{Z}$-valued 2-cochain.888First let us
clarify why ${b^{\mathbb{Z}}\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}}+({\beta_{2}}{\rm
w}_{2})a^{{\mathbb{R}/\mathbb{Z}}^{f}}+b^{\mathbb{Z}}(\frac{1}{2}{\rm w}_{2})$
is well-defined in $\mathbb{R}/\mathbb{Z}$-valued, for the cup product between
a $\mathbb{Z}$-valued cohomology class (e.g., here $b^{\mathbb{Z}}$,
${\beta_{2}}{\rm w}_{2}$, etc.) and a ${\mathbb{R}/\mathbb{Z}}$-valued
cohomology class (e.g., here $\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}$,
$a^{{\mathbb{R}/\mathbb{Z}}^{f}}$, $\frac{1}{2}{\rm w}_{2}$, etc.).
Generally, for $z\in\mathbb{Z}$ and the equivalence class
$[{x}]\in\mathbb{R}/\mathbb{Z}$ where the representative $x\in\mathbb{R}$, we
have that the definition $z[{x}]\equiv[{zx}]\in\mathbb{R}/\mathbb{Z}$ is well-
defined since if $[x]=[y]\in\mathbb{R}/\mathbb{Z}$, then $x-y\in\mathbb{Z}$
and $z(x-y)\in\mathbb{Z}$. So $[zx]=[zy]\in\mathbb{R}/\mathbb{Z}$, thus we
prove that $z[x]\in\mathbb{R}/\mathbb{Z}$ is well-defined. Thus we prove that
${b^{\mathbb{Z}}\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}}+({\beta_{2}}{\rm
w}_{2})a^{{\mathbb{R}/\mathbb{Z}}^{f}}+b^{\mathbb{Z}}(\frac{1}{2}{\rm w}_{2})$
is well-defined in $\mathbb{R}/\mathbb{Z}$-valued.  We add a gauge invariant
term $\hskip 1.0pt\mathrm{e}^{-\frac{1}{2}g\int_{B^{4}}|b+\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}+\frac{1}{2}{\rm w}_{2}|^{2}}$,
which will produce the Maxwell term after we integrating out the
$b^{\mathbb{Z}}$ field. We find that a small $g$ leads to a semiclassical
$\mathrm{U}(1)^{f}$ gauge theory. Thus, the above describes a
$\mathrm{U}(1)^{f}$ gauge theory,999We can motivate better about the 4d action
$\int_{B^{4}}{b^{\mathbb{Z}}\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}}$. Schematically, without any
other source or operator insertions in the path integral, we have the equation
of motion $\hskip 1.0pt\mathrm{d}{b^{\mathbb{Z}}}=0$. Although the cocycle
(closed) is not necessarily coboundary (exact), locally we can write
${b^{\mathbb{Z}}}=\hskip 1.0pt\mathrm{d}{\rm
v}^{{\mathbb{R}/\mathbb{Z}}^{f}}$, since the $b^{\mathbb{Z}}\in
C^{2}(B^{4};\mathbb{Z})$ has the integer quantized electric flux, then ${\rm
v}^{{\mathbb{R}/\mathbb{Z}}^{f}}\in
C^{1}(B^{4};{{\mathbb{R}/\mathbb{Z}}^{f}})$ is the dual gauge field (the ’t
Hooft magnetic gauge field). Since the pure abelian U(1) gauge theory has the
action $\hskip
1.0pt\mathrm{d}{a}^{{\mathbb{R}/\mathbb{Z}}^{f}}\wedge\star\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}=\star\hskip 1.0pt\mathrm{d}{\rm
v}^{{\mathbb{R}/\mathbb{Z}}^{f}}\wedge\hskip 1.0pt\mathrm{d}{\rm
v}^{{\mathbb{R}/\mathbb{Z}}^{f}}$ with the U(1) gauge coupling suppressed, we
can also regard $b^{\mathbb{Z}}$ as the integer quantized electric flux
$\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int\hskip
1.0pt\mathrm{d}{\rm
v}^{{\mathbb{R}/\mathbb{Z}}^{f}}=\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int\star\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}\in\mathbb{Z}$ on a closed
2-cycle, while $\star b^{\mathbb{Z}}$ as the integer quantized magnetic flux
$\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int\star\hskip
1.0pt\mathrm{d}{\rm
v}^{{\mathbb{R}/\mathbb{Z}}^{f}}=\displaystyle\int\mkern-7.0mu\begin{picture}(0.0,3.0)\put(0.0,3.0){\oval(10.0,8.0)}\end{picture}\mkern-8.0mu\int\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}\in\mathbb{Z}$ on a closed
2-cycle. Thus in this special case, we may also treat
$\int_{B^{4}}{b^{\mathbb{Z}}\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}}$ as $\int_{B^{4}}\hskip
1.0pt\mathrm{d}{a}^{{\mathbb{R}/\mathbb{Z}}^{f}}\wedge\star\hskip
1.0pt\mathrm{d}{a}^{{\mathbb{R}/\mathbb{Z}}^{f}}=\int_{B^{4}}\star\hskip
1.0pt\mathrm{d}{\rm v}^{{\mathbb{R}/\mathbb{Z}}^{f}}\wedge\hskip
1.0pt\mathrm{d}{\rm v}^{{\mathbb{R}/\mathbb{Z}}^{f}}$. By the Maxwell
equations, $\hskip 1.0pt\mathrm{d}\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}=\hskip
1.0pt\mathrm{d}\star\hskip 1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}=0$,
so $\hskip 1.0pt\mathrm{d}a$ is a harmonic form (closed and co-closed) and the
Hodge dual $\star\hskip 1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}=\hskip
1.0pt\mathrm{d}{\rm v}^{{\mathbb{R}/\mathbb{Z}}^{f}}$ is also a harmonic form.
By the Hodge theorem, each cohomology class has a unique harmonic
representative. So Hodge star is an isomorphism from the harmonic
representatives of $H^{k}_{DR}(M)$ to the the harmonic representatives of
$H^{n-k}_{DR}(M)$, here we can take $M=B^{4}$. Poincaré duality says that
$\langle\alpha,\beta\rangle=\int_{M}\alpha\wedge\beta$ is a perfect pairing
between $H^{k}_{DR}(M)$ and $H^{n-k}_{DR}(M)$. So
$\int_{M}\alpha\wedge\star\alpha>0$ for any nonzero $\alpha$ and we can define
$\int_{M}|\alpha|^{2}$ as $\int_{M}\alpha\wedge\star\alpha$. The sum of
$|\alpha|^{2}$ on a 2-simplex over the spacetime simplex is also the action
$\int_{M}|\alpha|^{2}$.
For a cohomology class $\upalpha$, the integral $\int_{M}|\upalpha|^{2}$ is
defined to be $\int_{M}|\alpha|^{2}$ for any cocycle representative $\alpha$
of the cohomology class $\upalpha$. In particular, we can choose the harmonic
representative. For a harmonic form $\alpha$, we have
$\int_{M}|\alpha|^{2}=\int_{M}\alpha\wedge\star\alpha$ since
$\alpha\wedge\star\alpha=|\alpha|^{2}$.  with a gravitational anomaly of ${\rm
w}_{2}{\rm w}_{3}$.

