[7]G^ #1,#2_#3,#4(#5 #6| #7) aainstitutetext: William H. Miller III Department
of Physics and Astronomy, Johns Hopkins University, 3400 North Charles Street,
Baltimore, MD 21218, U.S.A.

# Non-Invertible Symmetries, Brane Dynamics, and Tachyon Condensation

Ibrahima Bah a Enoch Leung a and Thomas Waddleton<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the Symmetry Topological Field Theory in holography associated with
4d $\mathcal{N}=1$ Super Yang-Mills theory with gauge algebra
$\mathfrak{su}(M)$. From this, all the bulk symmetry operators are computed
and matched to various D-brane configurations. The fusion algebra of the
operators emerges from brane dynamics. In particular, we show that the
symmetry operators are purely determined from the center-of-mass modes of the
branes. We identify the TQFT fusion coefficients with the relative motion of
the branes. We also establish the origin of condensation defects, arising from
fusion of non-invertible operators, as the consequence of tachyon condensation
in brane-anti-brane pairs.

##  1 Introduction

In recent times, the study of Quantum Field Theory (QFT) has localized to its
surprisingly rich kinematic sector. This follows from the novel perspective,
due to Gaiotto:2014kfa , that such aspects of QFT can be characterized by
various topological structures acting on them. It has led to several
generalizations of the notion of symmetry such as higher-form symmetries,
higher groups, non-invertible symmetries and inevitably to categorical
symmetries. In all of these cases there continues to be rapid development on
the nature of topological symmetries, and the many ways they provide a
fundamental scaffolding of QFT. A priori these emerging symmetries seem
exotic, however in the last two years they have been discovered in many
“bread-and-butter” QFTs such as Yang-Mills theories in four dimensions, and
even for the Standard Model of particle physics.

Two central themes of research now are to understand the scope of topological
symmetries, broadly, and the mathematical framework needed to characterize
them. It has come to focus that a useful way to describe topological
symmetries associated to a QFT in a $d$-dimensional space, $M_{d}$, is to
consider a topological field theory on a $(d+1)$-dimensional space $X_{d+1}$
with $\partial X_{d+1}=M_{d}$ Gaiotto:2014kfa ; Gaiotto:2020iye ;
Freed:2022qnc . Such TQFTs are referred to as Symmetry Topological Field
Theories (SymTFTs) Apruzzi:2021nmk ; they naturally emerge via anomaly inflow
methods for QFTs engineered from string theory and holography Bah:2019rgq ;
Bah:2020uev ; Bah:2020jas ; Apruzzi:2021nmk . From this point of view, string
theory provides an important avenue for studying topological symmetries and
their role in the dynamics of QFTs.

An important result appeared recently in the context of non-invertible
symmetries Apruzzi:2022rei ; GarciaEtxebarria:2022vzq . In these works, it was
demonstrated how to realize them in string theory and holography from
supergravity and brane dynamics. Such developments seeded many other
interesting results where non-invertible symmetries are constructed and
analyzed from various D-brane setups Apruzzi:2022rei ;
GarciaEtxebarria:2022vzq ; Heckman:2022muc ; Heckman:2022xgu ;
Etheredge:2023ler ; Apruzzi:2023uma ; Cvetic:2023plv ; Dierigl:2023jdp ;
Bashmakov:2022uek ; Bashmakov:2022jtl .

In this paper, we will explore the SymTFT associated to 4d $\mathcal{N}=1$
Super Yang-Mills (SYM) theory with gauge algebra $\mathfrak{su}(M)$. This QFT
emerges in the IR limit of a stack of $N$ $D3$ branes probing the conifold in
IIB string theory, with $M$ units of $G_{3}$ flux where $M$ is a divisor of
$N$. This is famously described by the Klebanov-Tseytlin Klebanov:2000nc  and
Klebanov-Strassler Klebanov:2000hb  (KT-KS) backgrounds in IIB supergravity;
these are holographically dual to cascading field theories. A relevant review
for these backgrounds is provided in the Appendix A. Our main interest is to
obtain the full SymTFT as initiated in Apruzzi:2022rei , and then construct
all the symmetry operators and their fusion rules. We will identify all the
brane configurations that are associated to the operators, and then
characterize their fusion rules from brane dynamics. Another objective will be
to sharply understand what data captures the topological operators and their
associated physics derived from the SymTFT.

We start with a brief discussion of our main results. First we complete the
derivation of the SymTFT of the KT-KS background discussed in Apruzzi:2022rei
. Here we identify all the couplings relevant to the TQFT and its associated
discrete gauge symmetries. In particular we emphasize how continuum fields in
supergravity can be used to construct and describe the discrete gauge fields
in a TQFT. By using the (classical) Gauss Law constraints, we derive all the
topological operators which generate a bulk gauge symmetry from the SymTFT
Apruzzi:2022rei ; Belov:2004ht . We then match each symmetry generator with a
specific D$p$-brane configuration by considering the reduction of the Wess-
Zumino action for the branes. The details are the focus of Sections 2, 4, and
5, but a quick summary can be found in Table 1.

Symmetry Action | Symmetry Generator | Brane Configuration
---|---|---
$\mathbb{Z}_{2M}^{(0)}$ | $\mathcal{N}_{1}(M^{3})$ | D5 wrapping $S^{3}\subset T^{1,1}$
$\widehat{\mathbb{Z}}_{2M}^{(2)}$ | $\mathcal{V}_{1}(\gamma^{1})$ | D1 wrapping Hopf fiber $S^{1}\subset S^{3}$
$\mathbb{Z}_{M}^{(1)}$ | $\mathcal{O}_{1}(W^{2})$ | D3-D1 bound state wrapping $S^{2}\subset T^{1,1}$
$\widehat{\mathbb{Z}}_{M}^{(1)}$ | $\mathcal{U}_{1}(\Sigma^{2})$ | D3 wrapping Hopf base $\widetilde{S^{2}}\subset S^{3}$
Table 1:  Symmetry generators/defects and their D-brane realizations.222We
assume a Euclidean spacetime throughout this paper, and use the terminology
“generator”, “operator”, and “defect” interchangeably.

