††institutetext: Department of Applied Mathematics and Theoretical Physics
(DAMTP),††institutetext: University of Cambridge,††institutetext: Wilberforce
Road, CB3 0WA, Cambridge, UK

# Categorical Symmetries and Fiber Functors from Multiple Condensing
Homomorphisms from 6D ${\cal N}=(2,0)$ SCFTs

Veronica Pasquarella<EMAIL_ADDRESS>
###### Abstract

Exploiting the symmetry topological field theory/topological order
correspondence (SymTFT/TO), together with the higher-categorical structure of
6D ${\cal N}=$(2,0) SCFTs, we prove that the total quantum dimension of the
relative condensation algebra leading to intrinsic non-invertible symmetries
between class ${\cal S}$ theories is greater with respect to the non-intrinsic
case. From a higher-categorical perspective, this supports the idea that
multiplicity is allowed to exceed unity in some superselection sectors.

###### Keywords:

higher categories, representation theory, relative QFTs, gauge theory,
condensing homomorphisms

##  1 Introduction

Since first encountered, Witten:1995zh , 6D ${\cal N}=(2,0)$ SCFTs, also
referred to as _theory X_ , have attracted great interest in, both, the
mathematics and high-energy physics community alike. From the high-energy
physics point of view, they are essential for deriving lower-dimensional
SCFTs, Witten:2007ct ; Chacaltana:2012zy ; Tachikawa:2013hya ; Moore , whose
features are much better understood. On the other hand, from a purely
mathematical perspective, they provide an extremely interesting and rich setup
where to exploit the present-day knowledge of representation theory,
Witten:2009at ; DBZ .

In recent years, the two communities have been brought together by fascinating
developments on either side, Bah:2022wot , highlighting the importance of
bridging the gap in between different formalisms, with category theory playing
a key role in pursuing this task. Indeed, the pioneering works of Moore:1988qv
; Moore:1988ss ; Verlinde:1988sn  led to the discovery of modular tensor
categories in anyon condensation theory. Since then, such deep mathematical
language inspired applicability to higher-dimensional QFTs, by identifying
their symmetries with topological defects, Gaiotto:2014kfa . Of particular
importance to us is the richness of higher-categorical theory applied to
topological orders (TOs), Kong:2020cie ; Kong:2022cpy ; Kong:2019byq ;
Kong:2019cuu ; Kong:lastbutone ; Gaiotto:2019xmp ; Johnson-Freyd:2021tbq ;
Johnson-Freyd:2020usu ; MYu , and symmetry topological field theories
(SymTFTs), Freed:2012bs ; Freed:2022qnc ; Freed:2022iao .

Embracing the thrust of recent developments in the field, the present work is
meant to be the first of a series of papers where the author applies the
techniques of category theory to furthering the understanding of 6D ${\cal
N}=(2,0)$ SCFTs and theories derived from them. In doing so, we will attempt,
wherever possible, to provide an understanding of the procedure as well as of
the outcome of our findings, from, both, a mathematician’s and a theoretical
physicist’s point of view. Hopefully, this will further motivate both
communities to benefit form each other’s influence.

This article aims at furthering the understanding of the categorical structure
arising from dimensional reduction of 6D ${\cal N}=(2,0)$ SCFTs. In
particular, our aim is that of identifying a criterion to distinguish between
intrinsic and non-intrinsic non-invertible symmetries111See, for example,
Bashmakov:2022jtl ; Bashmakov:2022uek ; Bhardwaj:2022kot ; Bhardwaj:2022yxj ;
9 ; Kaidi:2021xfk ; Kaidi:2022uux ; Kaidi:2022cpf ; Choi:2022zal ;
Choi:2021kmx  for explanation on the current state of the art. in terms of the
quantum dimension222The total quantum dimension is defined from the braiding
and fusion rules of the underlying categorical structure of the topological
symmetries of the theory in question. Further explanation will be provided in
due course. of condensible algebras. In doing so, we rely upon their
description in terms of relative field theories, as described in
Bashmakov:2022uek  in terms of the Freed-Moore-Teleman setup, Freed:2012bs ;
Freed:2022qnc ; Freed:2022iao . Our main proposal is that the total quantum
dimension of the condensible algebra implementing the gauging in the bulk Sym
TFT is a key quantity to distinguish in between the two cases. Specifically,
for the intrinsic case, multiplicity is greater w.r.t. the non-intrinsic case,
thereby signalling the possibility for additional d.o.f. to be stored in
certain superselection sectors of the resulting absolute theory.

Our results extend arguments proposed in Kaidi:2022cpf , where the authors
also proposed a way of distinguishing intrinsic from non-intrinsic non-
invertibility in 2D by means of the quantum dimension of the non-invertible
defect.

The present work is organised as follows: section 2 provides a brief overview
of some key features of higher-categorical structures that are relevant for
our analysis. In the first part of this section, we briefly recall the
relation between the total quantum dimension, fusion and braiding structures.
As an example of a higher-categorical structure, we describe the Chern-Simons
theory for a flux-charge-quasiparticle setup, and how gauging a subgroup of
the anyonic symmetry affects the total quantum dimension.

Section 3 opens with an explanation for how to gauge a higher-categorical
structure by means of algebraic condensation, TJF ; Kong:2013aya ; Yu:2021zmu
, emphasising the richer structure resulting from the case in which
condensation of multiple subalgebras takes place. We then turn to implementing
such techniques to theories descending from 6D ${\cal N}=(2,0)$ SCFTs upon
dimensional reduction over a Riemann surface. At first, we briefly recall the
realisation of 6D theories with maximum amount of supersymmetry as a relative
QFT in the Freed-Teleman setup, and how to recover intrinsic non-invertible
symmetries in class ${\cal S}$ theories by performing a double floodding of
the Sym TFT according to the SymTFT/TO correspondence. We show that the
quantum dimension of the relative condensation algebra interpolating between
different absolute 4D gauged theories in the intrinsic case always exceeds
that featuring in the non-intrinsic case. We thereby propose this as a general
criterion to be applied to class ${\cal S}$ theories descending from 6D ${\cal
N}=(2,0)$ SCFTs. En-passing, we comment on a proposal regarding the
realisation of such composite absolute theories and the definition of a fiber
functor intrinsically related to the notion of a partition function for a 3D
theory rather than a 4D theory, which is current work in progress by the same
author, VP .

##  2 Higher-categorical structures

For a QFT to be well defined, its spectrum of allowed operators must be
compatible with the symmetries of the given theory, Aharony:2013hda . This
statement has been a very active area of research, and mostly motivated
furthering a full mathematical formulation of QFTs. Category theory has been
pointed out as a promising candidate in pursuing such aim.

Following suite, the present work aims at exploring the higher-categrical
theory of and descending from 6D ${\cal N}=(2,0)$ SCFTs with the aim of
identifying a categorical theory quantity enabling to distinguish between
intrinsic and non-intrinsic non-invertible symmetries in class ${\cal S}$
theories obtained by dimensional reduction of their parent theory.

Before delving into the specifics of the 6D theory, to which section 3 is
devoted, the current section aims at providing a brief overview of some
essentail tools we will need in the remainder of the paper, as well as further
motivating the use of suc hmathematical tools within the following theoretical
physics treatment. This section is structured as follows:

  * •

In the first part of this section we briefly overview some key definitions of
higher-categories.

  * •

As a specific example, we outline the flux-charge-quasiparticle treatment in
terms of Chern-Simons (CS) theory, and how this is related to representation
theory.

###  2.1 Defect fusion categories

_Categories_ are defined as sets of objects related to each other by
morphisms. An $n$-category is a collection of categories related to each other
by _functors_. Groups constitute a special type of categories, where only one
element is present, and every morphism is an isomorphism. A _higer_ -category,
or $n$-category, is an $(\infty,1)$-category enriched with $(n-1)$-categories,
with $n$ denoting the dimensionality.

A category, ${\cal C}$, is _monoidal_ (MC) if it comes equipped with a bi-
functor

$\otimes\ :\ {\cal C}\times{\cal C}\ \rightarrow\ {\cal C}.$ (1)

For ${\cal C}$ semisimple, the tensor product of any two simple objects
$x,y\in{\cal C}$ is

$x\ \otimes\ y\ \simeq\ \bigoplus_{z\in\text{Irr}({\cal C})}\
N_{{}_{xy}}^{{}^{z}}\cdot z\ \ \ \ ,\ \ \ \
N_{{}_{xy}}^{{}^{z}}\in\mathbb{N},$ (2)

where $\left\\{N_{{}_{xy}}^{{}^{z}}\right\\}_{{}_{x,y,z\in\text{Irr}({\cal
C})}}$ denote the _fusion rules_ of ${\cal C}$. Equivalently, the isomorphism
classes of simple objects ${\cal C}$ generate the _fusion ring_ or
_Grothendieck ring_ of ${\cal C}$, where multiplication is given by (1), and
$N_{{}_{xy}}^{{}^{z}}$ as structure constants.

Given $x,y\in\ {\cal C}$

$\text{Hom}:\ x,y\ \ \rightarrow\ \text{Hom}(x,y)$ (3)

is an $(n-1)$-category, and for $\mathbb{1}\in{\cal C}$

$\text{Hom}:\ \mathbb{1},\mathbb{1}\ \ \rightarrow\ \text{Hom}(\mathbb{1}\
,\mathbb{1})\ \overset{def.}{=}\ \Omega\ {\cal C},$ (4)

$\Omega\ {\cal C}$ is a monoidal $(n-1)$-category.

### Braiding and ribbon structure

A modular category (MC) is _braided_ if, in addition to (1) it is also
equipped with a natural isomorphism

$c_{{}_{x,y}}:x\otimes y\ \rightarrow\ y\otimes x,$ (5)

called _braiding_. Given two simple objects $x,y\in{\cal C}$, the _S-matrix_
is defined as

$\boxed{\ \ S\ \overset{def.}{=}\
\left(S_{{}_{xy}}\right)_{{}_{z\in\text{Irr}({\cal
C})}},\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }$ (6)

with components defined from (5)

$S{{}_{xy}}\ \overset{def.}{=}\ \text{Tr}\ \left(c_{{}_{y,x^{*}}}\ \circ\
c_{{}_{x^{*},y}}\right)\ \equiv\ \text{Tr}\ \left(c_{{}_{y^{*},x}}\ \circ\
c_{{}_{x,y^{*}}}\right).$ (7)

Figure 1: The S-matrix corresponds to the Hopf link, and encodes the
information of mutual-statistics. As shown in these pictures, the linking
between particle worldlines encodes information of mutual-statistics.