In the presence of the 1d worldline of $\mathrm{U}(1)^{f}$-electric charge
(i.e. Wilson line) and the worldline of $\mathrm{U}(1)^{f}$-magnetic monopole
(i.e. ’t Hooft line) on the boundary, the boundary theory becomes

$\displaystyle
Z=\hskip-8.99994pt\sum_{a^{\mathbb{R}/\mathbb{Z}^{f}},b^{\mathbb{Z}}}\hskip-6.00006pt$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}{\rm w}_{2}\beta_{2}{\rm w}_{2}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}a^{{\mathbb{R}/\mathbb{Z}}^{f}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}}+a^{{\mathbb{R}/\mathbb{Z}}^{f}}\beta_{2}{\rm
w}_{2}+\frac{1}{2}{\rm w}_{2}b^{\mathbb{Z}}}$ $\displaystyle\hskip
1.0pt\mathrm{e}^{-\frac{1}{2}g\int_{B^{4}}|b+\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}+\frac{1}{2}{\rm w}_{2}|^{2}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}l_{3}^{e\;\mathbb{Z}}a^{\mathbb{R}/\mathbb{Z}^{f}}+\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}b^{\mathbb{Z}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}^{e\;\mathbb{Z}}+l_{3}^{e\;\mathbb{Z}}{\rm
w}_{2}+\text{Sq}^{2}{\hskip
1.0pt\mathrm{d}}\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}+\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}{\hskip
1.0pt\mathrm{d}}{\rm w}_{2}}$ (62)

where $l_{3}^{e\;\mathbb{Z}}$ is a $\mathbb{Z}$-valued 3-cocycle – the
Poincaré dual of the Wilson line $\hskip 1.0pt\mathrm{e}^{\hskip
1.0pt\mathrm{i}\hskip 1.0pt\oint_{S^{1}}a^{\mathbb{R}/\mathbb{Z}^{f}}}$ (the
worldline of the $\mathrm{U}(1)^{f}$ electric charge), and
$\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}$ is a $\mathbb{R}/\mathbb{Z}$-valued
2-cocycle, i.e. it satisfies

$\displaystyle\hskip
1.0pt\mathrm{d}\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}=l^{m\;\mathbb{Z}}_{3}.$
(63)

Here $l_{3}^{m}$ is a $\mathbb{Z}$-valued cocycle, which is the Poincaré dual
of the ’t Hooft line $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\oint_{S^{1}}v^{\mathbb{R}/\mathbb{Z}^{f}}}$ (the worldline of
$\mathrm{U}(1)^{f}$ magnetic monopole) written in terms of the dual gauge
field.101010Note that $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}l_{3}^{e\;\mathbb{Z}}a^{\mathbb{R}/\mathbb{Z}^{f}}+\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}b^{\mathbb{Z}}}$
in (III.3) can be schematically rewritten as  $\displaystyle\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}l_{3}^{e\;\mathbb{Z}}a^{\mathbb{R}/\mathbb{Z}^{f}}+\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}\hskip
1.0pt\mathrm{d}{\rm v}^{{\mathbb{R}/\mathbb{Z}}^{f}}}$ $\displaystyle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}l_{3}^{e\;\mathbb{Z}}a^{\mathbb{R}/\mathbb{Z}^{f}}+(\hskip
1.0pt\mathrm{d}\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}){\rm
v}^{{\mathbb{R}/\mathbb{Z}}^{f}}}$ $\displaystyle=\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}l_{3}^{e\;\mathbb{Z}}a^{\mathbb{R}/\mathbb{Z}^{f}}+l_{3}^{m\;\mathbb{Z}}{\rm
v}^{{\mathbb{R}/\mathbb{Z}}^{f}}}$ (64) via $b^{\mathbb{Z}}=\hskip
1.0pt\mathrm{d}{\rm v}^{{\mathbb{R}/\mathbb{Z}}^{f}}=\star\hskip
1.0pt\mathrm{d}a^{{\mathbb{R}/\mathbb{Z}}^{f}}$. Hence we can identify the
Poincaré dual of the Wilson line ($a^{\mathbb{R}/\mathbb{Z}^{f}}$) as
$l_{3}^{e\;\mathbb{Z}}$, while the Poincaré dual of the ’t Hooft line
($v^{\mathbb{R}/\mathbb{Z}^{f}}$) as $l_{3}^{m\;\mathbb{Z}}$.  We note that
$\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}$ (of $\mathbb{R}/\mathbb{Z}$ or U(1)
value) in (III.3) is related to $\frac{1}{2}s_{2}$ where $s_{2}$ (of
$\mathbb{Z}_{2}$ or $\mathbb{Z}$ value) appears in (III.2.1).111111If we map
$\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}\mapsto\frac{1}{2}s_{2}$, both are
$\mathbb{R}/\mathbb{Z}$-valued, then we can map between the expressions in
(III.3) and (III.2.1):  $\text{Sq}^{2}{\hskip
1.0pt\mathrm{d}}\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}+\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}{\hskip
1.0pt\mathrm{d}}{\rm w}_{2}\mapsto\text{Sq}^{2}{\hskip
1.0pt\mathrm{d}}\frac{1}{2}s_{2}+\frac{1}{2}s_{2}{\hskip 1.0pt\mathrm{d}}{\rm
w}_{2}\\\ =\text{Sq}^{2}\text{Sq}^{1}s_{2}+s_{2}\text{Sq}^{1}{\rm
w}_{2}=\text{Sq}^{2}(\beta_{2}s_{2})+s_{2}{\rm w}_{3}$ (65)

The theory (III.3) is invariant under the gauge transformation (III.3). Also,
the term

$\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}^{e\;\mathbb{Z}}+l_{3}^{e\;\mathbb{Z}}{\rm
w}_{2}+\text{Sq}^{2}\hskip
1.0pt\mathrm{d}\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}+\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}\hskip
1.0pt\mathrm{d}{\rm w}_{2}}$

only depends on the fields on the boundary $B^{4}=\partial M^{5}$. Based on
the relation (63), this term can be also read schematically as

$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}(\text{Sq}^{2}+{\rm
w}_{2})(l_{3}^{e\;\mathbb{Z}}+l_{3}^{m\;\mathbb{Z}})}.$ (66)