Once we have the symmetry generators, the next question is to understand their
fusion algebra. When considering the parallel fusion of two symmetry
generators, the corresponding D$p$-branes become coincident and are subject to
non-Abelian dynamics involving the worldvolume gauge fields. In this work we
demonstrate that the non-Abelian effects are excluded from the topological
data of the symmetry defects; only the Abelian modes in the D$p$-brane
worldvolume theory participate in the fusion. It is kinematical information,
rather than dynamical, captured in the fusion of the corresponding operators.
Specifically, the center-of-mass mode for a given stack of branes yields the
symmetry operator. For two stacks brought together, the resulting operator is
associated to the total center-of-mass mode. The modes describing any relative
motion between the two stacks decouple from the defects but survive as
nontrivial fusion coefficients in the form of decoupled TQFTs.

Another interesting feature we point out, elaborated upon in the main text, is
the non-invertibility of operators can be attributed to induced brane charges
due to bulk-brane dynamics. For example, the operator $\mathcal{O}_{1}(W^{2})$
presented in Table 1 takes the explicit form,

$\mathcal{O}_{1}(W^{2})=\int\mathcal{D}\lambda_{1}\mathcal{D}\varphi\exp\left(2\pi
i\int_{W^{2}}\hat{c}_{2}+\lambda_{1}\mathcal{A}_{1}+\varphi\mathcal{B}_{2}-M\varphi
d\lambda_{1}\right)\,,$ (1.1)

where $\lambda_{1}$ and $\varphi$ are modes localized on the D3-D1 bound
state. The first term in this action encodes all the information contained in
the D3- and D1-branes themselves. The second and third terms describe induced
charges present on the D3-D1 worldvolume, while the final term is a standard
BF-term describing a $\mathbb{Z}_{M}\times\mathbb{Z}_{M}$ gauge theory. Figure
3 provides a picture to keep in mind during this discussion. As will be seen
later in the paper, the induced charges are associated with D1-branes and
F1-strings in the worldvolume.

Brane Fusion | Defect Fusion
---|---
D5 $\otimes$ D5 | $\mathcal{N}_{p}(M^{3})\otimes\mathcal{N}_{p^{\prime}}(M^{3})=\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}\mathcal{N}_{p+p^{\prime}}(M^{3})$
D5 $\otimes$ $\overline{{\rm D}5}$ | $\mathcal{N}_{p}(M^{3})\otimes\mathcal{N}_{-p}(M^{3})=\mathcal{C}(M^{3})$
D3 $\otimes$ D3 | $\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{U}_{q^{\prime}}(\Sigma^{2})=\mathcal{U}_{q+q^{\prime}}(\Sigma^{2})$
D3 $\otimes$ D5 | $\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{N}_{p}(M^{3})=\mathcal{N}_{p}(M^{3})$
D3 $\otimes$ ( $\sum$ D3) | $\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{C}(M^{3})=\mathcal{C}(M^{3})$
D5 $\otimes$ ( $\sum$ D3) | $\mathcal{N}_{p}(M^{3})\otimes\mathcal{C}(M^{3})=(\mathcal{Z}_{M})_{0}\,\mathcal{N}_{p}(M^{3})$
( $\sum$ D3) $\otimes$ ( $\sum$ D3) | $\mathcal{C}(M^{3})\otimes\mathcal{C}(M^{3})=(\mathcal{Z}_{M})_{0}\,\mathcal{C}(M^{3})$
(D3-D1) $\otimes$ (D3-D1) | $\mathcal{O}_{q}(W^{2})\otimes\mathcal{O}_{q^{\prime}}(W^{2})=\mathcal{X}^{M,qq^{\prime}(q+q^{\prime})^{-1}}\mathcal{O}_{q+q^{\prime}}(W^{2})$
(D3-D1) $\otimes$ $(\overline{{\rm D}3\text{-}{\rm D}1})$ | $\mathcal{O}_{q}(W^{2})\otimes\mathcal{O}_{-q}(W^{2})=\widetilde{\mathcal{C}}(W^{2})$
D1 $\otimes$ D1 | $\mathcal{V}_{p}(\gamma^{1})\otimes\mathcal{V}_{p^{\prime}}(\gamma^{1})=\mathcal{V}_{p+p^{\prime}}(\gamma^{1})$
D1 $\otimes$ (D3-D1) | $\mathcal{V}_{p}(\gamma^{1})\otimes\mathcal{O}_{q}(W^{2})=\mathcal{O}_{q}(W^{2})$
D3 $\otimes$ (D3-D1) | $\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{O}_{q^{\prime}}(W^{2})=\mathcal{O}_{q^{\prime}}(W^{2})$
D1 $\otimes$ ( $\sum$ (D1-F1)) | $\mathcal{V}_{p}(\gamma^{1})\otimes\widetilde{\mathcal{C}}(W^{2})=\widetilde{\mathcal{C}}(W^{2})$
D3 $\otimes$ ( $\sum$ (D1-F1)) | $\mathcal{U}_{q}(\Sigma^{2})\otimes\widetilde{\mathcal{C}}(W^{2})=\widetilde{\mathcal{C}}(W^{2})$
(D3-D1) $\otimes$ ( $\sum$ (D1-F1)) | $\mathcal{O}_{q}(W^{2})\otimes\widetilde{\mathcal{C}}(W^{2})=\mathcal{X}^{M,-1}\mathcal{X}^{M,-1}\mathcal{O}_{q}(W^{2})$
( $\sum$ (D1-F1)) $\otimes$ ( $\sum$ (D1-F1)) | $\widetilde{\mathcal{C}}(W^{2})\otimes\widetilde{\mathcal{C}}(W^{2})=\mathcal{X}^{M,-1}\mathcal{X}^{M,-1}\widetilde{\mathcal{C}}(W^{2})$
Table 2:  Fusion rules for the symmetry defects. The left column indicates the
schematic fusion in terms of the associated D$p$-branes, while the right
column shows the explicit fusion rules. The coefficients
$\mathcal{A}^{M,p},\mathcal{X}^{M,q}$, and $(\mathcal{Z}_{M})_{0}$ denote
decoupled TQFTs that are elaborated upon in the main text.

