††institutetext: Department of Mathematical Sciences, Durham University,
Durham, DH1 3LE, United Kingdom

# Branes and Non-Invertible Symmetries

Iñaki García Etxebarria<EMAIL_ADDRESS>
###### Abstract

$\mathcal{N}=4$ supersymmetric Yang-Mills theories with algebra
$\mathfrak{so}(4N)$ and appropriate choices of global structure can have non-
invertible symmetries. We identify the branes holographically dual to the non-
invertible symmetries, and derive the fusion rules for the symmetries from the
worldvolume dynamics on the branes.

##  1 Introduction

The notion of symmetry is undergoing rapid evolution: during the last few
years a number of works have convincingly argued that the classical textbook
definition of symmetry as a group of transformations acting on local operators
can (and should) be extended to include higher form symmetries acting on
extended operators Gaiotto:2014kfa , higher groups structures Sharpe:2015mja ;
Tachikawa:2017gyf ; Benini:2018reh  and more generally higher categorical
structures.

The importance of such higher categorical structures in two dimensions has
been realised for a long time, where they often appear from discrete gauging
Frohlich:2009gb ; Carqueville:2012dk ; Brunner:2013xna ; Bhardwaj:2017xup . A
number of recent works have shown that symmetry operators without inverses
(and which are therefore not elements of any group, but should rather be
thought of in categorical terms) are also very common in higher dimensional
theories Gaiotto:2019xmp ; Heidenreich:2021xpr ; Choi:2021kmx ; Kaidi:2021xfk
; Roumpedakis:2022aik ; Bhardwaj:2022yxj ; Arias-Tamargo:2022nlf ;
Choi:2022zal ; Choi:2022jqy ; Cordova:2022ieu ; Kaidi:2022uux ;
Antinucci:2022eat ; Bashmakov:2022jtl ; Damia:2022bcd ; Bhardwaj:2022lsg ;
Lin:2022xod ; Bartsch:2022mpm . In this paper we will focus on one class of
theories where such non-invertible symmetries appear: $\mathcal{N}=4$ theories
with gauge group111In this note we do not aim to analyse fully the mapping
from boundary conditions to global structures, so we will ignore the existence
of discrete choices of $\theta$ angles in some of the theories we discuss
Aharony:2013hda . A careful analysis of the mapping from global structures to
properties of the holographic duals will be provided in EGEHR .
$\mathrm{Pin}^{+}(4N)$, $Sc(4N)$ and $PO(4N)$ Bhardwaj:2022yxj . The details
are a little different in the three cases, so in this introduction we will
focus on the $Sc(4N)$ case for concreteness. This theory has three 2-surface
symmetry generators, which we will call $D_{2}^{\mathsf{c},e}(\Sigma_{2})$,
$D_{2}^{\mathsf{s},m}(\Sigma_{2})$ and their product
$D_{2}^{\mathsf{c},e}(\Sigma_{2})D_{2}^{\mathsf{s},m}(\Sigma_{2})$. There is
additionally a three-surface operator $\mathcal{N}(\mathcal{M}^{3})$. The
terms in the fusion algebra involving $\mathcal{N}(\mathcal{M}^{3})$ are

$\displaystyle\mathcal{N}(\mathcal{M}^{3})\times\mathcal{N}(\mathcal{M}^{3})$
$\displaystyle=\sum_{\Sigma_{2},\Sigma_{2}^{\prime}\in
H_{2}(\mathcal{M}^{3};\mathbb{Z})}D_{2}^{\mathsf{c},e}(\Sigma_{2})D_{2}^{\mathsf{s},m}(\Sigma_{2}^{\prime})\,,$
(1a) $\displaystyle\mathcal{N}(\mathcal{M}^{3})\times
D_{2}^{\mathsf{c},e}(\Sigma_{2})$
$\displaystyle=\mathcal{N}(\mathcal{M}^{3})\,,$ (1b)
$\displaystyle\mathcal{N}(\mathcal{M}^{3})\times
D_{2}^{\mathsf{s},m}(\Sigma_{2})$
$\displaystyle=\mathcal{N}(\mathcal{M}^{3})\,.$ (1c)

The right hand side of (1a) is generically a sum of operators, and therefore
$\mathcal{N}(\mathcal{M}^{3})$ is not invertible.

All these theories can be obtained from the $\mathcal{N}=4$ $SO(4N)$ theories
by suitable gaugings of discrete symmetries. Whether we have performed the
gauging or not is not visible for a local observer measuring processes on a
topologically trivial (but arbitrarily large) neighbourhood of a point. This
suggests that the holographic dual of all these theories is the same, which is
indeed the case: the holographic dual is in all cases IIB on
$\mathrm{AdS}_{5}\times{\mathbb{R}\mathbb{P}}^{5}$. The different theories
arise from different choices of asymptotic behaviour for discrete gauge fields
in the bulk, as discussed in related examples in Witten:1998xy ;
Aharony:2016kai .

Since all these theories share the same bulk description, it should be
possible to describe the non-invertible symmetry generators (in the cases
where they are present in the field theory) in terms of objects living on the
holographic IIB dual. The goal of this note is to identify these objects, and
to derive their fusion rules using IIB techniques.222The techniques we use in
our analysis do not require knowledge of the Lagrangian of the boundary SCFT
(although the choice of theories to study is certainly informed by the field
theory results in Bhardwaj:2022yxj , and we will chose our notation to
dovetail the field theory analysis), so they apply equally well to the study
of non-Lagrangian theories realised either holographically or via geometric
engineering. See Bashmakov:2022jtl  for a recent study of non-invertible
symmetries in non-Lagrangian theories using a different approach.,333We refer
the reader to Damia:2022bcd  for a holographic study of a different class of
non-invertible defects. Surprisingly, given the perhaps unfamiliar fusion
relations (1), it will transpire that the symmetry generators are represented
holographically by ordinary branes wrapping torsional cycles in the internal
${\mathbb{R}\mathbb{P}}^{5}$.

In order to explain how this is possible, it is useful to review briefly how
the fusion relations (1) are derived in Bhardwaj:2022yxj  (see also
Kaidi:2021xfk ). We start with the $SO(4N)$ theory, which has a
$\mathbb{Z}_{2}$ outer automorphism 0-form symmetry and a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}$ 1-form symmetry. We will denote the
background for the 0-form symmetry $A_{1}$, and the backgrounds for the two
$\mathbb{Z}_{2}$ factors $B_{2}^{m}$ and $C_{2}^{e}$. There is a cubic ’t
Hooft anomaly represented by an anomaly theory with action

$\begin{split}i\pi\int A_{1}B_{2}^{e}C_{2}^{m}\,.\end{split}$ (2)

We obtain the $Sc(4N)$ theory by gauging both 1-form symmetries
simultaneously.444There is a discrete choice when gauging, related to the
precise way in which we sum over $B_{2}^{m}$ backgrounds. A slightly different
choice (related by the outer automorphism) gives the $Ss(4N)$ global form
instead, which also has non-invertible symmetries. The analysis of both cases
is essentially identical, so we will focus on the $Sc(4N)$ case. (The
$\mathrm{Pin}^{+}$ and $PO$ cases are obtained by gauging other pairs of
symmetries involved in the cubic anomaly.) Naively, we would say that the
0-form symmetry is broken due to the cubic anomaly (2). The more precise
statement is that due to the anomaly the generator
$D_{3}^{(0)}(\mathcal{M}^{3})$ of the 0-form symmetry is not invariant under
combined gauge transformations of $B_{2}^{m}$ and $C_{2}^{e}$. But as argued
in Kaidi:2021xfk ; Bhardwaj:2022yxj  it is possible to “dress” (or stack)
$D_{3}^{(0)}(\mathcal{M}^{3})$ with an anomalous TQFT $\mathcal{T}$ depending
on $B_{2}^{m}$ and $C_{2}^{e}$. The combined topological operator is gauge
invariant, and survives as a topological operator of the gauged theory. The
price to pay is that the fusion rules for $\mathcal{T}$ are more involved, and
lead to non-invertibility of the dressed operator
$D_{3}^{(0)}(\mathcal{M}^{3})\times\mathcal{T}$ (the details will be reviewed
below).

Coming back to the IIB holographic setup, the main observation of this paper
is that $D_{3}^{(0)}(\mathcal{M}^{3})\times\mathcal{T}$ is precisely the IR
limit of the theory on branes wrapping suitable torsional cycles in the
holographic dual background. For instance, we will see that in the $Sc(4N)$
case the non-invertible operator arises from a D3 brane wrapping
$\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}$, where
${\mathbb{R}\mathbb{P}}^{1}\subset{\mathbb{R}\mathbb{P}}^{5}$. Reducing on the
${\mathbb{R}\mathbb{P}}^{5}$ leaves an effective 3-dimensional brane wrapping
$\mathcal{M}^{3}$ inside the five dimensional bulk, which becomes
$D_{3}^{(0)}(\mathcal{M}^{3})$ when pushed to the boundary.

