11institutetext: Department of Physics, The University of Tokyo, 7-3-1 Hongo,
Bunkyo-ku, Tokyo 113-0033, Japan22institutetext: Yukawa Institute for
Theoretical Physics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku,
Kyoto 606-8502, Japan

# Solitonic symmetry as non-invertible symmetry: cohomology theories with TQFT
coefficients

Shi Chen<EMAIL_ADDRESS>2 Yuya Tanizaki
<EMAIL_ADDRESS>
###### Abstract

Originating from the topology of the path-integral target space $Y$, solitonic
symmetry describes the conservation law of topological solitons and the
selection rule of defect operators. As Ref. Chen:2022cyw  exemplifies, the
conventional treatment of solitonic symmetry as an invertible symmetry based
on homotopy groups is inappropriate. In this paper, we develop a systematic
framework to treat solitonic symmetries as non-invertible generalized
symmetries. We propose that the non-invertible solitonic symmetries are
generated by the partition functions of auxiliary topological quantum field
theories (TQFTs) coupled with the target space $Y$. We then understand
solitonic symmetries as non-invertible cohomology theories on $Y$ with TQFT
coefficients. This perspective enables us to identify the invertible solitonic
subsymmetries and also clarifies the topological origin of the non-
invertibility in solitonic symmetry. We finally discuss how solitonic symmetry
relies on and goes beyond the conventional wisdom of homotopy groups. This
paper is aimed at a tentative general framework for solitonic symmetry,
serving as a starting point for future developments.

††preprint: YITP-23-59

##  1 Introduction

Symmetry provides one of the guiding principles when studying strongly-coupled
physics in quantum field theories (QFTs), and astonishingly, the notion of
symmetry itself has been vastly generalized in these recent years. The
generalization is achieved under the motto that the topologicalness of
operators should always represent conservation laws, and we identify the
algebraic structure of topological operators in QFTs with the generalized
symmetries. Such generalizations mainly include two directions: One is the
higher-form symmetry (Gaiotto:2014kfa, ) and higher-group symmetry
(Sharpe:2015mja, ; Benini:2018reh, ; Cordova:2018cvg, ; Tanizaki:2019rbk, ;
Hidaka:2020izy, ; Hidaka:2021kkf, ; Hidaka:2021mml, ), where the symmetry
operators are defined on various codimensions, and the more recent one is the
non-invertible symmetry, where the fusion rule obeys a suitable algebraic
structure beyond the usual group multiplications. The non-invertible
symmetries in $(1+1)$-dim are now well-understood, and the fusion category
captures their algebraic structure (when finitely generated) (Verlinde:1988sn,
; Fuchs:2002cm, ; Carqueville:2012dk, ; Aasen:2016dop, ; Bhardwaj:2017xup, ;
Tachikawa:2017gyf, ; Chang:2018iay, ; Thorngren:2019iar, ; Lichtman:2020nuw, ;
Komargodski:2020mxz, ; Aasen:2020jwb, ; Thorngren:2021yso, ; Nguyen:2021naa, ;
Huang:2021nvb, ; Vanhove:2021zop, ). The non-invertible symmetries in higher
dimensions start to be realized in various QFTs (Nguyen:2021yld, ;
Heidenreich:2021xpr, ; Koide:2021zxj, ; Choi:2021kmx, ; Kaidi:2021xfk, ;
Roumpedakis:2022aik, ; Hayashi:2022fkw, ; Choi:2022zal, ; Kaidi:2022uux, ;
Choi:2022jqy, ; Cordova:2022ieu, ; Choi:2022rfe, ; Radhakrishnan:2023zcq, ;
Chen:2022cyw, ; Yokokura:2022alv, ; Kaidi:2023maf, ; Koide:2023rqd, ;
Cao:2023doz, ; Inamura:2023qzl, ; vanBeest:2023dbu, ; Damia:2023ses, ;
Copetti:2023mcq, ), and there is also a massive endeavor to identify the
precise mathematical structures behind generalized symmetries (Ji:2019jhk, ;
Kong:2020cie, ; Gaiotto:2020iye, ; Bhardwaj:2022yxj, ; Kaidi:2022cpf, ;
Freed:2022qnc, ; Bartsch:2022mpm, ; Bartsch:2022ytj, ; Bhardwaj:2022lsg, ;
Bhardwaj:2022kot, ; Bhardwaj:2022maz, ; Bartsch:2023pzl, ; Bartsch:2023wvv, ;
Bhardwaj:2023wzd, ; Bhardwaj:2023ayw, ).

In this paper, we shall investigate non-invertible symmetries organizing
conservation laws of topological solitons, which we call (non-invertible)
solitonic symmetries (Chen:2022cyw, ). Topological solitons are
nonperturbative objects in QFTs that appear due to the nontrivial topology of
the path-integral target space, such as kinks, vortices, monopoles, etc. They
are created/annihilated by defect operators. It has been common wisdom that
their topological stability, an intriguing aspect of topological solitons, is
captured by the homotopy group of the target space (Mermin:1979zz, ;
Coleman:1985rnk, ; Manton:2004tk, ). In the previous study (Chen:2022cyw, ),
the present authors revealed that solitonic symmetry also becomes a non-
invertible symmetry in general, and the solitonic symmetry generators are
given by the partition functions of auxiliary lower-dim topological QFTs
(TQFTs) coupled with the original system. This finding provides us an
opportunity to reconsider the foundations of solitonic symmetries, and,
surprisingly, deep mathematics turns out to be there behind the solitonic
symmetries, which echoes those behind classifying gapped phases
(Lurie:2009keu, ; Kong:2014qka, ; Freed:2016rqq, ; freed2019lectures, ;
Gaiotto:2019xmp, ; Johnson-Freyd:2020usu, ; Kong:2020jne, ).

The fact that solitonic symmetries become non-invertible implies that the
usual homotopy group of the target space $Y$ does not always correctly
characterize the topological conservation laws of solitons, which raises the
following questions:

  * •

What is the new foundation for solitonic symmetry? (Sec. 2)

  * •

What are the topological operators for solitonic symmetry? (Secs. 3)

  * •

What is the algebraic structure of solitonic symmetry? (Sec. 4)

  * •

What makes solitonic symmetry go beyond homotopy groups? (Sec. 5)

We give proposals and/or solutions to these questions in the indicated
sections, and let us summarize the results as follows.

When discussing symmetries, we need to specify the symmetry generators and
their action on charged objects. We showed in Sec. 2 that, in the case of the
topological conservation law of solitons, the charged objects are given by the
defect operators that create/annihilate solitons, and then the symmetry
generators should be given by topological functionals using the fundamental
fields in the path-integral formulation. Then in Sec. 3, we would like to find
the most general form of topological functionals to define the non-invertible
solitonic symmetries, and we clarify the physical requirements to be satisfied
by the topological functionals. In particular, it turns out that the essential
requirement comes from locality. As a natural ansatz satisfying such
requirements, we propose that the solitonic symmetries are generated by
partition functions of auxiliary fully-extended TQFT coupled to the
fundamental fields (Ansatz 3.2).

The rigorous treatment of Ansatz 3.2 requires us to employ the knowledge of
fully-extended TQFTs. We concisely review the relevant mathematical treatments
in Sec. 4. Also in that section, we clarify the algebraic structure of
solitonic symmetry, which is described by symmetric fusion higher-categories
$\mathsf{Rep}^{\bullet}(Y)$ for the bosonic case and
$\mathsf{sRep}^{\bullet}(Y)$ for the fermionic case. These observations are
consistent with the latest progress on generalized symmetry in the literature,
such as Refs. (Bhardwaj:2022lsg, ; Bhardwaj:2022kot, ; Bartsch:2022mpm, ;
Bartsch:2022ytj, ). We see that solitonic symmetry serves as non-invertible
generalizations of cohomology theories with TQFT coefficients. We also discuss
the invertible solitonic subsymmetry given by orthodox cohomology theories.

Armed with this mathematical guidance, in Sec. 5, we can systematically study
the origin of non-invertibility in the solitonic symmetry and how the
conventional wisdom of homotopy groups is surpassed. We shall see that
$\mathsf{(s)Rep}^{\bullet}(Y)$ can be decomposed into two parts. The first
part comes from $\mathsf{(s)Rep}^{\bullet-1}(Y)$ via formulating condensations
and contains topological functionals that are trivial on spheres. The second
part comes from the homotopy group $\pi_{\bullet}-$ and contains topological
functionals that are nontrivial on spheres. But according to the topological
data present in the theory, these spherically-nontrivial topological
functionals have to take a non-invertible fusion rule. This decomposition
unpacks the structure of solitonic structure inductively and provides us with
insight into the connection between the generalized solitonic symmetry from
the contemporary perspective and the conventional wisdom since Coleman etc.

###### Acknowledgements.

This work was initiated during S. C.’s visit to YITP with the Atom-type
visiting program, and the authors appreciate the hospitality of Yukawa
Institute. The authors are grateful to Mayuko Yamashita for bringing Ref.
Gaiotto:2019xmp  to their scope in a seminar. They also thank Kantaro Ohmori
for valuable comments on the manuscript. S. C. thanks Yuji Tachikawa for
cheerful discussions and useful comments on the idea developed in this paper.
This work was supported by JSPS KAKENHI Grant No. 21J20877 (S. C.), 22H01218
and 20K22350 (Y. T.), and also by the Center for Gravitational Physics and
Quantum Information (CGPQI).

##  2 Basic concepts

Solitonic symmetry is generated by topological functionals. It (i) describes
the conservation law in the solitonic sector, and (ii) prescribes the
selection rule in correlation functions between solitonic defects, and (iii)
determines the possible topological couplings with a background gauge field.
We explain these basic notions in this section. The spacetime dimension is
denoted by $d$ throughout.

###  2.1 Target space of the path integral

Let us consider a general situation where the quantum field theory (QFT) is
defined by the path integral, and its partition function is given by

$\mathcal{Z}=\int{\mathcal{D}}\sigma\exp(-S[\sigma]),$ (1)

and $\sigma$ is some field on the $d$-dim spacetime. We focus on the non-
Grassmann sector of this path integral, and thus $\sigma$ might be a scalar
field, a gauge field, a higher-form gauge field, or even a combination of them
coupled. We note that, except in the simplest case of pure scalar fields, all
other fields are not maps to a fixed space.

In this paper, we are interested in a series of nonperturbative phenomena
insensitive to continuous deformations of field configurations. Namely, we
only care about the deformation classes of field configurations on some closed
(smooth) manifold $M$. The actual choice of $M$ depends on specific problems;
it might be the spacetime itself, a submanifold like a space slice or a world
line, or a virtual submanifold like the normal sphere bundle of a defect
operator (see Sec. 2.2.2). Conveniently, we can always find a topological
space $Y$ such that there exists a one-to-one correspondence,

$\begin{split}\text{Deformation classes o}&\text{f field configurations on
$M$}\\\ &\parallel\\\ \text{Deformation classe}&\text{s of maps from $M$ to
$Y$}\,.\end{split}$ (2)

We shall refer to this topological space $Y$ as the (homotopy) target space of
the path integral and abuse the symbol $\sigma$ to denote also the auxiliary
maps to the target space, i.e.,

$\sigma|_{M}:M\mapsto Y.$ (3)

Formally, the set of deformation classes of field configurations are now
expressed by the set of homotopy classes of maps to the target space $Y$,

$[M,Y]\equiv\\{f:M\mapsto Y\\}/\text{homotopies}.$ (4)

Because the dimension of closed manifold $M$ cannot exceed spacetime dimension
$d$, the set $[M,Y]$ depends merely on the homotopy $d$-type of $Y$.

###### Proposition 2.1

In a $d$-dim QFT defined by a path integral, the deformation classes of field
configurations on any submanifold depend only on the homotopy $d$-type of the
target space $Y$.

Thus, $Y$ is understood up to $(d\\!+\\!1)$-connected maps in this paper.
Without loss of generality, we can always require $Y$ to be a $d$-aspherical
space.

###### Definition 2.2

Topological space $X$ is $n$-aspherical if $\pi_{\bullet}(X,x)\simeq 0$ for
all $\bullet>n$ and $x\in X$.

Namely, we can always replace a general $Y$ with its $d$-th Postnikov
truncation. Nevertheless, in actual practice, despite Proposition 2.1, we
often render more topological and even geometrical structures on $Y$ to keep
in touch with other physics that are sensitive to continuous deformations of
field configurations.

We now construct several common $Y$’s to illustrate the above abstract
concepts.

  1. 1.

$Y\simeq X$ for a $X$-valued scalar field, where $X$ is a topological space.

  2. 2.

$Y\simeq BG$, the delooping of $G$ (or the classifying space of $G$), for a
gauge field with gauge group $G$, where $G$ is a topological group (can be
discrete).

  3. 3.

If we couple the two fields above via a continuous $G$-action on $X$, then $Y$
fits into a fibration $X\to Y\to BG$ determined by the $G$-action111 If $G$
acts on $X$ freely (i.e., the action has no fixed point), we just have
$Y\simeq X/G$. If $X$ is contractible, we just have $Y\simeq BG$. Otherwise,
$Y$ has a quite complicated structure, such as many orbifolding examples. .

  4. 4.

$Y\simeq B^{p}G$, the $p$-th delooping of $G$, for a $p$-form gauge field with
Abelian $G$.

  5. 5.

$Y\simeq B\mathsf{G}$ for a gauge field of a local higher-group symmetry
$\mathsf{G}$ (see the discussions around Philosophies 4.1 and 4.2, and also
the remark in Sec. 6.1).

Concrete examples will appear later.

###  2.2 Solitonic symmetry

From a contemporary viewpoint of QFTs, the notion of symmetry is vastly
generalized and can be summarized as

$\text{Generalized symmetry}\equiv\text{Algebra of topological operators}.$
(5)

In the path-integral formalism, there are basically two constructions of
operators222 We note that these two constructions can be interchanged under
the duality operation, and thus the notion of the solitonic symmetry depends
on the explicit path-integral realization of a given QFT. , defects and
functionals. Both types of operators have the chance to be topological.
Topological defects are the most orthodox topological operators and all
symmetries “on the electric side” are generated by them. Topological
functionals are more or less atypical and symmetries “on the magnetic side”
are generated by them. In this paper, we shall refer to the symmetry generated
by topological functionals as solitonic symmetry, i.e.,

$\text{Solitonic symmetry}\equiv\text{Symmetry generated by topological
functionals}\,.$ (6)

The core purpose of this paper is to reveal the structure of solitonic
symmetry by studying the behavior of topological functionals.

####  2.2.1 Elementary properties

For a functional on an $n$-dim closed manifold $M$ to be topological, it must
factor through the deformation classes of field configurations. Namely, it
factors through $[M,Y]$ for the target space $Y$. Given that $[M,Y]$ depends
only on the homotopy $n$-type of $Y$, we see the following property of
topological functionals.

###### Proposition 2.3

The $n$-dim topological functionals depend only on the homotopy $n$-type of
the target space.

This is consistent with Proposition 2.1. As a prototypical example of
topological functionals, let us pick up a cohomology class
$\omega\in\mathbb{E}^{\bullet}(Y)$ for some multiplicative cohomology
$\mathbb{E}$333 Spectrum $\mathbb{E}$ can be assumed connective, i.e.
$\mathbb{E}_{\bullet}\simeq 0$ for $\bullet<0$, given Proposition 2.3. . It
can be an ordinary cohomology $\mathbb{H}R$ for some ring $R$ or an
extraordinary cohomology such as $\mathbb{K}$ and $\mathbb{KO}$. Then, on any
closed $n$-dim $\mathbb{E}$-orientable manifold $M$ that acquires a chosen
$\mathbb{E}$-orientation, we can define a topological functional as

$U_{g}(M)\equiv g\left(\int_{M}\sigma^{*}\omega\right)\,,\qquad\forall
g\in\mathrm{Hom}\bigl{(}\mathbb{E}_{n-\bullet},U(1)\bigr{)}\,.$ (7)

We can readily see the fusion rule $U_{g_{1}}U_{g_{2}}=U_{g_{1}g_{2}}$ and
thus we obtain a $p$-form symmetry with $p=d\\!-\\!n\\!-\\!1$. The operator
dimension $n$ can range from $d$ to $0$, corresponding to the symmetry form
$p$ ranging from $-\\!1$ to $d\\!-\\!1$. Note that $n$-dim topological
functionals are exactly $\theta$-angles, i.e. topological terms in the action
that depend on $Y$ only444 In contrast, topological terms that also depend on
the geometry beyond the mere topology of $Y$ are not $n$-dim topological
functionals. CS terms and WZW terms are such counter-examples. . We shall see
concrete examples of operator (7) as soon as in Sec. 3.1.

The solitonic symmetry defined above has an invertible fusion rule, which is
captured by an Abelian group, $\mathrm{Hom}(\mathbb{E}_{n-\bullet},U(1))$ or
one of its quotients. Actually, operator (7) gives the universal construction
of invertible solitonic symmetry. In this paper, we shall discuss the most
generalized connotation of topological functional and solitonic symmetry with
more complicated non-invertible fusion rules. In particular, non-invertible
$\theta$-angles mean couplings to topological orders, as explained in Refs.
(Freed:2017rlk, ; Chen:2022cyw, ). Nevertheless, in any case, the fusion rule
of topological functionals must still be commutative, because on each
supporting manifold, the fusion is just the multiplication of complex numbers:

###### Proposition 2.4

Solitonic symmetry is commutative.