In addition to the above fusion, we study further the condensation defect
present in many recent examples such as GarciaEtxebarria:2022vzq ;
Apruzzi:2022rei ; Choi:2022fgx ; Choi:2022jqy ; Kaidi:2021xfk ;
Heckman:2022muc . We find that these defects are described naturally in string
theory via tachyon condensation. When a D$p$\- and a $\overline{{\rm
D}p}$-brane are brought together, these branes will not completely annihilate
but leave behind a solitonic D$(p-2)$-brane in the original worldvolume,
determined by a choice of tachyon profile. By summing over all possible
solitonic D$(p-2)$-branes on the original worldvolume for a specific tachyon
profile, we can reproduce exactly the condensation defect that arise when
considering the operator fusion. There is a parallel discussion here as in the
other cases of fusion; we can see the condensate being given by the center-of-
mass mode for the original brane-anti-brane worldvolume. Including the
appropriate condensates, we work out the full fusion algebra for the symmetry
defects in the KT-KS background. The full list of the bulk fusion rules can be
found in Table 2. We also discuss the realization of these fusion rules in the
boundary field theory in terms of allowed boundary conditions on the bulk
supergravity fields.

This paper is organized as follows. In Section 2, we construct the SymTFT for
the KT-KS background, and use the associated Gauss Law constraints to derive
the gauge symmetry generators in this setup. In Section 3, we switch gears and
discuss how symmetry generators in a generic string theory background can also
be realized using D-branes. In Sections 4 and 5, we study in detail the fusion
rules for the $(\mathcal{N},\mathcal{U})$ and $(\mathcal{O},\mathcal{V})$
defect pairs respectively. These two pairs of defects share a strikingly
similar fusion algebra, e.g. the former undergoes non-invertible fusion to
produce a condensation defect containing the latter. We offer an
interpretation of these fusion rules in terms of the underlying brane
kinematics and dynamics. In Section 6, we derive the mixed fusion rules
between all the symmetry defects, as well as their braiding relations. In
Section 7, we analyze the role of the choice of boundary conditions in
realizing the bulk gauge symmetry generators as global symmetry generators in
the boundary field theory. Last but not least, we follow up by a discussion in
Section 8.

##  2 Bulk Supergravity Gauge Symmetries

###  2.1 Symmetry Topological Field Theory

In holographic constructions, one can use anomaly inflow methods to obtain a
Symmetry Topological Field Theory (SymTFT) that encodes the symmetry data of a
lower-dimensional QFT (e.g. Bah:2018gwc ; Bah:2019rgq ; Bah:2020jas ;
Bah:2020uev ; Apruzzi:2021nmk ). In the case of supergravity solutions in the
low energy limit of Type IIB string theory, the SymTFT can be obtained by
starting with the anomaly polynomial of the ten-dimensional supergravity
theory and dimensionally reducing over the internal space. This anomaly
polynomial was derived in Bah:2020jas . The resulting topological theory, once
augmented with kinetic terms for the relevant fields, is the SymTFT of the
boundary field theory.

In the present work, we expand upon the SymTFT of the KT-KS solution as
derived in Apruzzi:2022rei , and particularly study the effect of additional
terms that were previously neglected. A review of the KT-KS solution can be
found in Appendix A. We first start with the 11-form anomaly polynomial in
Type IIB string theory,333Unless otherwise specified, we choose the
normalization of all field strengths appearing in this work to be such that
they have integral periods, and we also work in dimensionless units, i.e.
$2\pi\alpha^{\prime}=1$ and $g_{\text{s}}=1$, to avoid cluttering of notation.
For the same reason, we will often suppress the wedge/cup products between
differential forms/cochains.

$\mathcal{I}_{11}=\frac{1}{2}\,\mathcal{G}_{5}\wedge
d\mathcal{G}_{5}+\mathcal{G}_{5}\wedge H_{3}\wedge G_{3}\,,$ (2.1)

where $G_{3}$ and $H_{3}$ are the field strengths for the RR 2-form and NSNS
2-form potentials respectively, and $\mathcal{G}_{5}$ is defined such that
$G_{5}=(1+\star_{10})\mathcal{G}_{5}$ is the self-dual RR 5-form. To
dimensionally reduce the above polynomial, we perform an expansion of the
supergravity field strengths onto the $U(1)_{R}$ -equivariant cohomology
classes defined in (A.2) and (A.3), namely,

$H_{3}=h_{3},\qquad
G_{3}=M\omega_{3}+g_{1}\omega_{2}+g_{3}\,,\qquad\mathcal{G}_{5}=N\Omega_{5}+g_{2}\omega_{3}\,.$
(2.2)

The fluctuating fields $h_{3},g_{1},g_{2},$ and $g_{3}$ are defined on the
external space $W^{5}$, while the coefficients $M$ and $N$ of $\omega_{3}$ and
$\Omega_{5}$ reflect the nontrivial fluxes threading $S^{3}$ and $T^{1,1}$
respectively in the KT-KS setup.