A pleasing consequence of the identification in this note is that anomaly
cancellation of the dressed operator follows automatically: the background
fields for the symmetries of the theory are given by asymptotic values for the
supergravity fields in the IIB dual, and the D3 brane action is necessarily
gauge invariant under all possible gauge transformations of these (although
the precise way in which this happens is often subtle). Since anomaly
cancellation is automatic once we start talking about branes, it is
illuminating to understand why non-invertible symmetries appear in the
holographic dual without referring to anomalous operators. This is also
desirable since the split between the bare $D(\mathcal{M}^{3})$ and its
“dressing” $\mathcal{T}$ is unnatural in terms of the brane theory,
particularly once we try to formulate things in the language of K-theory. We
provide such an explanation below in terms of incomplete cancellation of
induced brane charges due to quantum effects.

#### Note added

_I thank the authors ofABBS  for informing me of their related upcoming work,
where they give complementary evidence for the identification of non-
invertible symmetries with branes in holographic settings, and for agreeing to
coordinate submissions._

##  2 4d $\mathcal{N}=4$ $\mathfrak{spin}(4N)$ SYM and non-invertibles

The $\mathrm{Spin}(4N)$ SYM theory has a 2-group structure, with one-form
symmetry group555Our conventions are as follows: $\mathrm{Spin}(4n)$ has two
spinor irreps unrelated by complex conjugation, which we denote by
“$\mathsf{s}$” and “$\mathsf{c}$”. $\mathbb{Z}_{2}^{\mathsf{s}}$ acts on
$\mathsf{c}$, and leaves $\mathsf{s}$ invariant, while
$\mathbb{Z}_{2}^{\mathsf{c}}$ acts on $\mathsf{s}$ and leaves $\mathsf{c}$
invariant. This choice of notation is motivated by consistency with the fact
that the diagonal $\mathbb{Z}_{2}$ combination, traditionally denoted
$\mathbb{Z}_{2}^{V}$, does not act on the vector. We define
$Sc(4N)\coloneqq\mathrm{Spin}(4N)/\mathbb{Z}_{2}^{\mathsf{s}}$.

$\Gamma^{(1)}=\mathbb{Z}_{2}^{\mathsf{s}}\times\mathbb{Z}_{2}^{\mathsf{c}}\,,$
(3)

and a 0-form symmetry part $\mathbb{Z}_{2}^{(0)}$ which is an outer
automorphism that acts on the 1-form symmetry by exchanging the two factors:
$\mathbb{Z}_{2}^{\mathsf{s}}\leftrightarrow\mathbb{Z}_{2}^{\mathsf{c}}$. We
will now construct the topological defects that generate these symmetries in
the holographic dual.

This holographic dual is obtained as the near horizon limit of a stack of
D3-branes on top of an O3- orientifold, and is given by IIB string theory on
$\mathrm{AdS}_{5}\times{\mathbb{R}\mathbb{P}}^{5}$ Witten:1998xy . In general
we want to put the field theory on some spin666We will assume for simplicity
that neither $\mathcal{M}^{4}$ nor any of the submanifolds where we will wrap
defects contains torsion in homology. This is not physically required, but it
simplifies some of the formulas below. manifold $\mathcal{M}^{4}$ different
from $S^{4}$, so we will replace $\mathrm{AdS}_{5}$ by a non-compact manifold
$X^{5}$ which asymptotically becomes $\mathbb{R}\times\mathcal{M}^{4}$
Witten:1998wy . There is a non-trivial $SL(2,\mathbb{Z})$ duality fibration
over ${\mathbb{R}\mathbb{P}}^{5}$, which acts with the $-1\in
SL(2,\mathbb{Z})$ element as we go around the non-trivial generator of
$\pi_{1}({\mathbb{R}\mathbb{P}}^{5})=\mathbb{Z}_{2}$. (This element can be
represented alternatively as $\Omega F_{L}$ in worldsheet terms, but with
future generalisations in mind we will describe it as an $SL(2,\mathbb{Z})$
bundle instead.) The 2-form supergravity fields $B_{2}$ and $C_{2}$ get a sign
under this action, and project down to $\mathbb{Z}_{2}$ fields on
$\mathrm{AdS}_{5}$, while $C_{4}$ does not get a sign and survives as a
continuous field. We will find it useful to work in a democratic formulation,
where we also include the $B_{6}$ and $C_{6}$ fields magnetic dual to $B_{2}$
and $C_{2}$. $SL(2,\mathbb{Z})$ is a gauge symmetry of the theory on the
(orientable) space $\mathrm{AdS}_{5}\times{\mathbb{R}\mathbb{P}}^{5}$, so in
order for the action to be well defined we need $B_{6}$ and $C_{6}$ to also
transform with a minus sign under $-1\in SL(2,\mathbb{Z})$.

What this means is that $H_{3}$ and $F_{3}$ are elements of the cohomology
group with local coefficients
$H^{3}(X^{5}\times{\mathbb{R}\mathbb{P}}^{5};\widetilde{\mathbb{Z}})$ (we
refer the reader to appendix 3.H of Hatcher:478079  for details), and
similarly their magnetic duals $H_{7}$ and $F_{7}$ are elements of
$H^{7}(X^{5}\times{\mathbb{R}\mathbb{P}}^{5};\widetilde{\mathbb{Z}})$. On the
other hand $F_{5}$ is classified by
$H^{5}(X^{5}\times{\mathbb{R}\mathbb{P}}^{5};\mathbb{Z})$. In what follows we
will focus on the structure on ${\mathbb{R}\mathbb{P}}^{5}$, as the
$SL(2,\mathbb{Z})$ bundle is trivial on $X^{5}$. The untwisted cohomology
groups of ${\mathbb{R}\mathbb{P}}^{5}$ are standard, and the twisted ones can
be derived easily from the results in Thom1952 :

$\displaystyle H^{\ast}({\mathbb{R}\mathbb{P}}^{5},\mathbb{Z})$
$\displaystyle=\\{\mathbb{Z}\,,\ 0\,,\ \mathbb{Z}_{2}\,,\ 0\,,\
\mathbb{Z}_{2}\,,\ \mathbb{Z}\\}$ (4) $\displaystyle
H^{\ast}({\mathbb{R}\mathbb{P}}^{5},\widetilde{\mathbb{Z}})$
$\displaystyle=\\{0\,,\ \mathbb{Z}_{2}\,,\ 0\,,\ \mathbb{Z}_{2}\,,\ 0\,,\
\mathbb{Z}_{2}\\}\,.$

Similar considerations hold for homology: $(p,q)$ 1-branes (such as
fundamental strings and D1 branes) are elements of
$H_{2}(X^{5}\times{\mathbb{R}\mathbb{P}}^{5};\widetilde{\mathbb{Z}})$, $(p,q)$
5-branes are elements of
$H_{6}(X^{5}\times{\mathbb{R}\mathbb{P}}^{5};\widetilde{\mathbb{Z}})$, and D3
branes are elements of
$H_{4}(X^{5}\times{\mathbb{R}\mathbb{P}}^{5};\mathbb{Z})$. The relevant
homology groups are (by Poincaré duality, which holds since
${\mathbb{R}\mathbb{P}}^{5}$ is orientable)

$\displaystyle H_{\ast}({\mathbb{R}\mathbb{P}}^{5},\mathbb{Z})$
$\displaystyle=\\{\mathbb{Z}\,,\ \mathbb{Z}_{2}\,,\ 0\,,\ \mathbb{Z}_{2}\,,\
0\,,\ \mathbb{Z}\\}$ (5) $\displaystyle
H_{\ast}({\mathbb{R}\mathbb{P}}^{5},\widetilde{\mathbb{Z}})$
$\displaystyle=\\{\mathbb{Z}_{2}\,,\ 0\,,\ \mathbb{Z}_{2}\,,\ 0\,,\
\mathbb{Z}_{2}\,,\ 0\\}\,.$

With an understanding of the cycles that the branes can wrap, it is
straightforward to identify the charged operators of the $\mathrm{Spin}(4N)$
theory Witten:1998xy : the vector Wilson line $W_{V}$ is a fundamental string
on a point of ${\mathbb{R}\mathbb{P}}^{5}$, the $\mathsf{s}$-spinor Wilson
line $W_{\mathsf{s}}$ is a D5-brane on ${\mathbb{R}\mathbb{P}}^{4}$, and
finally the $\mathsf{c}$-spinor Wilson line $W_{\mathsf{c}}$ is the
combination of both previous lines: a D5-brane/F1 bound state, again wrapped
on ${\mathbb{R}\mathbb{P}}^{4}$. In all cases the branes wrap a surface on
$X^{5}$ extending to the boundary, where they end on a line.