This term makes the $\mathrm{U}(1)^{f}$ electric charge (Wilson line $\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\oint_{S^{1}}a^{\mathbb{R}/\mathbb{Z}^{f}}}$ as Poincaré dual of
$l_{3}^{e\;\mathbb{Z}}$) and the $\mathrm{U}(1)^{f}$ magnetic monopole (’t
Hooft line $\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\oint_{S^{1}}v^{\mathbb{R}/\mathbb{Z}^{f}}}$ as Poincaré dual of
$l_{3}^{m\;\mathbb{Z}}$) to be fermions, via a higher dimensional bosonization
Wen (2017); Kapustin and Thorngren (2017); Lan  _et al._ (2019). Thus, we show
that the boundary of ${\rm w}_{2}{\rm w}_{3}$ topological order has the
$\mathrm{U}(1)^{f}$ electric charge has the fermionic statistics, the
$\mathrm{U}(1)^{f}$ magnetic monopole has the fermionic statistics, and their
bound object $\mathrm{U}(1)^{f}$ dyon also has the fermionic statistics. This
is known as the all fermion quantum electrodynamics (QED4).

Before ending this section, we like to write down two bosonic theories with no
gravitational anomaly. The first one is given by the following path integral

$\displaystyle Z=\sum_{a^{\mathbb{R}/\mathbb{Z}^{f}},b^{\mathbb{Z}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}a^{{\mathbb{R}/\mathbb{Z}}^{f}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}}-\frac{1}{2}g\int_{B^{4}}b\star b}$ (67)
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}l_{3}^{e\;\mathbb{Z}}a^{\mathbb{R}/\mathbb{Z}^{f}}+\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}b^{\mathbb{Z}}}$

which is invariant under the following gauge transformation

$\displaystyle a^{{\mathbb{R}/\mathbb{Z}}^{f}}\to
a^{{\mathbb{R}/\mathbb{Z}}^{f}}+\hskip 1.0pt\mathrm{d}\alpha_{0},\ \ \ \
b^{\mathbb{Z}}\to b^{\mathbb{Z}}.$ (68)

When $g$ is small, the above bosonic model realizes a U(1) gauge theory at low
energies, where both electric charge described by $l_{3}^{e\;\mathbb{Z}}$ and
magnetic charge described by $\hskip
1.0pt\mathrm{d}\eta_{2}^{m\;{\mathbb{R}/\mathbb{Z}}}$ are bosons.

The second bosonic model is given by

$\displaystyle Z=$
$\displaystyle\hskip-8.99994pt\sum_{a^{\mathbb{R}/\mathbb{Z}^{f}},b^{\mathbb{Z}}}\hskip-6.00006pt\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}a^{{\mathbb{R}/\mathbb{Z}}^{f}}\hskip
1.0pt\mathrm{d}b^{\mathbb{Z}}+\frac{1}{2}{\rm w}_{2}b^{\mathbb{Z}}}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{-\frac{1}{2}g\int_{B^{4}}(b+\hskip
1.0pt\mathrm{d}a^{\mathbb{R}/\mathbb{Z}^{f}}+\frac{1}{2}{\rm
w}_{2})\star(b+\hskip
1.0pt\mathrm{d}a^{\mathbb{R}/\mathbb{Z}^{f}}+\frac{1}{2}{\rm w}_{2})}$
$\displaystyle\hskip 1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt2\pi\int_{B^{4}}l_{3}^{e\;\mathbb{Z}}a^{\mathbb{R}/\mathbb{Z}^{f}}+\eta_{2}^{\mathbb{R}/\mathbb{Z}^{f}}b^{\mathbb{Z}}}\hskip
1.0pt\mathrm{e}^{\hskip 1.0pt\mathrm{i}\hskip
1.0pt\pi\int_{M^{5}}\text{Sq}^{2}l_{3}^{e\;\mathbb{Z}}+l_{3}^{e\;\mathbb{Z}}{\rm
w}_{2}}$ (69)

which is invariant under the following gauge transformation

$\displaystyle a^{{\mathbb{R}/\mathbb{Z}}^{f}}$ $\displaystyle\to
a^{{\mathbb{R}/\mathbb{Z}}^{f}}+\hskip
1.0pt\mathrm{d}\alpha_{0}+\frac{1}{2}v_{1},\ \ \ b^{\mathbb{Z}}\to
b^{\mathbb{Z}},$ $\displaystyle{\rm w}_{2}$ $\displaystyle\to{\rm
w}_{2}-\hskip 1.0pt\mathrm{d}v_{1}.$ (70)

When $g$ is small, the above bosonic model realizes a U(1) gauge theory at low
energies, where the electric charge described by $l_{3}^{e\;\mathbb{Z}}$ is a
fermion, and the magnetic charge described by $\hskip
1.0pt\mathrm{d}\eta_{2}^{m\;{\mathbb{R}/\mathbb{Z}}}$ is a boson.

##  IV 4d Boundary of SU(2) or Other Spin($n$) Internal Symmetric Theories

So far we have formulated two kinds of 4d boundary theories of 5d ${\rm
w}_{2}{\rm w}_{3}$: the 4d $\mathbb{Z}_{2}$ gauge theories (where the local
$\mathbb{Z}_{2}$ gauge field is more precisely the dynamical Spin structure
summed over in the path integral) and the 4d $\mathrm{U}(1)$ gauge theories
(where the local $\mathrm{U}(1)$ gauge field is more precisely the dynamical
$\mathrm{U}(1)$ gauge connection of Spinc structure summed over in the path
integral). We also have provided these boundary gauge theory constructions as
the trivialization of ${\rm w}_{2}{\rm w}_{3}$ of SO structure via its
pullback $p$ to the $p^{*}{\rm w}_{2}{\rm w}_{3}=0$ in Spin or Spinc
structures (Namely, $1\to\mathbb{Z}_{2}\to{\rm Spin}\overset{p}{\to}{\rm
SO}\to 1$ and $1\to\mathrm{U}(1)\to{\rm Spin}^{c}\overset{p}{\to}{\rm SO}\to
1$ see further details in Appendix D). The ${\rm Spin}^{c}={\rm
Spin}\times_{\mathbb{Z}_{2}}\mathrm{U}(1)$ structure constrains that the
fermions carry odd U(1) charge while the bosons carry even U(1) charge. In
this section, we consider the ${\rm Spin}^{h}={\rm
Spin}\times_{\mathbb{Z}_{2}}{\rm SU}(2)$ structure such that the fermions are
in even dimensional representation (e.g., ${\bf 2},{\bf 4},\dots$; or isospin
$\frac{1}{2},\frac{3}{2},\dots$) of SU(2) while the bosons are in odd
dimensional representation (e.g., ${\bf 1},{\bf 3},\dots$; or isospin
$0,1,\dots$) of SU(2). More generally, we can consider the ${\rm
Spin}\times_{\mathbb{Z}_{2}}{\rm Spin}(n)$ structure. For example, for $n=10$,
fermions are in the spinor representation of Spin(10) (e.g., ${\bf 16},\dots$)
while bosons are in other tensor representation of Spin(10) (e.g., ${\bf
10},\dots$). We list down the cobordism invariants from $\mathrm{TP}_{5}({{\rm
Spin}\times_{\mathbb{Z}_{2}}{\rm Spin}(n)})$ in Appendix D. In this section,
we summarize and enumerate other 4d boundary theories of 5d ${\rm w}_{2}{\rm
w}_{3}$, based on the ${\rm Spin}\times_{\mathbb{Z}_{2}}{\rm Spin}(n)$
structure construction, in particular $n=3$ and $10$.