For ${\cal C}$ a braided fusion category and $x\in{\cal C}$, the 2 morphisms

$x\ \xrightarrow{\text{id}_{{}_{x}}\otimes b_{{}_{x}}}\ x\otimes x\otimes
x^{*}\ \xrightarrow{c_{{}_{x,x}}\otimes\text{id}_{{}_{x^{*}}}}\ x\otimes
x\otimes x^{*}\xrightarrow{\text{id}_{{}_{x}}\otimes b_{{}_{x}}^{{\dagger}}}\
x$ (8)

$x\ \xrightarrow{d_{{}_{x}}^{{\dagger}}\otimes\text{id}_{{}_{x}}}\
x^{*}\otimes x\otimes x\ \xrightarrow{\text{id}_{{}_{x^{*}}}\otimes
c_{{}_{x,x}}}\ x^{*}\otimes x\otimes
x\xrightarrow{b_{{}_{x}}\otimes\text{id}_{{}_{x}}}\ x$ (9)

are both equivalent to

$\theta_{{}_{x}}:\ x\rightarrow x,$ (10)

defining the _twist_ or _topological spin of x_. Importantly,
$\theta_{{}_{x}}\neq\text{id}_{{}_{x}}$, since the worldline should be viewed
as a ribbon, hence a line equipped with a framing. Both processes described in
figure 2 are therefore equivalent to twisting the ribbon counterclockwise. The
twsisting rotates the topological defect by 360∘, reason why $\theta_{{}_{x}}$
is called the topological spin of x.

Figure 2: The ribbon structure describing self-statistics is here drawn as the
worldline of a parrtcle excitation $x$ winding around itself. Such information
is encoded in the T-matrix.

$\forall x\in{\cal C}$, since Hom${}_{{}_{{\cal C}}}(x,x)\simeq\mathbb{C}$,

$\theta_{{}_{x}}\equiv T_{{}_{x}}\cdot\text{id}_{{}_{x}}\ \ \ ,\ \ \
T_{{}_{x}}\in\mathbb{C},$ (11)

where the $T$-matrix of the unitary braided fusion category is defined as

$T\overset{def.}{=}\left(T_{{}_{x}}\delta_{{}_{xy}}\right)_{{}_{x,y\in\text{Irr}({\cal
C})}},$ (12)

encoding the information of self-statistics.

### Quantum double category of a finite group

$\forall$ finite group $G$, there is a unitary MTC ${\cal D}_{{}_{G}}$ called
_double quantum category of G_ , Kong:2022cpy , such that:

  1. 1.

An object in ${\cal D}_{{}_{G}}$ is a vector space $V$ equipped with a
$G$-action

$\rho:\ G\ \longrightarrow\ GL(V),$ (13)

and a $G-grading$ $V\equiv\underset{g\in G}{\bigoplus}V_{{}_{g}}$ such that

$\rho(g):V_{{}_{h}}\ \longrightarrow\ V_{{}_{ghg^{{}^{-1}}}}\ \ \ ,\ \ \
\forall g,h\ \in\ G.$ (14)

  2. 2.

A morphism in ${\cal D}_{{}_{G}}$ is a $\mathbb{C}$-linear map that is, both,
a morphism in Rep$(G)$ and a morphism in Vec$(G)$.

  3. 3.

Given $g\in G$, we denote its conjugacy class as $[g]$, and its centraliser as
$Z(g)$. If $\pi$ is an irrep $Z(g)$-representation, the induced representation
Ind${}_{{}_{Z(g)}}^{{}^{G}}(\pi)\overset{def.}{=}X_{{}_{(g,\pi)}}$ is a
$G$-representation admitting a canonical $G/Z(g)\simeq[g]$-grading.
$X_{{}_{(g,\pi)}}$ is a simple object in ${\cal D}_{{}_{G}}$.

  4. 4.

The isomorphism classes of objects in ${\cal D}_{{}_{G}}$ are labelled by
pairs $([g],[\rho])$ where $[g]$ is the conjugacy class in $G$ and $[\rho]$ is
an isomorphism class of irreps of $Z(g)$, where $Z[g]$ only depends on the
conjugacy class of $g$.

The properties of the $S$ and $T$ matrices defined in (6) and (12), and their
generalisations to the case of quantum double categories have been extensively
described in previous works (see, for example, Kong:2022cpy ). For the purpose
of the present work, it is important to stress that the $S$ and $T$ matrices
are related to the generators of the $SL(2,\mathbb{Z})$ group

$s\overset{def.}{=}\left(\begin{matrix}0\ \ -1\\\ 1\ \ \ \ 0\\\
\end{matrix}\right)\ \ \ ,\ \ \ t\overset{def.}{=}\left(\begin{matrix}1\ \ \
1\\\ 0\ \ \ 1\\\ \end{matrix}\right),$ (15)

such that

$(st)^{{}^{3}}\ \equiv\ s^{{}^{2}}\ \ \ ,\ \ \ s^{{}^{4}}\equiv 1.$ (16)

under a projective $SL(2,\mathbb{Z})$-representation

$s\ \mapsto\ \frac{S}{\ \sqrt{\text{dim}({\cal C})\ }}\ \ \ ,\ \ \ t\ \mapsto\
\ T.$ (17)

The quantity ${\cal D}\overset{def.}{=}$dim$({\cal C})$ is the _total quantum
dimension_ of the category ${\cal C}$. For the case in which ${\cal C}$ is
made up of purely simple objects, the total quantum dimension of a given
superselection sector simply reads

${\cal D}_{{}_{x}}\overset{def.}{=}\sqrt{\ \sum_{i}\
\text{dim}^{{}^{2}}(x_{{}_{i}})\ }=\sqrt{\ \sum_{i}\
S_{{}_{\mathbb{1}x_{i}}}^{{}^{2}}\ }.$ (18)

###  2.2 Example of a higher-categorical structure

We now turn to describing the main example of MTC of interest to us. We will
assume that the total category associated to a theory ${\mathfrak{T}}$ with
gauge group $G$, ${\cal C}_{{}_{\mathfrak{T}}}$ is given by

${\cal C}_{{}_{\mathfrak{T}}}\ \overset{def.}{=}\\{{\cal C}_{{}_{point}},{\cal
C}_{{}_{flux}},{\cal C}_{{}_{Hopf}}\ \\},$ (19)

where each subcategory is a different superseclection sector of the theory,
defined as follows:

  1. 1.

${\cal C}_{{}_{point}}\ \equiv\ \\{1,a,...\\}$, $a\equiv R\in\
(G)_{{}_{IRREP}}$, are point excitations, corresponding to the irreducible
representations of the group $G$. The quantum dimension of each element of the
category is $d_{{}_{a}}\ \overset{def.}{=}\ \text{dim}\ R$.

  2. 2.

${\cal C}_{{}_{flux}}\ \equiv\ \\{1,\mu,...\\}$ is the superselection sector
of pure fluxes, and therefore corresponds to the conjugacy classes of the
finite group $G$, $C\in\ (G)_{{}_{cj}}$. Their quantum dimensions are
$d_{{}_{\mu}}\ \overset{def.}{=}\ \sqrt{|C|\ }$, with $C$ denoting the number
of group elements in a given conjugacy class.

  3. 3.

${\cal C}_{{}_{Hopf}}\ \equiv\ \\{1,\eta,...\\}$ is the collection of Hopf
link excitations,

with $\eta\overset{def.}{=}\ \left(C_{{}_{(g,h)}},R\right),gh\equiv
hg,C_{{}_{(g,h)}}\equiv\ \\{(tgt^{-1},tht^{-1}|t\in G\\},R\in\ E_{{}_{(g,h)}}\
\equiv\ \\{t\in G\ |(g,h)\equiv\left(tgt^{-1},tht^{-1}\right)\\}$, and quantum
dimensions

$d_{{}_{\eta}}\ \overset{def.}{=}\ \frac{|G|}{\ |E_{{}_{(g,h)}}|\ }\
\text{dim}\ R$ (20)

The 1-morphisms between these categories are

$\varphi\ :\ {\cal C}_{{}_{flux}}\ \hookrightarrow\ {\cal C}_{{}_{loop}}\
\subset\ {\cal C}_{{}_{Hopf}}$ (21)

$\phi\ :\ {\cal C}_{{}_{point}}\ \hookrightarrow\ {\cal C}_{{}_{loop}}\
\subset\ {\cal C}_{{}_{Hopf}}$ (22)

$v\ :\ {\cal C}_{{}_{Hopf}}\
\xrightarrow{\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}aaaa\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}}\
{\cal C}_{{}_{Hopf}}\ \ \ ,\ \ \ v(\eta)\ \equiv\eta^{{}^{\text{V}}}$ (23)

The aforementioned properties suggest the phisical interpretation of the
superselection sectors in (19) as the electric, magnetic and dyonic charges
characterising the spectrum of the theory $\mathfrak{T}$. As a result, the
objects living in ${\cal C}_{{}_{\mathfrak{T}}}$ are defined by a 3-tuple
$\chi\equiv([M],\lambda,\rho)$, corresponding to a flux-quasiparticle-charge
composite.

As mentioned earlier on in the introduction, application of categories in
physics began in the study of 2D CFTS and $D+1$ TQFTs. Their mathematical
formulation was originally motivated by the idea of formalising the
factorisation property of path-integrals as a certain monoidal functor defined
on a cobordism category. Thanks to the pioneering work of Moore and Seiberg on
_modular tensor categories_ in 2D RCFTs, Rashetikhin and Turaeev reformulated
their discovery in present-day categorical formulation, and used it to give
the 2+1 D RT TQFT formulation, eventually leading to the study of topoplogical
excitations (or anyons) in a 2D topological order. Anyons are quasiparticles,
and as such can be thought of as being the physical counterpart of (19).
Systems exhibiting anyon symmetries with gauge group $G$ in a given
topological phase efficiently be related to Chern-Simons (CS) theory.

For such correspondence to hold, the CS theory needs to account for the
topological information defined by the braiding statistics and quasiparticle
fusion rules (cf. section 2.1). For example, an abelian phase in (2+1)D can be
characterised by a QFT with partition function defined as follows

${\cal Z}[{\cal J}]\ \equiv\ \int\left[{\cal
D}\alpha(\overset{\rightarrow}{r})\right]\ \exp\left(iS[{\cal J}]\right],$
(24)

involving an $N$-component set of $U(1)$ gauge fields
$\alpha\equiv(\alpha_{{}_{1}},...,\alpha_{{}_{N}})$ with action

$S[{\cal J}]\equiv\frac{1}{4\pi}\ \int\left(K_{{}_{IJ}}\alpha^{{}^{I}}\wedge
d\alpha^{{}^{J}}+\alpha^{{}^{I}}\ {\cal J}_{{}_{I}}\right).$ (25)

Quasiparticles, $\psi^{{}^{\overset{\rightarrow}{a}}}$, are sources for ${\cal
J}_{{}_{1}}^{{}^{a_{{}_{1}}}},...,{\cal J}_{{}_{N}}^{{}^{a_{{}_{N}}}}$,
labelled by $\overset{\rightarrow}{a}\equiv(a_{{}_{1}},...a_{{}_{N}})$ on a
lattice $\Gamma^{*}\equiv\mathbb{Z}^{{}^{N}}$. At long distance, nearby
quasiparticles can form a single entity, leading to the definition of a fusion
structure. The $K$-matrix in (25) dictates the braiding statistics of quasi-
particles

${\cal D}\ S_{{}_{ab}}\ \equiv\ e^{{}^{2\pi ia^{{}^{T}}K^{-1}b}}\ \ \ ,\ \ \
{\cal D}\equiv\sqrt{|\text{det}(K)|\ }\equiv\sqrt{{\cal A}\ }$ (26)

with ${\cal D}$ the total quantum dimension. The exchange statistics, instead,
is defined by

$\theta_{{}_{a}}\ \equiv\ e^{{}^{2\pi ia^{{}^{T}}K^{-1}b}},$ (27)

corresponding to the spin of the quasi particle. The topological phase
generated by gauging the anyonic symmetry, is referred to as _twisted liquid_.
The latter are generalisations of (2+1)D discrete gauge theories. Their
quasiparticles are compositions of fluxes and charges associated to the gauged
anyonic symmetry, as well as a superselection sector of the original
topological state. Their total quantum dimensions are related as follows

${\cal D}_{{}_{TL}}\ \equiv\ {\cal D}_{{}_{o}}\ |G|,$ (28)

with $|G|$ the order of the anyonic symmetry group $G$ being gauged.