We can go to the $SO(4N)$ theory by gauging the diagonal factor
$\mathbb{Z}_{2}^{V}\subset\mathbb{Z}_{2}^{\mathsf{s}}\times\mathbb{Z}_{2}^{\mathsf{c}}$.
The vector line $W_{V}$ is unaffected by the gauging, so it survives, but the
$W_{\mathsf{s}}$ and $W_{\mathsf{c}}$ lines are no longer gauge invariant, and
become non-genuine (that is, boundaries of surface operators). A non-genuine
line $H_{V}$ of the $\mathrm{Spin}(4N)$ theory, with
$w_{2}^{\mathsf{s}}=w_{2}^{\mathsf{c}}$ flux around it, now becomes a genuine
line operator in the $SO(4N)$ theory. Holographically this operator
corresponds to a D1 brane wrapping a point in
$H_{0}({\mathbb{R}\mathbb{P}}^{5};\widetilde{\mathbb{Z}})=\mathbb{Z}_{2}$.777The
simplest derivation of this fact follows from recalling that the $SO(4N)$
field theory is invariant under $SL(2,\mathbb{Z})$, which maps to an
$SL(2,\mathbb{Z})$ action on the holographic dual. We refer the reader to
EGEHR  for a systematic analysis. We denote the generators for these two
symmetries $D_{2}^{B,e}(\mathcal{M}^{2})$ (acting on fundamental strings) and
$D_{2}^{C,m}(\mathcal{M})$ (acting on D1 branes), and the corresponding
background fields $B_{2}^{e}$ and $C_{2}^{m}$.

Starting from the $SO(4N)$ theory we can gauge various pairs of global
symmetries, an operation that, due to the cubic anomaly (2), results in
theories with non-invertible symmetries Kaidi:2021xfk ; Bhardwaj:2022yxj :

$\displaystyle\mathrm{Pin}^{+}(4N)\colon$ gauge $D_{3}^{(0)}$ and
$D_{2}^{C,m}$ (6) $\displaystyle Sc(4N)\colon$ gauge $D_{2}^{B,e}$ and
$D_{2}^{C,m}$ $\displaystyle PO(4N)\colon$ $\displaystyle\quad\text{gauge
$D_{3}^{(0)}$ and $D_{2}^{B,e}$}\,.$

In these expressions $D_{3}^{(0)}$, or more precisely
$D_{3}^{(0)}(\mathcal{M}^{3})$ is the generator for the outer automorphism
0-form symmetry of the $SO(4N)$ theory.

We are thus led to the crucial question in this paper: having identified the
_charged_ operators in the field theory in terms of the holographic dual, what
is the holographic description of the _charge_ operators implementing the
global symmetries in the $SO(4N)$ theory?

For concreteness, let us specialise to the holographic dual of the symmetry
generator $D_{2}^{C,m}(\mathcal{M}^{2})$ of the $SO(4N)$ theory, measuring how
many ’t Hooft lines (mod 2) $H_{V}$ are linked by $\mathcal{M}^{2}$, without
taking into account the Wilson lines $W_{V}$. Given our identification of
lines above, a natural guess would be

$D_{2}^{C,m}(\mathcal{M}^{2})\stackrel{{\scriptstyle?}}{{\to}}e^{i\pi\int_{\mathcal{M}_{2}\times{\mathbb{R}\mathbb{P}}^{4}}C_{6}}\,,$
(7)

where $\mathcal{M}^{2}$ lives on $\mathcal{M}^{4}$, and becomes the symmetry
operator when pushed to the boundary. This holonomy certainly measures the
number of D1 branes linked by $\mathcal{M}^{2}$ (the basic argument is given
below in case of the outer automorphism 0-form symmetry), but it cannot be the
right answer for a number of reasons. First, we know that in IIB string theory
fluxes are not measured by cohomology, but rather K-theory Moore:1999gb ;
Freed:2000tt ; Freed:2000ta . A way of capturing the right K-theoretic formula
is to phrase the answer in terms of the Wess-Zumino coupling in the D5 brane
action:

$D_{2}^{C,m}(\mathcal{M}^{2})\stackrel{{\scriptstyle?}}{{\to}}e^{\text{WZ}(\mathcal{M}_{2}\times{\mathbb{R}\mathbb{P}}^{4})}\,.$
(8)

where Cheung:1997az ; Minasian:1997mm ; Freed:2000ta

$\mathrm{WZ}(X)=2\pi
i\int_{X}e^{F_{2}-B_{2}}\sqrt{\frac{\hat{A}(TX)}{\hat{A}(NX)}}(C_{0}+C_{2}+\ldots)$
(9)

A second reason why we expect neither (7) nor (8) to be the full answer is
that in string theory there are no local operators, only dynamical objects. So
we should aim to represent the charge generator by a dynamical object, and not
simply a defect. The dynamical objects that are electrically charged under
$C_{6}$, and would arise when fixing the insertion of the defect as a boundary
condition, are D5 branes.

While neither argument is conclusive, they both suggest that the holographic
description of the symmetry generator is a full D5, pushed to the
boundary:888The authors of ABBS  provide complementary evidence for the same
proposal.

$D_{2}^{C,m}(\mathcal{M}^{2})\to\text{D5}(\mathcal{M}_{2}\times{\mathbb{R}\mathbb{P}}^{4})\,.$
(10)

This ansatz has the additional virtue of restoring the common origin between
lines and charge generators, familiar from the formulation of symmetries in
terms of relative field theories Freed:2012bs .

An objection one might raise about (10) is that branes are not topological,
while charge operators should be. As we will see in a moment, the worldvolume
theory on the branes, when reduced to $\mathcal{M}^{2}\subset X^{5}$, is a
discrete $\mathbb{Z}_{2}$ gauge theory. Therefore the potential lack of
deformation-invariance coming from the gauge fields on the brane is not an
issue. There is still an overall factor of the volume, but it does not couple
to the dynamical fields of the field theory on the boundary, so it can be
absorbed into a counterterm.

A subtle feature of (10) is that the worldvolume theories on the brane are
quantum field theories, so we should sum over them. As we will argue, the sum
over worldvolume degrees of freedom provides precisely the minimal anomalous
TQFT “dressing” the bare symmetry generator identified in Bhardwaj:2022yxj .
This is a very non-trivial test of the identification (10).

Clearly, if the ansatz (10) is correct, the holographic dual of the operator
counting ’t Hooft lines $H_{V}$ is the S-dual of (10):

$D_{2}^{B,e}(\mathcal{M}^{2})\to\text{NS5}(\mathcal{M}_{2}\times{\mathbb{R}\mathbb{P}}^{4})\,.$
(11)

Additionally, the $SO(4M)$ theory has the 0-form parity symmetry discussed
above. The point operator charged under this symmetry is known as the Pfaffian
operator. As discussed in Witten:1998xy  the Pfaffian operator is represented
holographically by a D3 brane wrapping the ${\mathbb{R}\mathbb{P}}^{3}$ cycle
inside ${\mathbb{R}\mathbb{P}}^{5}$, and extending to a point on the boundary.
We will refer to this brane as the “Pfaffian brane”.

We now argue that the holographic dual of the generator of this symmetry is

$D_{3}^{(0)}(\mathcal{M}^{3})\to\text{D3}(\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1})\,.$
(12)

More precisely, we will show that the Pfaffian brane is charged under this D3
in the Hamiltonian formalism, so we take the boundary to be of the form
$\mathcal{M}^{3}\times\mathbb{R}_{t}$, with the last component the time
direction along the boundary. We choose coordinates so that the endpoint of
the Pfaffian operator is at $t=0$. Now we wrap the putative symmetry generator
D3 on $\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}$, where
$\mathcal{M}^{3}$ is at the boundary at $t=0$. Because the $F_{5}$ RR flux is
self-dual, the two D3 branes that we have introduced do not commute
Gukov:1998kn ; Moore:2004jv ; Freed:2006ya ; Freed:2006yc :

$\begin{split}\text{D3}(\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1})\operatorname{Pf}(\mathrm{pt})=e^{2\pi
i\,\mathsf{L}({\mathbb{R}\mathbb{P}}^{1},{\mathbb{R}\mathbb{P}}^{3})}\operatorname{Pf}(\mathrm{pt})\text{D3}(\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1})=-\operatorname{Pf}(\mathrm{pt})\text{D3}(\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1})\end{split}$
(13)

where $\operatorname{Pf}(\mathrm{pt})$ is the D3 brane representing the
Pfaffian operator,
$\mathsf{L}({\mathbb{R}\mathbb{P}}^{1},{\mathbb{R}\mathbb{P}}^{3})=\frac{1}{2}$
is the linking pairing between the given cycles of
${\mathbb{R}\mathbb{P}}^{5}$, and we have used that the given branes intersect
at a point on the $t=0$ spatial slice on $X^{5}$. Equation (13) is the
Hamiltonian version of the statement that the Pfaffian operator is charged
under $\text{D3}(\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1})$, as
claimed. This discussion can be generalised straightforwardly to show that the
branes (10) and (11) do indeed give the expected charges to the $W_{V}$ and
$H_{V}$ lines of the $SO(4N)$ theory, as claimed.