  1. 1.

Boundary with the SU(2) and ${\rm Spin}^{h}={\rm
Spin}\times_{\mathbb{Z}_{2}}{\rm SU}(2)={\rm Spin}\times_{\mathbb{Z}_{2}}{\rm
Spin}(3)$ structures:

    1. (a)

When the SU(2) is a global symmetry,

$\bullet$ An odd number of the fundamental 2-dimensional representation (Rep)
of SU(2) of the spacetime Weyl spinor $\psi$ cannot be gapped by quadratic
mass term while preserving the Lorentz and SU(2) symmetries. This is due to
that the only quadratic mass term
$\epsilon^{\alpha\beta}\epsilon^{ij}\psi_{\alpha i}\psi_{\beta j}=0$ (where
$\alpha,\beta$ are the Lorentz indices and ${i,j}\in\\{1,2\\}$ are SU(2)
indices; we take both the singlet 1 out of ${\bf 2}\otimes{\bf 2}={\bf
1}\oplus{\bf 3}$) vanish due to the fermi statistics. This suggests a possible
anomaly — another hint is that the fermion spectrum under the SU(2) gauge
bundle over $S^{4}$ with an instanton number 1 background gives an odd number
of fermion zero mode. More generally, an odd number of ${\bf 4}r+{\bf 2}$ Rep
of SU(2) Weyl spinor has the same anomaly known as Witten anomaly as a ’t
Hooft anomaly of the ${\rm Spin}^{h}$-symmetry (See Table 1). But these 4d
theories live on the boundary of another 5d cobordism invariant, known as
$\tilde{\eta}\mathrm{PD}(c_{2}(V_{{\rm SU}(2)}))$.121212The $\tilde{\eta}$ is
a mod 2 index of 1d Dirac operator from $\mathrm{TP}_{1}({\rm
Spin})=\mathbb{Z}_{2}$ or $\Omega_{1}^{{\rm Spin}}=\mathbb{Z}_{2}$. A 1d
manifold generator for the cobordism invariant $\tilde{\eta}$ is a 1d $S^{1}$
for fermions with periodic boundary condition, so called the Ramond circle. A
4d manifold generator for the $c_{2}(V_{{\rm SU}(2)})$ is the nontrivial SU(2)
bundle over the $S^{4}$, such that the instanton number is 1. The PD is
Poincaré dual.  These 4d theories do not live on the boundary of the 5d ${\rm
w}_{2}{\rm w}_{3}$.

$\bullet$ An odd number of the 4-dimensional representation (Rep) of SU(2) of
the spacetime Weyl spinor $\Psi$ also cannot be gapped by quadratic mass term
while preserving the Lorentz and SU(2) symmetries. The singlet of both Lorentz
and SU(2) symmetry requires any quadratic mass term vanishes:
$\epsilon^{\alpha\beta}C^{IJ}\Psi_{\alpha I}\Psi_{\beta J}=0$.131313Here
${I,J}\in\\{1,2,3,4\\}$ are SU(2) indices which correspond to
isospin-$\frac{3}{2}$ indices
$\\{\frac{3}{2},\frac{1}{2},-\frac{1}{2},-\frac{3}{2}\\}$. We take the singlet
1 out of the tensor product of 4 of SU(2): ${\bf 4}\otimes{\bf 4}={\bf
1}\oplus{\bf 3}\oplus{\bf 5}\oplus{\bf 7}$. Based on the Clebsch-Gordan
coefficients
$\frac{1}{2}(|\frac{3}{2},\frac{3}{2}\rangle|\frac{3}{2},-\frac{3}{2}\rangle-|\frac{3}{2},\frac{1}{2}\rangle|\frac{3}{2},-\frac{1}{2}\rangle+|\frac{3}{2},-\frac{1}{2}\rangle|\frac{3}{2},\frac{1}{2}\rangle-|\frac{3}{2},-\frac{3}{2}\rangle|\frac{3}{2},\frac{3}{2}\rangle)=|0,0\rangle$,
we have $C^{ij}\Psi_{\alpha i}\Psi_{\beta j}=\frac{1}{2}\big{(}(\Psi_{\alpha
1}\Psi_{\beta 4}-\Psi_{\alpha 4}\Psi_{\beta 1})-(\Psi_{\alpha 2}\Psi_{\beta
3}-\Psi_{\alpha 3}\Psi_{\beta 2})\big{)}$. Since $\epsilon^{\alpha\beta}$ and
$C^{ij}$ are both anti-symmetric, while all $\Psi$ are Grassmannian variables
due to fermi statistics, thus $\epsilon^{\alpha\beta}C^{ij}\Psi_{\alpha
i}\Psi_{\beta j}=0$. Ref. Wang  _et al._ , 2019 shows that on a 4d non-spin
manifold, the complex projective space ${\mathbb{CP}}^{2}$, with an
appropriate large diffeomorphism by complex conjugation the
${\mathbb{CP}}^{2}$ coordinates $z_{j}\to\bar{z}_{j}$ and an appropriate SU(2)
large gauge transformation, we can construct a mapping torus 5d Dold manifold
${\mathbb{CP}}^{2}\rtimes S^{1}$ such that the large gauge-diffeomorphism is
transformed along the fifth dimension. Moreover, together with the SU(2)
bundle, the whole theory is compatible with the ${\rm Spin}^{h}$ structure.
But the path integral gets a $(-1)$ sign under this large gauge-diffeomorphism
transformation. This odd $(-1)$ non-invariance shows the new SU(2) anomaly.
More generally, an odd number of ${\bf 8}r+{\bf 4}$ Rep of SU(2) Weyl spinor
has the same anomaly known as the new SU(2) anomaly as a ’t Hooft anomaly of
the ${\rm Spin}^{h}$-symmetry (See Table 1).