An important remark is of order, though. A CS description of a topological
phase (cf. (25)) is not unique. This follows from the fact that the $K$ matrix
encodes the same fusion and braiding structure even after undergoing a basis
transformation. The set of automorphisms, Aut$(K)$, preserving $K$ correspond
an anyonic symmetry operation permuting the anyons with the same fusion
properties and spin-statistics. As such, it classifies the global symmetries
of the TQFT associated to the CS action, $S_{{}_{CS}}$. Once having modded-out
the trivial relabellings, Inn$(K)$, we are left with the outer automorphisms

$\text{Outer}(K)\ \overset{def.}{=}\ \frac{\text{Aut}(K)}{\text{Inner}(K)},$
(29)

which will play a key role in the remainder of the present work.

These results about the total quantum dimension in gauged topological orders
have been thoroughly investigated in recent years. In the next section, we
shall see how the presence of intrinsic non-invertibles arising in the gauge
theory lead to a different expression for the total quantum dimension, thereby
signalling that the multiplicity of some superselection sectors can be greater
that unity.

##  3 Fiber functors from non-invertible symmetries

The main motivation of the present section is that of studying the defects
interpolating different 4D class ${\cal S}$ theories descending from
dimensional reduction of 6D ${\cal N}$=(2,0) SCFTs. In particular, explaining
how such defects emerge under condensation of multiple subalgebras, we propose
the quantum dimension of the relative condensable subalgebra as the quantity
needed to distinguish whether such interpolating defect is intrinsically or
non-intrinsically non-invertible, thereby proposing the 4D counterpart of
arguments proposed in Kaidi:2022cpf  for the 2D case.

The present section is therefore structured into 4 main parts:

  * •

At first, we describe the so-called gauging-by-condensation procedure, TJF ;
Kong:2013aya ; Yu:2021zmu , and its adaptation to the Freed-Teleman,
Freed:2012bs , and Freed-Moore-Teleman construction, Freed:2022qnc ;
Freed:2022iao , of relative field theories and topological symmetries of QFTs,
respectively, with particular emphasis on the case of multiple simultaneous
condensation of different subalgebras.

  * •

We then briefly overview the procedure adopted by Bashmakov:2022uek  for
realising absolute 4D SCFTs from ${\cal N}=(2,0)$ SCFTs, explicitly rephrasing
it in terms of the multiple gauging-by-condensation procedure described in
section 3.1.

  * •

When describing such interpolating defects in terms of condensation of
homomorphisms, they naturally admit an interpretation in terms of fusion
tensor categories interpolating between different 2-categories, to a total
quantum dimension can be assigned. Section 3.3 closes with the relation
between the issue in defining a fiber functor in presence of non-invertible
defects interpolating between different absolute theories, which is current
work in progress by the same author, VP .

  * •

Making use of the gauging-by-condensation procedure explained in section 3.1,
we propose a criterion for distinguishing between the two types of non-
invertible defects in terms of the total quantum dimension of the relative
condensation algebra implementing the gauging.

###  3.1 Gauging-by-condensation in relative QFTs

In the formulation of Freed:2012bs , a _relative_ field theory, $\tilde{F}$,
requires additional topological data in order to be fully specified. Such data
is encoded in a pair $(\sigma,\rho)$, referred to as _quiche_. $\sigma$ is the
_symmetry topological field theory_ (SymTFT), wheras $\rho$ geometrises the
choice of boundary conditions for the fields defining the relative theory
$\tilde{F}$. The overall system, depicted in figure 3, gives rise to an
_absolute_ QFT, $F_{{}_{\rho}}$.

Figure 3: The Freed-Moore-Teleman setup, with $\tilde{F}$ denoting a relative
QFT. Specifying the topological data $(\sigma,\rho)$, the resulting theory,
$\tilde{F}_{{}_{\rho}}$ is absolute.

In mathematical terms, the description outlined above can be formulated in
terms of bordism in the following way. Fixing $N\in\mathbb{Z}^{{}^{\geq 0}}$,
then a quiche is a pair $(\sigma,\rho)$ in which
$\sigma:\text{Bord}_{{}_{N+1}}(F)\rightarrow{\cal C}$ is an $N+1$-dimensional
TFT and $\rho$ is a right topological $\sigma$-module. The quiche is
$N$-dimensional, hence it shares the same dimensionality as the theory on
which it acts. Let $F$ be an $N$-dimensional field theory. A
$(\sigma,\rho)$-_module structure_ on $F$ is a pair $(\tilde{F},\theta)$, in
which $\tilde{F}$ is a left $\sigma$-module and $\theta$ is an isomorphism

$\theta\ :\rho\ \otimes_{{}_{\sigma}}\ \tilde{F}\xrightarrow{\ \ \simeq\ \ }\
F_{{}_{\rho}},$ (30)

of absolute $N$-dimensional theories, with (30) defining the dimensional
reduction leading to the absolute theory. $\sigma$ needs only be a _once-
categorified_ $N$-dimensional theory, whereas $\rho$ and $\tilde{F}$ are
relative theories.

In theories admitting a higher-categorical structure as (19), topological
defects in different superselection sectors have different dimensionality.
Consequently, the fusion rules are categorical, in the sense that they usually
don’t follow group-like composition laws. Because of this these defects are
also called categorical or non-invertible.

Prior to delving into the specific case of interest to us, namely relative
QFTs, we first wish to provide some further explanation for what gauging a
categorical structure actually means. In doing so, we refer to the work of
many experts in the field, and, in particular TJF . As explained in such
reference, for any fusion n-category $\mathfrak{G}$, any fiber functor

${\cal F}:\ \mathfrak{G}\ \rightarrow\ \text{nVec},$ (31)

selects nVec as the image of a condensation algebra living in $\mathfrak{G}$,
corresponding to a projection on the identity. The gauging process, can
therefore be defined as a map

$\mu:\ \mathfrak{G}\ \rightarrow\ {\cal A},$ (32)

with ${\cal A}$ the algebra of invertible topological operators in
$\mathfrak{G}$. Given (32), the norm element

$N\overset{def}{=}\bigoplus_{g\in\mathfrak{G}}\ \mu(g).$ (33)

carries the structure of an n-categorical idempotent, also known as
_condensation algebra_ , depicted in black in figure 4. The requirement for
(33) to be a higher-idempotent is needed to ensure the flooding doesn’t depend
on the specific features of the network being adopted to perform the gauging.
The algebra of topological operators that are left are denoted by ${\cal
A}//^{{}^{\mu}}\mathfrak{G}$. The equivalence of the second and third picture
from the left in figure 4 follows from $N$ being a higher-condensation
algebra. As we shall see, this pattern emerges when gauging the 5D SymTFT of
class ${\cal S}$ theories, where the objects of the 2-category in question
will be Wilson surfaces charged under the 1-form symmetry being gauged.

Figure 4: Gauging corresponds to condensing an algebra in a TFT. Idempotency
ensures the resulting theory can be effectively thought of as featuring a
unique defect, as shown on the RHS. Figure 5: Condensing two different
subalgebras, ${\cal A}_{{}_{1}},{\cal A}_{{}_{2}}\ \subset\ {\cal A}$, the
resulting theory corresponds to one with a changed phase with a condesating
defect resulting from a relative condensable algebra, ${\cal A}_{{}_{12}}$
ending in the bulk. The defect at the endpoint is nontrivial, and can
therefore be thought of as a Hom$(\mathbb{1}_{{}_{{\cal C}}},{\cal
A}_{{}_{12}})$.

Unlike 4, 5 does not admit a straightforward expression as (31).

Figure 6: Adaptation of Freed-Moore-Teleman to the case involving twisted
condensation defects. The figure on the far right corresponds to the case of
interest to us, namely a configuration involving two different absolute 4D
gauge theories separated by a defect. As we shall see, such defect is
intrinsically non-invertible, corresponding to the presence of a relative
uncondensed subalgebra, dressing ${\cal N}_{{}_{3}}$ in a nontrivial way. In
higher-categorical terms, it corresponds to a fusion tensor category
implementing the morphisms between the operators charged under the gauged
symmetry.

###  3.2 Absolute 4D theories from 6D ${\cal N}=(2,0)$ SCFTs

The procedure described in section 3.1 cannot be applied to the case of
maximally supersymmetric 6D theories. The reason being that, while a bulk
SymTFT can be defined, thanks to the Milnor-Moore theorem, specifying any
topological boundary conditions would inevitably imply supersymmetry breaking,
Freed:2012bs . Nevertheless, such 6D theories provide the natural realm from
which one can realise lower-dimensional absolute theories, as well as
constituting an extremely rich setting where to explore and put to practice
mathematical techniques provided by representation theory, DBZ .

The main features of 6D ${\cal N}$=(2,0) SCFTs are that:

  * •

They admit no Lagrangian formulation, Freed:2012bs ; Witten:2009at .

  * •

The theory depends on the Lie algebra, $\mathfrak{g}$, rather than the Lie
group, $G$.

  * •

Ordinary333By this we mean that the lower-dimensional theories depend on the
specific choice of the Lie group. 5D and 4D QFTs can be obtained by
dimensional reduction on a Riemann surface, $\Sigma_{{}_{g,n}}$.

  * •

It is _relative_ w.r.t. an _extended_ TFT in 1-dimension higher, Freed:2012bs
.

Assuming the 6D ${\cal N}=(2,0)$ SCFT is defined on
$Y=X\times\Sigma_{{}_{g}}$, with $X$ a compact 4D spacetime, torsion-free, and
with trivial 1${}^{{}^{st}}$ cohomology, $H^{{}^{1}}(X,\mathbb{Z}_{{}_{N}})$,
the 6D self-dual fluxes belong to444$\mathbb{Z}_{{}_{N}}$ denotes the _defect
group_ of the 6D theory.