Consistency of the IIB Bianchi identities for the RR field strengths, which in
this background take the form $dH_{3}=0,dG_{3}=0$, and $dG_{5}=-H_{3}\wedge
G_{3}$, impose constraints on the external field strengths written above.
These extra constraints reduce to the 5d Bianchi identities

$dh_{3}=dg_{3}=d\mathsf{F}_{2}=0\,,\qquad dg_{1}=2M\mathsf{F}_{2},\qquad
dg_{2}=-Mh_{3}\,.$ (2.3)

We may solve the leftmost equations via

$h_{3}=db_{2}-\mathcal{H}_{3}\,,\qquad
g_{3}=dc_{2}+\mathcal{C}_{3}\,,\qquad\mathsf{F}_{2}=d\mathsf{A}_{1}+\mathcal{F}_{2}\,,$
(2.4)

where $\mathcal{H}_{3},\mathcal{C}_{3},\mathcal{F}_{2}\in
H^{*}(\mathcal{W}^{5};\mathbb{Z})$ and $b_{2},c_{2}$, and $\mathsf{A}_{1}$ are
globally well-defined gauge potentials.444The fact that these are valued in
integer cohomology follows from the flux quantization of the RR field
strengths Freed:2000ta . In the later computations $\mathcal{C}_{3}$ will play
no role, so without any loss of generality we will set it to zero. On the
other hand, the nontrivial Bianchi identities for $g_{1}$ and $g_{2}$ indicate
the presence of discrete gauge fields in the bulk theory. Specifically, these
putative $U(1)$ gauge fields are respectively Higgsed down to
$\mathbb{Z}_{2M}$ and $\mathbb{Z}_{M}$ gauge fields via the Stückelberg
mechanism. As a result, we may solve the Bianchi identities as

$g_{1}=dc_{0}+2M\mathsf{A}_{1}+\mathcal{A}_{1}\,,\qquad
g_{2}=d\beta_{1}-Mb_{2}+\mathcal{B}_{2}\,,$ (2.5)

where $\mathcal{A}_{1}\in H^{1}(\mathcal{W}^{5};\mathbb{Z}_{2M})$ and
$\mathcal{B}_{2}\in H^{2}(\mathcal{W}^{5};\mathbb{Z}_{M})$, while $c_{0}$ and
$\beta_{1}$ are globally well-defined gauge potentials.555More precisely,
$\mathcal{A}_{1}$ and $\mathcal{B}_{2}$ are cohomology class representatives.
The difference between two representatives of the same class is exact, i.e.
$\mathcal{A}_{1}-\mathcal{A}_{1}^{\prime}=d\lambda_{0}$ for some globally
well-defined 0-cochain $\lambda_{0}$ on $\mathcal{W}^{5}$. We see that the
choice of representative is immaterial since $g_{1}$ is invariant under the
redefinitions $\mathcal{A}_{1}\to\mathcal{A}_{1}+d\lambda_{0}$ and $c_{0}\to
c_{0}-\lambda_{0}$. Similarly, $g_{2}$ is invariant under
$\mathcal{B}_{2}\to\mathcal{B}_{2}+d\lambda_{1}$ and
$\beta_{1}\to\beta_{1}-\lambda_{1}$. See Appendix B for a related discussion.
The quantities $g_{1}$ and $g_{2}$, containing both free and torsional
components, carry the same information as differential cochains. See
Hopkins:2002rd ; Freed:2006yc ; Cordova:2019jnf ; Hsieh:2020jpj  for a
collection of reviews.

The coefficients of the cohomology groups in which $\mathcal{A}_{1}$ and
$\mathcal{B}_{2}$ live can be understood as follows. The Bianchi identites in
(2.3) put constraints on the basepoint fluxes $\mathcal{F}_{2}$ and
$\mathcal{H}_{3}$ that we have written above, i.e. $2M\mathcal{F}_{2}$ and
$M\mathcal{H}_{3}$ must vanish identically for the right-hand side of the
relations to be an exact cocycle.666In general one might demand
$2M\mathcal{F}_{2}$ to be simply cohomologically trivial rather than to vanish
identically, but such an exact piece can be removed via a field redefinition
of $\mathsf{A}_{1}$. This requirement implies that $\mathcal{F}_{2}$ and
$\mathcal{H}_{3}$ are each contained in the image of a Bockstein homomorphism.
Specifically, we may express

$\mathcal{F}_{2}=\widetilde{\beta}(\mathcal{A}_{1})\,,\qquad\mathcal{H}_{3}=\widetilde{\beta}^{\prime}(\mathcal{B}_{2})\,.$
(2.6)

In the above $\widetilde{\beta}$ and $\widetilde{\beta}^{\prime}$ are
respectively the Bockstein maps777A brief review of Bockstein homomorphisms
can be found in Appendix B.

$\widetilde{\beta}:H^{*}(\mathcal{W}^{5};\mathbb{Z}_{2M})\to
H^{*+1}(\mathcal{W}^{5};\mathbb{Z})\,,\qquad\widetilde{\beta}^{\prime}:H^{*}(\mathcal{W}^{5};\mathbb{Z}_{M})\to
H^{*+1}(\mathcal{W}^{5};\mathbb{Z})\,.$ (2.7)

With this notation we can now compute the 5d SymTFT by reducing (2.1) over the
conifold $T^{1,1}$. The topological action $S_{\rm top}$ takes the following
form,

$S_{\rm
top}=2\pi\int_{W^{5}}\bigg{(}Mb_{2}d\hat{c}_{2}-2Mc_{3}d\mathsf{A}_{1}+b_{2}\mathcal{A}_{1}\mathcal{B}_{2}-\mathsf{A}_{1}\mathcal{B}_{2}^{2}-\frac{\mathcal{K}}{M}\,\mathcal{A}_{1}\beta(\mathcal{A}_{1})\beta(\mathcal{A}_{1})\bigg{)}\,,$
(2.8)

where $\mathcal{K}$ is defined in (A.5), and we have defined the field
$d\hat{c}_{2}=kdc_{2}+da_{2}$ in terms of a field $a_{2}$ that arises when
computing the SymTFT.888Technically, at the level of cochains, the term
$\mathcal{B}_{2}\cup\mathcal{B}_{2}$ is more approrpiately written as
$\mathcal{P}(\mathcal{B}_{2})$, where $\mathcal{P}$ denotes the Pontryagin
square operation defined in, e.g. Whitehead:1949 ; Aharony:2013hda ;
Benini:2018reh ; Cordova:2019uob . The details of this computation can be
found in Appendix A.