Namely, topological functionals never care about their order. Solitonic
symmetry is also insensitive to the theory details, including the action and
ambient spacetime, because topological functionals and their fusions do not
care about these theory details. Therefore, when a $d$-dim and a
$(d\\!+\\!1)$-dim theory share the same target space $Y$ (i.e., the homotopy
$d$-types of the target spaces are identical), we have for $0\leq n\leq d$,

$\begin{split}\text{$n$-dim topological fu}&\text{nctional in $d$-dim QFT}\\\
&\parallel\\\ \text{$n$-dim topological func}&\text{tional in
$(d\\!+\\!1)$-dim QFT}\,,\end{split}$ (8)

and accordingly, for $d\\!-\\!1\geq p\geq-\\!1$, we have the following
equivalence,

$\begin{split}\text{$p$-form solitonic s}&\text{ymmetry in $d$-dim QFT}\\\
&\parallel\\\ \text{$(p\\!+\\!1)$-form solitonic s}&\text{ymmetry in
$(d\\!+\\!1)$-dim QFT}\,.\end{split}$ (9)

From this point of view, the algebraic structure of solitonic symmetry is
supposed to be a sort of cohomology theory on the target space $Y$; we shall
justify this in Sec. 4.3. Thus we shall neglect to mention spacetime when
discussing topological functionals. The spacetime dimension implicitly enters
as an upper limit for the possible dimension of topological functionals,
according to Prop. 2.1.

Before continuing the journey to more details about topological functionals,
let us pause here to acquaint readers with solitonic symmetry’s physical
consequences.

####  2.2.2 Physical significance

First, a symmetry shows the presence of certain conserved charges. To find
charged objects of solitonic symmetry, let us consider a correlation function
involving a topological functional. Then this topological functional can
produce a nontrivial number as long as the field configuration on its
supporting manifold cannot be continuously deformed to a trivial
configuration. This can be achieved only if the correlation function includes
proper solitonic defects or the spacetime has a special topology.

Let us now introduce the notion of solitonic defects. For some $0\leq p\leq
d\\!-\\!1$, we take a $p$-dim submanifold $N$ of the spacetime and excise its
infinitesimal neighborhood. Then a $p$-dim Dirichlet defect operator on $N$,
which we call a solitonic defect operator, is defined by putting the Dirichlet
boundary condition on the boundary of the excised region. More formally, it is
a Dirichlet boundary condition on the normal sphere bundle $\mathcal{S}N$ of
$N$ (Locally, $\mathcal{S}N\simeq N\times S^{d-p-1}$). It can be virtually
viewed as a $(d\\!-\\!1)$-dim submanifold in the spacetime via a tubular
neighborhood. According to Sec. 2.1, the deformation classes of such Dirichlet
boundary conditions can be expressed by deformation classes of maps

$\sigma|_{\mathcal{S}N}:\mathcal{S}N\mapsto Y.$ (10)

Namely, the deformation classes of solitonic defects on $N$ are classified by

$[\mathcal{S}N,Y]\,.$ (11)

Solitonic defects can be either topological or non-topological operators,
depending on the details of the theory. If topological, they also generate
symmetry, but “on the electric side” and thus non-solitonic.

Solitonic defects couple to a nonperturbative sector called the solitonic
sector which appears because of the nontrivial topology of the target space555
The solitonic sector discussed in this paper can exist on closed spacetimes.
There are also other types of solitonic sectors that inhabit non-closed
spacetimes with boundaries only. They also originate from the path-integral
topology but are not captured by the target space. Nevertheless, in many
cases, they can still be understood from the target space of another related
theory; See Sec. 6.1 for more discussions. . When we put solitonic defects in
the spacetime and evaluate the correlation function via the path integral, the
configurations to be integrated are topological solitons bounded by those
defects (see Sec. 2.2.3 for a few examples). In particular, the tree-level
contribution to the correlation functions comes from solitonic solutions of
the classical equation of motion. Therefore, the solitonic sector in quantum
theory can be viewed as the quantization of classical solitons, and the
solitonic defects are the creation/annihilation operators for quantum
solitons. Aside from putting solitonic defects, arranging a nontrivial
spacetime topology is also a common method to visualize the solitonic sector.

It is now clear that the charged objects of solitonic symmetry are exactly
those solitonic objects introduced above. Solitonic symmetry puts conserved
charges on these solitonic objects, which we call topological charges. The
conservation law of topological charges constrains the correlation functions
among solitonic defects and prescribes the selection rule in the physical
processes in which topological solitons are involved.

Second, a symmetry prescribes the possible couplings to background gauge
fields. This coupling is topological for solitonic symmetry, namely a
topological term in the action. Typically, the coupling is directly related to
the solitonic sector. However, some couplings are not related to any authentic
solitonic sector. For example, a U(1) Chern-Simons path integral

$\int\mathcal{D}a\,\exp\left\\{\frac{\mathrm{i}k}{4\pi}\int
a\mathrm{d}a\right\\}$ (12)

has no physical solitonic sector since monopoles are not gauge invariant as
point-like local operators. However, we can still couple a $U(1)$ background
gauge field $A$ topologically via

$\int\mathcal{D}a\,\exp\left\\{\frac{\mathrm{i}k}{4\pi}\int
a\mathrm{d}a+\frac{\mathrm{i}}{2\pi}\int a\mathrm{d}A\right\\}\,.$ (13)

The information of the existence of such possible topological couplings with a
background $U(1)$ gauge field is also encoded in a 0-form $U(1)$ solitonic
symmetry from $Y\simeq BU(1)$, although the corresponding solitonic objects
are unphysical.

Another general case is that, due to dimensional reasons, no solitonic defect
carries a $(-\\!1)$-form topological charge. Instantons, the charged objects
under $(-\\!1)$-form solitonic symmetry, are just classical objects. Thus
$(-\\!1)$-form solitonic symmetry, generated by $\theta$-angles, never really
rules a quantum solitonic sector but instead prescribes the topological
couplings to a background “0-form gauge field”, i.e., a background axion
field.

####  2.2.3 Conventional wisdom and homotopy groups

In the conventional discussion on topological solitons, their stability and
the selection rules are usually discussed using the homotopy group of the
target space. To be concrete, let us consider two simple examples:

  * (1)

An $S^{1}$ sigma model has a $(d\\!-\\!2)$-dim solitonic defect, on which a
kink can end. Their $(d\\!-\\!2)$-form topological charge is related to
$\pi_{1}S^{1}\simeq\mathbb{Z}$. This gives rise to a $(d\\!-\\!2)$-form $U(1)$
solitonic symmetry.

  * (2)

A pure U(1) gauge theory or a U(1) Higgs model has a $(d\\!-\\!3)$-dim ’t
Hooft defect, on which a magnetic flux or a gauge vortex can end. Their
$(d\\!-\\!3)$-form topological charge is related to
$\pi_{2}BU(1)\simeq\mathbb{Z}$. This gives rise to a $(d\\!-\\!3)$-form $U(1)$
solitonic symmetry.

In these examples, the solitons and their conservation laws are controlled by
$\pi_{\bullet}Y$, the homotopy groups of the target space $Y$. This
conventional treatment suggested a conventional wisdom that the $p$-form
solitonic symmetry is described by

$\mathrm{Hom}\Bigl{(}\pi_{d-p-1}Y,\,U(1)\Bigr{)}\,.$ (14)

Indeed, in the two examples above, the relevant topological functionals can be
constructed via the universal invertible form (7). Thus the solitonic
symmetries in these two examples are indeed invertible and described by Eq.
(14).

We would like to note that the above examples are actually selected rare
cases. In general, most of the topological charges prescribed by homotopy
groups cannot be detected by the universal invertible construction (7), and
the solitonic symmetry is not given by Eq. (14). The key problem is the
following.

  * •

The notion of the dimension of a solitonic configuration is, in general,
ambiguous. A generic soliton may be a mixture of solitons of clear dimensions.

This is especially often the case for the solitonic configuration bounded by
solitonic defects, which determines the correlation function of these
solitonic defects.

  * •

A $p$-dim solitonic defect may be able to create/annihilate not only
$(p+1)$-dim solitons but also solitons of all dimensions $<p+1$.

  * •

A $p$-dim solitonic defect can carry $q$-form topological charges for all
$0\leq q\leq p$.

This is not surprising: Any conserved $q$-form charge can be carried by
operators of dimension $\geq q$, which has already been noticed since the
beginning of generalized symmetry (Gaiotto:2014kfa, ). This phenomenon is the
starting point that led us to topological charges beyond homotopy groups in
the $\mathbb{C}P^{1}$ model (Chen:2022cyw, ), which provides an example of
non-invertible solitonic symmetry. The entire Sec. 5 will be devoted to
discussing how general solitonic symmetry goes beyond homotopy groups and
becomes non-invertible.

##  3 Topological functional

From the contemporary perspective, understanding solitonic symmetry exactly
means understanding topological functionals. We shall carefully inspect the
notion of topological functionals in this section. At first glance, functional
operators look more orthodox and simpler than defect operators. However, we
will find that they are much subtler than defects, and several physical
requirements vastly constrain a well-behaved notion of topological
functionals. Pursuing a well-behaved notion of topological functionals, we
will eventually bring ourselves to Ansatz 3.2 which claims that topological
functionals are best understood as the partition functions of auxiliary fully-
extended TQFTs.

###  3.1 The identity problem

As we have mentioned at the beginning of Sec. 2.2.1, a topological functional
on a manifold $M$ has to factor through $[M,Y]$. However, not every
$[M,Y]\mapsto\mathbb{C}$ gives rise to topological functionals. A priori, only
the topology of $M$ matters for a topological functional. We cannot
distinguish the original $M$ and a new $M$ transformed by a self-
diffeomorphism $M\stackrel{{\scriptstyle f}}{{\to}}M$. Accordingly, let us
consider two different configurations $M\stackrel{{\scriptstyle a}}{{\to}}Y$
and $M\stackrel{{\scriptstyle b}}{{\to}}Y$ such that $f$ transforms one to the
other, say $b=f\circ a$. Then, a topological functional we can construct with
our bare hands must be blind to the difference between $a$ and $b$. More
formally, let us consider the mapping class group $\pi_{0}\mathrm{Diff}(M)$,
i.e., the group of the isotopy classes of self-diffeomorphisms on $M$. This
group acts on $[M,Y]$ as described above. Then, a topological functional
should actually factor through the equivalence classes,

$[M,Y]\Bigr{/}\pi_{0}\mathrm{Diff}(M)\,.$ (15)

Unfortunately, in most interesting cases, the $\pi_{0}\mathrm{Diff}(M)$-action
on $[M,Y]$ results in vast degeneracies. Many elements in $[M,Y]$ must be
considered identical. We thus call this the identity problem. This problem
terribly prevents us from having sufficiently many meaningful topological
functionals.

There is one way out, to add some extra structure to $M$. The self-
diffeomorphism $f$ may transform this structure into a different structure of
the same type. If this happens, we can distinguish $a$ and $b$ by their
relative relationship to the extra structure. More formally, now we should
consider the mapping class group
$\pi_{0}\mathrm{Diff}(M,\gamma)\subseteq\pi_{0}\mathrm{Diff}(M)$ of isotopy
classes of self-diffeomorphisms that preserve the extra structure $\gamma$.
Then a topological functional that relies on $\gamma$ should factor through
the equivalence classes,

$[M,Y]\Bigr{/}\pi_{0}\mathrm{Diff}(M,\gamma)\,.$ (16)

In the consistent practice of physicists, we almost always unconsciously
assume some extra structure; recall “$\mathbb{E}$-orientation” for Eq. (7). It
is precisely the identity problem that motivates our subconscious to do so.
Such $\mathbb{E}$-orientations are primary examples of the structure $\gamma$.
Thus we shall just call $\gamma$ a generalized orientation. Consequently, a
topological functional that can detect sufficiently many deformation classes
of field configurations needs to rely on a generalized orientation $\gamma$.

#### Example: orientation

For example, let us consider $Y\simeq S^{2}$ and $M\simeq S^{2}$. Then we have
$[M,Y]\simeq\pi_{2}(S^{2})\simeq\mathbb{Z}$. Let us consider the reflection
self-diffeomorphism $f$ defined by $f(\vec{n})=-\vec{n}$ where we view
$S^{2}\subseteq\mathbb{R}^{3}$. This self-diffeomorphism generates

$\pi_{0}\mathrm{Diff}(S^{2})\simeq\mathbb{Z}_{2}\,.$ (17)

Clearly, $f$ acts on $[M,Y]$ via $\mathbb{Z}\to-\mathbb{Z}$. Thus with bare
hands, we cannot distinguish two configurations labeled by opposite integers.
To break the ice, we note that $M$ is orientable. Recall that orientations
form a $H^{0}(-;\mathbb{Z}_{2})$-torsor. Thus there are two orientations on
$M$, $\xi$ and $\xi^{\prime}$, which are exchanged under the transformation of
$f$. That is,

$\pi_{0}\mathrm{Diff}(S^{2},\xi)\simeq 0\,.$ (18)

Combining configurations with orientations, we see that for an
$n\in\mathbb{Z}\simeq[M,Y]$, $(n,\xi)\stackrel{{\scriptstyle
f}}{{\longleftrightarrow}}(-n,\xi^{\prime})$ and
$(-n,\xi)\stackrel{{\scriptstyle f}}{{\longleftrightarrow}}(n,\xi^{\prime})$
are distinct from each other and cannot be mixed by $f$. Based on an
orientation, we can construct the following topological functional,

$U_{\theta}(M)\ \equiv\
\exp\left\\{\mathrm{i}\theta\int_{M}\sigma^{*}b\right\\}\,,\qquad\forall\theta\in\mathbb{R}/2\pi\mathbb{Z}\,,$
(19)

where $b$ denotes the canonical 2-form on $S^{2}$ that integrates to $1$. This
operator can distinguish any two elements in $[M,Y]$. We can also recast
operator (19) into the universal form (7) by choosing
$\mathbb{E}\simeq\mathbb{H}\mathbb{Z}$. Concretely, in Eq. (7), we take
$\omega$ as a generator of $H^{2}(Y;\mathbb{Z})\simeq\mathbb{Z}$, and
$g\in\mathrm{Hom}(\mathbb{Z},U(1))\simeq\mathbb{R}/2\pi\mathbb{Z}$.

#### Example: spin structure

Besides orientations, other structures may also be needed, such as spin
structures. An interesting example was presented in our earlier work
(Chen:2022cyw, ) (also implicitly in Ref. (Freed:2017rlk, )). We consider
$Y\simeq S^{2}$ and $M\simeq S^{2}\times S^{1}$. With some efforts we can
compute

$[S^{2}\\!\times\\!S^{1},S^{2}]\,\simeq\,\bigl{\\{}(m,\ell)\,\bigl{|}\,m\in\mathbb{Z},\,\ell\in\mathbb{Z}_{2|m|}\bigr{\\}}\,,$
(20)

where $\mathbb{Z}_{0}$ means $\mathbb{Z}$. For a configuration $\sigma$, the
$m$ labels $\sigma|_{S^{2}\\!\times\\!\\{p\\}}$ in
$\pi_{2}(S^{2})\simeq\mathbb{Z}$ for an arbitrary $p\in S^{1}$. Now consider
the twist diffeomorphism $f$ defined by
$f(\vec{n},t)\equiv(\mathrm{e}^{t\hat{\vec{z}}\times}\vec{n},t)$, where we
view $S^{2}\subseteq\mathbb{R}^{3}$ and
$S^{1}\simeq\mathbb{R}/2\pi\mathbb{Z}$, as well as another diffeomorphism $g$
defined by $g(\vec{n},t)\equiv(-\vec{n},-t)$. Both $f$ and $g$ preserve an
orientation. Actually, for an orientation $\xi$, they generate

$\pi_{0}\mathrm{Diff}(S^{2}\\!\times\\!S^{1},\xi)\simeq\mathbb{Z}_{2}\times\mathbb{Z}_{2}\,.$
(21)

Both $f$ and $g$ induce an almost double degeneracy on $[M,Y]$:
$(m,\ell)\stackrel{{\scriptstyle g}}{{\longleftrightarrow}}(-m,\ell)$ while
$f$ exchanges $(m,\ell_{1})$ and $(m,\ell_{2})$ if $2\ell_{1}=2\ell_{2}$. For
example, $f$ transforms $(\vec{n},t)\mapsto\vec{n}$ into
$(\vec{n},t)\mapsto\mathrm{e}^{t\hat{\vec{z}}\times}\vec{n}$, and these two
maps belong to different classes $(1,0)$ and $(1,1)$ in $[M,Y]$, respectively.

We would like to lift the $f$-degeneracy. We note that $M$ is spinnable.
Recall that, on top of a given orientation, spin structures form a
$H^{1}(-;\mathbb{Z}_{2})$-torsor. Since
$H^{1}(S^{2}\\!\times\\!S^{1};\mathbb{Z}_{2})\simeq\mathbb{Z}_{2}$, $M$ have
two different spin structures $\rho$ and $\rho^{\prime}$ on top of an
orientation $\xi$. They are exchanged by $f$. That is,

$\pi_{0}\mathrm{Diff}(S^{2}\\!\times\\!S^{1},\xi,\rho)\simeq\mathbb{Z}_{2}\,,$
(22)

where only the $g$-degeneracy remains. Based on a spin structure, we can
construct the following topological functional as a spin Chern-Simons
integral,

$U_{k}(M)\ \equiv\
\exp\left\\{\mathrm{i}k\int_{M}\frac{\sigma^{*}\\!a\,\mathrm{d}\sigma^{*}\\!a}{4\pi}\right\\}\,,$
(23)

where $k\in\mathbb{Z}$, $k\sim k+2$, and $a$ denotes the $U(1)$ gauge field on
$S^{2}$ associated with the Hopf fibration $S^{1}\to S^{3}\to S^{2}$, which is
related to $b$ in the former example (19) via $b=\mathrm{d}a/2\pi$. We can
also recast operator (23) into the universal form (7) by choosing
$\mathbb{E}\simeq\mathbb{KO}$ since a $\mathbb{KO}$-orientation is exactly a
spin structure. Concretely, in Eq. (7), we take $\omega$ as a generator of
$KO^{2}(Y)\simeq\mathbb{Z}$, and
$g\in\mathrm{Hom}(KO_{1},U(1))\simeq\mathrm{Hom}(\mathbb{Z}_{2},U(1))$. The
operator (23) can lift the almost double degeneracy in $[M,Y]$ caused by $f$.
Nevertheless, distinguishing other elements in $[M,Y]$ (except the
$g$-degeneracy) requires non-invertible topological functionals based on a
spin structure as the present authors showed in Ref. (Chen:2022cyw, ), which
will also be discussed in Sec. 5.2.2 in this paper.