$H^{{}^{3}}(Y,\mathbb{Z}_{{}_{N}})\ \simeq\
H^{{}^{1}}(\Sigma_{{}_{g}},\mathbb{Z}_{{}_{N}})\ \otimes\
H^{{}^{2}}(X,\mathbb{Z}_{{}_{N}}).$ (34)

As far as the 6D theory is concerned, this is the only information at our
disposal, given that the theory is non-Lagrangian, Freed:2012bs ;
Witten:2009at . However, for the 4D theory to be fully defined, additional
data is needed. In particular, one needs to identify a _maximal isotropic
sublattice_ $L$ of $H^{{}^{1}}(\Sigma_{{}_{g}},\mathbb{Z}_{{}_{N}})$ w.r.t.
the canonical pairing induced by the intersection pairing on
$\Sigma_{{}_{g}}$, thereby leading to

${\cal L}\ \simeq\ L\ \otimes\ H^{{}^{2}}(X,\mathbb{Z}_{{}_{N}})$ (35)

as the maximal isotropic lattice of $H^{{}^{3}}(Y,\mathbb{Z}_{{}_{N}})$. (35)
enables to distinguish between different global forms of class ${\cal S}$
theories. Having fixed ${\cal L}$, the fluxes in its complement

${\cal L}^{\perp}\ \overset{def.}{=}\
H_{{}_{3}}(X_{{}_{6}},\mathbb{Z}_{{}_{N}})/{\cal L}$ (36)

parametrise the possible partition functions for the 4D theory with
inequivalent 1-form symmetry backgrounds along $X$.

The 6D ${\cal N}$=(2,0) theory, assigns to the 6D manifold $Y$ a partition
vector rather than a partition function, with the former belonging to the
Hilbert space constituting a representation of a Heisenberg algebra of non-
commuting discrete 3-form fluxes in $\mathbb{Z}_{{}_{N}}$.

For fixed ${\cal L}$, the partition vector takes the form

$|{\cal Z}(Y)>\ \equiv\ \sum_{v\ \in\ {\cal L}^{{}^{\perp}}}\ {\cal
Z}_{{}_{v}}(Y)\ |{\cal L};v>,$ (37)

with ${\cal Z}_{{}_{v}}(Y)$ denoting the 6D conformal blocks, which also
encode the _global structures_ of the 4D theory. Under a change of basis, (37)
turns into

$|{\cal Z}(Y)>\ \equiv\ \sum_{v^{\prime}\ \in\ {\cal L}^{{}^{\prime\perp}}}\
{\cal Z}_{{}_{v}}(Y)\ \sum_{v\ \in\ {\cal L}^{{}^{\perp}}}\
R_{{}_{v}}^{{}^{v^{\prime}}}\ |{\cal L};v>,$ (38)

leading to the following relation between conformal blocks

${\cal Z}_{{}_{v}}(Y)\ \equiv\ \sum_{v^{\prime}\ \in\ {\cal
L}^{{}^{\prime\perp}}}\ R_{{}_{v}}^{{}^{v^{\prime}}}\ {\cal Z}_{{}_{v}}(Y).$
(39)

## The 7D SymTFT

The Milnor-Moore Theorem states that, the universal enveloping algebra of a
Lie algebra is a Hopf algebra whose primitive elements are the elements of the
original Lie algebra. A Hopf algrebra on an associative algebra turns the
category of modules into a monoidal category equipped with a fiber functor.
This is a statement of Tannaka duality.

Putting this together for the case of 6D ${\cal N}=(2,0)$ SCFTs, the bulk
SymTFT is a pointed braided tensor category where the braiding is in bijection
with 3-cocycles.

Following the arguments outlined in section 2.2, one can take the 7D TQFT in
the bulk of the relative 6D ${\cal N}=$(2,0) theory, to be a CS theory with
action

$S_{{}_{7D}}=\frac{N}{4\pi}\ \int_{{}_{W_{{}_{7}}}}\ dc\ \wedge\ c\ \ \ ,\ \ \
c\in H^{{}^{3}}(W_{{}_{7}},U(1)),$ (40)

and Wilson surfaces

$\Phi_{{}_{q}}({\cal M}_{{}_{3}})\overset{def.}{=}\ e^{{}^{iq\oint_{{}_{{\cal
M}_{{}_{3}}}}c}}\ \ \ ,\ \ \ q\in\mathbb{Z}_{{}_{N}}\ \ \ ,\ \ \ {\cal
M}_{{}_{3}}\ \in H^{{}^{3}}(W_{{}_{7}},U(1)).$ (41)

Taking two such Wilson surfaces $\Phi_{{}_{q}}({\cal
M}_{{}_{3}}),\Phi_{{}_{q^{\prime}}}({\cal M}_{{}_{3}}^{\prime})$, with ${\cal
M}_{{}_{3}},{\cal M}_{{}_{3}}^{\prime}$ forming a Hopf link in $W_{{}_{7}}$,
and inserting them in the path integral, amounts to changing the action (40)
by adding the holonomy terms associated to the operator insertions

$\displaystyle S_{{}_{7D}}$ $\displaystyle=\frac{N}{4\pi}\
\int_{{}_{W_{{}_{7}}}}\ dc\ \wedge\ c+q\int_{{}_{{\cal M}_{{}_{3}}}}c\
+q^{\prime}\int_{{}_{{\cal M}_{{}_{3}}^{\prime}}}c$ (42)
$\displaystyle=\frac{N}{4\pi}\ \int_{{}_{W_{{}_{7}}}}\ dc\ \wedge\ c+\
\int_{{}_{W_{{}_{7}}}}\ \left(q\omega_{{}_{{\cal
M}_{{}_{3}}}}+q^{\prime}\omega_{{}_{{\cal M}_{{}_{3}}^{\prime}}}\right)\
\wedge\ c,$

with $\omega_{{}_{{\cal M}_{{}_{3}}}},\omega_{{}_{{\cal
M}_{{}_{3}}^{\prime}}}$ denoting the Poincaré duals of ${\cal
M}_{{}_{3}},{\cal M}_{{}_{3}}^{\prime}$, respectively. Integrating-out $c$,
and imposing

$dc\equiv-\frac{2\pi}{N}\left(q\omega_{{}_{{\cal
M}_{{}_{3}}}}+q^{\prime}\omega_{{}_{{\cal M}_{{}_{3}}^{\prime}}}\right),$ (43)

defining $V_{{}_{4}}$ such that

$\partial V_{{}_{4}}\equiv\left(q\omega_{{}_{{\cal
M}_{{}_{3}}}}+q^{\prime}\omega_{{}_{{\cal M}_{{}_{3}}^{\prime}}}\right),$ (44)

implies

$c\equiv-\frac{2\pi}{N}\ \text{PD}(V_{{}_{4}}).$ (45)

From this, (42) reduces to

$\displaystyle S_{{}_{7D}}$ $\displaystyle=\frac{2\pi}{N}\ qq^{\prime}\
\text{link}\left({\cal M}_{{}_{3}},{\cal M}_{{}_{3}}^{\prime}\right),$ (46)

where the linking is the Hopf link occurring in $W_{{}_{7}}$. It features in
(46) due to the fact that $V_{{}_{4}}$ is a Seifert surface for the
combination $q{\cal M}_{{}_{3}}+q^{\prime}{\cal M}_{{}_{3}}^{\prime}$, meaning
that, every time ${\cal M}_{{}_{3}}$ pierces $V_{{}_{4}}$, it links ${\cal
M}_{{}_{3}}^{\prime}$ once. It thereby follows that, the Hopf link of Wilson
surfaces supported on ${\cal M}_{{}_{3}}$ and ${\cal M}_{{}_{3}}^{\prime}$ in
$W_{{}_{7}}$ can be thought of as an operator insertion in the 7D path
integral,

$<\Phi_{{}_{q}}({\cal M}_{{}_{3}})\Phi_{{}_{q^{\prime}}}({\cal
M}_{{}_{3}}^{\prime})...>\equiv e^{{}^{\frac{2\pi
i}{N}qq^{\prime}\text{link}({\cal M}_{{}_{3}},{\cal M}_{{}_{3}}^{\prime})}}\
<...>,$ (47)

where the phase on the RHS being the analog of the $S$-matrix element
$S_{{}_{ab}}$ between anyons of charges $a,b$ in $U(1)_{{}_{k}}$ CS theory in
3D, namely $S_{{}_{ab}}\equiv e^{{}^{2\pi iab/k}}$.

For a given $X_{{}_{6}}\equiv\Sigma_{{}_{g,n}}\times X_{{}_{4}}$, there is a
particular class of 7-manifolds $W_{{}_{7}}$ obtained by taking
$W_{{}_{7}}\equiv V_{{}_{g,n}}\times X_{{}_{4}}$, with $V_{{}_{g,n}}$ a
3-manifold with $\partial V_{{}_{g,n}}\equiv\Sigma_{{}_{g,n}}$.

For any Riemann surface $\Sigma_{{}_{g,0}}$, there are many inequivelent
3-manifolds with $\Sigma_{{}_{g,0}}$ as its boundary. One such example are
handlebodies. To construct a handlebody, choose a set $g$ of _meridians_ ,
$\\{\mu_{{}_{i}}\\},i\equiv 1,...g$, i.e. a set of generators of
$H_{{}_{1}}(\Sigma_{{}_{g,0}},\mathbb{Z})$ which become trivial as elements of
$H_{{}_{1}}(V_{{}_{g,0}},\mathbb{Z})$. The remaining $g$ generators of
$H_{{}_{1}}(\Sigma_{{}_{g,0}},\mathbb{Z})$ lift to generators of
$H_{{}_{1}}(V_{{}_{g,0}},\mathbb{Z})$, and are referred to as _longitudes_ ,
$\\{\lambda_{{}_{i}}\\},i\equiv 1,...g$. Not every choice of $g$ generators
gives rise to a legitimate set of meridians (i.e. vanishing cycles). For a
specific choice of meridians to be legitimate, it must correspond to a maximal
isotropic sublattice

$L\ \in\ H_{{}_{1}}(\Sigma_{{}_{g,0}},\mathbb{Z}).$ (48)

The handlebody specified by the choice of meridians $L$ will be denoted by
$V_{{}_{g,0}}^{{}^{L}}$. Given $V_{{}_{g,0}}^{{}^{L}}$, there are multiple
choices of longitudes, differing by shifts in meridians

$\lambda_{{}_{i}}^{\prime}\equiv\lambda_{{}_{i}}+\sum_{j=1}^{3}k_{{}_{ij}}\mu_{{}_{j}}\
\ \ ,\ \ k_{{}_{ij}}\in\mathbb{Z},$ (49)

satisfying the following constraint

$<\mu_{{}_{i}},\lambda_{{}_{j}}>\equiv-<\lambda_{{}_{j}},\mu_{{}_{i}}>\equiv\delta_{{}_{ij}}.$
(50)

7D CS theory on a handlebody can be formulated by starting with a definition
of the fields which reads

$b_{{}_{i}}\overset{def.}{=}\oint_{{}_{\mu_{{}_{i}}}}c\ \ \ ,\ \ \
\hat{b}_{{}_{i}}\overset{def.}{=}\oint_{{}_{\lambda_{{}_{i}}}}c\ \ \ ,\ \ \
i=1,...,g.$ (51)

The meridians are naively contractible by definition. However, this does not
necessarily imply the vanishing of $b_{{}_{i}}$, since there might be Wilson
surfaces supported on $\lambda_{{}_{i}}$ and $\mu_{{}_{i}}$ that are
nontrivially linking, effectively making $\mu_{{}_{i}}$ non-contractible.