###  2.2 5d Bulk Gauge Symmetries

The topological action $S_{\rm top}$ in (2.8) details the symmetry structure
of the 5d bulk gauge theory, and it acts as the SymTFT for the 4d QFT living
on the boundary. The first two terms of act as BF terms, so for instance, the
fluctuating $U(1)$ gauge fields $b_{2}$ and $\hat{c}_{2}$ in the former term
are Higgsed down to $\mathbb{Z}_{M}$ gauge fields at low energies via the
Stückelberg mechanism Banks:2010zn . As a consequence, the holonomies of
$\mathsf{A}_{1}$ and $b_{2}$ in the IR turn into

$\exp\left(2\pi
i\int_{\gamma^{1}}\mathsf{A}_{1}\right)\xrightarrow{\text{IR}}\exp\left(\frac{2\pi
i}{2M}\int_{\gamma^{1}}\mathcal{A}_{1}\right)\,,\quad\exp\left(2\pi
i\int_{\Sigma^{2}}b_{2}\right)\xrightarrow{\text{IR}}\exp\left(-\frac{2\pi
i}{M}\int_{\Sigma^{2}}\mathcal{B}_{2}\right)\,.$ (2.9)

We shall remark that the $U(1)$ gauge symmetry associated with
$\mathsf{A}_{1}$ comes from the $S^{1}$ isometry circle inside $T^{1,1}$. This
isometry, as was discussed in the introduction, corresponds to the
$R$-symmetry in the 4d $\mathcal{N}=1$ boundary field theory. The Higgsing of
the gauge symmetry down to a $\mathbb{Z}_{2M}$ subgroup is exactly the same as
how the $U(1)$ global $R$-symmetry of the boundary theory is broken down to a
$\mathbb{Z}_{2M}$ subgroup by the ABJ anomaly. Reading off the form degrees of
the gauge fields, the first term in $S_{\rm top}$ corresponds to
$\mathbb{Z}_{M}^{(1)}$ and $\widehat{\mathbb{Z}}_{M}^{(1)}$ gauge symmetries
in the 5d bulk theory, where the hat denotes the Pontryagin-dual group.999Here
the form degree of a symmetry (as labeled by the superscript) is defined with
respect to a 4d “time” slice in the bulk resulting from Hamiltonian
quantization. In this case, a $p$-form gauge field is associated with a
$(3-p)$-form symmetry. Similarly, the latter term corresponds to
$\mathbb{Z}_{2M}^{(0)}$ and $\widehat{\mathbb{Z}}_{2M}^{(2)}$ gauge
symmetries.

The third and fourth terms in $S_{\rm top}$ are interaction terms describing
an interplay between the $\mathbb{Z}_{M}^{(1)}$ and $\mathbb{Z}_{2M}^{(0)}$
gauge symmetries present in the bulk theory.101010The reason for $b_{2}$ being
a gauge field for the $\mathbb{Z}_{M}^{(1)}$ symmetry rather than the
$\widehat{\mathbb{Z}}_{M}^{(1)}$ symmetry will be made clear when we consider
the boundary field theory later in Section 7. The interplay, when considered
from the perspective of the 4d boundary theory, is a mixed ’t Hooft anomaly
between the two symmetries. Similarly, the final term in $S_{\rm top}$ is a
cubic interaction term in the bulk gauge theory, and becomes a ’t Hooft
anomaly purely for the $\mathbb{Z}_{2M}^{(0)}$ symmetry on the boundary.

###  2.3 Gauss Laws and Symmetry Generators

For each of the discrete gauge symmetries present in the SymTFT, we may
construct a symmetry operator that generates the corresponding symmetry. This
can be done by treating the radial direction of the external space
$\mathcal{W}^{5}$ as a time coordinate and performing a Hamiltonian analysis
of the 5d bulk action Witten:1998wy ; Gukov:2004id ; Belov:2004ht . In this
formalism, the time components of the bulk fields have vanishing conjugate
momentum and thus impose (classical) Gauss Law constraints. Each of the fields
$b_{2},\hat{c}_{2},\mathsf{A}_{1},$ and $c_{3}$ will impose a Gauss Law
constraint, which were originally computed in Apruzzi:2022rei  and we restate
them here for convenience,

$\displaystyle\mathcal{G}_{b_{2}}$
$\displaystyle=Md\hat{c}_{2}+\mathcal{A}_{1}\mathcal{B}_{2}\,,$
$\displaystyle\mathcal{G}_{{\sf A}_{1}}$
$\displaystyle=2Mdc_{3}+\mathcal{B}_{2}^{2}\,,$
$\displaystyle\mathcal{G}_{\hat{c}_{2}}$ $\displaystyle=Mdb_{2}\,,$
$\displaystyle\mathcal{G}_{c_{3}}$ $\displaystyle=2Md{\sf A}_{1}\,.$ (2.10)

The quantities $\mathcal{G}_{i}$ may be thought of as $dP_{i}$ where $P_{i}$
is a “Page charge” generating the corresponding gauge transformation. For a
continuous symmetry, e.g. $U(1)$, the operator $\exp\big{(}2\pi
i\int_{M_{i}}P_{i}\big{)}$ acts as a symmetry generator supported on some
submanifold $M_{i}\subset\mathcal{W}^{5}$ of the appropriate dimension. In
contrast, if the symmetry is broken down to a finite group, e.g.
$\mathbb{Z}_{k}$, then the proper generator to consider is the “$k$-th” root
of the aforementioned one, i.e. $\mathcal{D}\sim\exp\big{(}(2\pi
i/k)\int_{M_{i}}P_{i}\big{)}$, such that $\mathcal{D}^{\otimes
k}\cong\mathds{1}$ corresponds to the identity element of the finite group (up
to condensation defects in the case of non-invertible symmetries).

To illustrate this, let us look at the Gauss Law constraint
$\mathcal{G}_{\mathsf{A}_{1}}$. Using the prescription above, one would
naïvely construct an operator of the form,

$\mathcal{N}_{1}(M^{3},M^{4})=\exp\bigg{(}2\pi i\int_{M^{3}}c_{3}+\frac{2\pi
i}{2M}\int_{M^{4}}\mathcal{B}_{2}^{2}\bigg{)}\,.$ (2.11)

This operator is well-defined when placed on a 3-manifold $M^{3}$ that is the
boundary of some 4-manifold $M^{4}$. To define the operator solely on $M^{3}$,
we may rewrite the second term as a 3d action by coupling $\mathcal{B}_{2}$ to
a dynamical $U(1)$ gauge field $a_{1}$ localized on $M^{3}$ Choi:2022jqy ;
Apruzzi:2022rei , i.e.