###  3.2 The coherence problem

The identity problem concerns a topological functional on a single supporting
manifold only. An even more severe problem appears if we move from one
supporting manifold to another. Namely, for manifolds $M_{1}\not\simeq M_{2}$,
given a function on $[M_{1},Y]$ and another function on $[M_{2},Y]$, how can
we tell whether they are just different topological functionals or different
realizations of the same topological functional? An incorrect assignment would
lead to wrong solitonic physics. We call this the coherence problem.

We can learn hints from the universal construction (7) for invertible
solitonic symmetry. It is naturally defined on any closed
$\mathbb{E}$-orientable $\mathbb{E}$-oriented manifolds. Also, its concrete
incarnations in operators (19) and (23) are automatically defined on any
closed oriented orientable manifolds and any closed spin spinnable manifolds,
respectively. It is natural to regard the operators on different manifolds but
defined by the same expression Eq. (7) as the different realizations of the
same topological functional.

The above observation suggests that the solution to the coherence problem is
to require locality. Recall that defect operators are defined by infinitesimal
boundary conditions around the operator (see Sec. 2.2.2). This definition
concerns field configurations in the vicinity of each point on the supporting
manifold and thus satisfies a good sense of locality. However, a naively
defined functional operator may behave quite non-local and might not yield a
physically sensible operator. We propose that a functional operator satisfies
locality if it is the multiplication of piecewise data localized around each
point. The universal invertible construction (7) provides the special cases
where multiplication is given the “exponentiation” of summation, represented
by morphisms to $U(1)$. Furthermore, gluing these local data may require a
generalized orientation on supporting manifolds. In summary,

$\text{Locality}\equiv\text{Multiplying local data with respect to generalized
orientation}.$ (24)

A functional satisfying locality automatically renders coherence on a class of
manifolds with the prescribed generalized orientation.

A natural subsequent question is whether there is a universal construction of
multiplying local data, which goes beyond the exponentiation (7) and can
capture non-invertible cases. We propose a positive answer: The universal way
of multiplying local data is another path integral, and a most general
functional that satisfies locality is the partition function of another fully-
extended QFT. Namely, we can consider some auxiliary fields that inhabit the
operator manifold only, couple them to the dynamical quantum fields, and
perform a path integral of the auxiliary fields. The output of this auxiliary
path integral is the partition function of an auxiliary fully-extended QFT
that inhabits the operator manifold only. We optimistically assume that all
functional operators satisfying locality can be produced by the partition
functions of some auxiliary fully-extended QFT. In particular, we require all
topological functionals to satisfy locality, i.e., all topological functionals
$\mathcal{T}[M,\sigma]$ are assumed to be produced by the partition functions
of auxiliary topological fully-extended QFTs (TQFTs) coupled to the target
space $Y$:

$\mathcal{T}[M,\sigma]=\text{TQFT partition function that couples with
$\sigma|_{M}:M\to Y$}.$ (25)

In particular, this construction includes Eq. (7) as its special cases for
invertible topological functionals. Also, this construction is consistent with
Propositions 2.3 and 2.1. We now flesh out the precise connotation of “TQFT”
in our proposal.

###  3.3 Our ansatz

In a specific theory, which generalized orientation the topological
functionals rely on should be determined by the theory itself. Therefore,
instead of studying topological functionals for an arbitrary generalized
orientation, we focus on the most common generalized orientations in this
paper. The most fundamental property of a theory that determines the
generalized orientation is particle statistics. As might be surprising at
first glance, a non-Grassmann path integral can be used to define not only a
bosonic QFT that inhabits any oriented spacetime but also a fermionic QFT that
inhabits any spin spacetime. These fermionic theories are not bosonic in
disguise, i.e., the $\mathbb{Z}_{2}$-grading $(-)^{F}$ on states is
nontrivial, as long as the action includes proper spin topological terms.

In a bosonic (resp. fermionic) theory, topological functionals are supposed to
inhabit oriented (resp. spin) manifolds. This is natural since the theory
itself inhabits oriented (resp. spin) spacetime. A more persuasive rationale
is that all the solitonic defects (see Sec. 2.2.2), the charged operators of
topological functionals, are defined by field configurations on oriented
(resp. spin) manifolds. To see this, one notes that the normal sphere bundle
$\mathcal{S}N$ of any closed submanifold $N$ in the spacetime naturally
inherits an orientation (resp. a spin structure) from the spacetime, even if
$N$ itself is not orientable or spinnable. Consequently, the example in Eq.
(23) cannot exist in bosonic theories. We focus on the elementary cases of
bosonic and fermionic theories in the paper while leaving the theories where
other interesting generalized orientations666 Such theories appear typically
when one wants to take into account (1) unorthodox statistics, (2) discrete
spacetime symmetry, (3) mixing between spacetime and internal (non-solitonic)
symmetry, and (4) conditions on higher objects than particles.  are involved
to future work.

We now have all the ingredients to formulate an Ansatz for topological
functionals, however, unfortunately, given that $Y$ satisfies a finiteness
condition.

###### Definition 3.1

Topological space $Y$ is $n$-finite if $\pi_{0}Y$ is finite and $\pi_{q}(Y,y)$
is finite for all $0\leq q\leq n$ and $y\in Y$.

If $M$ is a closed manifold of dimension $\leq n$, $[M,Y]$ is finite when $Y$
is $n$-finite. $[M,Y]$ has the chance to have infinitely many elements if $Y$
is not $n$-finite. Infinitely many topological charges are a sign of
continuous symmetry. The existence of infinitely many elements in $[M,Y]$ and
infinitesimal symmetry transformations cause tricky technical troubles. We
shall present a systematic treatment of discrete solitonic symmetry but only
an approximating treatment of continuous solitonic symmetry.

Let us describe our ansatz for topological functionals responsible for
discrete solitonic symmetry. To produce bosonic (resp. fermionic) topological
functionals, TQFTs themselves must be bosonic (resp. fermionic). Their
partition functions inhabit closed manifolds equipped with a map to the target
space $Y$, $\sigma|_{M}:M\to Y$. To justify “multiplication of local data”,
these TQFTs must be maximally local, i.e., they should be fully-extended
TQFTs, and we reach the following ansatz777 When the theory itself is
topological, there have been attempts to classify defect operators
(Freed:2009qp, ), which is consistent with our Ansatz. :

###### Ansatz 3.2

For an $n$-finite space $Y$, an $n$-dim bosonic (resp. fermionic) topological
functional to $Y$ is the partition function of an $n$-dim bosonic (resp.
fermionic) $Y$-enriched fully-extended TQFT.

We regard this ansatz as a complete characterization of topological
functionals. Based on the physical principle that a QFT is completely
determined by its partition functions (in the presence of various kinds of
background fields), we make the following conjecture.

###### Conjecture 3.3

For an $n$-finite space $Y$, inequivalent $n$-dim bosonic (resp. fermionic)
$Y$-enriched fully-extended TQFTs produce different $n$-dim bosonic (resp.
fermionic) topological functionals to $Y$.

Ansatz 3.2 and Conjecture 3.3 will be the foundation for our analysis of
solitonic symmetry in this paper. We note that similar TQFTs often appear in a
thriving contemporary theme of physics, the classification of gapped phases.
In particular, they are tightly related to the notion of symmetry-protected
topological phases and symmetry-enriched topological orders.

To conclude this section, we slightly discuss continuous solitonic symmetry.
There are basically two problems. First, although infinitesimal symmetry
generators can be unbounded (self-adjoint operators in the invertible case),
finite symmetry operators must be bounded (unitary operators in the invertible
case), which means we have to require a topological functional
$[M,Y]\mapsto\mathbb{C}$ to be bounded. Second, Conjecture 3.3 fails due to
the existence of non-semisimple TQFTs which superfluous produce duplicated
partition functions as semisimple TQFTs; recall that for group representations
$R\not\simeq A\oplus R/A$, we still have
$\mathrm{tr}_{R}=\mathrm{tr}_{A}+\mathrm{tr}_{R/A}=\mathrm{tr}_{A\oplus R/A}$.
This paper does not attempt a systematic treatment of continuous solitonic
symmetry. Instead, we shall be satisfied with the discrete subsymmetries of
continuous solitonic symmetry. This is realized by considering $n$-finite
homotopy quotients of $Y$.

###### Definition 3.4

$Z$ is a homotopy quotient of $Y$ if there is a map $f:Y\mapsto Z$ such that
$f_{*}:\pi_{0}Y\to\pi_{0}Z$ is surjective and
$f_{*}:\pi_{q}(Y,y)\to\pi_{q}(Z,f(y))$ is an epimorphism for all $q>0$ and
$y\in Y$.

Picking up an $n$-finite homotopy quotient $Z$ of $Y$ and considering
topological functionals that factor through $[-,Z]$, we obtain a discrete
solitonic subsymmetry. We believe that the colimit of all discrete
subsymmetries (see Sec. 4.3.1) leads to an almost faithful approximation to a
continuous symmetry, just like approximating $U(1)$ by
$\mathbb{Q}/\mathbb{Z}\simeq\bigcup_{n\in\mathbb{N}}\mathbb{Z}_{n}$. Besides,
we shall find it easy to describe the continuous solitonic symmetry directly
in some concrete examples.

##  4 Algebraic structure of solitonic symmetry

We are going to reveal the universal algebraic structure of solitonic symmetry
based on Ansatz 3.2 and Conjecture 3.3. First in Sec. 4.1, we present a short
mathematical preliminary on higher-categories. Then in Sec. 4.2, we shall
formulate the mathematical notion of fully-extended TQFTs to clarify the
accurate connotation of Ansatz 3.2. Finally in Sec. 4.3, we shall discuss the
mathematical structure that describes the algebraic structure of solitonic
symmetry.

This section aims to establish a more or less rigorous mathematical ground for
solitonic symmetry. Thus our expositions might inevitably look more or less
abstract. However, readers acquainted with the issue of classifying gapped
phases by fully-extended TQFTs will find the expositions familiar and
recognize tremendous echoes.

###  4.1 Preliminaries on higher-categories

The most efficient way to formulate fully-extended TQFTs needs the package of
higher-categories. The goal here is to overview this nice mathematical package
briefly. We are not attempting a self-contained exposition, but instead, we
present an oversimplified introduction following Sec. 1.3 of Ref.
(Lurie:2009keu, ). Unfamiliar readers could find good entrances in the
literature like Refs. (Lurie:2009keu, ; Baez:1995xq, ; Kapustin:2010ta, ) to
get started into this evolving contemporary discipline.

####  $n$-category

To start with, let us recollect the definition of categories. A category
comprises (i) objects, (ii) a set $\mathrm{Hom}(x,y)$ between any two objects
$x$ and $y$, and (iii) an associative unital composition map
$\mathrm{Hom}(x,y)\times\mathrm{Hom}(y,z)\to\mathrm{Hom}(x,z)$ for any three
objects $x$, $y$, and $z$. In particular, the composition map makes
$\mathrm{Hom}(x,x)$ a monoid. Elements of $\mathrm{Hom}(x,y)$ are called
morphisms. One of the tremendous reasons why categories are useful is that
they are regarded as soft instead of rigid. Namely, we regard two categories
as “the same” as long as they are equivalent rather than isomorphic. This is
similar to algebraic topology, which cares about (weak) homotopy equivalences
instead of homeomorphisms.

Generalizing the above definition, we can sketch the notion of $n$-categories
by an induction. At the root of this induction, a 0-category is a set, and its
elements are called objects or $0$-morphisms. The induction then goes as
follows. An $n$-category comprises (i) objects, also called $0$-morphisms,
(ii) a small $(n\\!-\\!1)$-category $\mathrm{Hom}(x,y)$ between any two
objects $x$ and $y$, and (iii) an associative unital composition functor
$\mathrm{Hom}(x,y)\times\mathrm{Hom}(y,z)\to\mathrm{Hom}(x,z)$ for any three
objects $x$, $y$, and $z$. In particular, the composition functor makes
$\mathrm{Hom}(x,x)$ a monoidal $(n\\!-\\!1)$-category. For all $0\leq p<n$,
$p$-morphisms of $\mathrm{Hom}(x,y)$ are called $(p\\!+\\!1)$-morphisms of
this $n$-category. Clearly, if we take $n=1$, we just recover the definition
for an ordinary category. We can conceive an $\infty$-category as a proper
limit of $n$-categories as $n$ approaches $\infty$. Its $\mathrm{Hom}(x,y)$’s
are also $\infty$-categories.

The above definition sketch looks promising but hides the subtlety in treating
the associativity and the unitality for compositions. We do not need the too
rigid notion of strict $n$-categories, where these conditions are satisfied
literally. Instead, we need weak $n$-categories, where these conditions are
satisfied up to specified equivalences. For example, in the case of $n=2$, we
want $\mathrm{Hom}(X,X)$ to be a (weak) monoidal category rather than a strict
monoidal category. However, as $n$ increases, characterizing accurately all
the axioms gets rapidly a formidable task. Different models for organizing
this have been proposed, and the equivalence between them, though widely
believed, is a matter of ongoing research. We shall drop the prefix “weak”
henceforth.

We now introduce two convenient notations. Let us consider an $n$-category
$\mathsf{C}$ with a distinguished object $1_{\mathsf{C}}$. For example,
$\mathsf{C}$ might be a monoidal $n$-category, which is an $n$-category with
the “tensor product” $\otimes$ and the unit object $1_{\mathsf{C}}$. We
conceive a monoidal $(n\\!-\\!1)$-category $\Omega\mathsf{C}$ via

$\Omega\mathsf{C}\equiv\mathrm{Hom}(1_{\mathsf{C}},1_{\mathsf{C}})\,.$ (26)

When $\mathsf{C}$ is a monoidal $n$-category, we conceive a one-object
$(n\\!+\\!1)$-category $B\mathsf{C}$ such that

$\Omega B\mathsf{C}=\mathsf{C}\,.$ (27)

Remarkably, if $\mathsf{C}$ is further symmetric, $B\mathsf{C}$ has a
canonical symmetric monoidal structure. In this particular case, we can
further define iterated $B^{n}\mathsf{C}$. The two operations $\Omega$ and $B$
are apparently borrowed from looping and delooping in algebraic topology. The
reason why they are adopted will be clear shortly.

#### Space and $n$-groupoid

We can extract an $n$-category $\pi_{\leq n}X$ from each topological space
$X$. Objects thereof are points in $X$, 1-morphisms are homotopies of objects
(paths), 2-morphisms are homotopies of 1-morphisms (based homotopies of
paths), 3-morphisms are homotopies of 2-morphisms (based homotopies of based
homotopies of paths), and so on inductively until that $n$-morphisms are the
homotopy classes of homotopies of $(n\\!-\\!1)$-morphisms. Note that we can
allow $n=\infty$ by canceling terminating the induction and deleting the
eventual command of taking the homotopy class in the above construction.

The $n$-category $\pi_{\leq n}X$ has a special property that all of its
morphisms are invertible up to equivalence. In general, an $n$-category with
such a property is called an $n$-groupoid. Thus $\pi_{\leq n}X$ is called the
fundamental $n$-groupoid of $X$. This $\pi_{\leq n}X$ knows a great deal about
the topology of $X$. First, the equivalence classes of objects in $\pi_{\leq
n}X$ exactly constitute $\pi_{0}X$. Second, for any base point $x\in X$, the
fusion monoid of the equivalence classes of objects in $\Omega^{q}\pi_{\leq
n}X$ is exactly the homotopy group $\pi_{q}(X,x)$ for $q\leq n$ and the
trivial group for $q>n$, accordingly. Furthermore, the entire first $n$-th
stages of the Postnikov tower of each path component of $X$ are encoded in
$\pi_{\leq n}X$. In other words, the fundamental $n$-groupoid completely
determines the homotopy $n$-type of the space. Namely, $\pi_{\leq
n}X\simeq\pi_{\leq n}Y$ as long as there is an $(n\\!+\\!1)$-connected map
between $X$ and $Y$.

It is a classical theorem that every groupoid is equivalent to the fundamental
groupoid of some space. Since Quillen (Quillen:1967, ) and Grothendieck
(Grothendieck:1983, ), it has also been generally accepted that every
$n$-groupoid is equivalent to the fundamental $n$-groupoid of some space.
Therefore, based on the homotopical property discussed above, we have the
following equivalence.

###### Philosophy 4.1

An $n$-groupoid is equivalent to a homotopy $n$-type. That is, an $n$-groupoid
is an $n$-aspherical space, and the equivalence between $n$-groupoids is the
weak homotopy equivalence between $n$-aspherical spaces.

One can interpret this equivalence as saying that $n$-categories are non-
invertible generalizations of homotopy $n$-types. Let us look at the
particular case of $n=\infty$. Then $\pi_{\leq\infty}X$ encodes the entire
Postnikov tower of $X$ and completely determines the homotopy type of $X$. The
above equivalence then suggests that we can model $\infty$-groupoids by
spaces.

###### Philosophy 4.2

An $\infty$-groupoid is equivalent to a homotopy type. That is, an
$\infty$-groupoid is a space, and the equivalence between $\infty$-groupoids
is the weak homotopy equivalence between spaces.

An invertible monoidal $(n\\!-\\!1)$-groupoid is called an $n$-group. An
$\infty$-group is equivalent to a loop space. $\Omega$ and $B$ establish a
one-to-one correspondence between $n$-groups and one-object
$(n\\!+\\!1)$-groupoids.