The Wilson lines wrapping meridians and longitudes can be expressed as follows

$\Phi_{{}_{i}}({\cal M}_{{}_{2}})\ \overset{def.}{=}\Phi_{{}_{i}}({\cal
M}_{{}_{2}}\times\mu_{{}_{i}})\equiv e^{{}^{\frac{2\pi i}{N}\oint_{{}_{{\cal
M}_{{}_{2}}}}b_{{}_{i}}}}\ \ \ ,\ \ \ \hat{\Phi}_{{}_{i}}({\cal M}_{{}_{2}})\
\overset{def.}{=}\hat{\Phi}_{{}_{i}}({\cal
M}_{{}_{2}}\times\lambda_{{}_{i}})\equiv e^{{}^{\frac{2\pi
i}{N}\oint_{{}_{{\cal M}_{{}_{2}}}}\hat{b}_{{}_{i}}}},$ (52)

which might be nontrivially linking in the 5D theory resulting from
dimensional reduction of the 7D TFT on $\Sigma_{{}_{g,0}}$. Compactifying the
theory on $\Sigma_{{}_{g,0}}$ in presence of a series of Wilson 3-surfaces on
$\lambda_{{}_{j}}\times{\cal M}_{{}_{2,j}}^{\prime}$, the linking equation
becomes an operator equation in 4D

$\Phi_{{}_{i}}({\cal M}_{{}_{2}})\ \equiv\ e^{{}^{\frac{2\pi
i}{N}\sum_{j=1}^{g}<\mu_{{}_{i}},\lambda_{{}_{i}}>\oint_{{}_{{\cal
M}_{{}_{2}}}}B_{{}_{j}}}},$ (53)

where $B_{{}_{j}}\in H^{{}^{2}}(X_{{}_{4}},\mathbb{Z}_{{}_{N}})$ denotes the
Poincaré dual of ${\cal M}_{{}_{2,j}}^{\prime}$. Hence, in any 4D calculation
involving $\Phi_{{}_{i}}$, the latter can be moved to the right of
$\hat{\Phi}_{{}_{j}}$, and subsequently be shrunk to a point by introducing
the phase (53). Hence, the insertion of longitudinal Wilson lines allows us to
make $b_{{}_{i}}\neq 0$, while still keeping it constant, thereby defining a
background field.

At this point, the Sym TFT can be obtained by splitting the handlebody in the
following way:

$V_{{}_{g,0}}\ \overset{def.}{\ }\begin{cases}V_{{}_{g,0}}^{{}^{L,in}}\ \ \
\text{for}\ \ \ y\geq y_{{}_{*}}\\\ \\\
V_{{}_{g,0}}^{{}^{ext}}\overset{def}{=}\Sigma_{{}_{g,0}}\times[0,y_{{}_{*}}]\
\ \ \text{for}\ \ \ 0\leq y\leq y_{{}_{*}}\\\ \end{cases}.$ (54)

Importantly, it features two boundaries (cf. figure 7):

  * •

One, at $y=0$, is where the 6D ${\cal N}=$(2,0) theory lives.

  * •

The other, at $y=y_{{}_{*}}$, is where the longitudinal Wilson surfaces fixing
the background fields are placed.

Figure 7: In presence of the splitting (54), the state $<L|$ separates the
anomaly TFT from the Sym TFT. The picture above is valid for $\forall g$, but
has been drawn with $g=1$ for simplicity.

###  3.3 Intrinsic non-invertible symmetries and fiber functors

On the handlebody, one can perform two different operations:

  1. 1.

A _modular transformation_ of $\Sigma_{{}_{g,0}}$, acting on the entire
handlebody and generically changing the period matrix of $\Sigma_{{}_{g,0}}$
at each cross section. The theory in the bulk of the handlebody is
topological, implying only $\Sigma_{{}_{g,0}}$ is affected by such
transformations, which is where the 6D ${\cal N}=$(2,0) theory lives.

  2. 2.

Excising an _inner_ handlebody, in the setup where the splitting (54) takes
place. The operation of interest consists in gluing together the 2 parts
$V_{{}_{g,0}}^{{}^{L,in}}$ and $V_{{}_{g,0}}^{{}^{ext}}$ with a nontrivial
element of the modular group $Sp(2g,\mathbb{Z}_{{}_{N}})$, corresponding to a
surgery operation on the 3-manifolds.

Given that the latter takes place in the interior of the handlebody, it
doesn’t affect $\Sigma_{{}_{g,0}}$ where the 6D theory lives, hence, the
period matrix remains unchanged in the region the theory is sensitive to.
However, this surgery operation changes the global form of $L$, since it
changes which are the contractible cycles in the handlebody. Hence, surgery is
implemented by an element $F$ of the modular group which only affects the
inner boundary of the handlebody, thereby changing the global form. This
operation may in turn be combined with a modular transformation acting on the
entire handlebody, reverting the interior of the handlebody back to its
original form, but now changing the period matrix of the Riemann surface.

Choosing a period matrix that remains invariant under the action of $F$, the
geometry with the combined surgery plus modular transformation has the same
boundaries as the original geometry. Importantly, though, the full geometry is
no longer the same, due to the internal twist.

Under dimensional reduction on $\Sigma_{{}_{g,0}}$, (42) becomes

$S_{{}_{5D}}\equiv\frac{2\pi}{N}\sum_{i=1}^{g}\int_{{}_{X_{{}_{4}\times[0,y_{{}_{*}}]}}}b_{{}_{i}}\
\cup\ \delta\hat{b}_{{}_{i}}.$ (55)

The fact that $b_{{}_{i}},\hat{b}_{{}_{i}}$ are dynamical, implies
$\lambda_{{}_{i}},\mu_{{}_{i}}$ are both nontrivial in
$V_{{}_{g,0}}^{{}^{ext}}$. This follows from the fact that $L$ specifies which
directions of the Riemann surface become trivial in the handlebody, namely the
meridians, and which of the dynamical fields of the $BF$-theory become
background. To each $L$ corresponds a different choice of Dirichlet boundary
conditions for different sets of fields in the $BF$-theory. As explained in
Bashmakov:2022uek , the 7D CS theory on a Riemann surface can only capture the
full Sym TFT iff all the non-invertible defects are non-intrinsic.

The 5D Wilson lines are defined from the 7D Wilson surfaces as follows

$\Phi_{\overset{\rightarrow}{n}}\left({\cal M}_{{}_{2}}\right)\
\overset{def.}{=}\ \Phi\left({\cal
M}_{{}_{2}}\times\gamma_{{}_{\overset{\rightarrow}{n}}}\right),$ (56)

with 1-cycles

$\gamma_{{}_{\overset{\rightarrow}{n}}}\ \overset{def.}{=}\
\sum_{j=1}^{g}e_{{}_{j}}\lambda_{{}_{j}}+\sum_{j=1}^{g}\
m_{{}_{j}}\mu_{{}_{j}}\ \ \ ,\ \ \ \overset{\rightarrow}{n}\
\overset{def.}{=}\
\left(e_{{}_{1}},...,e_{{}_{g}};m_{{}_{1}},...,m_{{}_{g}}\right).$ (57)

Explicitly, (56) can be expressed as follows

$\Phi_{\overset{\rightarrow}{n}}\left({\cal M}_{{}_{2}}\right)\ \equiv\
e^{{}^{\frac{2\pi i}{N}\ \frac{1}{2}e\cdot m({\cal M}_{2},{\cal M}_{2})}}\
\prod_{i=1}^{g}\Phi^{{}^{e_{{}_{i}}}}\left({\cal
M}_{{}_{2}}\times\gamma_{{}_{\overset{\rightarrow}{n}}}\right)\ \
\prod_{i=1}^{g}\Phi^{{}^{m_{{}_{i}}}}\left({\cal
M}_{{}_{2}}\times\gamma_{{}_{\overset{\rightarrow}{n}}}\right),$ (58)

with each electric and magnetic component, in turn descending from a 3D Wilson
surface in 7D associated to a cycle $\gamma_{{}_{\overset{\rightarrow}{n}}}$.

Surgery defects in 7D correspond to loci across which $F$ acts on
$\gamma_{{}_{\overset{\rightarrow}{n}}}$ as follows

$F\left(\gamma_{{}_{\overset{\rightarrow}{n}}}\right)\ \equiv\
\gamma_{F\overset{\rightarrow}{n}}.$ (59)

In 5D, they reduce to condensation defects implementing morphisms in a higher-
categorical structure

${\cal C}_{{}_{F}}(X_{{}_{4}})\ :\ \Phi_{\overset{\rightarrow}{n}}\left({\cal
M}_{{}_{2}}\right)\ \mapsto\ \Phi_{F\overset{\rightarrow}{n}}\left({\cal
M}_{{}_{2}}\right)\ \ \ \ ,\ \ \ {\cal M}_{{}_{2}}\ \subset\ X_{{}_{4}}.$ (60)

Figure 8: The picture on the LHS shows the identification of a morphism
between objects as a defect in a higher-categorical structure. On the RHS, the
interface turns into a boundary defect once having performed the folding
trick.

As suggested by (60), $F$ acts on the labellings of the objects living in the
theory. These objects are dyonic, with nontrivial linking. An explicit
expression for the morphism (60) as a defect follows by performing the folding
trick on the surgery point in the 5D configuration depicted in figure 8

${\cal C}(X_{{}_{4}})\ =\ \frac{\
|H^{{}^{0}}(X_{{}_{4}},\mathbb{Z}_{{}_{p}})|\
}{|H^{{}^{1}}(X_{{}_{4}},\mathbb{Z}_{{}_{p}})|}\ \sum_{{}_{{\cal
M}_{{}_{2}}\in
H_{{}_{2}}\left(X_{{}_{4}},\mathbb{Z}_{{}_{p}}^{{}^{2g}}\right)}}\
\exp\left(\frac{2\pi i}{p}\ <F{\cal M}_{{}_{2}},{\cal
M}_{{}_{2}}>\right)\Phi\left((\mathbb{1}-F){\cal M}_{{}_{2}}\right),$ (61)

implying that, $\forall\ F\in\ Sp(2g,\mathbb{Z}_{{}_{p}})$, there is a
codimension-1 defect implementing the symmetry associated to $F$ in the bulk
5D TQFT. Such defect is invertible, and therefore its fusion rules simply read

$\boxed{\ \ \ {\cal C}(X_{{}_{4}})\ \times\ \bar{\cal C}(X_{{}_{4}})\ \equiv\
{\cal A}\ \equiv\
\mathbb{1}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }.$ (62)

where ${\cal A}$ corresponds to the condensed algebra implementing the
gauging. Notice that the defect (61) is equivalent to the one arising from
gauging by algebraic condensation in a topological order, as explained in
section 3.1 and depicted in figure 4. The crucial meaning of equation (62) is
the fact that, gauging-by-condensation is effectively acting as a redefinition
of the identity from the mother theory to the gauged one. In particular, the
last equality claims that the condensed algebra in the mother theory is
projected to the identity of the gauged theory. Pictorially, this corresponds
to the configuration on the LHS of 9.