$\mathcal{N}_{1}(M^{3})=\int\mathcal{D}a_{1}\exp\bigg{(}2\pi
i\int_{M^{3}}c_{3}+\frac{M}{2}a_{1}da_{1}+a_{1}\mathcal{B}_{2}\bigg{)}\,.$
(2.12)

As an aside, one may recognize the latter two terms as an effective action for
a fractional quantum Hall state. We claim that such an operator generates a
$\mathbb{Z}_{2M}^{(0)}$ symmetry, and as we will show in Section 4, stacking
$2M$ copies of (2.12) indeed produces a trivial operator up to a condensation
defect.

It is instructive to point out that the defect $\mathcal{N}_{1}(M^{3})$ can be
schematically decomposed into two pieces, i.e.

$\mathcal{N}_{1}(M^{3})=\exp\left(2\pi
i\int_{M^{3}}c_{3}\right)\otimes\mathcal{A}^{M,1}[\mathcal{B}_{2}]\coloneqq\widetilde{\mathcal{N}}_{1}(M^{3})\otimes\mathcal{A}^{M,1}[\mathcal{B}_{2}]\,,$
(2.13)

where $\widetilde{\mathcal{N}}_{1}(M^{3})$ generates a $\mathbb{Z}_{2M}^{(0)}$
$R$-symmetry rotation, and $\mathcal{A}^{M,1}[\mathcal{B}_{2}]$ is a 3d
minimal TQFT associated with the $\widehat{\mathbb{Z}}_{M}^{(1)}$ symmetry
Hsin:2018vcg . Note that the simultaneous appearance of these two operators in
$\mathcal{N}_{1}(M^{3})$ results from the presence of the anomaly term
$\mathsf{A}_{1}\mathcal{B}_{2}^{2}$ in (2.8). An analogous decomposition of
the chiral symmetry defect can be found in Choi:2022jqy  in the context of 4d
massless QED (see also, e.g. Kaidi:2021xfk ; Cordova:2022ieu  for similar
constructions). There, the defect is similarly composed of a naïve chiral
rotation operator stacked with a minimal TQFT.

Applying the same philosophy, we can construct symmetry defects generating the
remaining symmetries encoded by the SymTFT, namely,
$\widehat{\mathbb{Z}}_{2M}^{(2)},\mathbb{Z}_{M}^{(1)},$ and
$\widehat{\mathbb{Z}}_{M}^{(1)}$, using the Gauss Law constraints
$\mathcal{G}_{c_{3}},\mathcal{G}_{b_{2}},$ and $\mathcal{G}_{\hat{c}_{2}}$.
Explicitly, these defects take the respective forms,

$\displaystyle\mathcal{V}_{1}(\gamma^{1})$ $\displaystyle=\exp\left(\frac{2\pi
i}{2M}\int_{\gamma^{1}}\mathcal{A}_{1}\right)\,,$ (2.14)
$\displaystyle\mathcal{O}_{1}(W^{2},W^{3})$ $\displaystyle=\exp\left(2\pi
i\int_{W^{2}}(kc_{2}+a_{2})+\frac{2\pi
i}{M}\int_{W^{3}}\mathcal{A}_{1}\mathcal{B}_{2}\right)\,,$ (2.15)
$\displaystyle\mathcal{U}_{1}(\Sigma^{2})$ $\displaystyle=\exp\left(\frac{2\pi
i}{M}\int_{\Sigma^{2}}\mathcal{B}_{2}\right)\,.$ (2.16)

When constructing $\mathcal{V}_{1}$ and $\mathcal{U}_{1}$, we have made use of
the discussion surrounding (2.9).

Analogously to the previous case of $\mathcal{N}_{1}$, the operator
$\mathcal{O}_{1}(W^{2},W^{3})$ as written in (2.15) is well-defined when
placed on a 2-manifold $W^{2}$ that is the boundary of some 3-manifold
$W^{3}$. To define the operator solely on $W^{2}$, we may rewrite the last
term in (2.15) as a 2d action by coupling $\mathcal{A}_{1}$ and
$\mathcal{B}_{2}$ respectively to dynamical $U(1)$ gauge fields $\lambda_{1}$
and $\varphi$ localized on $W^{2}$, i.e.

$\mathcal{O}_{1}(W^{2})=\int\mathcal{D}\lambda_{1}\mathcal{D}\varphi\exp\bigg{(}2\pi
i\int_{W^{2}}\hat{c}_{2}+\lambda_{1}\mathcal{A}_{1}+\varphi\mathcal{B}_{2}-M\varphi
d\lambda_{1}\bigg{)}\,.$ (2.17)

For convenience, let us define the shorthand notation,

$\mathcal{X}^{M,q}[\mathcal{A}_{1},\mathcal{B}_{2}]\coloneqq\int\mathcal{D}\lambda_{1}\mathcal{D}\varphi\exp\bigg{(}2\pi
iq\int\lambda_{1}\mathcal{A}_{1}+\varphi\mathcal{B}_{2}-M\lambda_{1}d\varphi\bigg{)}\,,$
(2.18)