##  3 TQFT stacking from Wess-Zumino couplings

Our task in this section will be to deduce the non-invertibility of the
symmetry generators of the theories in (6) from our assumption that symmetry
generators are represented holographically by branes.

###  3.1 Fluxes and twisted differential cohomology

Our basic tool will be differential cohomology. We refer the reader to
Apruzzi:2021nmk  for a review of the basic techniques and notation that we
use. The analysis in this paper has some novelties with respect to the
discussion in Apruzzi:2021nmk , which we now discuss.

The main difference is that we will be working with _twisted_ differential
cohomology. The twisted and untwisted cohomology groups of
${\mathbb{R}\mathbb{P}}^{5}$ were given in (4) above. The ring structure
induced by the cup product for ${\mathbb{R}\mathbb{P}}^{5}$ can be obtained by
adapting the discussion in Lemma 1 of twisted-cohomology-ring  (see also
Thom1952 ). It is most easily described by adjoining the twisted and untwisted
cohomology groups

$\displaystyle\mathsf{H}^{*}({\mathbb{R}\mathbb{P}}^{5})$
$\displaystyle=H^{*}({\mathbb{R}\mathbb{P}}^{5};\mathbb{Z})\oplus
H^{*}({\mathbb{R}\mathbb{P}}^{5};\widetilde{\mathbb{Z}})$ (14)
$\displaystyle=\mathbb{Z}[t_{1},u_{5}]/(2t_{1},t_{1}^{6},u_{5}^{2})\,.$

That is, we have free components of degree 0 and 5, and $\mathbb{Z}_{2}$
torsional components of degrees 1 to 5, generated by $t_{1}^{n}$. In
particular, taking an even number of powers of $t_{1}$ gives an untwisted
class, while taking an odd number of powers gives a twisted one. In what
follows we will use the notation $t_{2n+1}=t_{1}^{2n+1}$ for twisted classes
and $u_{2n}=t_{1}^{2n}$ together with $u_{5}$ for untwisted ones.

We denote by $\breve{t}_{k}$ a flat differential cohomology class with
characteristic class $t_{k}$, which we denote by $I(\breve{t}_{k})=t_{k}$, and
similarly for $\breve{u}_{k}$. We note that $I(\breve{t}_{1}^{2})=u_{2}$, and
similarly $I(\breve{t}_{1}^{4})=u_{4}$, so perfectness of the linking pairing
on $H^{2}({\mathbb{R}\mathbb{P}}^{5};\mathbb{Z})\times
H^{4}({\mathbb{R}\mathbb{P}}^{5};\mathbb{Z})=\mathbb{Z}_{2}\times\mathbb{Z}_{2}$
implies that

$\int_{{\mathbb{R}\mathbb{P}}^{5}}\breve{t}_{1}^{6}=\int_{{\mathbb{R}\mathbb{P}}^{5}}\breve{u}_{2}\star\breve{u}_{4}=\frac{1}{2}\mod
1\,.$ (15)

This equation together with the ring structure (14) will be our workhorses in
what follows.

Finally, before moving on to the examples, we need to know how to represent
background fluxes in terms of differential cohomology. To lighten notation, in
this section we introduce $b_{2}\coloneqq B_{2}^{e}$ and $c_{2}\coloneqq
C_{2}^{m}$. Recall that the objects charged under these backgrounds are F1 and
D1 branes, respectively, so the holographic fluxes encoding these backgrounds
are $\breve{H}_{3}$ and $\breve{F}_{3}$, which are asymptotically of the form
$\breve{H}_{3}=b_{2}\star\breve{t}_{1}$ and
$\breve{F}_{3}=c_{2}\star\breve{t}_{1}$. By imposing this asymptotic form we
ensure that the charged lines in the field theory acquire the right
holonomies, see Garcia-Etxebarria:2019cnb ; Apruzzi:2021nmk  for analysis of
similar examples. We could also include terms proportional to $\breve{t}_{3}$
in these expansions, but they would correspond to a change of the gauge
algebra to $\mathfrak{so}(4N+1)$ (for $\breve{F}_{3}$) of $\mathfrak{usp}(4N)$
(for $\breve{H}_{3}$) Witten:1998xy  so we will not consider these terms
further.999When doing this sort of expansion there is an additional subtlety
involving topologically trivial differential characters that is discussed at
length in Apruzzi:2021nmk . It will not affect our considerations, so we will
ignore such terms. Finally, a field theory 0-form symmetry background $a_{1}$
for $D_{2}^{(0)}$ is represented by $\breve{F}_{5}=a_{1}\star\breve{u}_{4}$.
There are additional terms possible in the expansion for $\breve{F}_{5}$, we
will discuss these below.

###  3.2 Non-invertibles in $Sc(4N)$ from D3 branes

We start with the case of the $Sc(4N)$ theory, where following the analysis in
Kaidi:2021xfk ; Bhardwaj:2022yxj , we expect to get the topological defect
that is non-invertible from the generator of the 0-form symmetry in the
$SO(4N)$ theory. We identified this generator above with the D3-brane wrapped
on $\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}=\mathcal{M}^{3}\times
S^{1}$. The worldvolume $U(1)$ field on the D3 brane is odd under $-1\in
SL(2,\mathbb{Z})$, because it is a trivialisation of $B_{2}$, which is odd.
Therefore it takes values in the twisted cohomology group
$H^{2}(\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1};\widetilde{\mathbb{Z}})$.
We have
$H^{*}({\mathbb{R}\mathbb{P}}^{1};\widetilde{\mathbb{Z}})=\\{0,\mathbb{Z}_{2}\\}$.

When computing the path integral on the D3, the field strength $F_{2}$ on the
brane will induce D1 charge due to the Wess-Zumino term (9), while the
magnetic field strength $F_{2}^{D}$ will induce F1 charge. The Wess-Zumino
action written in terms of the electric variable $F_{2}$ is

$S_{\text{D3},e}=2\pi
i\int_{\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}}\breve{F}_{5}+\breve{F}_{3}\star(\breve{\mathscr{F}}_{2})+\breve{F}_{1}\star(\frac{1}{2}\breve{\mathscr{F}}_{2}\star\breve{\mathscr{F}}_{2}+\frac{1}{24}\breve{e})\,,$
(16)

with $\breve{\mathscr{F}}_{2}=\breve{F}_{2}-\breve{B}_{2}$, $\breve{F}_{1}$ a
differential cohomology uplift of $C_{0}$, and $\breve{e}$ is a differential
cohomology uplift of the Euler class of
$\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}$. The term proportional to
$\breve{F}_{1}$ will be relevant only for analysis of anomalies in the space
of coupling constants, which we do not analyse in this note (although this is
certainly an interesting direction to explore further).

We also need to consider couplings of the form
$\breve{F}_{3}\star\breve{B}_{2}$. As argued in Freed:2000ta  (elaborating on
results of Taylor:2000za ; Alekseev:2000ch ), these couplings are not present
when measuring the actual K-theory charges, which is what we are ultimately
interested in, so we will simply set $\breve{B}_{2}$ to 0. A more careful
treatment of this issue would be desirable, but given that inclusion of these
background fields would not change our conclusions (since they would provide
overall invertible prefactors on the brane action in any case, even if we
included them), we will postpone a more careful treatment of this point to
future work.

With these simplifications taken into account, the relevant part of the Wess-
Zumino action for the D3 becomes

$S_{\text{D3},e}=2\pi
i\int_{\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}}\breve{F}_{5}+\breve{F}_{3}\star\breve{F}_{2}\,.$
(17)

The first term is the differential cohomology avatar of the naive guess
$\int_{\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}}C_{4}$ for the flux
operator in the field theory. $\breve{F}_{5}$ is even under the $-1\in
SL(2,\mathbb{Z})$ action, so its general decomposition is of the form
$\breve{F}_{5}=N\star\breve{u}_{5}+a_{1}\star\breve{u}_{4}+a_{3}\star\breve{u}_{2}+N\star\breve{1}$.
Here $N$ are the number of units of RR 5-form flux on the
${\mathbb{R}\mathbb{P}}^{5}$, and we have used that $F_{5}$ is self-dual to
relate the components of degrees 5 and 0.

In terms of this decomposition we have an effective operator in
$\mathrm{AdS}_{5}$ of the form

$\begin{split}D(\mathcal{M}^{3})&=\exp\left(2\pi
i\int_{\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}}\breve{F}_{5}\right)=\exp\left(2\pi
i\int_{\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{5}}\breve{F}_{5}\star\breve{u}_{4}\right)\\\
&=\exp\left(\pi i\int_{\mathcal{M}^{3}}a_{3}\right)\,,\end{split}$ (18)

where in the second equality we have used Poincaré duality on
${\mathbb{R}\mathbb{P}}^{5}$ to relate ${\mathbb{R}\mathbb{P}}^{1}$ to
$u_{4}$, and in the third used that the only non-trivial pairing in
${\mathbb{R}\mathbb{P}}^{5}$ appearing after the expansion of $\breve{F}_{5}$
is (15). This is the expected formula for the operator measuring discrete
electric flux for the outer automorphism symmetry in the $SO(4N)$ theory.