Figure 9: Gauging the defects living in the SymTFT upon implementing an outer
automorphisms, changes the boundary conditions of the resulting absolute
theory. In presence of a unique condensation, the resulting configuration is
depicted on the LHS. On the other hand, upon condensing two different
subalgebras of the original bulk theory, leads to a bulk condensation defect
effectively ending in the bulk, as shown in the figure in the middle. This is
equivalent to realising a configuration comprising two different boundary
conditions separated by a topological defect (as shown on the RHS).

Condensation defects associated to $F$, can in turn give rise to $|F|$-ality
defects in the boundary 4D gauge theory. Such defects can be constructed from
(61) by admitting Dirichlet boundary conditions for the condensate, allowing
them to terminate in the bulk, Kaidi:2022cpf ; Teo:2015xla ; Barkeshli:2014cna
. Practically, this corresponds to a non-genuine 3D-manifold attached to a 4D
manifold, which, combined together, give rise to a _twist defect_ ,
$T_{{}_{F}}({\cal M}_{{}_{3}},{\cal M}_{{}_{4}})$, whose expression is
obtained from (61) once rewritten in terms of relative cohomology. From the
perspective of the gauging-by-condensation procedure, this amounts to
cconsidering simulatneous condensation of two different subalgebras within the
condensable subalgebra of the mother theory. In terms of its SymTFT
characterisation, this would correspond to the condensation defect effectively
ending in the bulk SymTFT as shown in the middle of figure 9. Equivalently,
this is equivalent to describing a relative field theory with two boundary
conditions interpolated by a topological defect.

Figure 10: The condensation defect ${\cal C}_{{}_{F}}(X_{{}_{4}})$ implements
the outer automorphism in the 5D TQFT. Applying twisted BCs, the condensation
defect can effectively terminate in the 5D bulk, and is therefore topological.
At this point, the width of the 5D theory can be taken to vanish, resulting in
the configuarion on the RHS, with the two phases of the 4D theory separated by
a topological non-invertible defect, ${\cal N}_{{}_{F}}({\cal M}_{{}_{3}})$.

The condensation defect can be reexpressed in terms of the quantum dimensions
of the underlying anyon description by noticing that the action of $F$ is
equivalent to that of the outer automorphism keeping track of the anyonic
symmetry being gauged555As explained in section 2.2 and 3.1.. This follows
from the fact that the elements of the sum appearing in the condensation
defect, (61), can be broken down into two main parts, namely

$\exp\left(\frac{2\pi i}{p}\ <F{\cal M}_{{}_{2}},{\cal
M}_{{}_{2}}>\right)\Phi\left((\mathbb{1}-F){\cal M}_{{}_{2}}\right)\ \equiv\
S_{{}_{\overset{\rightarrow}{a}\ F\overset{\rightarrow}{a}}}\ {\cal
G}_{{}_{F\overset{\rightarrow}{a}}},$ (63)

with $S_{{}_{\overset{\rightarrow}{a},\ F\overset{\rightarrow}{a}}}$ and
${\cal G}_{{}_{F\overset{\rightarrow}{a}}}$ denoting the $S$-matrix entries
for the associated anyonic charges, and the exceptional fibration operator666A
deeper explanation of this is the key topic of an upcoming work VP .,
respectively. Extending these arguments, the sum over the 2-manifolds ${\cal
M}_{{}_{2}}$ can be replaced by a sum over the anyonic charges compatible with
the symmetries of the given theory. This way, (61) can be rewritten as follows

$T_{{}_{F}}({\cal M}_{{}_{3}},{\cal M}_{{}_{4}})\ \overset{def.}{=}\ \frac{\
|H^{{}^{0}}(X_{{}_{4}},\mathbb{Z}_{{}_{p}})|^{{}^{2g}}\
}{|H^{{}^{1}}(X_{{}_{4}},\mathbb{Z}_{{}_{p}})|^{{}^{2g}}}\
\sum_{{}_{\overset{\rightarrow}{a}}}\ S_{{}_{\overset{\rightarrow}{a},\
F\overset{\rightarrow}{a}}}\ {\cal G}_{{}_{F\overset{\rightarrow}{a}}},$ (64)

where $\overset{\rightarrow}{a}\ \overset{def.}{=}\ \left([a],\rho(a)\right)$,
with $[a],\rho(a)$ denoting the conjugacy class and the representation of the
corresponding anyon charges, respectively. Given that the defect is now
topological, it can be moved to the 4D boundary, defining an interface for the
4D gauge theory

${\cal N}({\cal M}_{{}_{3}})\ \overset{def.}{=}\ \underset{y\rightarrow
0}{\lim}\ T_{{}_{F}}({\cal M}_{{}_{3}},{\cal M}_{{}_{4}}),$ (65)

obeying non-invertible fusion rules. From the 4D perspective of the absolute
theory on the RHS of figure 10, ${\cal N}({\cal M}_{{}_{3}})$ is
_intrinsically_ -non-invertible, namely, it cannot be rendered invertible by
acting with $SL(2,\mathbb{Z})$-transformations. This is equivalent to stating
that the action of $F$ cannot be reversed simply by acting with a modular
transformation. Non-invertibility implies its fusion rules now read

$\boxed{\ \ \ {\cal N}({\cal M}_{{}_{3}})\ \times\bar{\cal N}({\cal
M}_{{}_{3}})\ =\ {\cal
A}_{{}_{X_{{}_{4}}}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ },$ (66)

with ${\cal A}_{{}_{X_{{}_{4}}}}$ denoting the relative condensable algebra in
the absolute theory.

Crucially, the RHS of equation (66) is not simply the identity of a 4D
absolute theory, corresponding to the fact that there is no direct counterpart
of equation (31) upon performing a double condensation involving two distinct
subalgebras of the condensable algebra characterising the mother SymTFT. Given
that the latter is basically equivalent to the existence of a well-defined
partition function for the resulting absolute theory, we claim that, if might
be able to define a counterpart of the functor (31) for the case depicted at
the centre and RHS of figure 9, by composing the following chain of functors

Figure 11: A qualitative diagram depicting the counterpart of the fiber
functor’s domain for the case of figure 9. The black arrors indicate the
choice of different condensable subalgebras, leading to different 4D absolute
theories, labelled by 2-categories ${\cal C,D}$.

where ${\cal A}_{{}_{4D}}$ is the condensable algebra leading to a single 4D
absolute theory, whereas ${\cal A}_{{}_{1}}$ and ${\cal A}_{{}_{2}}$
correspond to different subalgebras of ${\cal A}_{{}_{4D}}$ leading to
different 2-categories of underlying charges associated to the gauged symmetry
characterising the two distinct 4D absolute theories. This is current work in
progress by the same author, and we plan to report about it in a forthcoming
work. The main message we want to convey with this is that the fiber functor
that can be defined for the configuration at the centre and RHS of figure 9
leads to the partition function of a 3D theory. We plan to report about this
in due course, VP .

###  3.4 Quantum dimension from relative condensable algebra

In the concluding part of this section, we make use of the tools outlined in
the previous parts of the present treatment, leading to the proposal of a
probe quantity enabling to distinguish whether the defect interpolating in
between class ${\cal S}$ theories is intrinsically or non-intrinsically non-
invertible. The starting point will be the setting of section 3.1, where we
revised the gauging-by-condensation procedure, implemented by flooding the
topological order corresponding to the 2-category of Wilson surfaces living in
the bulk SymTFT with a tensor network of an idempotent element, namely the
norm element (33).

Practically, the two phases (prior and after gauging) can be thought of as
arising from a phase transition in between two MTCs, ${\cal B}_{{}_{ung}}$ and
${\cal B}_{{}_{g}}$, corresponding to the ungauged and the gauged phases,
respectively, Kong:2013aya . Imposing natural physical requirenents, it is
possible to derive a relation between the anyons in the ${\cal B}_{{}_{g}}$
phase and those in the ${\cal B}_{{}_{ung}}$ phase. The vacuum, or tensor unit
${\cal A}$ in ${\cal B}_{{}_{g}}$ is necessarily a connected commutative
separable algebra in ${\cal B}_{{}_{ung}}$, and the category ${\cal
B}_{{}_{g}}$ is equivalent to the category of local ${\cal A}$-modules as
MTCs. This condensation produces a gapped domain wall (DW) with wall
excitations given by the category of ${\cal A}$-modules in ${\cal
B}_{{}_{ung}}$. The domain wall separating them, is, in turn a category,
${\cal A}\equiv{\cal C}(X_{{}_{4}})$, as shown in figure 12.

Figure 12: This picture shows the categories of anyons associated to the
ungauged and gauged theories, denoted by ${\cal B}_{{}_{ung}}$ and ${\cal
B}_{{}_{g}}$, respectively. Gauging-by-condensation can be visualised as the
domain wall effectively separating two different topological phases.

For any pair of elements, $M,N$,

$\text{Hom}_{{}_{{\cal B}_{{}_{g}}}}(M,N)\ \hookrightarrow\
\text{Hom}_{{}_{{\cal B}_{{}_{ung}}}}(M,N).$ (67)

Assuming vacuum degeneracy being trivial,

$\text{Hom}_{{}_{{\cal A}}}(\mathbb{1}_{{}_{\cal A}},\mathbb{1}_{{}_{\cal
A}})\ \simeq\ \mathbb{C},$ (68)

${\cal A}$ must be a UFC with a unique spherical structure. All objects in
${\cal A}$ should come from objects in ${\cal B}_{{}_{ung}}$, and should
satisfy the following properties:

  1. 1.

${\cal A}$ is a subcategory of ${\cal B}_{{}_{ung}}$

  2. 2.

$\text{Hom}_{{}_{{\cal A}}}(M,N)\ \hookrightarrow\ \text{Hom}_{{}_{{\cal
B}_{{}_{ung}}}}(M,N)$ (69)

  3. 3.

$\mathbb{1}_{{}_{ung}}$ should condense into $B$, such that

$\iota_{{}_{B}}:\ \mathbb{1}_{{}_{{\cal B}_{{}_{ung}}}}\ \rightarrow\ B\ \in\
{\cal B}_{{}_{ung}}$ (70)

$\mathbb{1}_{{}_{{\cal B}_{{}_{g}}}}\equiv{\cal A}$ should fuse into the
vacuum on the wall when moving ${\cal A}$ close to the wall, such that, in
${\cal B}_{{}_{ung}}$, the corresponding map reads

$\iota_{{}_{A}}^{{}^{B}}:\ A\ \rightarrow\ B$ (71)

What we have just said can be succinctly re-expressed as follows

$\boxed{\ \ \mathbb{1}_{{}_{{\cal B}_{{}_{g}}}}\ \equiv\ {\cal A}\ \subset\
{\cal
B}_{{}_{ung}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }.$ (72)

After what we have just said, the key point to focus on is the fact that the
gauging-by-condensation procedure, the identity operator changes, in the sense
that, from being a simple element in the category ${\cal B}_{{}_{ung}}$, it is
nontrivial in the category of the gauged theory, ${\cal B}_{{}_{g}}$, since
$\mathbb{1}_{{}_{{\cal B}_{{}_{g}}}}\equiv{\cal A}$.