$\begin{array}[]{c |c c c c c c c c c |c c c c}\hline\cr\text{SU(2)
isospin}&0&\frac{1}{2}&1&\frac{3}{2}&2&\frac{5}{2}&3&\frac{7}{2}&\text{mod
}4&2r+\frac{1}{2}&4r+\frac{3}{2}&\text{mod }4\\\ \hline\cr\text{SU(2) Rep {\bf
R} (dim)}&{\bf 1}&{\bf 2}&{\bf 3}&{\bf 4}&{\bf 5}&{\bf 6}&{\bf 7}&{\bf
8}&\text{mod }8&{\bf 4}r+{\bf 2}&{\bf 8}r+{\bf 4}&\text{mod }8\\\
\hline\cr\text{Witten SU(2)
anomaly}&&\checkmark&&&&\checkmark&&&&\checkmark&&\\\ \hline\cr\text{New SU(2)
anomaly}&&&&\checkmark&&&&&&&\checkmark&&\\\ \hline\cr\end{array}$

Table 1: For a single spacetime Weyl spinor ${\bf 2}_{L}$ in 4d, it has a
Witten SU(2) anomaly if the spacetime Weyl spinor is also the ${\bf 4}r+{\bf
2}$-dimensional representation (Rep) of SU(2) (or isospin $2r+\frac{1}{2}$),
for some nonnegative integer $r$. It has a new SU(2) anomaly if the spacetime
Weyl spinor is also the ${\bf 8}r+{\bf 4}$-dimensional Rep of SU(2) (or
isospin $4r+\frac{3}{2}$), for some nonnegative integer $r$. The checkmark
$\checkmark$ means the fermion theory has the corresponding anomaly. These
SU(2) anomalies can be interpreted as either a ’t Hooft anomaly of global
symmetry (if the SU(2) is global symmetry not gauged), or dynamical gauge
anomaly (if the SU(2) is dynamically gauged).

    2. (b)

When the SU(2) is dynamically gauged,

$\bullet$ Witten anomaly gives a dynamical gauge anomaly constraint such that
an odd number of ${\bf 4}r+{\bf 2}$ Rep of SU(2) Weyl spinor coupled to
dynamical SU(2) gauge fields are ill-defined. It is not physically sensible to
study its gauge dynamics.

$\bullet$ The gauge theories with new SU(2) anomalies are still well-defined
theories with well-defined gauge dynamics on Spin manifolds, because ${\rm
w}_{2}=0$ means no ${\rm w}_{2}{\rm w}_{3}$ anomaly on the Spin manifolds.
However, their gauge dynamics become ill-defined in 4d on non-Spin manifolds.

    3. (c)

Let us discuss further about the SU(2) theory with a ${\bf 4}$ Rep of SU(2)
Weyl spinor.

$\bullet$ Explicit symmetry breaking: If we are allowed to break this SU(2)
theory with the new SU(2) anomaly, for example by choosing the quadratic
fermion mass term via the $\epsilon^{\alpha\beta}C^{\prime IJ}\Psi_{\alpha
I}\Psi_{\beta J}$ such that the pairing $C^{\prime IJ}\Psi_{\alpha
I}\Psi_{\beta J}$ selects the 3-dimensional Rep of SU(2) (which is also the
vector Rep of SO(3), and the isospin-1 Rep of SU(2) and SO(3)), then
$C^{\prime IJ}$ is symmetric under $I\leftrightarrow J$, and such a mass term
does not vanish under fermi statistics.141414Here we take the 3 out of the
tensor product of 4 of SU(2): ${\bf 4}\otimes{\bf 4}={\bf 1}\oplus{\bf
3}\oplus{\bf 5}\oplus{\bf 7}$. Based on the Clebsch-Gordan coefficients, we
may choose
$\big{(}\sqrt{\frac{9}{20}}(|\frac{3}{2},\frac{3}{2}\rangle|\frac{3}{2},-\frac{3}{2}\rangle+|\frac{3}{2},-\frac{3}{2}\rangle|\frac{3}{2},\frac{3}{2}\rangle)-\sqrt{\frac{1}{20}}(|\frac{3}{2},\frac{1}{2}\rangle|\frac{3}{2},-\frac{1}{2}\rangle+|\frac{3}{2},-\frac{1}{2}\rangle|\frac{3}{2},\frac{1}{2}\rangle))=|1,0\rangle$,
we have $C^{\prime IJ}\Psi_{\alpha I}\Psi_{\beta
J}=\big{(}\sqrt{\frac{9}{20}}(\Psi_{\alpha 1}\Psi_{\beta 4}+\Psi_{\alpha
4}\Psi_{\beta 1})-\sqrt{\frac{1}{20}}(\Psi_{\alpha 2}\Psi_{\beta
3}+\Psi_{\alpha 3}\Psi_{\beta 2})\big{)}$. Since $C^{\prime IJ}$ are
symmetric, $\epsilon^{\alpha\beta}C^{\prime IJ}\Psi_{\alpha I}\Psi_{\beta
J}\neq 0$. This mass term explicitly break the SU(2) down to U(1) symmetry.

$\bullet$ Spontaneous symmetry breaking: Instead we can consider the
spontaneous symmetry breaking by introducing the Yukawa-Higgs term
$\epsilon^{\alpha\beta}(\vec{C^{\prime}}^{IJ}\Psi_{\alpha I}\Psi_{\beta
J})\vec{\Phi}$ such that not merely $(\vec{C^{\prime}}^{IJ}\Psi_{\alpha
I}\Psi_{\beta J})$ is the 3 of SU(2) but also the Higgs scalar $\vec{\Phi}$ is
also the 3 of SU(2), while we again select the SU(2) singlet ${\bf 1}$ out of
their tensor product pairing ${\bf 3}\otimes{\bf 3}={\bf 1}\oplus{\bf
3}\oplus{\bf 5}$. Then the Higgs condensation $\langle\vec{\Phi}\rangle\neq 0$
only breaks the SU(2) _spontaneously_ down to U(1).

$\bullet$ SU(2) to all-fermion U(1) gauge theories: Ref. Wang  _et al._ , 2019
shows that the spherical rotationally symmetric ’t Hooft-Polyakov monopole (up
to SU(2) gauge transformation) in the presence of ${\bf 4}$ Rep of SU(2) Weyl
spinor traps fermionic zero modes in the spectrum. Ref. Wang  _et al._ , 2019
shows that the ’t Hooft-Polyakov monopole becomes fermion. This means under
the breaking from SU(2) to U(1), such that

the isospin-$\frac{3}{2}$ Weyl fermion of SU(2) becomes the fermionic electric
charge 1 of U(1). the fermionic ’t Hooft-Polyakov monopole of SU(2) becomes
the fermionic magnetic monopole of U(1).