We also unify the symbols for spaces and higher-groupoids. We shall write
$\pi_{\leq\infty}X$ just as $X$ and abandon the now tautological symbol
$\pi_{\leq\infty}X$. We shall also write the homotopy $n$-type of $X$,
incarnated by the $n$-th Postnikov truncation of $X$, just as $\pi_{\leq n}X$.
It is reasonable to regard the higher-category theory as the non-invertible
generalization of algebraic topology, in the sense that any correct model for
higher-categories should produce philosophies 4.1 and 4.2 as theorems.

####  $(n,r)$-category

We now learned that the invertibility of morphisms is a distinguished property
for higher-categories. Thus people introduced the notion of
$(n\\!+\\!r,n)$-categories to specify the information about invertibilities.
From one perspective, $(n\\!+\\!r,n)$-category is just an
$(n\\!+\\!r)$-category whose $p$-morphisms are invertible up to equivalence
for all $p>n$. For example, an $(r,0)$-category just means an $r$-groupoid and
an $(n,n)$-category just means an $n$-category. For $n_{1}\leq n_{2}\leq n$,
an $(n,n_{1})$-category is also an $(n,n_{2})$-category.

From another perspective, an $(n\\!+\\!r,n)$-category is an $n$-category
enriched over $r$-groupoids. Namely, in the previous definition sketch of
$n$-categories, we now choose to start the induction from $r$-groupoids
instead of the mere sets. These $r$-groupoids are directly modeled by
topological spaces according to philosophies 4.1 (and 4.2). This second
perspective has bonus advantages in some aspects and has thus drawn vast
attention. In particular, researches on $(\infty,n)$-categories are thriving.

###  4.2 Fully-extended TQFT

A general $n$-dim fully-extended TQFT is formulated as a symmetric monoidal
functor between two symmetric monoidal $(\infty,n)$-categories, which
axiomatizes the “results” of the path integral. The domain is a bordism
$(\infty,n)$-category; the codomain is a fully-dualizable
$(\infty,n)$-category. The specific physics context determines the choice of
domains and codomains.

####  4.2.1 Bordism domain $(\infty,n)$-category

An $n$-tangential structure is a map
$X\stackrel{{\scriptstyle\Gamma}}{{\to}}BO(n)$. The simplest examples include

$n$-framing: $\displaystyle\qquad\\{*\\}\stackrel{{\scriptstyle
fr}}{{\longrightarrow}}BO(n)\,,$ (28a) $n$-orientation:
$\displaystyle\qquad\text{canonical }BSO(n)\stackrel{{\scriptstyle
SO}}{{\longrightarrow}}BO(n)\,,$ (28b) $n$-spin structure:
$\displaystyle\qquad\text{canonical }BSpin(n)\stackrel{{\scriptstyle
Spin}}{{\longrightarrow}}BO(n)\,.$ (28c)

For an $n$-tangential structure $\Gamma$ and for $q\leq n$, a $q$-dim $\Gamma$
manifold means a $q$-dim manifold $M$ equipped with a map $M\to X$, such that
the composition $M\to X\stackrel{{\scriptstyle\Gamma}}{{\to}}BO(n)$ classifies
the $n$-stabilized tangent bundle of $M$ (i.e.
$TM\oplus\underline{\mathbb{R}}^{n-q}$).

Given an $n$-tangential structure $\Gamma$, Lurie (Lurie:2009keu, , Sec. 2.2)
conceives a symmetric monoidal $(\infty,n)$-category
$\mathsf{Bord}^{\Gamma}_{n}$ as follows. In the non-invertible-morphism region
of $q\leq n$, a $q$-morphism is a $q$-dim $\Gamma$ bordism, i.e., a $q$-dim
$\Gamma$ manifold with corners. In the invertible-morphism region, the
homomorphism $\infty$-groupoids between $n$-morphisms are the spaces of
boundary-fixed diffeomorphisms along the philosophy 4.2. The symmetric
monoidal structure on $\mathsf{Bord}^{\Gamma}_{n}$ is prescribed by the
disjoint union. Such $\mathsf{Bord}^{\Gamma}_{n}$’s give the domains of a
fully-extended TQFT. However, given that the codomains we will be considering
are essentially $n$-categories, the spaces of diffeomorphisms will not
contribute to our results.

We want to equip all manifolds with a map to the target space $Y$. The
simplest way to implement this is to adopt a special $n$-tangential structure
$e_{Y}\\!\times\\!\Gamma:Y\\!\times\\!X\to BO(n)$, the product of the collapse
map $Y\stackrel{{\scriptstyle e_{Y}}}{{\to}}\\{*\\}$ and another
$n$-tangential structure $X\stackrel{{\scriptstyle\Gamma}}{{\to}}BO(n)$, such
as those listed in Eq. (28). In this case, we shall particularly call
$e_{Y}\\!\times\\!\Gamma$ manifolds $Y$-enriched $\Gamma$ manifolds, and
particularly write

$\mathsf{Bord}^{\Gamma}_{n}(Y)\equiv\mathsf{Bord}^{e_{Y}\times\Gamma}_{n}\,.$
(29)

Such $\mathsf{Bord}^{\Gamma}_{n}(Y)$’s give the domains for $Y$-enriched
fully-extended TQFTs. This notion of enriched TQFT is tightly related to
symmetric TQFT (or say equivariant TQFT). Due to the dimensional reason, a map
from a manifold of dimension $\leq n$ to $Y$ factors into $Y$’s homotopy
$n$-type, $\pi_{\leq n}Y$. That is, we have the following equivalence,

$\mathsf{Bord}^{\Gamma}_{n}(Y)\simeq\mathsf{Bord}^{\Gamma}_{n}(\pi_{\leq
n}Y)\,.$ (30)

If $Y$ is path connected, an $n$-dim $Y$-enriched TQFT is exactly an $n$-dim
TQFT that acquires an action by the higher-group $\Omega\pi_{\leq n}Y$, i.e.
an $\Omega\pi_{\leq n}Y$-symmetric TQFT. If $Y$ has multiple path components,
we obtain a TQFT consisting of a $\pi_{0}Y$ worth of universes, each of which
is a higher-group-symmetric TQFT.

Physically, a priori, any TQFT suffers from a framing anomaly and requires a
framing dependence. Therefore, $\mathsf{Bord}^{fr}_{n}(Y)$ would become a
suitable choice of the domain for all TQFTs. A universal $Y$-enriched fully-
extended TQFT with framing is a symmetric monoidal functor

$\mathcal{Z}:\mathsf{Bord}_{n}^{fr}(Y)\ {\to}\ \mathsf{D}_{n}\,,$ (31)

where the target category is a fully-dualizable symmetric monoidal
$(\infty,n)$-category $\mathsf{D}_{n}$ as we shall see later. The maximal
$\infty$-groupoid inside $\mathsf{D}_{n}$ acquires a natural homotopy
$O(n)$-action according to the cobordism hypothesis (Lurie:2009keu, , Theorem
2.4.6 and Corollary 2.4.10). This homotopy $O(n)$-action can be lifted to a
homotopy $\Omega X$-action by an $n$-tangential structure
$X\stackrel{{\scriptstyle\Gamma}}{{\to}}BO(n)$. If this $\Omega X$-action is
(canonically) trivializable, the a priori framing anomaly turns out to be
merely a $\Gamma$ anomaly, and then the TQFT $\mathcal{Z}$ does not really
depend on an $n$-framing but an $n$-tangential structure $\Gamma$ instead.
Consequently, in such a situation, we can (canonically) extend the domain of
$\mathcal{Z}$ to any $Y$-enriched $\Gamma$ manifolds so that

$\mathcal{Z}^{\Gamma}:\mathsf{Bord}_{n}^{\Gamma}(Y)\ {\to}\ \mathsf{D}_{n}\,.$
(32)

Along this philosophy of treatment, what manifolds a TQFT can inhabit are
determined by the property of its codomain888 This treatment may be phrased as
codomain-dominated. There is also a domain-dominated treatment, where we take
a universal codomain $\mathsf{D}_{n}$ (for each $n$) and ask $\Gamma$ to vary.
Then the logic will be reversed, i.e., the choice of $\Gamma=SO$ (resp.
$\Gamma=Spin$) implies that bosonic (resp. fermionic) state spaces will be
picked up in the universal codomain. Since a universal codomain is challenging
to construct, we take the codomain-dominated approach. .

####  4.2.2 Physical codomain $(\infty,n)$-category

Let $\mathsf{D}_{n}$ denote a fully-dualizable symmetric monoidal
$(\infty,n)$-category that we want to use as the codomain for TQFTs (see Sec.
2.3 of Ref. (Lurie:2009keu, ) for the definition of full dualizability). For a
physical TQFT from domain $\mathsf{Bord}^{\Gamma}_{n}(Y)$, we want to assign a
complex number to each closed $n$-dim $Y$-enriched $\Gamma$ manifold as its
partition function, and assign a vector space to each closed $(n\\!-\\!1)$-dim
$Y$-enriched $\Gamma$ manifolds as its state space. In other words, we want
$\Omega^{n-1}\mathsf{D}_{n}$ to be an $(\infty,1)$-completion of a symmetry
monoidal category of proper vector spaces. There are two inequivalent natural
$(\infty,1)$-completions of a category of vector spaces.

  * •

Only identity higher morphisms are added. $\mathrm{Hom}(-,-)$ has the discrete
topology.

  * •

Iterated isotopies are added. $\mathrm{Hom}(-,-)$ has the subspace topology
from some $\mathbb{C}^{m}$.

The second completion is appropriate for classifying gapped phases because it
identifies TQFT deformation classes999 It captures deformation classes of not
only orthodox TQFTs but also half-geometric-half-topological QFTs. These
unorthodox “TQFTs” depend on geometric data of background fields
$\sigma:M\mapsto Y$ but are invariant under spacetime diffeomorphisms. This is
exactly what is needed for classifying gapped phases. . However, the first
completion is appropriate for our purpose because we want to speak of the
partition function of each individual TQFT. Thus in this paper, we regard the
first completion as the canonical $(\infty,1)$-completion and abuse the same
symbol of the category to denote also its canonical $(\infty,1)$-completion.

The choice of $\Omega^{n-1}\mathsf{D}_{n}$ is determined by what state spaces
we want. In the bosonic case, we consider
$\Omega^{n-1}\mathsf{D}_{n}\simeq\mathsf{Vect}^{fd}$ whose objects are finite-
dim $\mathbb{C}$-linear spaces and morphisms are $\mathbb{C}$-linear maps. The
tensor product gives rise to a monoidal structure on $\mathsf{Vect}^{fd}$. A
swap map comes from the index exchange,

$x\otimes y\,\mapsto\,y\otimes x\,,$ (33)

which makes $\mathsf{Vect}^{fd}$ a symmetric monoidal category. All objects
have duals due to the finite-dim condition. It also has finite biproducts
given by the direct sum, is semisimple with respect to it, and has a unique
class of simple objects. It is further $\mathbb{C}$-linear. All these
structures make $\mathsf{Vect}^{fd}$ a symmetric fusion category.

As for the fermionic case, the state space $V$ has a $\mathbb{C}$-linear
involution $\varepsilon$ called fermionic parity, which makes the state space
$\mathbb{Z}_{2}$-graded. Such a pair $(V,\varepsilon)$ is called a super
vector space. The $(+\\!1)$-eigenspace of $\varepsilon$ is the bosonic sector
and the $(-\\!1)$-eigenspace is the fermionic sector. Let
$\mathsf{sVect}^{fd}$ denote the category whose objects are finite-dim super
$\mathbb{C}$-linear spaces and morphisms are $\mathbb{C}$-linear maps that
commute with fermionic parities. Its monoidal structure also comes from the
tensor product,

$\bigl{(}X,\varepsilon_{X}\bigr{)}\otimes\bigl{(}Y,\varepsilon_{Y}\bigr{)}\simeq\bigl{(}X\otimes
Y,\,\varepsilon_{X}\otimes\varepsilon_{Y}\bigr{)}\,.$ (34)

$\mathsf{sVect}^{fd}$ is also $\mathbb{C}$-linear, has finite biproducts, and
has duals for all objects, making it a fusion category. As fusion categories,
$\mathsf{sVect}^{fd}$ is equivalent to $\mathsf{Rep}(\mathbb{Z}_{2})$, the
representation category of $\mathbb{Z}_{2}$. It is the swap map that
distinguishes them as inequivalent symmetric fusion categories. The swap map
in $\mathsf{sVect}^{fd}$ encodes the fermionic statistics following the Koszul
sign rule. Namely, for fermionic parity eigenstates $x\in X$ and $y\in Y$, the
swap map sends

$x\otimes y\,\mapsto(-)^{|x||y|}\,y\otimes x\,,$ (35)

where $|\bullet|\\!=\\!0$ if $\bullet$ is bosonic and $|\bullet|\\!=\\!1$ if
$\bullet$ is fermionic. We take
$\Omega^{n-1}\mathsf{D}_{n}\simeq\mathsf{sVect}^{fd}$ for the fermionic case.

The final step is to determine the entire $\mathsf{D}_{n}$. Since we do not
have any requirement on higher objects than particles, our guiding principle
is that no superfluous data on lower-dim manifolds, except when necessary,
should be introduced. Gaiotto and Johnson-Freyd found an elegant solution in
Ref. (Gaiotto:2019xmp, ) (see also Ref. (douglas2018fusion, )). They noticed
the Karoubi-completeness among various other features and clarified that the
minimal necessary complexity is exactly the $n$-categorical generalization of
being Karoubi-complete. Roughly speaking, being Karoubi-complete means that
every idempotent comes from a splitting, among all $p$-morphisms. Given a
Karoubi-complete monoidal $n$-category $\mathsf{C}$, they defined its “stable
suspension” $\Sigma\mathsf{C}$ as the Karoubi completion of its delooping
$B\mathsf{C}$. Symbolically,

$\Sigma\mathsf{C}\equiv\mathrm{Kar}(B\mathsf{C})\,.$ (36)

When the above $n$-category $\mathsf{C}$ is further symmetric,
$\Sigma\mathsf{C}$ turns out to be a Karoubi-complete symmetric monoidal
$(n+1)$-category (Gaiotto:2019xmp, , Theorem 4.1.1). Given that both
$\mathsf{Vect}^{fd}$ and $\mathsf{sVect}^{fd}$ are Karoubi-complete, the
codomain $\mathsf{D}_{n}$ we are looking for can be given by

(bosonic) $\displaystyle\Sigma^{n-1}\mathsf{Vect}^{fd}\,,$ (37a) (fermionic)
$\displaystyle\Sigma^{n-1}\mathsf{sVect}^{fd}\,.$ (37b)

We shall flexibly regard them as either $n$-categories or
$(\infty,n)$-categories (with identity higher morphisms) according to the
context. Gaiotto and Johnson-Freyd also conjecture (Gaiotto:2019xmp, ,
Conjecture 1.4.6) that the canonical homotopy $SO(n)$-action [resp.
$Spin(n)$-action] on the maximal $\infty$-groupoid inside
$\Sigma^{n-1}\mathsf{Vect}^{fd}$ (resp. $\Sigma^{n-1}\mathsf{sVect}^{fd}$) is
canonically trivializable. If these conjectured properties are true, according
to the discussion around Eq. (32), bosonic (resp. fermionic) topological
functionals indeed inhabit any closed oriented (resp. spin) manifold.

####  4.2.3 Summary of the formulation

We now summarize all the ingredients found above to present accurate
formulations of the relevant TQFTs to clarify the connotation of Ansatz 3.2. A
priori, any TQFT suffers from a framing anomaly and requires a framing
dependence.

###### Definition 4.3

An $n$-dim bosonic (resp. fermionic) $Y$-enriched fully-extended TQFT is a
symmetric monoidal functor between symmetric monoidal $(\infty,n)$-categories,

$\mathcal{B}:\mathsf{Bord}_{n}^{fr}(Y)\ {\to}\
\Sigma^{n-1}\mathsf{Vect}^{fd}\,,\qquad\Bigl{[}\text{resp. }\
\mathcal{F}:\mathsf{Bord}_{n}^{fr}(Y)\ {\to}\
\Sigma^{n-1}\mathsf{sVect}^{fd}\Bigr{]}\,.$ (38)

It is reasonable to anticipate that having bosonic (resp. fermionic) state
spaces logically implies inhabiting any oriented (resp. spin) spacetime, i.e.,
the framing anomaly should merely be an orientation (resp. spin) anomaly. This
anticipation is realized by the following conjectured property of
$\Sigma^{n-1}\mathsf{Vect}^{fd}$ and $\Sigma^{n-1}\mathsf{sVect}^{fd}$
(Gaiotto:2019xmp, ):

###### Conjecture 4.4

The canonical homotopy $SO(n)$-action [resp. $Spin(n)$-action] on the maximal
$\infty$-groupoid inside $\Sigma^{n-1}\mathsf{Vect}^{fd}$ (resp.
$\Sigma^{n-1}\mathsf{sVect}^{fd}$) is canonically trivializable.

By the cobordism hypothesis combined with this conjecture, every $\mathcal{B}$
(resp. $\mathcal{F}$) has no genuine framing dependence but just an
orientation (resp. spin) dependence, so every $\mathcal{B}$ or $\mathcal{F}$
can be canonically extended to

$\displaystyle\mathcal{B}^{SO}:$ $\displaystyle\ \mathsf{Bord}_{n}^{SO}(Y)\
{\to}\ \Sigma^{n-1}\mathsf{Vect}^{fd}\,,$ (39a)
$\displaystyle\mathcal{F}^{Spin}:$ $\displaystyle\
\mathsf{Bord}_{n}^{Spin}(Y)\ {\to}\ \Sigma^{n-1}\mathsf{sVect}^{fd}\,.$ (39b)

Therefore, we will talk about the partition functions of $\mathcal{B}$ (resp.
$\mathcal{F}$) on any closed oriented (resp. spin) $n$-manifold $M$, which
really mean those of $\mathcal{B}^{SO}$ (resp. $\mathcal{F}^{Spin}$).