assuming $\mathop{\mathrm{gcd}}(M,q)=1$. The anomaly label $q\sim q+M$
specifies the braiding statistics between $\lambda_{1}$ and $\varphi$. Note
that in the absence of the coupling to $\mathcal{A}_{1}$ and
$\mathcal{B}_{2}$, the TQFT $\mathcal{X}^{M,1}$ is an untwisted 2d
$\mathbb{Z}_{M}\times\mathbb{Z}_{M}$ BF theory.111111By construction, the
Wilson-like operators $e^{2\pi i\oint\lambda_{1}}$ and $e^{2\pi i\varphi}$
generate the two $\mathbb{Z}_{M}$ symmetries on $W^{2}$. The cocycles
$\mathcal{A}_{1}$ and $\mathcal{B}_{2}$ are respectively the BF-duals of
$\lambda_{1}\in H^{1}(W^{2};\mathbb{Z}_{M})$ and $\varphi\in
H^{0}(W^{2};\mathbb{Z}_{M})$ Kapustin:2014gua . This can be seen by rewriting
$\mathcal{X}^{M,1}[\mathcal{A}_{1},\mathcal{B}_{2}]$ in terms of continuum
fields $\hat{\mathsf{A}}_{1}=\mathcal{A}_{1}/M$ and $b_{2}=\mathcal{B}_{2}/M$,
and path-integrating over them instead of $\lambda_{1}$ and $\varphi$, which
results in the equations of motion, $Md\lambda_{1}=Md\varphi=0$. Similarly to
the defect $\mathcal{N}_{1}(M^{3})$, we observe that $\mathcal{O}_{1}(W^{2})$
decomposes into

$\mathcal{O}_{1}(W^{2})=\widetilde{\mathcal{O}}_{1}(W^{2})\otimes\mathcal{X}^{M,1}[\mathcal{A}_{1},\mathcal{B}_{2}]\,,$
(2.19)

where $\widetilde{\mathcal{O}}_{1}(W^{2})=\exp\big{(}2\pi
i\int_{W^{2}}\hat{c}_{2}\big{)}$ is an invertible operator generating a
$\mathbb{Z}_{M}^{(1)}$ transformation, and the appearance of the 2d TQFT
$\mathcal{X}^{M,1}[\mathcal{A}_{1},\mathcal{B}_{2}]$ is a consequence of the
interaction term $b_{2}\mathcal{A}_{1}\mathcal{B}_{2}$ in (2.8).

##  3 Symmetries from Branes

###  3.1 Bulk SymTFT vs. Brane Worldvolume Action

We derived earlier the SymTFT for the bulk supergravity theory in the KT-KS
background, from which the Gauss Law constraints give us a complete set of
symmetry generators in the 5d bulk. In this and the subsequent sections, we
show that each of these symmetry defects can be engineered as a certain probe
D-brane configuration in the KT-KS geometry. As alluded to in Section 1,
recently in the literature there have been field-theoretic constructions of
non-invertible symmetry defects with dimension $d\geq 2$. The hallmark of such
defects is two-fold:

  1. 1.

$\text{(Defect A)}\otimes\text{(Defect B)}=\text{(Decoupled TQFT
Coefficient)}\otimes\text{(Defect C)}$,

  2. 2.

$\text{(Defect)}\otimes\text{(Oppositely Oriented Defect)}=\sum\text{(Lower-
dimensional Defect)}$.

The significance of the brane-defect correspondence in our work is that it
allows us to interpret both of these properties purely in terms of brane
kinematics and dynamics in the bulk. In fact, as we will discuss in Section 7,
there exist obstructions to realize certain bulk-symmetry-generating branes
(or combinations of them) on the boundary field theory. The obstructions are
determined by the anomaly terms in the SymTFT and choices of boundary
conditions for the 5d bulk gauge fields. A comprehensive analysis of such
branes therefore realizes a much richer structure of the topological data
encoded by the SymTFT in the bulk, which would otherwise not be appreciated
from the perspective of the boundary field theory.

In Type II string theory, the worldvolume action of a general D$p$-brane
consists of two parts: a Dirac-Born-Infeld (DBI) action encoding the dynamics
of worldvolume gauge fields, and a Wess-Zumino (WZ) action describing the
topological coupling of the D$p$-brane to the bulk RR and NSNS fields. To
construct a topological operator out of a probe D$p$-brane, one can push it to
the conformal boundary of the external spacetime. The dynamical degrees of
freedom are frozen out since the brane tension scales as $T_{p}\sim r^{p}$,
causing the DBI term to decouple. This leaves us with the WZ action (see, e.g.
Polchinski:1995mt ; Polchinski:1996fm ), given by

$S_{\text{WZ}}^{\text{D$p$}}=\mu_{p}\int_{\mathcal{M}^{p+1}}I^{\text{D$p$}}_{p+1}\coloneqq\mu_{p}\int_{\mathcal{M}^{p+1}}\sum_{q\leq
p}C_{q+1}\wedge\text{ch}_{B}(\mathcal{E})\wedge\sqrt{\frac{\hat{\mathcal{A}}(\mathcal{R}_{T})}{\hat{\mathcal{A}}(\mathcal{R}_{N})}}\,\Bigg{|}_{\text{$(p+1)$-form}}\,,$
(3.1)

where $\mu_{p}$ is the charge of a single D$p$-brane, and $\mathcal{E}$ is the
Chan-Paton bundle over the worldvolume $\mathcal{M}^{p+1}$. We sum over even
$q$ in Type IIA and odd $q$ in Type IIB string theory. The quantities
$\mathcal{R}_{T}=2\pi R_{T}$ and $\mathcal{R}_{N}=2\pi R_{N}$ denote
respectively the (normalized) curvature 2-forms of the tangent and normal
bundles of the worldvolume $\mathcal{M}^{p+1}$ of the D$p$-branes, while
$\hat{\mathcal{A}}$ is the $A$-roof genus. In the analysis that follows, we
will omit the gravitational couplings in the WZ action for simplicity.