Together with the fermionic electric charge of U(1), the dyon of U(1) is also
a fermion. Ref. Wang  _et al._ , 2019 suggests that a possible renormalization
group (RG) flow such that the high energy theory is an asymptotic free SU(2)
gauge theory while the low energy theory is the all-fermion U(1) gauge theory.

The subtlety is that this 4d SU(2) gauge theory is ill-defined on a non-spin
manifold and cannot be put on the boundary of 5d ${\rm w}_{2}{\rm w}_{3}$. But
once we break SU(2) down to U(1), the 4d U(1) gauge theory can be put on a
non-spin manifold on the boundary of 5d ${\rm w}_{2}{\rm w}_{3}$.

$\bullet$ All-fermion U(1) to $\mathbb{Z}_{2}$ gauge theories: If we introduce
another Higgs also the 3 of SU(2), with a different vacuum expectation value,
we can further Higgs down the U(1) down to $\mathbb{Z}_{2}$, such that

the fermionic electric charge 1 of U(1) becomes the fermionic electric charge
1 of $\mathbb{Z}_{2}$. the fermionic monopole’s 1d ’t Hooft line of U(1)
becomes the orientation-reversal 1d fermionic worldline on an unorientable 2d
worldsheet of $\mathbb{Z}_{2}$ gauge theory.

  2. 2.

Boundary with the Spin(10) and ${\rm Spin}\times_{\mathbb{Z}_{2}}{\rm
Spin}(10)$ structures:

    1. (a)

The standard $so(10)$ Grand Unification Georgi and Glashow (1974); Fritzsch
and Minkowski (1975) with Spin(10) internal symmetry group and with Weyl
fermions in the 16 of Spin(10) does not have the ${\rm w}_{2}{\rm w}_{3}$
anomaly (more precisely, it is the ${\rm w}_{2}{\rm w}_{3}={\rm
w}_{2}^{\prime}{\rm w}_{3}^{\prime}$ mixed gauge-gravitational anomaly, which
is also a nonperturbative global anomaly, with ${\rm w}_{j}={\rm w}_{j}(TM)$
and ${\rm w}_{j}^{\prime}={\rm w}_{j}(V_{{\rm SO}(n=10)})$). This is the only
5d cobordism invariant from $\mathrm{TP}_{5}({{\rm
Spin}\times_{\mathbb{Z}_{2}}{\rm Spin}(10)})$, thus the only 4d global anomaly
for ${\rm Spin}\times_{\mathbb{Z}_{2}}{\rm Spin}(10)$ structure. The absence
of ${\rm w}_{2}{\rm w}_{3}={\rm w}_{2}^{\prime}{\rm w}_{3}^{\prime}$ anomaly
means that the standard $so(10)$ Grand Unification is free from all
perturbative local and nonperturbative global anomalies within ${\rm
Spin}\times_{\mathbb{Z}_{2}}{\rm Spin}(10)$ structure Wang and Wen (2020);
Wang  _et al._ (2019); Wan and Wang (2020).

    2. (b)

However, it is possible to construct a modified $so(10)$ Grand Unification
with Spin(10) internal symmetry group, also with Weyl fermions in the 16 of
Spin(10), but with additional discrete torsion class of Wess-Zumino-Witten
like term written on the 4d boundary and 5d bulk coupled system Wang and You
(2021a, 2021); You and Wang (2022). Here we summarize the results in Wang and
You (2021a, 2021); You and Wang (2022):

$\bullet$ When Spin(10) internal symmetry group is treated as a global
symmetry, this modified $so(10)$ Grand Unification can live on the boundary of
5d ${\rm w}_{2}{\rm w}_{3}={\rm w}_{2}^{\prime}{\rm w}_{3}^{\prime}$
invertible topological order. The ${\rm w}_{2}{\rm w}_{3}={\rm
w}_{2}^{\prime}{\rm w}_{3}^{\prime}$ anomaly is saturated by the discrete
torsion class of Wess-Zumino-Witten like term alone. The discrete torsion
class of Wess-Zumino-Witten like term gives rise to various possible gauge
theory realizations of low energy dynamics in 4d. The various possible gauge
theory realizations are the emergent gauge theories (similar to the emergent
dynamical Spin structure of the $\mathbb{Z}_{2}$ gauge theory and emergent
dynamical ${\rm Spin}^{c}$ structure of the all-fermion $\mathrm{U}(1)$ gauge
theory that we studied earlier).

$\bullet$ When Spin(10) internal symmetry group is dynamically gauged, the
Spin(10) gauge field in the 5d bulk ${\rm w}_{2}{\rm w}_{3}={\rm
w}_{2}^{\prime}{\rm w}_{3}^{\prime}$ is also gauged. The 5d bulk is no longer
a gapped invertible topological order; the 5d bulk becomes gapless and further
long-range entangled. Thus the Spin(10) gauge fields can live only on the 4d
boundary, but also propagate into the 5d bulk.

##  V Acknowledgements

This work started in the summer of 2018 during X.-G.W.’s visit at IAS
Princeton, and is based on the unpublished notes of 2018. We thank Edward
Witten for important discussions and previous collaborations Wang  _et al._
(2019) in 2018. JW thanks Dan Freed and Ryan Thorngren for discussions. ZW is
supported by the Shuimu Tsinghua Scholar Program. JW is supported by NSF Grant
DMS-1607871 “Analysis, Geometry and Mathematical Physics” and Center for
Mathematical Sciences and Applications at Harvard University. XGW is partially
supported by NSF DMR-2022428 and by the Simons Collaboration on Ultra-Quantum
Matter, which is a grant from the Simons Foundation (651446, XGW).

##  Appendix A Spacetime Complex with Branch Structure, Cochains, Higher Cup
Product

###  A.1 Spacetime complex and branch structure

In this article, we consider models defined on a spacetime lattice. A
spacetime lattice is a triangulation of the $d$-dimensional spacetime, which
is denoted by $M^{d}$. We will also call the triangulation $M^{d}$ as a
spacetime complex, which is formed by simplices – the vertices, links,
triangles, etc . We will use $i,j,\cdots$ to label vertices of the spacetime
complex. The links of the complex (the 1-simplices) will be labeled by
$(i,j),(j,k),\cdots$. Similarly, the triangles of the complex (the
2-simplices) will be labeled by $(i,j,k),(j,k,l),\cdots$.

In order to define a generic lattice theory on the spacetime complex $M^{d}$
using local Lagrangian term on each simplex, it is important to give the
vertices of each simplex a local order. A nice local scheme to order the
vertices is given by a branch structure.Costantino (2005); Chen  _et al._
(2013, 2012) A branch structure is a choice of orientation of each link in the
$d$-dimensional complex so that there is no oriented loop on any triangle (see
Fig. 2).