When $Y$ is path-connected, the evaluation of $\mathcal{B}$ (resp.
$\mathcal{F}$) on a point gives a linear $n$-group action on objects in
$\Sigma^{n-1}\mathsf{Vect}^{fd}$ (resp. $\Sigma^{n-1}\mathsf{sVect}^{fd}$)
imposed by the $n$-group $\Omega\pi_{\leq n}Y$. Namely, an $n$-dim
$Y$-enriched TQFT is the same as an $n$-dim $\Omega\pi_{\leq n}Y$-symmetric
TQFT. When $Y$ has multiple path components, we have a $\pi_{0}(Y)$ worth of
universes of higher-group symmetric TQFTs. The cobordism hypothesis
(Lurie:2009keu, ) asserts that such a characterization by evaluation on a
point is always faithful and complete. In general, $\mathsf{Bord}^{fr}_{n}(Y)$
is the free fully-dualizable symmetric monoidal $(\infty,n)$-category
generated by $\infty$-groupoid $Y$.

###### Definition 4.5

A $n$-representation (resp. super $n$-representation) of space $Y$ is a
functor between $(\infty,n)$-categories,

$\underline{\mathcal{B}}:Y\to\Sigma^{n-1}\mathsf{Vect}^{fd}\,,\qquad\Bigl{(}\text{resp.
}\ \underline{\mathcal{F}}:Y\to\Sigma^{n-1}\mathsf{sVect}^{fd}\Bigr{)}\,.$
(40)

Note that since $\Sigma^{n-1}\mathsf{Vect}^{fd}$ (resp.
$\Sigma^{n-1}\mathsf{sVect}^{fd}$) is essentially an $n$-category,
$\underline{\mathcal{B}}$ (resp. $\underline{\mathcal{F}}$) actually factors
through the homotopy $n$-type of $Y$. Namely, $\underline{\mathcal{B}}$ (resp.
$\underline{\mathcal{F}}$) is in essence a functor between $n$-categories,
from $\pi_{\leq n}Y$ to $\Sigma^{n-1}\mathsf{Vect}^{fd}$ (resp.
$\Sigma^{n-1}\mathsf{sVect}^{fd}$). The cobordism hypothesis specialized for
our purpose asserts the following equivalence (see (Lurie:2009keu, , Theorem
2.4.18)):

###### Proposition 4.6

Via point evaluation, an $n$-dim bosonic (resp. fermionic) $Y$-enriched fully-
extended TQFT (Definition 4.3) is equivalent to a $n$-representation (resp.
super $n$-representation) of $Y$ (Definition 4.5).

Higher-representations at lower dimensions in the context of generalized
symmetries are extensively discussed by, e.g., Refs. (Bartsch:2023pzl, ;
Bhardwaj:2023wzd, ).

As a preliminary example of topological functionals from these TQFTs, let us
consider 1D topological functionals to path-connected $1$-finite $Y$. Namely,
$\pi_{1}Y$ is finite. We start with the bosonic case. A $1$-representation
$\underline{\mathcal{B}}$ of $Y$ is just a representation $R$ of $\pi_{1}Y$.
An $S^{1}$ bosonic topological functional $\mathcal{B}(g)$ for $g\in[S^{1},Y]$
is a $\mathbb{C}$-valued function on

$[S^{1},Y]\,\simeq\,[S^{1},B\pi_{1}Y]\,\simeq\,\\{\text{conjugacy classes of
}\pi_{1}Y\\}\,.$ (41)

Note that $[S^{1},Y]\not\simeq\pi_{1}Y$ as long as $\pi_{1}Y$ is not
commutative101010 This discrepancy between $[S^{1},Y]$ and $\pi_{1}Y$ comes
from the difference between free homotopy and based homotopy. As we shall see
in Sec. 5.2.1, this difference accounts for a large class of non-invertible
fusion rules in solitonic symmetry. . Therefore, an $S^{1}$ topological
functional is equivalent to a class function of $\pi_{1}Y$, such as a
character. Note that $g\in[S^{1},Y]$ prescribes a $\pi_{1}Y$ gauge field on
$S^{1}$ whose holonomy conjugacy class is $g$. Therefore, the partition
function is indeed given by a character, i.e.,

$\mathcal{B}(g)=\mathrm{tr}_{R}(g)\,.$ (42)

One can generalize this relation to $T^{n}$ bosonic topological functionals
and define the notion of $n$-characters111111 Ideas of $n$-characters
originate from Ref. (hopkins2000generalized, ). 2-characters are defined in
Ref. (ganter2008representation, ) and are developed in, e.g., Refs.
(OSORNO2010369, ; bartlett2009unitary, ). Unlike ordinary characters, the
input of an $n$-character is a set of $n$ mutually commutative elements in
$G$, and the $n$-character is invariant under simultaneous conjugations. One
can immediately recognize that this input describes the isomorphism classes of
$G$-bundles on $T^{n}$, i.e., $[T^{n},BG]$. It is then natural to connect
these $n$-characters with $T^{n}$ bosonic topological functionals to $BG$. .
The traces of representations amount to all characters, and the space of
characters is $\mathbb{N}$-linearly spanned by simple characters. The
characters from $1$-dim representations factor through $H_{1}(Y;\mathbb{Z})$,
the Abelianization of $\pi_{1}Y$, and thus can be constructed via the
universal invertible form (7). The characters from higher-dim representations,
especially the irreducible ones, go beyond Eq. (7).

For the fermionic case, a super $1$-representation $\underline{\mathcal{F}}$
of $Y$ is just a super representation of $\pi_{1}(Y)$, which is further a pair
of ordinary representations $(B,F)$ on the bosonic and the fermionic sectors,
respectively. There are two spin structures on $S^{1}$, $0$ and $1$,
corresponding to the spin bordism group $\Omega^{Spin}_{1}=\mathbb{Z}_{2}$.
Then a similar analysis to the bosonic case shows that the $S^{1}$ fermionic
topological functional for
$(g,\varepsilon)\in[S^{1},Y]\times\Omega^{Spin}_{1}$ is given by [recall that
$g$ represents a conjugacy class in $\pi_{1}Y$]

$\begin{split}{\mathcal{F}}(g,0)=\mathrm{tr}_{B}(g)+\mathrm{tr}_{F}(g)\,,\\\
{\mathcal{F}}(g,1)=\mathrm{tr}_{B}(g)-\mathrm{tr}_{F}(g)\,.\end{split}$ (43)

We thus learned that $S^{1}$ fermionic topological functionals are virtual
characters of $\pi_{1}Y$. They are $\mathbb{Z}$-linearly spanned by simple
characters rather than $\mathbb{N}$-linearly. The virtual characters from
$1$-dim representations factor through the super integral homology
$\mathbb{SZ}_{1}(Y)$ (see Sec. 4.3.2) and thus can also be constructed via the
universal invertible form (7). Tremendous examples of higher-dim topological
functionals will be discussed in Sec. 5.

###  4.3 Cohomology with TQFT coefficients

As we mentioned in Sec. 2.2.1, a crucial feature of solitonic symmetry is its
independence of system details such as the action and the ambient spacetime.
It is just the algebra of topological functionals and is determined by the
target space $Y$ only. Formally, it gives homotopy-invariant contravariant
functors on the topological space $Y$. Thus the algebraic structure of
solitonic symmetry can be interpreted as a cohomology theory on $Y$, in a
vastly generalized sense.

####  4.3.1 Non-invertible: (Super) solitonic cohomology

Given two TQFTs $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$,
$\mathcal{Z}_{1}(-)\otimes\mathcal{Z}_{2}(-)$ also defines a TQFT denoted by
$\mathcal{Z}_{1}\otimes\mathcal{Z}_{2}$. We can transfer the fusion of two
topological functionals to the fusion of the two TQFTs beneath. To see the
tensor product between TQFTs, instead of looking at each individual TQFT, we
should consider the functor $(\infty,n)$-category containing all
$n$-representations121212 One may first come up with
$\mathsf{Fun}^{\otimes}\,\Bigl{(}\,\mathsf{Bord}_{n}^{fr}(Y)\,,\,\Sigma^{n-1}\mathsf{Vect}^{fd}\,\Bigr{)}\quad\text{
and
}\quad\mathsf{Fun}^{\otimes}\,\Bigl{(}\,\mathsf{Bord}_{n}^{fr}(Y)\,,\,\Sigma^{n-1}\mathsf{sVect}^{fd}\,\Bigr{)}\,.$
(44) According to the cobordism hypothesis, they are the maximal
$\infty$-groupoids (in essence $n$-groupoids) inside $\mathsf{Rep}^{n}(Y)$ and
$\mathsf{sRep}^{n}(Y)$, respectively. They have too few morphisms to support
the rich structures we shall discuss shortly. . It is actually an $n$-category
because of the $n$-category nature of the codomain.

###### Definition 4.7

The $n$-th solitonic cohomology [resp. super solitonic cohomology] of an
$n$-finite space $Y$ is a symmetric multi-fusion $n$-category,

$\mathsf{Rep}^{n}(Y)\,\equiv\,\mathsf{Fun}\,\bigl{(}\,Y,\,\Sigma^{n-1}\mathsf{Vect}^{fd}\,\bigr{)}\,,\quad\left[\text{resp.}\,\,\mathsf{sRep}^{n}(Y)\,\equiv\,\mathsf{Fun}\,\bigl{(}\,Y,\,\Sigma^{n-1}\mathsf{sVect}^{fd}\,\bigr{)}\right].$
(45)

They are symmetric fusion $n$-categories when $Y$ is further path-connected.
In this case, they can be understood as higher-representation higher-
categories of the $\infty$-group $\Omega Y$. In particular, the $n$-category
nature of the codomains suggests the following.

  * •

When $Y$ is path-connected, $\mathsf{Rep}^{1}(Y)$ [resp.
$\mathsf{sRep}^{1}(Y)$] is equivalent to the representation (resp. super-
representation) category of the group $\pi_{1}Y$.

  * •

When $Y$ is path-connected, $\mathsf{Rep}^{n}(Y)$ [resp.
$\mathsf{sRep}^{n}(Y)$] is equivalent to the representation (resp. super-
representation) $n$-category of the $n$-group $\Omega\pi_{\leq n}Y$.

The $1$-dim case reproduces the discussion on $1$-dim topological functionals
at the end of Sec. 4.1. The higher-group representation higher-category has
caught vast attention in recent literature on generalized symmetry, see, e.g.,
Refs. (Bhardwaj:2022lsg, ; Bhardwaj:2022kot, ; Bartsch:2022mpm, ;
Bartsch:2022ytj, ; Bhardwaj:2023wzd, ; Bartsch:2023pzl, ). When $Y$ is not
path-connected, the solitonic cohomologies are just the Cartesian product of
the solitonic cohomologies of each path component.

The fusion monoid of bosonic (resp. fermionic) topological functionals can be
equated with the fusion monoid of the equivalence classes of objects in
$\mathsf{Rep}^{n}(Y)$ [resp. $\mathsf{sRep}^{n}(Y)$], which gives what we have
been pursuing. However, we should not quickly throw away the far richer
structures contained in $\mathsf{Rep}^{n}(Y)$ and $\mathsf{sRep}^{n}(Y)$ than
their mere fusion monoids. To see their significance, let us analyze the
physical meaning of morphisms in $\mathsf{Rep}^{n}(Y)$ and
$\mathsf{sRep}^{n}(Y)$. Here $1$-morphisms are natural transformations between
$n$-representations. We can readily note that the natural endo-transformations
of the trivial $n$-representation are simply functors from $Y$ to
$\Sigma^{n-2}\mathsf{Vect}^{fd}$ and $\Sigma^{n-2}\mathsf{sVect}^{fd}$,
respectively. Namely, we arrive at the following relations between different
$n$.

###### Proposition 4.8

$\Omega\mathsf{Rep}^{n}(Y)\simeq\mathsf{Rep}^{n-1}(Y)$ and
$\Omega\mathsf{sRep}^{n}(Y)\simeq\mathsf{sRep}^{n-1}(Y)$.

This result hints at the physical meaning of other morphisms in
$\mathsf{Rep}^{n}(Y)$ and $\mathsf{sRep}^{n}(Y)$:

  * •

1-morphisms are $(n\\!-\\!1)$-dim topological interfaces between $n$-dim
TQFTs.

  * •

$(p\\!+\\!1)$-morphisms are $(n\\!-\\!p\\!-\\!1)$-dim topological interfaces
between $(n\\!-\\!p)$-dim topological interfaces.

Topological functionals, which are auxiliary-TQFT partition functions defined
on proper closed manifolds, cannot capture these richer data, which concern
auxiliary TQFTs themselves and inhabit networks made by non-closed manifolds
via connections and junctions. In recent literature on generalized symmetry,
people have recognized the significance of such networks of non-closed
topological operators as a formulation of background gauge fields. It starts
to become customary to recognize the algebraic structure of such networks as
the total generalized symmetry of a theory, given that it contains literally
all information about a symmetry, not just the fusion rule. We have no reason
not to follow this custom.

###### Proposition 4.9

Consider a $d$-dim bosonic (resp. fermionic) theory defined by a path integral
with $d$-finite target space $Y$.

  * •

The total solitonic symmetry is described by $\mathsf{Rep}^{d}(Y)$ [resp.
$\mathsf{sRep}^{d}(Y)$], a symmetric fusion $d$-category.

  * •

For $-\\!1\leq p\leq d\\!-\\!1$, the $(\geq\\!p)$-form solitonic symmetry is
described by $\mathsf{Rep}^{d-p-1}(Y)$ [resp. $\mathsf{sRep}^{d-p-1}(Y)$], a
symmetric fusion $(d\\!-\\!p\\!-\\!1)$-category.

  * •

For $-\\!1\leq p\leq d\\!-\\!1$, the $p$-form solitonic symmetry is described
by the fusion monoid of $\mathsf{Rep}^{d-p-1}(Y)$ [resp.
$\mathsf{sRep}^{d-p-1}(Y)$], a commutative rig.

Note that we also include $(-\\!1)$-form symmetry in the total solitonic
symmetry. The result here echoes recent progresses on categorical generalized
symmetry.

We may understand $\mathsf{Rep}^{\bullet}(-)$ and $\mathsf{sRep}^{\bullet}(-)$
as non-invertible generalizations of cohomology theories. We may view the
collections,

$\left\\{\Sigma^{n-1}\mathsf{Vect}^{fd}\right\\}_{n\in\mathbb{N}}\ \text{and}\
\left\\{\Sigma^{n-1}\mathsf{sVect}^{fd}\right\\}_{n\in\mathbb{N}},$ (46)

as non-invertible generalizations of spectra. We shall shortly see in Sec.
4.3.2 which orthodox spectra they generalize. The coefficients of these non-
invertible cohomologies are non-enriched fully-extended TQFTs:

$\begin{split}\mathsf{Rep}^{\bullet}&\,\equiv\,\mathsf{Rep}^{\bullet}\bigl{(}\\{*\\}\bigr{)}\,\simeq\,\Sigma^{\bullet-1}\mathsf{Vect}^{fd}\,,\\\
\mathsf{sRep}^{\bullet}&\,\equiv\,\mathsf{sRep}^{\bullet}\bigl{(}\\{*\\}\bigr{)}\,\simeq\,\Sigma^{\bullet-1}\mathsf{sVect}^{fd}\,.\end{split}$
(47)

$\mathsf{Rep}^{\bullet}(-)$ and $\mathsf{sRep}^{\bullet}(-)$ are indeed
homotopy-invariant contravariant functors from topological spaces because a
map $Y\to Z$ induces pullbacks
$\mathsf{Rep}^{\bullet}(Z)\to\mathsf{Rep}^{\bullet}(Y)$ and
$\mathsf{sRep}^{\bullet}(Z)\to\mathsf{sRep}^{\bullet}(Y)$ simply by their
definitions.

When $Y$ is not $n$-finite, Def. 4.7 contains too much superfluous data to
capture the authentic algebraic structure of solitonic symmetry. For example,
recall the discussion about $1$-dim topological functionals at the end of Sec.
4.2.3. Recall that $\mathsf{Rep}^{1}(Y)$ and $\mathsf{sRep}^{1}(Y)$ are simply
representation categories of $\pi_{1}Y$. If we allowed $\pi_{1}Y$ to be
infinite like $\mathbb{Z}$ or $SL(2,\mathbb{Z})$, non-semisimple
representations would appear since the Maschke theorem does not apply. They
are not unitarizable and produce superfluous duplicated topological
functionals. There are also semisimple non-unitarizable representations whose
$S^{1}$ partition functions are not bounded. It is natural to expect that only
unitarizable representations capture the solitonic symmetry. Nevertheless, for
convenience, we still formally use $\mathsf{Rep}^{\bullet}(Y)$ or
$\mathsf{sRep}^{\bullet}(Y)$ to indicate the algebraic structure of solitonic
symmetry even when $Y$ is not $\bullet$-finite.