The analogy we have drawn so far with the example shown in figure (72) taken
from Kong:2013aya , corresponds to the case of a single gauging of the 5D
SymTFT, leading to the LHS of our figure 9. For the case involving a double
algebraic condensation, instead, such prescription needs to be suitably
adapted. The motivation still follows the arguments presented in Kong:2013aya
. We now argue why this is indeed the case specifying to the example described
in section 3.2 and 3.3.

Recall that ${\cal C}(X_{{}_{4}})$ is the operator implementing the outer
automorphism on the category of defects defining the 5D SymTFT, and
corresponds to the idempotent object that is constituted of the algebra
elements that are floodding the SymTFT in order to implement the gauging.
Denoting by ${\cal C}_{{}_{g}},{\cal C}_{{}_{ung}}$ the 2-category of defects
characterising the 4D absolute theories, if the two phases are related by a
single bulk gauging-by-condensation, it follows that

$\text{dim}_{{}_{{\cal C}_{{}_{g}}}}\ \mathbb{1}_{{}_{{\cal C}_{{}_{g}}}}\
\equiv\ \frac{\ \text{dim}_{{}_{{\cal C}_{{}_{ung}}}}{\cal A}_{{}_{4D}}\
}{\text{dim}_{{}_{{\cal C}_{{}_{ung}}}}{\cal A}_{{}_{4D}}}\ \equiv\ 1.$ (73)

But this is only from the point of view of the gauged theory, ${\cal
C}_{{}_{g}}$. On the other hand, ${\cal A}_{{}_{4D}}$ is a graded object in
${\cal C}_{{}_{ung}}$, and therefore

$\text{dim}_{{}_{{\cal C}_{{}_{ung}}}}\ {\cal A}_{{}_{4D}}\ \neq\ 1.$ (74)

However, our aim is that of describing a composite configuration for the
absolute theory of the kind depicted on the LHS of figure 15, where, as
explained in sections 3.2 and 3.3, both phases really descend form the
condensation of different subalgebras within the condensable algebra of the
original ungauged SymTFT777This is our proposal on the basis of TJF ;
Kong:2013aya .. The different 4D absolute gauge theories are interpolated by a
non-invertible defect defined as

${\cal N}({\cal M}_{{}_{3}})\ \overset{def.}{\equiv}\ {\cal D}_{{}_{3}}\ {\cal
A}_{{}_{\epsilon\rho}}$ (75)

where ${\cal A}_{{}_{\epsilon\rho}}$ plays the role of a relative condensable
algebra that cannot be identified with the vacuum on either side.

Figure 13: Composite 4D absolute theories separated by an intrinsic non-
invertible defect on the LHS. Rephrased in terms of the higher-categorical
structure underlying the resulting theory, the algebra dressing the
interpolating defects ${\cal N}_{{}_{3}}$ is a homomorphism between the
objects of a 2-category.

Consistency of the configuration associated to the LHS of figure 13, is mapped
to a statement of Witt equivalence between the underlying 2-categories
depicted on the RHS of the same figure, Kong:2013aya ,

${\cal C}\ \boxtimes\ \bar{\cal D}\ \simeq\ \mathfrak{Z}({\cal{\cal
A}_{{}_{\epsilon\rho}}}),$ (76)

where ${\cal A}_{{}_{\epsilon\rho}}$ denotes the relative condensable algebra
featuring in (75).

The relative condensable algebra, ${\cal A}_{{}_{\epsilon\rho}}$, constitutes
a fusion tensor category888With ${\cal A}_{{}_{1}}$ and ${\cal A}_{{}_{0}}$
denoting the object and the morphisms, respectively. defining morphisms
between any pair of elements $L,L^{\prime}\ \in\ {\cal C,D}$.

${\cal A}_{{}_{\epsilon\rho}}^{{}^{L,L^{{}^{\prime}}}}\ \overset{def.}{=}\
\left\\{{\cal A}_{{}_{1}}^{{}^{L,L^{{}^{\prime}}}},{\cal
A}_{{}_{0}}^{{}^{L,L^{{}^{\prime}}}}\ \right\\},$ (77)

whose total quantum dimension can be compared to that of the identity, i.e. of
the algebra that has been condensed to achieve the reference gauge theory,
with 2-category ${\cal C}$.

Figure 14: The algebra for the underlying category, denoted by ${\cal A}$ can
be partially condensed, under suitable choice of subalgebras ${\cal
A}_{{}_{1}},{\cal A}_{{}_{2}}\ \subset\ {\cal A}$, such that they lead to
different gauge theories, whose 2-categories are labelled by ${\cal C,D}$.
Figure 15: The composite system (14) can be thought of as describing a
thickened domain wall, ${}_{{}_{{\cal A}_{{}_{1}}}}{\cal A}_{{}_{{\cal
A}_{{}_{2}}}}\ \overset{def.}{\equiv}\ {\cal A}_{{}_{1}}\ \otimes_{{}_{\cal
A}}\ {\cal A}_{{}_{2}}$.

From the point of view of the absolute 4D theory, the non-invertible defect
can be thought of as a thickened domain wall obtained by condensing two
different subalgebras within the original condensing algebra ${\cal A}$, as
shown in figure 15. Given that the target absolute theory we want to achieve
is that of figure 14, then the corresponding 2-categories of defects can be
obtained starting from

${\cal A}\ \equiv\ {\cal D}\ \boxtimes\ \mathfrak{Z}({\cal E}),$ (78)

$G:\ {\cal C}\ \boxtimes\ \mathfrak{Z}({\cal D})\ \simeq\ {\cal D}\ \boxtimes\
\mathfrak{Z}({\cal E}),$ (79)

and the fiber functors

${\cal F}_{{}_{\cal E}}\ :\ \mathfrak{Z}({\cal E})\ \rightarrow\ {\cal E}\ \ \
,\ \ \ {\cal F}_{{}_{\cal D}}\ :\ \mathfrak{Z}({\cal D})\ \rightarrow\ {\cal
D}$ (80)

with corresponding duals ${\cal F}_{{}_{\cal E}}^{{}^{V}},{\cal F}_{{}_{\cal
D}}^{{}^{V}}$.

The fact that ${\cal N}({\cal M}_{{}_{3}})$ is intrinsically non-invertible
corresponds to saying that the outer automorphism implemented by the action of
$F$ cannot be undone by a combination of modular transformations, as described
in section 3.3. In particular, ${\cal A}_{{}_{\epsilon\rho}}$ is different
from the identity on either side of the defect, and, therefore, ${\cal
D}_{{}_{{\cal A}_{{}_{\epsilon\rho}}}}\neq{\cal D}_{{}_{\mathbb{1}}}$.

## Proof of our main result

The concluding part of this section outlines the identification of a quantity
we wish to propose as a probe for asserting whether a given non-invertible
defect is either intrinsic or non-intrinsic. In doing so, we will be relying
on most of the tools outlined in the previous parts of the present work.

Take ${\cal B}$ to be a semisimple abelian category over $\mathbb{C}$;
assuming it is a rigid braided tensor category. $I$ is the set of isomorphism
classes of irreducible objects in ${\cal B}$, with representatives
$V_{{}_{i}},\ \forall i\in I$. Assuming the spaces of morphisms are finite-
dimensional. $\mathbb{1}\in{\cal B}$ is assumed to be a simple object
corresponding to the labelling $i\equiv 0\in I\ \Rightarrow\
V_{{}_{0}}\overset{def.}{\equiv}\mathbb{1}$. A ${\cal B}$-algebra is an object
${\cal A}\in{\cal C}$ with morphisms $\mu:\ {\cal A}\otimes{\cal
A}\rightarrow{\cal A}$ and $\iota_{{}_{{\cal
A}}}:\mathbb{1}\hookrightarrow{\cal A}$, with

$\text{dim \ Hom}_{{}_{{\cal B}}}(\mathbb{1},{\cal A})\ \equiv\ \mathbb{1}.$
(81)

Given ${\cal B}$ and a ${\cal B}$-algebra ${\cal A}$, Rep${\cal A}$ is defined
as pairs $(V,\mu_{{}_{V}})$, where $V\in{\cal B}$ and $\mu_{{}_{V}}:{\cal
A}\otimes V\rightarrow V$ morphisms in ${\cal B}$.

For example, if $G$ is a finite group and ${\cal B}$ is the category of
finite-dimensional complex representations of $G$, semisimple ${\cal
B}$-algebras correspond to different semigroups in $G$ (this is what we need
to implement gauging-by-condensation in the Freed-Moore-Teleman setup).

Here, in Rep${\cal A}$, ${\cal A}$ is the unit object. For the case where
there are two different absolute theories separated by an intrinsically non-
invertible defect, the estup of joint absolute theories results from a double
condensation procedure Kong:2013aya  and it makes sense to consider the
composite object $X\otimes_{{}_{{\cal A}}}Y$ with ${\cal A}$ the condensable
algebra in ${\cal C}$; so, we have that $X,Y\in$ Rep${\cal A}$. But, at the
same time, $X$ and $Y$ are also elements of two different subalgebras within
${\cal A}$ corresponding to two different gauging-by-condensations. E.g.
$X\in$ Rep${\cal A}_{{}_{1}}$ and $Y\in$ Rep${\cal A}_{{}_{2}}$, with ${\cal
A}{{}_{{}_{1}}},{\cal A}_{{}_{2}}\subset{\cal A}$. In turn, ${\cal
A}_{{}_{1}}$ is the unit element in Rep${\cal A}_{{}_{1}}$ and ${\cal
A}_{{}_{2}}$ id the unite element in Rep${\cal A}_{{}_{2}}$.