The second term is the more interesting one for our purposes. As explained
above, field theory backgrounds for the symmetry $D_{2}^{C,m}$ are described
holographically by fluxes with asymptotic form
$\breve{F}_{3}=c_{2}\star\breve{t}_{1}$. Similarly, we can expand
$\breve{F}_{2}=\gamma_{1}\star\breve{t}_{1}$. We then have (using the formulas
for integration on products reviewed in Apruzzi:2021nmk )

$2\pi
i\int_{\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}}\breve{F}_{3}\star\breve{F}_{2}=2\pi
i\int_{\mathcal{M}^{3}}c_{2}\gamma_{1}\int_{{\mathbb{R}\mathbb{P}}^{1}}\breve{t}_{1}\star\breve{t}_{1}=\pi
i\int_{\mathcal{M}^{3}}c_{2}\gamma_{1}$ (19)

where in the last step we have again used the fact that the linking pairing is
perfect, so

$\int_{{\mathbb{R}\mathbb{P}}^{1}}\breve{t}_{1}\star\breve{t}_{1}=\frac{1}{2}\mod
1.$ (20)

So far we have considered the charge induced on a D3 due to the gauge field
strength $F_{2}$. The computation above shows that it induces an effective
coupling on $\mathcal{M}^{3}$ to the background for $D_{2}^{C,m}$. By IIB
S-duality, this implies that a dual field strength
$F_{2}^{D}=\phi_{1}\star\breve{t}_{1}$ induces a coupling of the form

$\begin{split}\pi i\int_{\mathcal{M}^{3}}b_{2}\phi_{1}\end{split}$ (21)

on the effective operator on $\mathrm{AdS}_{5}$. The same result can be
obtained from the effective action presented in the magnetic variables
obtained in Kimura:1999jb .

In elementary terms, the two couplings (19) and (21) that we have just derived
can be understood as encoding the well known facts that worldvolume flux on
the D3 induces D1 charge, and magnetic worldvolume flux F1 charge. Recall that
the D1 and F1 are the charged objects in the $SO(4N)$ theory before gauging
their corresponding symmetries. After gauging, they will become the symmetry
generators for the dual magnetic symmetries in the $Sc(4N)$ theory (at least
if our general philosophy of identifying branes with symmetries is correct).
So what we have just shown, is that when doing the path integral on the D3 we
will have to sum over insertions of the symmetry generators for the 1-forms of
the theory. This is certainly suggestive that condensations Gaiotto:2019xmp ;
Choi:2022zal ; Roumpedakis:2022aik  are going to enter the picture after
gauging.

The precise details are nevertheless somewhat subtle. In general, when
performing the path integral the standard prescription is that we choose
whether we formulate the theory in terms of electric or magnetic variables,
and then sum over the specified variables only. From this point of view the
two couplings (19) and (21) seem somewhat at odds, and it is not clear which
one we should choose. What saves the day is that this standard prescription
has to be subtly modified whenever the cohomology groups where the electric
and magnetic fluxes live contain torsional components. In this case, as
originally pointed out by Moore:2004jv ; Freed:2006ya ; Freed:2006yc , the
electric and magnetic flux operators do not commute. As shown in
Apruzzi:2021nmk  (see also Camara:2011jg  for a different derivation of the
same result) this flux non-commutativity leads to the existence of a discrete
gauge theory when the theory is compactified on the space with torsion. The
argument, adapted to the system at hand, goes as follows.

Our initial theory is four dimensional $U(1)$ Maxwell theory on the D3,
compactified on $\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}$. We will
present a Hamiltonian quantisation analysis, so we assume that
$\mathcal{M}^{3}=\mathcal{N}^{2}\times\mathbb{R}$, and we identify the last
component with the time direction.101010A Lagrangian derivation will appear in
GEH . The spatial slice is of the form
$\mathcal{N}^{2}\times{\mathbb{R}\mathbb{P}}^{1}$. There is a non-trivial
$SL(2,\mathbb{Z})$ duality bundle along the ${\mathbb{R}\mathbb{P}}^{1}=S^{1}$
direction inherited from the ${\mathbb{R}\mathbb{P}}^{5}$ background, with
holonomy $-1$, which induces a $(F_{2},F_{2}^{D})\to(-F_{2},-F_{2}^{D})$
transformation of the worldvolume gauge field. Therefore, just as in the IIB
background itself, the worldvolume gauge fields on the D3 are valued in
twisted cohomology. In particular
$H^{1}({\mathbb{R}\mathbb{P}}^{1};\widetilde{\mathbb{Z}})=\mathbb{Z}_{2}$,
which justifies the statement above that there is torsion in this problem.

Consider the operators $\Phi_{e}(a\otimes t_{1})$, $\Phi_{m}(b\otimes t_{1})$
that measure electric and magnetic fluxes on the torsional sector. They are
associated with flat, topologically non-trivial elements of
$\operatorname{Tor}H^{2}(\mathcal{N}^{2}\times{\mathbb{R}\mathbb{P}}^{1};\widetilde{\mathbb{Z}})$
Freed:2006ya ; Freed:2006yc , which in our case are all of the form $a\otimes
t_{1}$, where $a\in H^{1}(\mathcal{N}^{2};\mathbb{Z})$ and $t_{1}$ is the
generator of $H^{1}({\mathbb{R}\mathbb{P}}^{1};\widetilde{\mathbb{Z}})$.
Alternatively, using Poincaré duality, we can view these operators as the
holonomy of the twisted fluxes $\breve{F}_{2}$ and $\breve{F}_{2}^{D}$ on
cycles $\alpha\times\widetilde{\mathrm{pt}}$ and
$\beta\times\widetilde{\mathrm{pt}}$, where $\alpha$ and $\beta$ are Poincaré
dual to $a$ and $b$ in $\mathcal{N}^{2}$, and $\widetilde{\mathrm{pt}}$ the
(twisted) point in $H_{0}(\mathcal{N}^{2};\widetilde{\mathbb{Z}})$, which is
Poincaré dual on ${\mathbb{R}\mathbb{P}}^{1}$ to $t_{1}$. So we have

$\begin{split}\Phi_{e}(a\otimes t_{1})=\exp\left(2\pi
i\int_{\alpha}\gamma_{1}\int_{\widetilde{\mathrm{pt}}}\breve{t}_{1}\right)=\exp\left(2\pi
i\int_{\alpha}\gamma_{1}\int_{{\mathbb{R}\mathbb{P}}^{1}}\breve{t}_{1}^{2}\right)=\exp\left(\pi
i\int_{\alpha}\gamma_{1}\right)\end{split}$ (22)

and similarly

$\begin{split}\Phi_{m}(b\otimes t_{1})=\exp\left(\pi
i\int_{\beta}\phi_{1}\right)\,.\end{split}$ (23)

Now, it follows from the general analysis of Moore:2004jv ; Freed:2006ya ;
Freed:2006yc  that

$\begin{split}\Phi_{e}(a\otimes t_{1})\Phi_{m}(b\otimes
t_{1})=(-1)^{\int_{\mathcal{N}^{2}}ab}\Phi_{m}(b\otimes
t_{1})\Phi_{e}(a\otimes t_{1})\,,\end{split}$ (24)

or equivalently, formulating everything in terms of homology on
$\mathcal{N}^{2}$ (and abusing notation slightly):

$\begin{split}\Phi_{e}(\alpha)\Phi_{m}(\beta)=(-1)^{\alpha\cdot\beta}\Phi_{m}(\beta)\Phi_{e}(\alpha)\,.\end{split}$
(25)

These commutation relations are precisely those of a $\mathbb{Z}_{2}$ theory.
We can represent this theory by a gauge theory on the two fields $\gamma_{1}$,
$\phi_{1}$ with action Banks:2010zn

$\begin{split}S_{\mathbb{Z}_{2}}=\pi
i\int_{\mathcal{M}^{3}}\gamma_{1}\delta\phi_{1}\,.\end{split}$ (26)

We have identified the fields appearing in the Lagrangian with $\gamma_{1}$
and $\phi_{1}$ since these are precisely the fields whole holonomies are
measured by the operators in the theory, by construction.