We do not attempt to generalize the above unitarizability condition to higher-
dim general cases in this paper. Instead, as we discussed at the end of Sec.
3.3, we shall just focus on $n$-finite homotopy quotients $Z$ of $Y$ and
discuss the solitonic subsymmetries that can be faithfully described by
$\mathsf{Rep}^{\bullet}(Z)$ or $\mathsf{sRep}^{\bullet}(Z)$. The collection of
them for all different choices of $Z$, together with natural functors between
them, form a diagram in the $(n\\!+\\!1)$-category of symmetric multi-fusion
$n$-categories. We expect that the colimit of this diagram exists and believe
that this colimit gives an almost complete approximation of the continuous
solitonic symmetry for $Y$. Furthermore, we expect that the continuous
solitonic symmetry may be constructed as a generalized Cauchy completion of
this colimit, provided we can cook up a well-behaved notion of
topologized/uniformized monoidal $n$-categories131313 The prototype of our
anticipation is just the Cauchy completion of $\mathbb{Q}/\mathbb{Z}$ to
$U(1)$. It is possible to prescribe a topology on $\mathbb{Q}/\mathbb{Z}$
through the colimit construction
$\mathbb{Q}/\mathbb{Z}\simeq\bigcup_{n\in\mathbb{N}}\mathbb{Z}_{n}$.
Topological groups are naturally uniformizable; thus, we can take the Cauchy
completion of $\mathbb{Q}/\mathbb{Z}$ to obtain $U(1)$. .

####  4.3.2 Invertible: (Super) unitary cohomology

As the first application, let us find out the invertible subsymmetry of the
generically non-invertible solitonic symmetry. Invertible topological
functionals are the partition functions of invertible fully-extended TQFTs. A
fully-extended TQFT $\mathcal{Z}:\mathsf{Bord}^{\Gamma}_{n}\ {\to}\
\mathsf{D}_{n}$ is said invertible if we can find another fully-extended TQFT
$\mathcal{Z}^{-1}:\mathsf{Bord}^{\Gamma}_{n}\ {\to}\ \mathsf{D}_{n}$ such that
$\mathcal{Z}\otimes\mathcal{Z}^{-1}\simeq 1$, the trivial TQFT. Therefore,
$\mathcal{Z}$ must factor through the following commutative diagram (see the
discussion around Sec. 6.2 of Ref. (freed2019lectures, )):

${\mathsf{Bord}^{\Gamma}_{n}}$${\mathsf{D}_{n}}$${\|\mathsf{Bord}^{\Gamma}_{n}\|}$${\mathsf{D}_{n}^{\times}}$$\scriptstyle{\mathcal{Z}}$$\scriptstyle{\mathcal{Z}^{\times}}$
(48)

Here, $\mathsf{D}_{n}^{\times}$ is the maximal Picard $\infty$-groupoid inside
$\mathsf{D}_{n}$, where a Picard $\infty$-groupoid means an invertible
symmetric monoidal $\infty$-groupoid (see Appendix A.4 of Ref.
(freed2019lectures, )). And $\|\mathsf{Bord}^{\Gamma}_{n}\|$ is the
$\infty$-groupoid completion of $\mathsf{Bord}^{\Gamma}_{n}$, which turns out
to have invertible objects only and becomes a Picard $\infty$-groupoid. Thus
$\mathcal{Z}^{\times}$ is a symmetric monoidal functor between two Picard
$\infty$-groupoids.

According to Philosophy 4.2, a Picard $\infty$-groupoid is an infinite loop
space, i.e., the $0$-space of a spectrum. Also, $\mathcal{Z}^{\times}$ is an
infinite loop map and the fusion between $\mathcal{Z}_{1}^{\times}$ and
$\mathcal{Z}_{2}^{\times}$ is induced by loop concatenation. Therefore,
classifying invertible TQFTs reduces to classifying infinite loop maps between
two infinite loop spaces, which further reduces to classifying spectrum maps
between two spectra. Therefore, the treatment of invertible TQFTs belongs to
the realm of stable homotopy theory.

The Galatius-Madsen-Tillmann-Weiss theorem (galatius2009homotopy, ), a theorem
derived out of the cobordism hypothesis (Lurie:2009keu, ), asserts that
$\|\mathsf{Bord}^{\Gamma}_{n}\|$ is weakly homotopy equivalent to the 0-space
of the Madsen-Tillmann spectrum $\Sigma^{n}\mathbb{MT}\Gamma$ (see also
(freed2019lectures, , Theorem 6.67)). In the case of our interest,
$\|\mathsf{Bord}^{fr}_{n}(Y)\|$ is independent of $n$ and is the free infinite
loop space generated by $Y_{+}\equiv Y\sqcup\\{*\\}$, i.e. $Y$ with an extra
base point. Namely,

$\|\mathsf{Bord}^{fr}_{n}(Y)\|\ \simeq\ \underset{q\to\infty}{\mathrm{colim}}\
\Omega^{q}\Sigma^{q}Y_{+}\,.$ (49)

It is the 0-space of the suspension spectrum of $Y_{+}$. The codomain is more
complicated due to the complexity of the higher-categorical Karoubi
completion. We thus leave the full analysis of the maximal invertible
subsymmetry to future works. Here we just focus on a sufficiently interesting
part we can obtain immediately by noticing

$\Omega^{n-1}\left(\Sigma^{n-1}\mathsf{D}\right)^{\times}\ \simeq\
\left(\Omega^{n-1}\Sigma^{n-1}\mathsf{D}\right)^{\times}\ \simeq\
\mathsf{D}^{\times}\,.$ (50)

In general, the spectrum
$\\{\left(\Sigma^{n-1}\mathsf{D}\right)^{\times}\\}_{n\in\mathbb{N}}$ may have
lousy connectivity, i.e., the Karoubi-completion procedure may add new
invertible objects and make many
$\left(\Sigma^{n-1}\mathsf{D}\right)^{\times}$ not path-connected. However,
here we shall neglect such contributions from the Karoubi-completion
procedure. In other words, we shall consider spectrum

$\\{B^{n-1}\mathsf{D}^{\times}\\}_{n\in\mathbb{N}}\,,$ (51)

which is the $(-\\!1)$-connective cover of spectrum
$\\{\left(\Sigma^{n-1}\mathsf{D}\right)^{\times}\\}_{n\in\mathbb{N}}$. We thus
obtain at least part of the maximal invertible subsymmetry141414 We cannot
help speculating a relationship between
$\\{\left(\Sigma^{n-1}\mathsf{Vect}^{fd}\right)^{\times}\\}_{n\in\mathbb{N}}$
[resp.
$\\{\left(\Sigma^{n-1}\mathsf{sVect}^{fd}\right)^{\times}\\}_{n\in\mathbb{N}}$]
and the $\mathbb{C}^{\times}$-dual of $\mathbb{M}SO$ (resp. $\mathbb{M}Spin$).
We leave the verification of our speculations to future works. .

Let us start with the bosonic case. Only the $1$-dim vector space is
invertible in $\mathsf{Vect}^{fd}$, and its invertible endomorphisms
constitute the group $\mathbb{C}^{\times}$. Therefore, we have

$B^{n-1}\left(\mathsf{Vect}^{fd}\right)^{\times}\,\simeq\,B^{n}\mathbb{C}^{\times}_{\delta}\,.$
(52)

When ambiguity may arise, we use the subscript $\delta$ to indicate the
discrete topology. These infinite loop spaces assemble to form the
Eilenberg–Maclane spectrum $\mathbb{H}\mathbb{C}^{\times}$. We then arrived at
the following theorem:

######  Proposition 4.10 (bosonic)

The fusion monoid of solitonic cohomology $\mathsf{Rep}^{\bullet}(Y)$ contains
a group $(\mathbb{H}\mathbb{C}^{\times})^{\bullet}(Y)\simeq
H^{\bullet}\bigl{(}Y;\mathbb{C}^{\times}\bigr{)}$.

We now turn to the more interesting fermionic case. ${\mathsf{sVect}^{fd}}$
has two classes of invertible objects, the bosonic and the fermionic $1$-dim
vector spaces. Their fusion monoid is $\mathbb{Z}_{2}$. The invertible
endomorphisms of each class constitute $\mathbb{C}^{\times}$. The symmetric
monoidal structure on $\mathsf{sVect}^{fd}$, especially the Koszul sign rule
(35), positions the infinite loop space
$\left(\mathsf{sVect}^{fd}\right)^{\times}$ into a Puppe sequence,

$\cdots\longrightarrow B^{n}\mathbb{C}^{\times}_{\delta}\longrightarrow
B^{n-1}\left(\mathsf{sVect}^{fd}\right)^{\times}\longrightarrow
B^{n-1}\mathbb{Z}_{2}\stackrel{{\scriptstyle\rho\circ\mathrm{Sq}^{2}}}{{\longrightarrow}}B^{n+1}\mathbb{C}^{\times}_{\delta}\longrightarrow\cdots\,,$
(53)

classified by the stable cohomology operation
$\rho\circ\mathrm{Sq}^{2}$.151515The direct determination of
$\rho\circ\mathrm{Sq}^{2}$ requires to evaluate the $E_{\infty}$-structure of
$\left(\mathsf{sVect}^{fd}\right)^{\times}$. The Koszul sign rule should
prescribe an $E_{\infty}$-structure that leads to a nontrivial stable
cohomology operation. We note however that $\rho\circ\mathrm{Sq}^{2}$ is the
only existing nontrivial stable cohomology operation from
$\mathbb{H}\mathbb{Z}_{2}$ to $\Sigma^{2}\mathbb{H}\mathbb{C}^{\times}$.  Here
$\mathrm{Sq}^{2}$ is the second Steenrod square, and $\rho$ is the change-of-
coefficient for the canonical inclusion
$\mathbb{Z}_{2}\to\mathbb{C}^{\times}$. These infinite loop spaces assemble to
form a spectrum and let us denote it as $\mathbb{S}\mathbb{C}$. We have thus
proved the following theorem:

######  Proposition 4.11 (fermionic)

The fusion monoid of super solitonic cohomology $\mathsf{sRep}^{\bullet}(Y)$
contains a group $\mathbb{SC}^{\bullet}(Y)$, where $\mathbb{SC}$ is defined
via Eq. (53).

For a $d$-finite target space $Y$, these theorems tell us that there are
invertible $(d\\!-\\!n\\!-\\!1)$-form solitonic symmetry given by
$H^{n}(Y;\mathbb{C}^{\times})$ in a $d$-dim bosonic theory and
$\mathbb{SC}^{n}(Y)$ in a $d$-dim fermionic theory. We can further ask what
higher-group they make up, which requires us to consider the entire map
spectrum $\mathrm{Map}(\mathbb{Y}_{+},\mathbb{HC})$ or
$\mathrm{Map}(\mathbb{Y}_{+},\mathbb{SC})$, where $\mathbb{Y}_{+}$ denotes the
suspension spectrum of $Y$, rather than merely its $\pi_{0}-$. We again leave
such analysis to future works.

When we focus on $\bullet$-finite spaces only, many other cohomology theories
can also provide the same results as above. For example, for $\bullet$-finite
space $Y$ we have

$\displaystyle H^{\bullet}(Y;\mathbb{C}^{\times})$
$\displaystyle\,\simeq\,H^{\bullet}\bigl{(}Y;U(1)\bigr{)}\,,$ (54a)
$\displaystyle\mathbb{SC}^{\bullet}(Y)$
$\displaystyle\,\simeq\,\mathbb{SU}^{\bullet}(Y)\,,$ (54b)

but they are different for general $Y$. Here $\mathbb{SU}$ is defined as the
spectrum obtained by substituting $\mathbb{C}^{\times}$ with $U(1)$ in Eq.
(53) and changing $\rho$ into the change-of-coefficient for $\mathbb{Z}_{2}\to
U(1)$. We shall call $\mathbb{H}U(1)$ the unitary spectrum and call
$\mathbb{SU}$ the super unitary spectrum. We conjecture that these two spectra
correctly capture invertible solitonic symmetry even when we remove the
finiteness condition on $Y$.

###### Conjecture 4.12

Consider a $d$-dim bosonic (resp. fermionic) theory defined by a path integral
with target space $Y$.

  * •

For $-\\!1\leq p\leq d\\!-\\!1$, the $p$-form solitonic symmetry contains the
unitary cohomology [resp. super unitary cohomology] group
$H^{d-p-1}\bigl{(}Y;U(1)\bigr{)}$ [resp. $\mathbb{SU}^{d-p-1}(Y)$].

(Note: This is a theorem mentioned above when $Y$ is $d$-finite.)

This conjecture is motivated by our discrete approximation discussed at the
end of Sec. 3.3 and Sec. 4.3.1. They are also consistent with physicists’ long
experience with invertible $\theta$-angles and the universal invertible
construction (7).

Before concluding this section, let us unpack these two cohomology theories.
$\mathbb{H}U(1)$ is the U(1)-dual of $\mathbb{H}\mathbb{Z}$ and its universal
coefficient theorem takes the naive form,

$H^{\bullet}\bigl{(}-;U(1)\bigr{)}\,\simeq\,\mathrm{Hom}\bigl{(}H_{\bullet}(-,\mathbb{Z}),U(1)\bigr{)}\,.$
(55)

As for the fermionic case, the modified version of the Puppe sequence (53)
leads to a long exact sequence161616 This is also the Atiyah–Hirzebruch
spectral sequence for $\mathbb{SU}^{\bullet}(-)$. Namely,
$d_{2}:E^{\bullet,1}_{2}\to E^{\bullet+2,0}_{2}=\rho\circ\mathrm{Sq}^{2}$. ,

$\cdots\longrightarrow
H^{\bullet}\bigl{(}-;U(1)\bigr{)}\longrightarrow\mathbb{SU}^{\bullet}(-)\longrightarrow
H^{\bullet-1}\bigl{(}-;\mathbb{Z}_{2}\bigr{)}\stackrel{{\scriptstyle\rho\circ\mathrm{Sq}^{2}}}{{\longrightarrow}}H^{\bullet+1}\bigl{(}-;U(1)\bigr{)}\longrightarrow\cdots\,.$
(56)

$\mathbb{SU}$ is the U(1)-dual of $\mathbb{S}\mathbb{Z}$, which we call the
super integral spectrum. Namely,

$\mathbb{SU}^{\bullet}(-)\,\simeq\,\mathrm{Hom}\bigl{(}\mathbb{SZ}_{\bullet}(-),U(1)\bigr{)}\,.$
(57)

$\mathbb{S}\mathbb{Z}$ can be defined as the first Postnikov truncation of the
sphere spectrum $\mathbb{S}$ or the Thom spectrum $\mathbb{M}Spin$. Namely,
when $Y$ is $(n\\!-\\!1)$-connected, we have

$\widetilde{\mathbb{SZ}}_{\bullet}(Y)\,\simeq\,{\pi}^{s}_{\bullet}(Y)\,\simeq\,\widetilde{\Omega}^{Spin}_{\bullet}(Y)\,,\qquad\text{for
}\bullet\leq n\\!+\\!1\,.$ (58)

Nonzero homotopy groups of the spectra mentioned here are summarized:

$\begin{tabular}[]{c||c|c||c|c|}$\mathbb{E}$&$\mathbb{H}\mathbb{Z}$&$\mathbb{H}U(1)$&$\mathbb{SZ}$&$\mathbb{SU}$\\\
\hline\cr\hline\cr$\mathbb{E}_{1}$&&&$\mathbb{Z}_{2}$&\\\
\hline\cr$\mathbb{E}_{0}$&$\mathbb{Z}$&$U(1)$&$\mathbb{Z}$&$U(1)$\\\
\hline\cr$\mathbb{E}_{-1}$&&&&$\mathbb{Z}_{2}$\\\ \hline\cr\end{tabular}\,.$
(59)

As a final remark, if we planned to classify gapped phases, we would take the
other $(\infty,1)$-completion at the beginning of Sec. 4.2.2. Then we would
reach spectra $\Sigma\mathbb{H}\mathbb{Z}$ and $\Sigma
I_{\mathbb{Z}}\mathbb{SZ}$ instead. Here $I_{\mathbb{Z}}\mathbb{SZ}$ denotes
the Anderson dual of $\mathbb{SZ}$ and is the spectrum studied by Freed
(Freed:2006mx, ) and Gu-Wen (Gu:2012ib, ).

##  5 Non-invertible structure beyond homotopy groups

Following Ansatz 3.2 and Conjecture 3.3, we have studied the algebraic
structure of solitonic symmetry in Sec. 4. These results are pretty formal,
and we would like to make some down-to-earth illustrations. For this purpose,
we adopt the following angle of looking at solitonic symmetry: What role does
the conventional wisdom $\mathrm{Hom}(\pi_{\bullet}-,U(1))$ (see Sec. 2.2.3)
play in the general solitonic symmetry? In this section, we focus on path-
connected target space $Y$ since different path components correspond just to
different universes171717 When we are interested in the relationship between
different universes, it is also helpful to render multiple path components. In
such a case, we can consider a point-like topological functional that
specifies one of the path-connected components, i.e., a local operator
$1_{i}(x)$ that takes $1$ if the field at $x$ is valued in $i\in\pi_{0}Y$ and
takes $0$ otherwise. Due to the continuity of the field, this operator is
indeed topological. It generates the $(d\\!-\\!1)$-form solitonic symmetry,
$\mathsf{Rep}^{0}(Y)\,\simeq\,\mathsf{sRep}^{0}(Y)\,\simeq\,\bigl{\\{}\pi_{0}Y\mapsto\mathbb{C}\bigr{\\}}\,,$
(60) which is non-invertible. The state space decomposes into different
sectors, i.e., universes, even at finite volumes (Pantev:2005zs, ;
Pantev:2005rh, ; Pantev:2005wj, ; Hellerman:2006zs, ; Hellerman:2010fv, ;
Tanizaki:2019rbk, ; Komargodski:2020mxz, ; Nguyen:2021naa, ; Lin:2022xod, ).
Interfaces between different universes, i.e. $(d\\!-\\!1)$-dim solitonic
defects, are the charged objects of the $(d\\!-\\!1)$-form solitonic symmetry.
.