The branch structure induces a _local order_ of the vertices on each simplex.
The first vertex of a simplex is the vertex with no incoming links, and the
second vertex is the vertex with only one incoming link, etc . So the simplex
in Fig. 2a has the following vertex ordering: $0,1,2,3$. We always use ordered
vertices to label a simplex. So the simplex in Fig. 2a and 2b are labeled as
$[0,1,2,3]$.

The branch structure also gives the simplex (and its sub-simplices) a
canonical orientation. Fig. 2 illustrates two $3$-simplices with opposite
canonical orientations compared with the 3-dimension space in which they are
embedded. The blue arrows indicate the canonical orientations of the
$2$-simplices. The black arrows indicate the canonical orientations of the
$1$-simplices.

###  A.2 Chain, cochain, cycle, cocycle

Given an Abelian group $(\mathbb{M},+)$, $\mathbb{M}$ can also be viewed a
$\mathbb{Z}$-module (i.e. a vector space with integer coefficient) that also
allows scaling by an integer:

$\displaystyle x+y$ $\displaystyle\in\mathbb{M},\ \ \ \ x*y\in\mathbb{M},\ \ \
\ mx=\underbrace{x+x+\cdots+x}_{m\text{ terms}}\in M,$ $\displaystyle x,y$
$\displaystyle\in\mathbb{M},\ \ \ m\in\mathbb{Z}.$ (71)

The direct sum of two modules $\mathbb{M}_{1}\oplus\mathbb{M}_{2}$ (as vector
spaces) is equal to the direct product of the two modules (as sets):

$\displaystyle\mathbb{M}_{1}\oplus\mathbb{M}_{2}\stackrel{{\scriptstyle\text{as
set}}}{{=}}\mathbb{M}_{1}\times\mathbb{M}_{2}$ (72)

An $n$-cochain $\alpha_{n}$ in $M^{d}$ is a formal combination of
$n$-simplexes in $M^{d}$ with coefficients in $\mathbb{M}$:

$\displaystyle\alpha_{n}=\hskip-5.0pt\sum_{[i,j,\cdots,k]}\hskip-5.0pt\alpha_{n;i,j,\cdots,k}[i,j,\cdots,k],\
\ \ \alpha_{n;i,j,\cdots,k}\in\mathbb{M},$ (73)

where $\sum_{[i,j,\cdots,k]}$ sums over all simplexes in $M^{d}$. The
collection of all such $n$-chains is denoted as $C_{n}(M^{d};\mathbb{M})$. For
example, a 2-chain can be a 2-simplex: $[i,j,k]$, a sum of two 2-simplices:
$[i,j,k]+[j,k,l]$, a more general composition of 2-simplices:
$[i,j,k]-2[j,k,l]$, etc .

Figure 2: Two branch simplices with opposite orientations. (a) A branch
simplex with positive orientation and (b) a branch simplex with negative
orientation.  Figure 3: A 1-cochain $a$ has a value $1$ on the red links:
$a_{ik}=a_{jk}=1$ and a value $0$ on other links: $a_{ij}=a_{kl}=0$. $\hskip
1.0pt\mathrm{d}a$ is non-zero on the shaded triangles: $(\hskip
1.0pt\mathrm{d}a)_{jkl}=a_{jk}+a_{kl}-a_{jl}$. For such 1-cohain, we also have
$a\smile a=0$. So when viewed as a $\mathbb{Z}_{2}$-valued cochain,
$\beta_{2}a\neq a\smile a$ mod 2.

An $n$-cochain $f_{n}$ in $M^{d}$ is an assignment of values in $\mathbb{M}$
to each $n$-simplex $M^{d}$. For example a value
$f_{n;i,j,\cdots,k}\in\mathbb{M}$ is assigned to $n$-simplex $(i,j,\cdots,k)$
(see Fig. 3). We also denote the value $f_{n;i,j,\cdots,k}$ as
$f_{n}([i,j,\cdots,k])$. So _a cochain $f_{n}$ can be viewed as a bosonic
field, $f_{n}([i,j,\cdots,k])$, on the spacetime lattice $M^{d}$_. To be more
precise $f_{n}$ is a linear map $f_{n}:n\text{-chain}\to\mathbb{M}$. We can
denote the linear map as $f_{n}(n\text{-chain})$, or

$\displaystyle
f_{n}(\alpha_{n})=\sum_{[i,j,\cdots,k]}\alpha_{n;i,j,\cdots,k}f_{n}([i,j,\cdots,k])\in\mathbb{M}.$
(74)

where $\sum_{[i,j,\cdots,k]}$ sums over all simplexes in $M^{d}$.

We will use $C^{n}(M^{d};\mathbb{M})$ to denote the set of all $n$-cochains on
$M^{d}$. $C^{n}(M^{d};\mathbb{M})$ can also be viewed as a set all
$\mathbb{M}$-valued fields (or paths) on $M^{d}$. Note that
$C^{n}(M^{d};\mathbb{M})$ is an Abelian group under the $+$-operation.

The total spacetime lattice $M^{d}$ correspond to a $d$-chain. We will use the
same $M^{d}$ to denote it:

$\displaystyle M^{d}=\sum_{[i,j,\cdots,k]}s_{i,j,\cdots,k}[i,j,\cdots,k],$
(75)

where $s_{i,j,\cdots,k}=\pm 1$, describing the relative orientation between
$M^{d}$ and $[i,j,\cdots,k]$. Viewing $f_{d}$ as a linear map of $d$-chains,
we can define an “integral” over $M^{d}$:

$\displaystyle\int_{M^{d}}f_{d}$ $\displaystyle\equiv
f_{d}(M^{d})=\sum_{[i,j,\cdots,k]}f_{d;i,j,\cdots,k}s_{i,j,\cdots,k}$
$\displaystyle=\sum_{[i,j,\cdots,k]}f_{d}([i,j,\cdots,k])s_{i,j,\cdots,k}.$
(76)

Figure 4: A 1-cochain $a$ has a value $1$ on the red links, Another 1-cochain
$a^{\prime}$ has a value $1$ on the blue links. On the left, $a\smile
a^{\prime}$ is non-zero on the shade triangles: $(a\smile
a^{\prime})_{ijl}=a_{ij}a^{\prime}_{jl}=1$. On the right, $a^{\prime}\smile a$
is zero on every triangle. Thus $a\smile a^{\prime}+a^{\prime}\smile a$ is not
a coboundary.