Assembling all the pieces together, we find that the effective partition
function on the D3, seen as an 3-surface dynamical object on $X^{5}$, is (up
to an overall normalisation)

$\begin{split}\mathcal{N}(\mathcal{M}^{3})=D_{3}^{(0)}(\mathcal{M}^{3})\cdot\int\mathcal{D}\gamma_{1}\mathcal{D}\phi_{1}\exp\left(\pi
i\int_{\mathcal{M}^{3}}\gamma_{1}\delta\phi_{1}+c_{2}\gamma_{1}+b_{2}\phi_{1}\right)\,.\end{split}$
(27)

The path integral over $\gamma_{1},\phi_{1}$ is the remnant of the $U(1)$ YM
path integral in this torsional setting. This is precisely the non-invertible
operator found in Bhardwaj:2022yxj .

#### Fusion rules

Now that we have a full description of the symmetry defect, including its TQFT
sector, we can derive the fusion rules for the extended operators in the
$Sc(4N)$ theory, in particular showing that $\mathcal{N}(\mathcal{M}^{3})$ is
a non-invertible operator of the $Sc(4N)$ theory. Since the TQFT that comes
out of the brane dynamics is identical to the one conjectured in
Bhardwaj:2022yxj , the rest of our derivation of the fusion rules can proceed
exactly as in that paper (and the similar analysis in Kaidi:2021xfk ). We
include the details of the argument for completeness and convenience for the
reader, and then offer some comments reinterpreting some of the features of
the computation from a brane perspective.

Consider first the fusion of two copies of $\mathcal{N}(\mathcal{M}^{3})$.
Each defect comes with its own $\mathbb{Z}_{2}$ TQFT, so we have two sets of
dynamical fields:

$\begin{split}\mathcal{N}(\mathcal{M}^{3})\times\mathcal{N}(\mathcal{M}^{3})=\int\mathcal{D}\gamma_{1}\mathcal{D}\phi_{1}\mathcal{D}\gamma_{1}^{\prime}\mathcal{D}\phi_{1}^{\prime}\exp\left(\pi
i\int_{\mathcal{M}^{3}}\gamma_{1}\delta\phi_{1}+\gamma_{1}^{\prime}\delta\phi_{1}^{\prime}+c_{2}(\gamma_{1}+\gamma_{1}^{\prime})+b_{2}(\phi_{1}+\phi_{1}^{\prime})\right)\,.\end{split}$
(28)

Switching to new variables $\gamma_{1}$,
$\hat{\gamma}_{1}\coloneqq\gamma_{1}+\gamma_{1}^{\prime}$, $\phi_{1}$,
$\hat{\phi}_{1}\coloneqq\phi_{1}+\phi_{1}^{\prime}$, the action becomes

$\begin{split}\mathcal{N}(\mathcal{M}^{3})\times\mathcal{N}(\mathcal{M}^{3})=\int\mathcal{D}\gamma_{1}\mathcal{D}\phi_{1}\mathcal{D}\hat{\gamma}_{1}\mathcal{D}\hat{\phi}_{1}\exp\left(\pi
i\int_{\mathcal{M}^{3}}\hat{\gamma}_{1}\delta\hat{\phi}_{1}+\hat{\gamma}_{1}\delta\phi_{1}+\gamma_{1}\delta\hat{\phi}_{1}+c_{2}\hat{\gamma}_{1}+b_{2}\hat{\phi}_{1}\right)\,.\end{split}$
(29)

We can integrate $\phi_{1}$ and $\gamma_{1}$ out, which imposes
$\delta\hat{\gamma}_{1}=\delta\hat{\phi}_{1}=0$, so
$\hat{\gamma}_{1}\delta\hat{\phi}_{1}=0$. We then have

$\begin{split}\mathcal{N}(\mathcal{M}^{3})\times\mathcal{N}(\mathcal{M}^{3})=\int\mathcal{D}\hat{\gamma}_{1}\mathcal{D}\hat{\phi}_{1}\exp\left(\pi
i\int_{\mathcal{M}^{3}}c_{2}\hat{\gamma}_{1}+b_{2}\hat{\phi}_{1}\right)\,.\end{split}$
(30)

Poincaré dualising $\hat{\gamma}_{1}$ and $\hat{\phi}_{1}$ to $\Gamma,\Phi\in
H_{2}(\mathcal{M}^{3};\mathbb{Z})$, this can be rewritten as

$\begin{split}\mathcal{N}(\mathcal{M}^{3})\times\mathcal{N}(\mathcal{M}^{3})=\sum_{\Gamma,\Phi\in
H_{2}(\mathcal{M}^{3};\mathbb{Z})}D_{2}^{\mathsf{c},e}(\Gamma)D_{2}^{\mathsf{s},m}(\Phi)\end{split}$
(31)

where $D_{2}^{\mathsf{c},e}$ and $D_{2}^{\mathsf{s},m}$ are the 1-form
symmetry generators of the $Sc(4N)$ theory. (The notation is explained below.)
So $\mathcal{N}$ is indeed a non-invertible defect in the $Sc(4N)$ theory,
since the right hand side is a sum of operators.

This was the derivation in Bhardwaj:2022yxj . Holographically, the physical
meaning of the computation can be understood as follows. We have argued that
the defect $\mathcal{N}(\mathcal{M}^{3})$ corresponds to a D3 wrapping
$\mathcal{M}^{3}\times{\mathbb{R}\mathbb{P}}^{1}$, including its quantum
dynamics. The effect of the quantum dynamics is to sum over induced charges,
which in this case means summing over D3/F1 and D3/F1 bound states. (The
precise way in which this sum happens involves, as shown above, a
$\mathbb{Z}_{2}$ gauge theory.) If there was no sum, but only a fixed induced
charge (the trivial one, say), then taking the square would lead to a complete
annihilation of the $\mathbb{Z}_{2}$ charges, and therefore a trivial
operator. Since there is a sum involved some of the cross-terms in the square
of the sum will lead to incomplete annihilations, leaving a sum over F1 and D1
insertions along the worldvolume of the D3. The D3 charge is always there no
matter the induced charge, and disappears, so only the sum over D1 and F1
insertions remains.

In order to show that this physical process does indeed produce (31), all we
need to verify is that the symmetry generators of the $Sc(4N)$ theory are the
F1 and D1. This is immediate, since they are the genuine lines in the $SO(4N)$
theory, and we are gauging the symmetry they are charged under, so they become
the magnetic symmetry generators in the gauged theory. It is also instructive
to derive it from the $\mathrm{Spin}(4N)$ starting point. The
$\mathsf{s}$-spinor Wilson line $W_{\mathsf{s}}$, given by a D5 brane wrapped
on ${\mathbb{R}\mathbb{P}}^{4}$, is neutral under
$\mathbb{Z}_{2}^{\mathsf{s}}$ (recall our conventions from footnote 5), which
is the symmetry that we gauge to go to $Sc(4N)$. So the corresponding charge
operator, the D1 on $\widetilde{\mathrm{pt}}$, survives as a charge operator
on the $Sc(4N)$ theory. We have denoted it above by $D_{2}^{\mathsf{c},e}$. On
the other hand the $\mathsf{c}$-spinor and vector Wilson lines are not
invariant, due to the presence of fundamental strings in them, which are not
invariant under $\mathbb{Z}_{2}^{\mathsf{s}}$. So after gauging
$\mathbb{Z}_{2}^{\mathsf{s}}$ the fundamental string on a twisted point in
${\mathbb{R}\mathbb{P}}^{5}$ becomes the second (magnetic) symmetry generator
in the $Sc(4N)$ theory, which we have denoted above by $D_{2}^{\mathsf{s},m}$.

We are finally left with the task of determining the fusion of
$\mathcal{N}(\mathcal{M}^{3})$ with the one-form symmetry generators
$D_{2}^{\mathsf{c},e}(\Gamma)$ and $D_{2}^{\mathsf{s},m}(\Phi)$. Consider for
example $D_{2}^{\mathsf{c},e}(\Gamma)$. We have just argued that it
corresponds to a D1 brane on $\Gamma\times\widetilde{\mathrm{pt}}$. Fusing it
with $\mathcal{N}(\mathcal{M}^{3})$, which involves a sum over induced D1
branes wrapping the Poincaré dual
$\operatorname{PD}[\gamma_{1}]\times\widetilde{\mathrm{pt}}$ to $\gamma_{1}$
amounts to shifting $\gamma_{1}\to\gamma_{1}+\operatorname{PD}[\Gamma]$ in
(27). But this can clearly be reabsorbed in a change of variables, giving back
$\mathcal{N}(\mathcal{M}^{3})$. So

$\begin{split}\mathcal{N}(\mathcal{M}^{3})\times
D_{2}^{\mathsf{c},e}(\Gamma)=\mathcal{N}(\mathcal{M}^{3})\,.\end{split}$ (32)

An identical argument shows

$\begin{split}\mathcal{N}(\mathcal{M}^{3})\times
D_{2}^{\mathsf{s},m}(\Phi)=\mathcal{N}(\mathcal{M}^{3})\,.\end{split}$ (33)