The conventional wisdom classifies topological charges by homotopy groups
$\pi_{\bullet}-$. From the perspective of topological functionals,
conventional wisdom concerns topological functionals only on spheres, and
solitonic symmetry has the structure $\mathrm{Hom}(\pi_{\bullet}-,U(1))$. Due
to Alexander’s trick, a sphere necessarily has

$\pi_{0}\mathrm{Diff}(S^{\bullet},\xi)\simeq 0$ (61)

for an orientation $\xi$. Thus spherical topological functionals do not suffer
from the identity problem discussed in Sec. 3.1. Therefore,
$\mathrm{Hom}(\pi_{\bullet}-,U(1))$ should not be regarded as just wrong.
Instead, it is the precursor of the non-invertible structure.

###  5.1 Rectification vs. condensation

Let us speculate the structure of $\mathsf{Rep}^{n}(Y)$ and
$\mathsf{sRep}^{n}(Y)$ inductively [We use $\mathsf{Rep}^{n}(Y)$ to illustrate
the idea]. We first suppose that we have already understood
$\mathsf{Rep}^{n-1}(Y)\simeq\mathsf{Rep}^{n-1}(\pi_{\leq n-1}Y)$, and we now
would like to understand $\mathsf{Rep}^{n}(Y)\simeq\mathsf{Rep}^{n}(\pi_{\leq
n}Y)$. Recall that $\pi_{\leq\bullet}Y$ denotes $Y$’s homotopy $\bullet$-type,
i.e., the $\bullet$-th Postnikov truncation of $Y$. The $n$-th floor of $Y$’s
Postnikov tower is a fibration

$B^{n}\pi_{n}Y\to\pi_{\leq n}Y\to\pi_{\leq n-1}Y\,.$ (62)

This fibration allows us to complete our mission in two steps:

$\mathsf{Rep}^{n-1}(Y)\ \stackrel{{\scriptstyle 1}}{{\Longrightarrow}}\
\mathsf{Rep}^{n}(\pi_{\leq n-1}Y)\ \stackrel{{\scriptstyle
2}}{{\Longrightarrow}}\ \mathsf{Rep}^{n}(Y)\,.$ (63)

In the first step, we need to construct $n$-dim topological functionals for
$\pi_{\leq n-1}Y$ from lower-dim topological functionals. Because $\pi_{\leq
n-1}Y$ is $(n\\!-\\!1)$-aspherical, i.e., it has no $n$-dim homotopical data
at all, the $n$-dim topological functionals can be obtained via condensation.
Physically, following Ref. (Roumpedakis:2022aik, ), condensation can be
formulated via higher-gauging the subsymmetries of solitonic symmetry on the
$n$-dim operator manifold only. Mathematically, following Ref.
(Gaiotto:2019xmp, ), condensation corresponds to formulating the Karoubi
completion. Thus we expect

$\mathsf{Rep}^{n}(\pi_{\leq n-1}Y)\ \simeq\ \Sigma\mathsf{Rep}^{n-1}(Y)\,.$
(64)

The higher-gauging procedure shows that condensation relies on nontrivial
lower-dim cycles on the operator manifold. Therefore, the $n$-dim topological
functionals obtained by condensation must take trivial values on $S^{n}$.

In the second step, the Postnikov fibration (62) induces an injective functor

$\Sigma\mathsf{Rep}^{n-1}(Y)\ \to\ \mathsf{Rep}^{n}(Y)\,.$ (65)

What new objects are added in this step? The homotopy fiber of the fibration
(62) points out the answer: the topological functionals that are nontrivial on
$S^{n}$. Namely, we expect that the Postnikov fibration induces a “short exact
sequence”,

$0\ \to\ \Sigma\mathsf{Rep}^{n-1}(Y)\ \to\ \mathsf{Rep}^{n}(Y)\ \to\
\widetilde{\mathsf{Rep}}^{n}(B^{n}\pi_{n}Y)\ \to\ 0\,.$ (66)

Here the “reduced” solitonic cohomology
$\widetilde{\mathsf{Rep}}^{n}(B^{n}\pi_{n}Y)$ is the $n$-category completion
of the commutative rig, the $\mathbb{N}$-span of
$\mathrm{Hom}\bigl{(}\pi_{n}Y,U(1)\bigr{)}$, by contractible morphisms. This
fits our experience with $n$-form gauge fields of gauge group $\pi_{n}Y$ and
can be directly verified when $\pi_{n}Y$ is finite. When the Postnikov
fibration (62) is a trivial product, the “short exact sequence” (66)
canonically splits, and the conventional wisdom makes the correct prediction.
But when the Postnikov fibration (62) is nontrivial, the map
$\mathsf{Rep}^{n}(Y)\to\widetilde{\mathsf{Rep}}^{n}(B^{n}\pi_{n}Y)$ has a non-
invertible kernel, and the fusion rule of spherical topological functionals
becomes non-invertible in general. This non-invertible rectification of
$\mathrm{Hom}\bigl{(}\pi_{n}Y,U(1)\bigr{)}$ is of our primary interest.

Using the invertible solitonic subsymmetry we discussed in Sec. 4.3.2, we can
measure the rectification of $\mathrm{Hom}(\pi_{\bullet}-,U(1))$ by
(generalized) Hurewicz maps,

$\begin{split}\text{(bosonic)}&\qquad\pi_{\bullet}-\ \stackrel{{\scriptstyle
h_{b}}}{{\longrightarrow}}\ {H}_{\bullet}(-;\mathbb{Z})\,,\\\
\text{(fermionic)}&\qquad\pi_{\bullet}-\ \stackrel{{\scriptstyle
h_{f}}}{{\longrightarrow}}\ {\mathbb{SZ}}_{\bullet}(-)\,.\end{split}$ (67)

Since these homology groups describe the invertible topological charges, the
rectification of conventional wisdom is measured by the non-injectivity of
Hurewicz maps.

  * •

Two elements in $\pi_{\bullet}-$ differed by an element in
$\mathrm{ker}\,h_{b}$ or $\mathrm{ker}\,h_{f}$ must share the same invertible
topological charge.

  * •

The image of the dual Hurewicz map
${H}^{\bullet}\bigl{(}-;U(1)\bigr{)}\to\mathrm{Hom}\bigl{(}\pi_{\bullet}-,U(1)\bigr{)}$
or
${\mathbb{SU}}^{\bullet}(-))\to\mathrm{Hom}\bigl{(}\pi_{\bullet}-,U(1)\bigr{)}$
characterizes the survivors in the conventional wisdom as invertible
operators.

Recall that the invertible solitonic symmetry we found in Sec. 4.3.2 is
probably not maximal. Thus we may overestimate the rectification and regard
some invertible operators as non-invertible. Nevertheless, when $Y$ is
$(n\\!-\\!1)$-connected, such overestimation does not happen for dimension
$\leq n+1$ because our spectra are the $(-\\!1)$-connective covers of the
authentic spectra. Besides, the non-rectified operators we find should be
truly invertible.

###  5.2 Examples of rectification

The failure of Hurewicz maps’ being injective originates from the non-
Abelianness in the homotopy structure of the target space $Y$. Namely, we say
$Y$ is Abelian if

$\text{all homotopy groups are Abelian and
}Y\simeq\prod_{n=1}^{\infty}B^{n}\pi_{n}Y\,.$

Hurewicz maps on Abelian spaces are always injective, and thus the
conventional wisdom $\mathrm{Hom}\bigl{(}\pi_{\bullet}-,U(1)\bigr{)}$ is not
rectified. However, a general topological space is far from Abelian, and its
Postnikov tower describes its non-Abelianness.

####  5.2.1 Spherical rectification

First of all, the homotopy classes of field configurations on a sphere are, in
general, not the homotopy group, i.e.,
$\pi_{\bullet}Y\not\simeq[S^{\bullet},Y]$. This comes from a distinction
between homotopies and based homotopies. Let us write based homotopy classes
of based maps between two based spaces as $[-,-]_{*}$. Homotopy groups are
precisely $\pi_{\bullet}Y\equiv[S^{\bullet},Y]_{*}$. Through the homotopy
extension property, $\pi_{1}Y$ always naturally acts on $[-,Y]_{*}$. Then
$[-,Y]$ is precisely the orbit space of this $\pi_{1}Y$-action, i.e. (see
Prop. 4A.2 of Ref. (Hatcher:478079, ))

$[-,Y]\ \simeq\ \frac{[-,Y]_{*}}{\pi_{1}Y\text{-action}}\,.$ (68)

On homotopy groups $\pi_{n}Y$, the $\pi_{1}Y$-action comprises automorphisms
of $\pi_{n}Y$. As long as $[S^{\bullet},Y]\not\simeq\pi_{\bullet}Y$, the
conventional wisdom $\mathrm{Hom}\bigl{(}\pi_{\bullet}-,U(1)\bigr{)}$ is
rectified.

In the simplest case of $\bullet=1$, $\pi_{1}Y$ acts on itself as inner
automorphisms and thus

$[S^{1},Y]\,\simeq\,\\{\text{conjugacy classes of }\pi_{1}Y\\}\,,$ (69)

We have already seen this around the end of Sec. 4.2.3. The Hurewicz map
$\pi_{1}Y\to{H}_{1}(Y;\mathbb{Z})\simeq\widetilde{\mathbb{SZ}}_{1}(Y)$ is just
the Abelianization. Therefore, we have

${H}^{1}\bigl{(}Y;U(1)\bigr{)}\simeq\mathrm{Hom}\bigl{(}\pi_{1}Y,U(1)\bigr{)}\,,\qquad\mathbb{SU}^{1}(Y)\simeq\mathrm{Hom}\bigl{(}\pi_{1}Y,U(1)\bigr{)}\times\mathbb{Z}_{2}\,,$
(70)

which comprises $1$-dim (super-)representations of $\pi_{1}Y$. The higher-dim
(super-)representations of $\pi_{1}Y$ constitute the non-invertible objects in
$\mathsf{Rep}^{1}(Y)$ and $\mathsf{sRep}^{1}(Y)$, the
$[\geq\\!(d\\!-\\!2)]$-form solitonic symmetry.

The $\pi_{1}Y$-action on higher homotopy groups181818 There is an illuminating
way to understand these actions. $Y$’s universal cover $\bar{Y}$ naturally
carries a free $\pi_{1}Y$-action. Each $\pi_{1}Y$ element as a homeomorphism
of $\bar{Y}$ induces a group automorphism on $\pi_{n}(\bar{Y})$. These
automorphisms assemble to a $\pi_{1}Y$-action on $\pi_{n}(\bar{Y})$, which is
then turned into a $\pi_{1}Y$-action on $\pi_{n}Y$ by the natural isomorphism
$\pi_{n}(\bar{Y})\simeq\pi_{n}Y$ for $n>1$.  has no chance to be inner since
higher $\pi_{\bullet}Y$ is Abelian. To describe topological functionals on

$[S^{\bullet},Y]\ \simeq\ \pi_{\bullet}Y\Bigl{/}\pi_{1}Y\text{-action}\,,$
(71)

we can prescribe a semidirect product from the $\pi_{1}Y$-action,

$\pi_{\bullet}Y\rtimes\pi_{1}Y\,.$ (72)

Then a $S^{\bullet}$ topological functional is a character of
$\pi_{\bullet}Y\rtimes\pi_{1}Y$ which vanishes for all group elements outside
$\pi_{\bullet}Y$. A nontrivial $\pi_{1}Y$-action implies the existence of such
representations of dimension $>1$ and accordingly, the $S^{\bullet}$
topological functionals have a non-invertible fusion rule191919 The fact that
the $\pi_{1}$-action can lead to a nontrivial interplay between solitons of
different dimensions and an unconventional topological conservation law was
also noticed by, e.g., Refs. Kobayashi:2011xb ; Kobayashi:2013rca . . The
simplest example for this effect is perhaps the $S^{2}$ topological
functionals for202020 The latter example $Y\simeq BO(2)$ was discussed as
semi-Abelian gauge theories in Refs. (Nguyen:2021yld, ; Heidenreich:2021xpr, ;
Arias-Tamargo:2022nlf, ). The non-invertible symmetries for the electric
$1$-form symmetries have been discussed, while our discussion here is devoted
to the magnetic one. In both cases, the mechanism for constructing non-
invertible symmetries is essentially the same, and we can be exchanged under a
duality transformation. We note a mixed ’t Hooft anomaly between the electric
and the magnetic symmetry, so we cannot find a path-integral description where
both symmetries are solitonic.

$Y\,\simeq\,\mathbb{R}P^{2}\,\simeq\,S^{2}/\mathbb{Z}_{2}\quad\text{or}\quad
Y\,\simeq\,BO(2)\,\simeq\,BU(1)/\mathbb{Z}_{2}\,,$ (73)

or any other $Y$ whose $\pi_{\leq 2}Y$ is given by the unique nontrivial split
fibration of the following form,

$B^{2}\mathbb{Z}\to\pi_{\leq 2}Y\to B\mathbb{Z}_{2}\,.$ (74)

In these examples, $\pi_{1}Y\simeq\mathbb{Z}_{2}$ acts on
$\pi_{2}Y\simeq\mathbb{Z}$ through $\mathbb{Z}\to-\mathbb{Z}$, which means

$[S^{2},Y]\,\simeq\,\mathbb{N}\,.$ (75)

We can find its homology to be

$H_{2}(Y;\mathbb{Z})\,\simeq\,0\,,\qquad\mathbb{SZ}_{2}(Y)\,\simeq\,0\,,$ (76)

which implies that the conventional wisdom
$\mathrm{Hom}\bigl{(}\pi_{\bullet}-,U(1)\bigr{)}$ is completely rectified into
non-invertible symmetry. $S^{2}$ topological functionals are given by
characters of $D_{\infty}\simeq\mathbb{Z}\rtimes\mathbb{Z}_{2}$ which vanish
outside $\mathbb{Z}$. Since there are infinitely many topological charges,
according to the discrete approximation, we should consider the
representations that are arbitrarily close to representations of quotient
$D_{n}\simeq\mathbb{Z}_{n}\rtimes\mathbb{Z}_{2}$ for any finite $n$.
Therefore, $S^{2}$ topological functionals are $\mathbb{N}$-linearly spanned
by the characters of $2$-dim irreducible representations of $D_{\infty}$,
namely $2\cos(\phi\bullet)$ with $\bullet\in\pi_{2}Y\simeq\mathbb{Z}$ for all
$\phi\in\mathbb{R}/2\pi\mathbb{Z}$. The non-invertible fusion rule follows
evidently,

$2\cos(\alpha\bullet)\,\times\,2\cos(\beta\bullet)\ \,=\
\,2\cos[(\alpha\\!+\\!\beta)\bullet]\,+\,2\cos[(\alpha\\!-\\!\beta)\bullet]\,.$
(77)

In summary, the the $\pi_{1}$-action
$\pi_{1}\times\pi_{\bullet}\to\pi_{\bullet}$ rectifies the conventional wisdom
by directly assigning a non-invertible fusion rule to spherical topological
functionals.

####  5.2.2 Non-spherical rectification

Let us now discuss subtler effects that cause non-injectivity beyond
$\pi_{1}$-actions. When $Y$ is $(k\\!-\\!1)$-connected with $k>1$, the
Hurewicz map, $h_{b}:\pi_{\bullet}Y\to H_{\bullet}(Y;\mathbb{Z})$, is
isomorphism for $\bullet\leq k$ (and same is true for $h_{f}$), which shows
that the solitonic symmetry is invertible at least up to $k$-dim topological
functionals. Non-invertibility can appear for larger dimensions, $\bullet\geq
k+1$. Since there is no $\pi_{1}$-action here, topological functionals must be
invertible as long as they inhabit spheres. Thus rectification happens because
a spherical topological functional can also inhabit non-spheres.

To see how the non-invertible symmetry appears in this situation, we need to
take into account the inter-dimensional effects between solitons. As we have
discussed in Sec. 2.2.3, for some $k\leq p<n$, a codimension-$(p+1)$ solitonic
defect may also carry not only the $p$-dim solitonic charge but also the
$n$-dim solitonic charge, and such $n$-dim solitonic charge needs to be
measured by non-spherical topological functionals. As the Hurewicz map of
degree $n$ is not injective, this may violate the selection rule given by
$\mathrm{Hom}(\pi_{n}Y,U(1))$, which suggests that the conventional selection
rule, $\mathrm{Hom}(\pi_{n}Y,U(1))$, is valid only if solitons with higher
dimensions are absent and selection rules are modified otherwise. Accordingly,
$n$-dim solitonic symmetry has to be non-invertible to capture this intriguing
selection rule.

As the present authors pointed out in the previous paper Ref. (Chen:2022cyw,
), this effect actually happens in the $4$-dim $\mathbb{C}P^{1}$ sigma model.
Let us discuss it here as an example. Homotopy groups of the simply-connected
$\mathbb{C}P^{1}\simeq S^{2}$ at low degrees are well-known to be

$\pi_{2}(\mathbb{C}P^{1})\simeq\mathbb{Z}\,,\qquad\pi_{3}(\mathbb{C}P^{1})\simeq\mathbb{Z}\,.$
(78)

The third Postnikov truncation of $\mathbb{C}P^{1}$ then sits in a principal
fibration

$B^{3}\mathbb{Z}\ \longrightarrow\ \pi_{\leq 3}\mathbb{C}P^{1}\
\longrightarrow\ B^{2}\mathbb{Z}\,.$ (79)

It is well-known that this fibration is classified by a generator of

$H^{4}(B^{2}\mathbb{Z};\mathbb{Z})\,\simeq\,\mathbb{Z}\,.$ (80)

This structure of the homotopy $3$-type of $\mathbb{C}P^{1}$ prescribes the
following homologies,

$\displaystyle H_{2}(\mathbb{C}P^{1};\mathbb{Z})\,\simeq\,\mathbb{Z}\,,$
$\displaystyle\qquad\mathbb{SZ}_{2}(\mathbb{C}P^{1})\,\simeq\,\mathbb{Z}\,,$
(81a) $\displaystyle H_{3}(\mathbb{C}P^{1};\mathbb{Z})\,\simeq\,0\,,$
$\displaystyle\qquad\mathbb{SZ}_{3}(\mathbb{C}P^{1})\,\simeq\,\mathbb{Z}_{2}\,,$
(81b)

and requires all the relevant Hurewicz maps to be epimorphic. The last map
$\pi_{3}(\mathbb{C}P^{1})\to\mathbb{SZ}_{3}(\mathbb{C}P^{1})$’s being
epimorphic can be understood via the Freudenthal suspension theorem due to the
natural
$\widetilde{\mathbb{SZ}}_{3}(\mathbb{C}P^{1})\simeq\pi_{3}^{s}(\mathbb{C}P^{1})$.