Hence, if we consider the original mother algebra ${\cal A}$ with its
characteristic elementary objects $V_{{}_{i}}$, and assuming every one of them
has multiplicity $1$ within ${\cal A}$, we get

$\text{dim}_{{}_{\cal A}}(X\otimes_{{}_{{\cal A}}}Y)\ \equiv\
\text{dim}_{{}_{\cal A}}(X)\ \text{dim}_{{}_{\cal A}}(Y)$ (82)

and

$\text{dim}_{{}_{\cal A}}(X)\ \equiv\ \frac{\text{dim}_{{}_{\cal
B}}({X})}{\text{dim}_{{}_{\cal B}}({\cal A})}.$ (83)

Furthermore, we have that

$\text{dim}_{{}_{\cal A}}(X\otimes_{{}_{\cal A}}Y)\ \equiv\
\frac{\text{dim}_{{}_{\cal B}}({X})\ \text{dim}_{{}_{\cal
B}}({Y})}{\text{dim}_{{}_{\cal B}}({\cal A})}.$ (84)

Extending equation (84) to the case in which $X,Y$ are replaced by the ${\cal
A}_{{}_{1}},{\cal A}_{{}_{2}}$ subalgebras themselves, we can look at the
following ratio

$\frac{\text{dim}_{{}_{\cal B}}({\cal A}_{{}_{1}}\otimes_{{}_{\cal A}}{\cal
A}_{{}_{2}})}{\text{dim}_{{}_{\cal B}}({\cal A}_{{}_{1}})}\ \equiv\
\frac{{\cal D}^{{}^{intr.}}_{{}_{{}_{{\cal A}_{{}_{1}}}}{\cal A}_{{}_{{\cal
A}_{{}_{2}}}}}}{{\cal D}^{{}^{non-intr.}}_{{\cal A}_{{}_{1}}}},$ (85)

where on the RHS we identified the numerator and denominator with the quantum
dimension of algebras associated to the intrinsic or non-intrinsic non-
invertible defects separating absolute theories resulting from the Freed-
Moore-Teleman construction discussed in the first part of section 3.1. Making
use of (82) and (83), (85), we get

$\frac{\text{dim}_{{}_{\cal A}}({\cal A}_{{}_{1}}\otimes_{{}_{\cal A}}{\cal
A}_{{}_{2}})\ \text{dim}_{{}_{\cal B}}({\cal A})}{\text{dim}_{{}_{\cal
A}}({\cal A}_{{}_{1}})\ \text{dim}_{{}_{\cal B}}({\cal A})}\ \equiv\
\frac{\text{dim}_{{}_{\cal A}}({\cal A}_{{}_{1}})\ \text{dim}_{{}_{\cal
A}}({\cal A}_{{}_{2}})}{\text{dim}_{{}_{\cal A}}({\cal A}_{{}_{1}})}\ \equiv\
\text{dim}_{{}_{\cal A}}({\cal A}_{{}_{2}}).$ (86)

Given that

$\text{dim}_{{}_{\cal A}}({\cal A}_{{}_{2}})\ \overset{def.}{=}\
\sqrt{\sum_{{}_{i}}d_{{}_{i}}\ }$ (87)

where

$d_{{}_{i}}\ \overset{def.}{=}\ \text{dim}_{{}_{\cal A}}V_{{}_{i}}\ \ >\ 1,$
(88)

with $V_{{}_{i}}\in{\cal A}_{{}_{2}}$ being simple objects999I.e. a basis. in
${\cal A}_{{}_{2}}$ (similar arguments hold for ${\cal A}_{{}_{1}}$, since
both are assumed to be subalgebras in ${\cal A}$), it therefore follows that

$\boxed{\ \ \ \frac{\text{dim}_{{}_{\cal B}}({\cal
A}_{{}_{1}}\otimes_{{}_{\cal A}}{\cal A}_{{}_{2}})}{\text{dim}_{{}_{\cal
B}}({\cal A}_{{}_{1}})}\ \equiv\ \frac{{\cal
D}^{{}^{intr.}}_{{}_{{}_{{}_{{\cal A}_{{}_{1}}}}{\cal A}_{{}_{{\cal
A}_{{}_{2}}}}}}}{{\cal D}^{{}^{non-intr.}}_{{\cal A}_{{}_{1}}}}\ >\
1\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ }.$ (89)

For the configuration of interest, namely two different absolute theories
obtained by gauging-by-condensation of two different subalgebras ${\cal
A}_{{}_{1}},{\cal A}_{{}_{2}}\subset{\cal A}$, separated by an intrinsic non-
invertible defect, the numerator in (89) corresponds to the quantum dimension
of the nontrivial fusion of an intrinsic non-invertible defect and its
orientation reverse, whereas the denominatorcorresponds to the quantum
dimension of the algebra that has been condensed in the SymTFT to obtain a
specific absolute theory. We are therefore led to conclude that the total
quantum dimension of the relative condensable algebra ${\cal
A}_{{}_{\epsilon\rho}}$ is greater w.r.t. that of the condensed algebra
leading to the single gauged phase, ${\cal C}$, and we propose (89) as a probe
quantity signalling whether a non-invertible defect is either intrinsic or
non-intrinsic.

For the specific case of class ${\cal S}$ theories, the algebras featuring in
(89) explicitly read

${\cal N}({\cal M}_{{}_{3}})\ \times\ \bar{\cal N}({\cal
M}_{{}_{3}})\equiv{\cal A}_{{}_{\epsilon\rho}}$ (90)

${\cal D}_{{}_{3}}\ \times\ \bar{\cal D}_{{}_{3}}\equiv\ \mathbb{1}\ \equiv\
{\cal A}_{{}_{1}}.$ (91)

Gathering these results together, we can rewrite equation (89) in terms of
(90) and (91), leading to the following result

$\displaystyle\boxed{\ \ \ \frac{<{\cal N}({\cal M}_{{}_{3}})\ \times\
\bar{\cal N}({\cal M}_{{}_{3}})>_{{}_{{\cal B}}}}{<{\cal D}_{3}\ \times\
\bar{\cal D}_{3}>_{{}_{{\cal B}}}}\ \equiv\ \frac{<{\cal
A}_{{}_{\epsilon\rho}}>_{{}_{{\cal B}}}}{\ <\mathbb{1}_{{}_{{\cal
C}}}>_{{}_{{\cal B}}}\ }\ \equiv\ \left(\frac{{\cal D}^{{}^{intr.}}_{{\cal
A}_{{}_{\epsilon\rho}}}}{{\cal D}^{{}^{non-intr.}}_{{\cal
A}_{{}_{1}}}}\right)_{{}_{X_{4}}}\ >\
1\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\
\ },$ (92)

with the last relation indicating that such ratio can be used as a parameter
probing whether the two gauge theories are separated by an intrinsic non-
invertible symmetry. Consequently, the inequality

$\boxed{\ \ \ {\cal D}_{{\cal
A}_{{}_{\epsilon\rho}}{(X_{{}_{4}})}}^{{}^{intr.}}\ >\ {\cal D}_{{\cal
A}_{{}_{\epsilon\rho}}{(X_{{}_{4}})}}^{{}^{non-
intr.}}\color[rgb]{1,1,1}\definecolor[named]{pgfstrokecolor}{rgb}{1,1,1}\pgfsys@color@gray@stroke{1}\pgfsys@color@gray@fill{1}\bigg{]}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@color@gray@fill{0}\
\ }$ (93)

implies that the fusion category associated to intrinsic non-invertible
defects is characterised by a greater quantum dimension, implying such
configuration (arising from double condensation) is able to store a higher
number of degrees of freedom w.r.t. the case in which a unique condensation is
taking place. 101010In (93) we explicitly used the pedix ${\cal
A}_{{}_{\epsilon\rho}}$ on both sides to indicate that the defect we are
considering is separating absolute theories with different choices of boundary
conditions on the topological boundary, according to the Freed-Moore-Teleman
prescription. The difference in between the intrinsic and non-intrinsic case
depends on whether such boundary conditions are isomorphic to each other or
not.

## Key points

  1. 1.

For class ${\cal S}$ theories, ${\cal D}_{{\cal A}_{{}_{\epsilon\rho}}}$
enables to probe whether a defect is intrinsically or non-intrinsically non-
invertible.

  2. 2.

${\cal D}_{{\cal A}_{{}_{\epsilon\rho}}{(X_{{}_{4}})}}^{{}^{intr.}}\ >\ {\cal
D}_{{\cal A}_{{}_{\epsilon\rho}}{(X_{{}_{4}})}}^{{}^{non-intr.}}$, and
therefore the former is a theory equipped with a higher-categorical structure
enabling to store more d.o.f. in certain superselection sectors.

  3. 3.

For theories descending from 6D ${\cal N}=(2,0)$ SCFTs, if the gauging
procedure leads to the presence of intrinsic non-invertible defects, the total
quantum dimension of the algebra implementing the non-invertibility of such
defects exhibits an increase in its total quantum dimension, signalling the
relative condensable algebra ${\cal A}_{{}_{\epsilon\rho}}$ is characterised
by superselection sectors admitting a richer structure w.r.t. the condensed
algebra leading to a single absolute 4D gauged theory.

##  4 Conclusions and Outlook

Key advancements towards achieving a rigorous mathematical formulation of QFTs
strongly rely upon a higher-categorical prescription, where the symmetries of
a QFT can be thought of as defects living in different categories connected by
functors.

In the present work we identify a criterion to distinguish between intrinsic
and non-intrinsic non-invertible symmetries arising from the categorical
structure of 6D ${\cal N}=(2,0)$ SCFTs. In particular we were able to derive a
relation in terms of the quantum dimension of the relative condensing algebra
implementing the gauging in the bulk SymTFT and leading to absolute 4D
theories interpolated by intrinsic non-invertible defects. In doing so, we
relied upon the description in terms of relative field theories, as described
in Bashmakov:2022uek  making use of the Freed-Moore-Teleman setup,
Freed:2012bs ; Freed:2022qnc ; Freed:2022iao , together with the gauging-by-
condensation prescription, TJF . For the intrinsic non-invertible case,
multiplicity for superselection sectors of the relative condensable algebra is
greater w.r.t. the non-intrinsic case, thereby signalling the possibility for
additional d.o.f. to be stored in certain superselection sectors of the
resulting composite absolute theory.

Our results extend arguments proposed in Kaidi:2022cpf , where the authors
also proposed a way of distinguishing intrinsic from non-intrinsic non-
invertibility in 2D by means of the quantum dimension of the non-invertible
defect, which in terms of the underlying higher-categorical structure of the
composite absolute 4D theory, plays the role of a fusion tensor category
defining the morphisms between objects of the 2-categories, with the latter
being the defects charged under the gauged symmetry.

In our analysis, we encountered two major setups built upon the notion of
higher-categories, namely symmetry TFTs (SymTFT) and topological orders (TO).
Plenty of effort has been made towards building a correspondence in between
the two, most recently in Ji:2019eqo . Our findings provide further support
towards strengthening the connection between the two description.

En-passing, we commented on a proposal regarding the realisation of such
composite absolute theories and the definition of a fiber functor
intrinsically related to the notion of a partition function for a 3D theory
rather than a 4D theory, which is current work in progress by the same author,
VP .

We conclude by stressing that our analysis is mostly motivated by furthering
the understanding of 6D ${\cal N}=(2,0)$ SCFTs, from, both, a mathematical and
physical point of view. To what extent these findings might be mapped to other
setups is currently under investigation. In an upcoming work VP , we will
report on further advancements building on these findings.

## Acknowledgements

The author wishes to thank Tudor Dimofte, I$\tilde{\text{n}}$aki García-
Etxebarria, David Jordan, Fernando Quevedo and Constantin Teleman for very
useful discussions and questions raised. A special acknowledgement is in order
to the Hodge Institute, where this work was completed and first presented.
This work is partially supported by an STFC scholarship through DAMTP.

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