We have shown that the $\mathcal{N}(\mathcal{M}^{3})$ operators of the
$Sc(4N)$ theory are non-invertible, and are represented holographically by D3
branes. A small puzzle remains: our starting point was that the bulk of the
holographic dual was the same for all global forms, so the same D3 brane
appears in the bulk of all theories with the same local dynamics, including
theories that are not expected to have non-invertible symmetries. The reason
that the D3 does not lead to non-invertible symmetries in some cases has to do
with boundary conditions (as it should, as this is the only thing that is
different in the various cases). Consider for instance the $SO(4N)$ theory,
where the D3 on ${\mathbb{R}\mathbb{P}}^{1}$ is also a symmetry operator,
implementing the outer automorphism. As we push to the boundary, we obtain an
operator of the form (27), but with a crucial difference: the IIB $B_{2}$ and
$C_{2}$ fields have a Dirichlet boundary condition in this case, so they are
not dynamical but instead they provide backgrounds for the global 1-form
symmetries for the $SO(4N)$ theory. So the term

$\begin{split}\int\mathcal{D}\gamma_{1}\mathcal{D}\phi_{1}\exp\left(\pi
i\int_{\mathcal{M}^{3}}\gamma_{1}\delta\phi_{1}+C_{2}\gamma_{1}+B_{2}\phi_{1}\right)\end{split}$
(34)

in (27) (where we have capitalised $B_{2}$ and $C_{2}$ to indicate that now
they are fixed background fields) does not depend on any dynamical field in
the $SO(4N)$ theory, so it is essentially trivial as an operator of the
$SO(4N)$ theory (it can be taken out of the path integral). In this case it is
consistent to split it off from the invertible part
$D_{3}^{(0)}(\mathcal{M}^{3})$, which can meaningfully be considered in
isolation.

###  3.3 4d $PO(4N)$ and $\mathrm{Pin}^{+}$ non-invertibles

The other two theories with non-invertible symmetries in (6) can be analysed
in a very similar way.

Let us start with the $PO(4N)$ case. Here we gauge $D_{3}^{(0)}$ and
$D_{2}^{B,e}$, so we expect the non-invertible 2-surface operator to be
associated with $D_{2}^{C,m}$, which we argued above is given by a D5 brane
wrapping ${\mathbb{R}\mathbb{P}}^{4}\subset{\mathbb{R}\mathbb{P}}^{5}$. We
will need the twisted and untwisted cohomology groups of
${\mathbb{R}\mathbb{P}}^{4}$, these are

$\begin{split}H^{*}({\mathbb{R}\mathbb{P}}^{4};\mathbb{Z})&=\\{\mathbb{Z},0,\mathbb{Z}_{2},0,\mathbb{Z}_{2}\\}\\\
H^{*}({\mathbb{R}\mathbb{P}}^{4};\widetilde{\mathbb{Z}})&=\\{0,\mathbb{Z}_{2},0,\mathbb{Z}_{2},\mathbb{Z}\\}\,.\end{split}$
(35)

(The second line follows from analysing the twisted Gysin sequence in Thom1952
.) As above, $\breve{F}_{2}$ is in the twisted sector, so it expands as
$\gamma_{1}\otimes\breve{t}_{1}$, but its magnetic dual $\breve{F}_{4}^{D}$ is
now _untwisted_ : this is needed to be able to write a kinetic term on the
twisted ${\mathbb{R}\mathbb{P}}^{4}$. It therefore has an expansion of the
form $\breve{F}_{4}^{D}=\phi_{4}\star
1+\phi_{2}\star\breve{u}_{2}+\phi_{0}\star\breve{u}_{4}$.

In the electric frame the action on the D5 is of the form

$\begin{split}S=\int_{\mathcal{M}^{2}\times{\mathbb{R}\mathbb{P}}^{4}}\breve{F}_{7}+\breve{F}_{2}\star\breve{F}_{5}+\left(\frac{1}{2}\breve{F}_{2}^{2}+\frac{1}{24}\breve{e}\right)\star\breve{F}_{3}+\ldots\end{split}$
(36)

where the missing terms are proportional to $\breve{F}_{1}$, so we will ignore
them. The term proportional to $\breve{F}_{2}^{2}\star\breve{F}_{3}$ does not
contribute for degree reasons, as it goes as $\breve{t}_{1}^{3}$. The
curvature term $\breve{e}\star\breve{F}_{3}$ could in principle contribute,
but it does not depend on the electric field so it will not enter our
considerations. We are left with the first two terms. The first one does
clearly contribute, and leads to the expected “naive” 2-surface holonomy
operator on $\mathcal{M}^{4}$, entirely analogously to the discussion around
(18). The second term is also interesting. Given our expansion of
$\breve{F}_{5}$ above, there is a single non-vanishing contribution of the
form

$\begin{split}2\pi
i\int_{\mathcal{M}^{2}\times{\mathbb{R}\mathbb{P}}^{4}}(\gamma_{1}\star\breve{t}_{1})\star(N\star\breve{u}_{5}+a_{1}\star\breve{u}_{4}+a_{3}\star\breve{u}_{2}+N\star\breve{1})=\pi
i\int_{\mathcal{M}^{2}}\gamma_{1}a_{1}\,,\end{split}$ (37)

where we have used that ${\mathbb{R}\mathbb{P}}^{4}$ is Poincaré dual to
$t_{1}$ in ${\mathbb{R}\mathbb{P}}^{5}$. This is the statement that
worldvolume flux $F_{2}$ induces D3 charge. The magnetic flux $F_{4}^{D}$ will
induce F1 charge (by a generalisation of the analysis in Kimura:1999jb ), via
a coupling of the form

$\begin{split}2\pi
i\int_{\mathcal{M}^{2}\times{\mathbb{R}\mathbb{P}}^{4}}\breve{F}_{4}^{D}\star\breve{H}_{3}=\pi
i\int_{\mathcal{M}^{2}}\phi_{0}b_{2}\,,\end{split}$ (38)

where we have used the expansion $\breve{H}_{3}=b_{2}\star\breve{t}_{1}$ as
above.

All that remains is to obtain the prescription for how to sum over electric
and magnetic fluxes. As above, flux non-commutativity can be used to argue
that there is an effective $\mathbb{Z}_{2}$ gauge theory on $\mathcal{M}^{2}$
with action

$\begin{split}i\pi\int_{\mathcal{M}^{2}}\gamma_{1}\delta\phi_{0}\,.\end{split}$
(39)

The only new subtlety in this derivation comes from the fact that on
${\mathbb{R}\mathbb{P}}^{4}$, being non-orientable, the perfect torsional
pairing is between a twisted class $\breve{t}_{1}$ and an untwisted one
$\breve{u}_{4}$. An easy way to verify the existence of such a coupling is to
use Poincaré duality on ${\mathbb{R}\mathbb{P}}^{5}$:

$\begin{split}\int_{{\mathbb{R}\mathbb{P}}^{4}}\breve{t}_{1}\breve{u}_{4}=\int_{{\mathbb{R}\mathbb{P}}^{5}}\breve{t}_{1}^{6}=\frac{1}{2}\mod
1\,.\end{split}$ (40)

Putting all these terms together we obtain the topological action

$\begin{split}S_{TFT}^{PO(4N)}=i\pi\int_{\mathcal{M}^{2}}\gamma_{1}\delta\phi_{0}+\phi_{0}b_{2}+\gamma_{1}a_{1}\end{split}$
(41)

which is precisely the action proposed in Bhardwaj:2022yxj . The fusion
algebra can be derived as above.

Finally, in the 4d $\mathrm{Pin}^{+}(4N)$ SYM theory we gauge $D_{3}^{(0)}$
and $D_{2}^{C,m}$, so the non-invertible surface defects are realised as
NS5-branes on ${\mathbb{R}\mathbb{P}}^{4}\times\mathcal{M}^{2}$. The
worldvolume theory on the NS5 is just as on the D5, but the gauge fields
couple to the S-dual supergravity fields. We can therefore write down the
answer immediately from (41):

$\begin{split}S_{TFT}^{\mathrm{Pin}^{+}(4N)}=i\pi\int_{\mathcal{M}^{2}}\gamma_{1}\delta\phi_{0}+\phi_{0}c_{2}+\gamma_{1}a_{1}\,.\end{split}$
(42)

###### Acknowledgements.

I thank Saghar Hosseini for related discussions, and Michele Del Zotto, Ben
Heidenreich and Sakura Schäfer-Nameki for initial collaboration and
discussions. This work is supported by the Simons Foundation via the Simons
Collaboration on Global Categorical Symmetries, and by the STFC consolidated
grant ST/T000708/1. I would also like to thank the Perimeter Institute, where
this work was initiated during the 2022 workshop on Global Categorical
Symmetries. Research at Perimeter Institute is supported in part by the
Government of Canada through the Department of Innovation, Science and
Economic Development Canada and by the Province of Ontario through the
Ministry of Colleges and Universities.