On the one hand, the conventional wisdom is not rectified at dimension $2$,
which gives a $U(1)$ $1$-form symmetry acting on the line solitonic defects.
The concrete topological functionals were constructed by Eq. (19) in Sec. 3.1.
On the other hand, the conventional wisdom is indeed rectified at dimension 3,
which gives non-invertible $0$-form symmetry acting on point solitonic
defects. The extent of rectification depends on the bosonic/fermionic nature
of the theory. The conventional wisdom is completely rectified and non-
invertible in the bosonic theory, while a $\mathbb{Z}_{2}$ part survives in
the fermionic theory. The concrete topological functional for this
$\mathbb{Z}_{2}$ part was constructed by Eq. (23) in Sec. 3.1.

We have also constructed non-invertible $3$-dim topological functionals in
Ref. (Chen:2022cyw, ) according to the discrete approximation. Concretely, we
construct the rectified operators for
$\mathbb{Z}_{2N}\subseteq\mathrm{Hom}(\pi_{3}(\mathbb{C}P^{1}),U(1))$ for each
$N\in\mathbb{N}$. In particular, a generator of $\mathbb{Z}_{2N}$ is rectified
into the following non-invertible form:

$\mathcal{H}_{\pi/N}(M)\equiv\int{\mathcal{D}}b\,\exp\left(-\mathrm{i}\int_{M}\frac{N}{4\pi}b\,\mathrm{d}b+\mathrm{i}\int_{M}\frac{1}{2\pi}b\,\mathrm{d}\sigma^{*}a\right)\,,$
(82)

where $\sigma^{*}a$ is the same as that in Eq. (23), i.e.,
$\sigma|_{M}:M\mapsto\mathbb{C}P^{1}$ denotes the field map and $a$ denotes
the $U(1)$ gauge field on $S^{2}$ associated with the Hopf fibration $S^{1}\to
S^{3}\to S^{2}$. In the fermionic theory, these operators (82) are the
building block of $\mathsf{sRep}^{3}(\mathbb{C}P^{1})$ beyond condensation
$\mathsf{sRep}^{3}(B^{2}\mathbb{Z})\simeq\Sigma\mathsf{sRep}^{2}(\mathbb{C}P^{1})$.
In the bosonic theory, $N$ needs to be restricted to even integers without
spin structures. These even-$N$ operators are also building blocks of
$\mathsf{Rep}^{3}(\mathbb{C}P^{1})$.

When $M_{3}=S^{3}$, these topological functionals recover a naive integral
expression for the Hopf invariant,

$\mathcal{H}_{\pi/N}(S^{3})=\frac{1}{\sqrt{N}}\exp\left(\mathrm{i}\frac{\pi}{N}\int_{S^{3}}\frac{\sigma^{*}a\,\mathrm{d}\sigma^{*}a}{4\pi^{2}}\right)=\frac{1}{\sqrt{N}}\exp\left(\mathrm{i}\frac{\pi}{N}\bullet\right)\,,$
(83)

for $\bullet\in\mathbb{Z}\simeq\pi_{3}(\mathbb{C}P^{1})$, which indeed echoes
the conventional wisdom. The non-invertible feature of Eq. (82) becomes
transparent by evaluating it around the line solitonic defect, which is also
the charged object of the $U(1)$ $1$-form solitonic symmetry. The line
solitonic defect is defined by setting the boundary condition for the
$\mathbb{C}P^{1}$ field on the normal sphere bundle around $S^{1}$, which is
nothing but $S^{2}\\!\times\\!S^{1}$. Based on to the structure of
$[S^{2}\\!\times\\!S^{1},S^{2}]$ described by Eq. (20), for
$(m,\ell)\in[S^{2}\\!\times\\!S^{1},S^{2}]$, we can evaluate to have

$\displaystyle\mathcal{H}_{\pi/N}(S^{2}\\!\times\\!S^{1})=\begin{cases}\exp\left(\mathrm{i}\frac{\pi}{N}\ell\right)\,,\quad
m=0\\!\\!\mod N\\\ 0\,,\quad m\neq 0\\!\\!\mod N\end{cases}.$ (84)

It does not lift the $g$-degeneracy discussed below Eq. (21) as we expected in
Sec. 3.1. We see that the topological functional becomes nonzero only if the
level $N$ divides the $1$-form topological charge $m$, which clarifies that
the presence of line solitonic defects causes the non-invertibility of the
$0$-form solitonic symmetry. Also, we see that different manifolds can support
coherent topological charges like that $\ell$ is correlated to
$\pi_{3}(\mathbb{C}P^{1})$.

###  5.3 Examples of condensation

Finally, we describe some examples of topological functionals that are trivial
on spheres, i.e., those that lie in the image of Eq. (65). In fact, Sec. 5.2.2
already implies examples when we try to fusion some operators there. However,
here we shall present several more clean examples where conventional wisdom
totally vanishes. We focus on the simplest $2$-dim case, so suppose $M$ is a
closed $2$-dim manifold.

Prototypical bosonic examples are the $2$-dim topological functionals for two
$\mathbb{Z}_{n}$ gauge fields $s_{1,2}\in H^{1}(M;\mathbb{Z}_{n})$, or for two
$S^{1}$-valued scalars $\phi_{1,2}:M\mapsto\mathbb{R}/2\pi\mathbb{Z}$. The
former case has $Y\simeq B\mathbb{Z}_{n}^{2}$ while the latter has $Y\simeq
T^{2}$. We see always that $\pi_{2}Y\simeq 0$ but

$H^{2}\bigl{(}B\mathbb{Z}_{n}^{2};U(1)\bigr{)}\,\simeq\,\mathbb{Z}_{n}\quad\text{
and }\quad H^{2}\bigl{(}T^{2};U(1)\bigr{)}\,\simeq\,U(1)\,.$ (85)

The corresponding $2$-dim bosonic topological functionals inhabit Riemann
surfaces $M$. They are trivial on $M\simeq S^{2}$ but non-trivial on higher-
genus surfaces such as $M\simeq T^{2}$. These topological functionals can be
explicitly constructed via the universal invertible form (7). For $Y\simeq
B\mathbb{Z}_{n}$, we have

$U_{k}(M)\ \equiv\ \exp\left\\{\mathrm{i}k\frac{2\pi}{n}\int_{M}s_{1}\cup
s_{2}\right\\}\,,\qquad k\in\mathbb{Z}\,,\ k\sim k+n\,.$ (86)

For $Y\simeq S^{1}$, we have

$U_{\theta}(M)\ \equiv\
\exp\left\\{\mathrm{i}\theta\int_{M}\frac{\mathrm{d}\phi_{1}}{2\pi}\wedge\frac{\mathrm{d}\phi_{2}}{2\pi}\right\\}\,,\qquad\theta\sim\theta+2\pi\,.$
(87)

They are apparently trivial on $S^{2}$. They can be defined just via field
configurations on several properly selected loops inside $M$.

Prototypical fermionic examples are the $2$-dim topological functionals for a
$\mathbb{Z}_{2n}$ gauge field $s\in H^{1}(M;\mathbb{Z}_{2n})$, or for a
$S^{1}$-valued scalar $\phi:M\mapsto\mathbb{R}/2\pi\mathbb{Z}$. The former has
$Y\simeq B\mathbb{Z}_{2n}$ while the later has $Y\simeq S^{1}$. We can readily
see $\pi_{2}Y\simeq H_{2}(Y;\mathbb{Z})\simeq 0$ and find via the long exact
sequence (56) that

$\mathbb{SZ}_{2}(Y)\,\simeq\,H_{1}(Y;\mathbb{Z}_{2})\,\simeq\,\mathbb{Z}_{2}\,.$
(88)

We thus see that the corresponding topological functionals are fermionic,
i.e., they inhabit spin Riemann surfaces $M$. Also, they are trivial on
$S^{2}$ but non-trivial on other spin Riemann surfaces. Via the universal
invertible form (7), we can construct these fermionic topological functionals
through the Arf invariant of spin structures. Namely, we pick up a spin
structure $\rho$ on $M$ such that $\mathrm{Arf}(\rho)=0$, i.e., $[\rho]\in
0\in\Omega^{Spin}_{2}$. Recall that, on top of an orientation, spin structures
form an $H^{2}(M;\mathbb{Z}_{2})$-torsor. Then for $Y\simeq B\mathbb{Z}_{2n}$,
we have

$U_{k}(M)\ \equiv\ (-)^{k\,\mathrm{Arf}\bigl{(}\rho+\bar{s}\bigr{)}}\,,\qquad
k\in\mathbb{Z}\,,\ k\sim k+2\,,$ (89)

where $\bar{s}$ denotes the mod-2 reduction of $s$. For $Y\simeq S^{1}$, we
just substitute $\bar{s}$ above with the mod-2 reduction of $[\phi]\in
H^{2}(M;\mathbb{Z})$. It is evident to see that this topological functional
vanishes on $S^{2}$.

We now carefully analyze the identity problem (see Sec. 3.1) of the above
example. Concretely, we consider torus topological functionals for a
$S^{1}$-valued scalar or a $\mathbb{Z}_{n}$-valued gauge field. Recall that
the target space takes the form $Y\simeq BR$ for $R\simeq\mathbb{Z}$ in the
former case and for $R\simeq\mathbb{Z}_{n}$ in the latter case. We readily
recognize

$[T^{2},BR]\simeq H^{1}(T^{2};R)\simeq R^{2}\,.$ (90)

The mapping class groups of $T^{2}$ are well-known, e.g.,

$\displaystyle\pi_{0}\mathrm{Diff}(T^{2})$ $\displaystyle\simeq
GL(2,\mathbb{Z})\,,$ (91a) $\displaystyle\pi_{0}\mathrm{Diff}(T^{2},\xi)$
$\displaystyle\simeq SL(2,\mathbb{Z})\,,$ (91b)
$\displaystyle\pi_{0}\mathrm{Diff}(T^{2},\xi,\rho)$ $\displaystyle\simeq
p^{-1}\Bigl{(}\mathbb{Z}_{2}\Bigr{)}\,,\quad\text{for }\mathbb{Z}_{2}\subseteq
SL(2,\mathbb{Z}_{2})\text{ and }SL(2,\mathbb{Z})\stackrel{{\scriptstyle
p}}{{\to}}SL(2,\mathbb{Z}_{2})\,.$ (91c)

where $\xi$ is an orientation and $\rho$ is still a spin structure such that
$\mathrm{Arf}(\rho)=0$. Without loss of generality, $\rho$ can be taken as
antiperiodic-periodic. Since $[T^{2},BR]$ is just a $2$-dim lattice, these
mapping class groups act on $[T^{2},BR]$ just as 2-by-2 $\mathbb{Z}$-valued
matrices. These actions are far from trivial, and concretely, the criteria for
classifying $(a,b)\in[T^{2},BR]$ into its action orbit are as follows:

$\displaystyle\pi_{0}\mathrm{Diff}(T^{2})$ $\displaystyle\text{: }\
\mathrm{gcd}(a,b)\,,$ (92a) $\displaystyle\pi_{0}\mathrm{Diff}(T^{2},\xi)$
$\displaystyle\text{: }\ \mathrm{gcd}(a,b)\,,$ (92b)
$\displaystyle\pi_{0}\mathrm{Diff}(T^{2},\xi,\rho)$ $\displaystyle\text{: }\
\mathrm{gcd}(a,b)\,,\ a\\!\\!\\!\\!\mod 2\,,$ (92c)

where we stipulate $\mathrm{gcd}(0,0)\equiv 0$. We see that a spin structure
slightly lifts the degeneracy when $R$ has an even characteristic. This tiny
degeneracy lifting is exactly realized by the invertible topological
functional (89). Despite huge degeneracy, equation (92) still gives tremendous
inequivalent topological charges, and they have to be distinguished by non-
invertible topological functionals. They are examples of non-invertible
condensations.

##  6 Discussion

In this paper, we discussed the general structure of the non-invertible
solitonic symmetry by attempting a precise mathematical formulation (Sec. 4)
after a pursuit of the proper physical constraints (Sec. 3). It allows us to
discuss how the non-invertible structure in general solitonic symmetry goes
beyond the conventional wisdom of homotopy groups (Sec. 5) in terms of
concrete examples. Nevertheless, our analysis in this paper is still far from
complete to be complemented by future studies. In this section, we are going
to make some remarks and discuss the outlooks for future studies.

###  6.1 Remarks

First, we point out an interesting connection between the solitonic symmetry
and the higher-group symmetry, which was actually already mentioned around the
end of Sec. 2.1. Assume we have a QFT $\mathcal{T}$ with an anomaly-free
discrete higher-group $\mathsf{G}$. Then we can consider another QFT
$\mathcal{T}/\mathsf{G}$ by dynamically gauging the higher-group $\mathsf{G}$.
This gauging procedure can be realized by adding a path integral over
$\mathsf{G}$ gauge fields. The homotopy target space of this path integral is
precisely the classifying space $B\mathsf{G}$. Thus the theory
$\mathcal{T}/\mathsf{G}$ acquires the solitonic symmetry
$\mathsf{(s)Rep}^{\bullet}(B\mathsf{G})$. Therefore, gauging a non-anomalous
higher-group symmetry produces a QFT with non-invertible solitonic symmetry.
In particular, this picture suggests the last property of solitonic symmetry
described in this paper.

###### Conjecture 6.1

Solitonic symmetry is free of ’t Hooft anomaly.

It comes from the belief that the dynamical gauging procedure is reversible.
Namely, if we could develop the proper gauging procedures of the solitonic
symmetry, then it would be natural to expect that gauging of
$\mathsf{(s)Rep}^{\bullet}(B\mathsf{G})$ in $\mathcal{T}/\mathsf{G}$ produces
the original theory $\mathcal{T}$. This picture echoes the ideas developed in
Refs. (Bhardwaj:2022lsg, ; Bhardwaj:2022kot, ; Bartsch:2022mpm, ;
Bartsch:2022ytj, ). One may consider a Tannaka duality between higher-group
symmetry and solitonic symmetry. This also suggests a wide adaption of
solitonic symmetry: A broad class of non-solitonic symmetry can also be
described by solitonic symmetry via a properly selected virtual target space
$Y$.

Second, we point out that QFTs with discrete higher-group symmetry also
possess solitonic sectors. Their solitonic sectors are the higher-domain-walls
of spontaneously breaking higher-group symmetry and inhabit only non-closed
manifolds (recall Footnote 5). However, via the dynamical gauging discussed
above, these non-closed solitonic sectors are the counterparts of our closed
solitonic sectors addressed in this paper. For example, when a discrete
symmetry is spontaneously broken, there are domain walls connecting different
vacua, and they are well-defined on the infinite space. However, when we
consider the closed space with the periodic boundary condition, a single
domain wall is inconsistent with the boundary condition. Even in this case,
the single domain wall becomes well-defined on closed manifolds by considering
the gauging of the broken symmetry or the symmetry-twisted sector. Therefore,
solitons in different theories can behave locally similarly but have different
global behaviors.

###  6.2 Outlooks

We present a list of outlooks based on the subtle points in the analysis of
fully-extended TQFTs in the present paper. They should also be of interest to
those concerned about classifying gapped phases.

  * •

Does the Karoubi completeness, as well as the operation $\Sigma$, provide a
fully satisfactory solution to the codomains for fully-extended TQFTs? Does
$\Sigma$ correctly capture the physical condensation?

  * •

The validity of Conjecture 4.4 on the homotopy actions should be confirmed.

  * •

The maximal invertible solitonic symmetry, i.e., the structure of the infinite
loop spaces $\left(\Sigma^{n-1}\mathsf{Vect}^{fd}\right)^{\times}$ and
$\left(\Sigma^{n-1}\mathsf{sVect}^{fd}\right)^{\times}$ for $n>1$, should be
clarified. The higher-group structure of invertible solitonic symmetry should
also be clarified.

As for the solitonic symmetry itself, we also have many prospects, such as
exploring the relation between solitonic symmetry and other generalized
symmetry, constructing explicitly more examples of topological functionals,
and presenting a comprehensive classification of low-dim topological
functionals. A list of intriguing general questions is as follows.

  * •

The validity of Conjecture 3.3 on the correspondence between TQFTs and their
partition functions should be confirmed.

  * •

A systematic treatment of continuous solitonic symmetry and infinitely many
topological charges should be developed. A proposal is to consider things like
$\Sigma^{n-1}\mathsf{Hilb}^{fd}$ (c.f. Ref. (bartlett2009unitary, )). Another
is to make the discrete approximation rigorous.

  * •

We should rigorously formulate the inductive decomposition into non-spherical
condensation and spherical rectification discussed in Sec. 5.1. Namely, we
should prove our expected “short exact sequence” (66) and determines its
structure by the Postnikov fibration.

  * •

Can we tackle solitonic symmetry from the side of topological charges? This
requires a complicated analysis of the homotopy classes of maps on manifolds,
including their relations via bordisms (see Ref. (Chen:2022cyw, )) to resolve
the coherence problem (see Sec. 3.2), as well as a direct treatment of the
identity problem (see Sec. 3.1).

If these problems can be solved, we will further strengthen our confidence in
Ansatz 3.2 or know how to modify it. This work thus serves as an outset in the
pursuit of a perfect description of solitonic symmetry.

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