###  A.3 Derivative operator on cochains

We can define a derivative operator $\hskip 1.0pt\mathrm{d}$ acting on an
$n$-cochain $f_{n}$, which give us an $(n+1)$-cochain (see Fig. 3):

$\displaystyle\ \ \ \ (\hskip 1.0pt\mathrm{d}f_{n})([i_{0}i_{1}i_{2}\cdots
i_{n+1}])$
$\displaystyle=\sum_{m=0}^{n+1}(-)^{m}f_{n}([i_{0}i_{1}i_{2}\cdots\hat{i}_{m}\cdots
i_{n+1}])$ (77)

where $i_{0}i_{1}i_{2}\cdots\hat{i}_{m}\cdots i_{n+1}$ is the sequence
$i_{0}i_{1}i_{2}\cdots i_{n+1}$ with $i_{m}$ removed, and
$i_{0},i_{1},i_{2}\cdots i_{n+1}$ are the ordered vertices of the
$(n+1)$-simplex $(i_{0}i_{1}i_{2}\cdots i_{n+1})$.

A cochain $f_{n}\in C^{n}(M^{d};\mathbb{M})$ is called a _cocycle_ if $\hskip
1.0pt\mathrm{d}f_{n}=0$. The set of cocycles is denoted by
$Z^{n}(M^{d};\mathbb{M})$. A cochain $f_{n}$ is called a _coboundary_ if there
exist a cochain $f_{n-1}$ such that $\hskip 1.0pt\mathrm{d}f_{n-1}=f_{n}$. The
set of coboundaries is denoted by $B^{n}(M^{d};\mathbb{M})$. Both
$Z^{n}(M^{d};\mathbb{M})$ and $B^{n}(M^{d};\mathbb{M})$ are Abelian groups as
well. Since $\hskip 1.0pt\mathrm{d}^{2}=0$, a coboundary is always a cocycle:
$B^{n}(M^{d};\mathbb{M})\subset Z^{n}(M^{d};\mathbb{M})$. We may view two
cocycles differ by a coboundary as equivalent. The equivalence classes of
cocycles, $[f_{n}]$, form the so called cohomology group denoted by

$\displaystyle
H^{n}(M^{d};\mathbb{M})=Z^{n}(M^{d};\mathbb{M})/B^{n}(M^{d};\mathbb{M}),$ (78)

$H^{n}(M^{d};\mathbb{M})$, as a group quotient of $Z^{n}(M^{d};\mathbb{M})$ by
$B^{n}(M^{d};\mathbb{M})$, is also an Abelian group.

For the $\mathbb{Z}_{N}$-valued cochain $x_{n}$, we lift $\mathbb{Z}_{N}$ to
$\mathbb{Z}$, via $\\{0,1,\cdots,N-1\\}\subset\mathbb{Z}_{N}$ to
$\\{0,1,\cdots,N-1\\}\subset\mathbb{Z}$, and define

$\displaystyle\beta_{N}x_{n}\equiv\frac{1}{N}\hskip 1.0pt\mathrm{d}x_{n}$ (79)

When $x_{n}$ is a cocycle, we have $\hskip
1.0pt\mathrm{d}x_{n}\overset{\scriptscriptstyle N}{=}0$. In this case,
$\beta_{N}x_{n}$ is a $\mathbb{Z}$-valued cocycle, and $\beta_{N}$ is
Bockstein homomorphism.

###  A.4 Cup product and higher cup product

From two cochains $f_{m}$ and $h_{n}$, we can construct a third cochain
$p_{m+n}$ via the cup product (see Fig. 4):

$\displaystyle p_{m+n}$ $\displaystyle=f_{m}\smile h_{n},$ (80) $\displaystyle
p_{m+n}([0\to{m+n}])$ $\displaystyle=f_{m}([0\to m])\rangle
h_{n}([m\to{m+n}]),$

where $i\to j$ is the consecutive sequence from $i$ to $j$:

$\displaystyle i\to j\equiv i,i+1,\cdots,j-1,j.$ (81)

Note that the the order of vertices in a simplex $(0\to m)$ and the notion of
consecutive sequence are determined by the branch structure. Thus the cup
product (and the higher cup product below) on a simplicial complex can be
defined only after we assign a branch structure to the simplicial complex. The
value of the cup product depends on the branch structure.

The cup product has the following property

$\displaystyle\hskip 1.0pt\mathrm{d}(h_{n}\smile f_{m})$
$\displaystyle=(\hskip 1.0pt\mathrm{d}h_{n})\smile
f_{m}+(-)^{n}h_{n}\smile(\hskip 1.0pt\mathrm{d}f_{m})$ (82)

for cochains with global or local values. We see that $h_{n}\smile f_{m}$ is a
cocycle if both $f_{m}$ and $h_{n}$ are cocycles. If both $f_{m}$ and $h_{n}$
are cocycles, then $f_{m}\smile h_{n}$ is a coboundary if one of $f_{m}$ and
$h_{n}$ is a coboundary. So the cup product is also an operation on cohomology
groups $\underset{{\scriptscriptstyle}}{\smile}:H^{m}(M^{D};\mathbb{M})\times
H^{n}(M^{D};\mathbb{M})\to H^{m+n}(M^{D};\mathbb{M})$. The cup product of two
_cocycles_ has the following property (see Fig. 4)

$\displaystyle f_{m}\smile h_{n}$ $\displaystyle=(-)^{mn}h_{n}\smile
f_{m}+\text{coboundary}$ (83)

We can also define higher cup product $f_{m}\underset{{\scriptscriptstyle
k}}{\smile}h_{n}$ which gives rise to a $(m+n-k)$-cochain Steenrod (1947):

$\displaystyle\ \ \ \ (f_{m}\underset{{\scriptscriptstyle
k}}{\smile}h_{n})([0,1,\cdots,m+n-k])$
$\displaystyle=\hskip-10.00002pt\sum_{0\leq i_{0}<\cdots<i_{k}\leq
n+m-k}\hskip-30.00005pt(-)^{p}f_{m}([0\to i_{0},i_{1}\to i_{2},\cdots])\times$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ h_{n}([i_{0}\to
i_{1},i_{2}\to i_{3},\cdots]),$ (84)

and $f_{m}\underset{{\scriptscriptstyle k}}{\smile}h_{n}=0$ for $k<0$ or for
$k>m\text{ or }n$. Here $i\to j$ is the sequence $i,i+1,\cdots,j-1,j$, and $p$
is the number of permutations to bring the sequence

$\displaystyle 0\to i_{0},i_{1}\to i_{2},\cdots;i_{0}+1\to i_{1}-1,i_{2}+1\to
i_{3}-1,\cdots$ (85)

to the sequence

$\displaystyle 0\to m+n-k.$ (86)

For example

$\displaystyle(f_{m}\underset{{\scriptscriptstyle 1}}{\smile}h_{n})([0\to
m+n-1])=\sum_{i=0}^{m-1}(-)^{(m-i)(n+1)}\times$ $\displaystyle f_{m}([0\to