## References

  * (1) D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, _Generalized Global Symmetries_ , _JHEP_ 02 (2015) 172, [1412.5148].
  * (2) E. Sharpe, _Notes on generalized global symmetries in QFT_ , _Fortsch. Phys._ 63 (2015) 659–682, [1508.04770].
  * (3) Y. Tachikawa, _On gauging finite subgroups_ , _SciPost Phys._ 8 (2020) 015, [1712.09542].
  * (4) F. Benini, C. Córdova and P.-S. Hsin, _On 2-Group Global Symmetries and their Anomalies_ , _JHEP_ 03 (2019) 118, [1803.09336].
  * (5) J. Frohlich, J. Fuchs, I. Runkel and C. Schweigert, _Defect lines, dualities, and generalised orbifolds_ , in _16th International Congress on Mathematical Physics_ , 9, 2009.  0909.5013.  DOI.
  * (6) N. Carqueville and I. Runkel, _Orbifold completion of defect bicategories_ , _Quantum Topol._ 7 (2016) 203–279, [1210.6363].
  * (7) I. Brunner, N. Carqueville and D. Plencner, _A quick guide to defect orbifolds_ , _Proc. Symp. Pure Math._ 88 (2014) 231–242, [1310.0062].
  * (8) L. Bhardwaj and Y. Tachikawa, _On finite symmetries and their gauging in two dimensions_ , _JHEP_ 03 (2018) 189, [1704.02330].
  * (9) D. Gaiotto and T. Johnson-Freyd, _Condensations in higher categories_ , 1905.09566.
  * (10) B. Heidenreich, J. McNamara, M. Montero, M. Reece, T. Rudelius and I. Valenzuela, _Non-invertible global symmetries and completeness of the spectrum_ , _JHEP_ 09 (2021) 203, [2104.07036].
  * (11) Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam and S.-H. Shao, _Noninvertible duality defects in 3+1 dimensions_ , _Phys. Rev. D_ 105 (2022) 125016, [2111.01139].
  * (12) J. Kaidi, K. Ohmori and Y. Zheng, _Kramers-Wannier-like Duality Defects in (3+1)D Gauge Theories_ , _Phys. Rev. Lett._ 128 (2022) 111601, [2111.01141].
  * (13) K. Roumpedakis, S. Seifnashri and S.-H. Shao, _Higher Gauging and Non-invertible Condensation Defects_ , 2204.02407.
  * (14) L. Bhardwaj, L. Bottini, S. Schafer-Nameki and A. Tiwari, _Non-Invertible Higher-Categorical Symmetries_ , 2204.06564.
  * (15) G. Arias-Tamargo and D. Rodriguez-Gomez, _Non-Invertible Symmetries from Discrete Gauging and Completeness of the Spectrum_ , 2204.07523.
  * (16) Y. Choi, C. Cordova, P.-S. Hsin, H. T. Lam and S.-H. Shao, _Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions_ , 2204.09025.
  * (17) Y. Choi, H. T. Lam and S.-H. Shao, _Non-invertible Global Symmetries in the Standard Model_ , 2205.05086.
  * (18) C. Cordova and K. Ohmori, _Non-Invertible Chiral Symmetry and Exponential Hierarchies_ , 2205.06243.
  * (19) J. Kaidi, G. Zafrir and Y. Zheng, _Non-invertible symmetries of $\mathcal{N}$ = 4 SYM and twisted compactification_, _JHEP_ 08 (2022) 053, [2205.01104].
  * (20) A. Antinucci, G. Galati and G. Rizi, _On Continuous 2-Category Symmetries and Yang-Mills Theory_ , 2206.05646.
  * (21) V. Bashmakov, M. Del Zotto and A. Hasan, _On the 6d Origin of Non-invertible Symmetries in 4d_ , 2206.07073.
  * (22) J. A. Damia, R. Argurio and E. Garcia-Valdecasas, _Non-Invertible Defects in 5d, Boundaries and Holography_ , 2207.02831.
  * (23) L. Bhardwaj, S. Schafer-Nameki and J. Wu, _Universal Non-Invertible Symmetries_ , 2208.05973.
  * (24) L. Lin, D. Robbins and E. Sharpe, _Decomposition, condensation defects, and fusion_ , 2208.05982.
  * (25) T. Bartsch, M. Bullimore, A. E. V. Ferrari and J. Pearson, _Non-invertible Symmetries and Higher Representation Theory I_ , 2208.05993.
  * (26) O. Aharony, N. Seiberg and Y. Tachikawa, _Reading between the lines of four-dimensional gauge theories_ , _JHEP_ 08 (2013) 115, [1305.0318].
  * (27) M. Etheredge, I. García Etxebarria, B. Heidenreich and S. Rauch, _To appear_ , .
  * (28) E. Witten, _Baryons and branes in anti-de Sitter space_ , _JHEP_ 07 (1998) 006, [hep-th/9805112].
  * (29) O. Aharony and Y. Tachikawa, _S-folds and 4d N=3 superconformal field theories_ , _JHEP_ 06 (2016) 044, [1602.08638].
  * (30) F. Apruzzi, I. Bah, F. Bonetti and S. Schäfer-Nameki, _To appear_ , .
  * (31) E. Witten, _AdS / CFT correspondence and topological field theory_ , _JHEP_ 12 (1998) 012, [hep-th/9812012].
  * (32) A. Hatcher, _Algebraic topology_.  Cambridge Univ. Press, Cambridge, 2000.
  * (33) R. Thom, _Espaces fibrés en sphères et carrés de steenrod_ , _Annales scientifiques de l’École Normale Supérieure_ 69 (1952) 109–182.
  * (34) G. W. Moore and E. Witten, _Selfduality, Ramond-Ramond fields, and K theory_ , _JHEP_ 05 (2000) 032, [hep-th/9912279].
  * (35) D. S. Freed and M. J. Hopkins, _On Ramond-Ramond fields and K theory_ , _JHEP_ 05 (2000) 044, [hep-th/0002027].
  * (36) D. S. Freed, _Dirac charge quantization and generalized differential cohomology_ , hep-th/0011220.
  * (37) Y.-K. E. Cheung and Z. Yin, _Anomalies, branes, and currents_ , _Nucl. Phys. B_ 517 (1998) 69–91, [hep-th/9710206].
  * (38) R. Minasian and G. W. Moore, _K theory and Ramond-Ramond charge_ , _JHEP_ 11 (1997) 002, [hep-th/9710230].
  * (39) D. S. Freed and C. Teleman, _Relative quantum field theory_ , _Commun. Math. Phys._ 326 (2014) 459–476, [1212.1692].
  * (40) S. Gukov, M. Rangamani and E. Witten, _Dibaryons, strings and branes in AdS orbifold models_ , _JHEP_ 12 (1998) 025, [hep-th/9811048].
  * (41) G. W. Moore, _Anomalies, Gauss laws, and Page charges in M-theory_ , _Comptes Rendus Physique_ 6 (2005) 251–259, [hep-th/0409158].
  * (42) D. S. Freed, G. W. Moore and G. Segal, _The Uncertainty of Fluxes_ , _Commun. Math. Phys._ 271 (2007) 247–274, [hep-th/0605198].
  * (43) D. S. Freed, G. W. Moore and G. Segal, _Heisenberg Groups and Noncommutative Fluxes_ , _Annals Phys._ 322 (2007) 236–285, [hep-th/0605200].
  * (44) F. Apruzzi, F. Bonetti, I. García Etxebarria, S. S. Hosseini and S. Schäfer-Nameki, _Symmetry TFTs from String Theory_ , 2112.02092.
  * (45) M. Čadek, _The cohomology of $BO(n)$ with twisted integer coefficients_, _Journal of Mathematics of Kyoto University_ 39 (1999) 277 – 286.
  * (46) I. García Etxebarria, B. Heidenreich and D. Regalado, _IIB flux non-commutativity and the global structure of field theories_ , _JHEP_ 10 (2019) 169, [1908.08027].
  * (47) W. Taylor, _D2-branes in B fields_ , _JHEP_ 07 (2000) 039, [hep-th/0004141].
  * (48) A. Alekseev, A. Mironov and A. Morozov, _On B independence of RR charges_ , _Phys. Lett. B_ 532 (2002) 350–356, [hep-th/0005244].
  * (49) T. Kimura and I. Oda, _Duality of superD-brane actions in general Type II supergravity background_ , _Int. J. Mod. Phys. A_ 16 (2001) 503–528, [hep-th/9904019].
  * (50) P. G. Camara, L. E. Ibanez and F. Marchesano, _RR photons_ , _JHEP_ 09 (2011) 110, [1106.0060].
  * (51) I. García Etxebarria and S. Hosseini, _To appear_ , .
  * (52) T. Banks and N. Seiberg, _Symmetries and Strings in Field Theory and Gravity_ , _Phys. Rev. D_ 83 (2011) 084019, [1011.5120].