For the purpose of our work, we would like to highlight two salient properties
of the WZ action. Firstly, a D$p$-brane is not only coupled to the
$(p+1)$-form RR field $C_{p+1}$, but also to the lower RR fields
$C_{p-1},C_{p-3},$ etc. via the (twisted) Chern character
$\text{ch}_{B}(\mathcal{E})=e^{B_{2}+F_{2}}$. Such couplings are physically
interpreted as D$q$-branes ($q<p$) being dissolved in $\mathcal{M}^{p+1}$
Douglas:1995bn . The numbers of these D$q$-branes are determined by the
worldvolume Chan-Paton flux on the D$p$-brane. As a simple example, suppose
$\mathcal{M}^{p+1}=\widetilde{\mathcal{M}}^{2}\times\mathcal{M}^{p-1}$, and
the Chan-Paton field strength decomposes as $F_{2}=\tilde{f}_{2}+f_{2}$ with
$\tilde{f}_{2}\in H^{2}(\widetilde{\mathcal{M}}^{2};\mathbb{Z})$ and $f_{2}\in
H^{2}(\mathcal{M}^{p-1};\mathbb{Z})$. The Chern character factorizes as

$e^{B_{2}+F_{2}}=e^{\tilde{f}_{2}}\wedge e^{B_{2}+f_{2}}=1+\tilde{f}_{2}\wedge
e^{B_{2}+f_{2}}+\cdots\,,$ (3.2)

and so the number, $k$, of D$(p-2)$-branes within the D$p$-brane is given by
$k=\int_{\widetilde{\mathcal{M}}^{2}}\tilde{f}_{2}$.

Let us now consider a single D$p$-brane with zero net D$p$ charge, i.e.
$\int_{\mathcal{M}^{p+1}}C_{p+1}=0$, but with $k$ units of worldvolume flux
through $\widetilde{\mathcal{M}}^{2}$. As far as the center-of-mass mode is
concerned, such an object can also be thought of as a stack of $k$ coincident
D$(p-2)$-branes. To see this, we use the fact that the Chan-Paton bundle
$\mathcal{E}$ on the brane stack has structure group $U(k)$. Suppose we
parametrize the $\mathfrak{u}(k)$-valued field strength
$F_{2}=\mathcal{F}_{2}+f_{2}\mathds{1}_{k}$, with $\mathcal{F}_{2}$ being
$\mathfrak{su}(k)$-valued and $f_{2}\in H^{2}(\mathcal{M}^{p+1};\mathbb{Z})$,
then the Chern character factorizes as

$e^{B_{2}}\,\text{Tr}(e^{F_{2}})=\text{Tr}(e^{\mathcal{F}_{2}})\wedge
e^{B_{2}+f_{2}}=ke^{B_{2}+f_{2}}+\cdots\,.$ (3.3)

We therefore recover the second term in (3.2) with
$k=\int_{\widetilde{\mathcal{M}}^{2}}\tilde{f}_{2}$. Note that the gauge field
$f_{2}$ here is associated with the center-of-mass mode of the stack of
D$(p-2)$-branes, while $\mathcal{F}_{2}$ is associated with the remaining
relative Chan-Paton modes. The relation between these two perspectives will
prove useful later as we study the fusion between D$p$-branes.

###  3.2 D-brane Anomaly Polynomial

As for any Chern-Simons action, we can naturally construct an anomaly
polynomial associated with the D$p$-brane via descent. It can be defined
locally as121212Formally, $\mathcal{I}^{\text{D$p$}}_{p+2}$ is defined on a
manifold $\mathcal{M}^{p+2}$ such that
$\partial\mathcal{M}^{p+2}=\mathcal{M}^{p+1}$ is the worldvolume of the
D$p$-brane. We also assume the gauge fields originally defined on
$\mathcal{M}^{p+1}$ can be extended to $\mathcal{M}^{p+2}$.

$\mathcal{I}^{\text{D$p$}}_{p+2}=dI^{\text{D$p$}}_{p+1}=\sum_{q\leq
p}G_{q+2}\,e^{B_{2}+F_{2}}\,.$ (3.4)

The RR and NSNS field strengths locally take the forms,

$G_{q+2}=dC_{q+1}+H_{3}C_{q-1}\,,\qquad H_{3}=dB_{2}\,.$ (3.5)

In the presence of backgroud D-branes, or when $[H_{3}]\in
H^{3}(\mathcal{M}^{p+1};\mathbb{Z})$ is nontrivial Freed:1999vc ;
Kapustin:1999di , the RR field strengths $G_{q+2}$ can be topologically
nontrivial. Globally, they should be defined by the Bianchi
identities,131313Equivalently, the Bianchi identities for the various RR
fields $G_{q+2}$ as differential forms can be collected and rewritten in a
more suggestive way, $(d+H)G=0$, which is one of the supporting arguments for
the claim that D-brane charges are formally classified by twisted K-theory
Minasian:1997mm ; Witten:1998cd .

$dG_{q+2}=-H_{3}G_{q}\,,$ (3.6)

in addition to the requirement that they are equivariant under all additional
gauge symmetries present in the system (e.g. in our setup we gauged the
$R$-symmetry circle in $T^{1,1}$). Throughout the rest of this work, we take
the RHS of (3.4) as the definition of the D-brane anomaly polynomial, subject
to $G_{q+2}$ satisfying the Bianchi identities (3.6). Specifically, we assume
the ansatzes for the RR fields as defined in (2.2) to account for global
topological data in the KS setup.

#### Bulk symmetry generators from wrapped D-branes.

Recall that the 5d topological action $\eqref{eq:SymTFT}$ of the SymTFT is
derived by reducing $\mathcal{I}_{11}$ of Type IIB string theory on the
conifold base $T^{1,1}$ of the KT-KS background. Similarly, one can reduce
$\mathcal{I}^{\text{D$p$}}_{p+2}$ of a D$p$-brane on some $n$-manifold
$X^{n}\subset T^{1,1}$, resulting in an effective action for a topological
defect defined on a $(p+1-n)$-submanifold $M^{p+1-n}$ of the 5d bulk
$\mathcal{W}^{5}$.

We claim that each of the four symmetry generators, $\mathcal{N}_{1}(M^{3})$,
$\mathcal{V}_{1}(\gamma^{1})$, $\mathcal{O}(W^{2})$, and
$\mathcal{U}_{1}(\Sigma^{2})$, derived in Section 2 with a Gauss Law
constraint admits a D-brane origin. The first symmetry generator we are
interested in is
$\mathcal{N}_{1}=\widetilde{\mathcal{N}}_{1}\otimes\mathcal{A}^{M,1}[\mathcal{B}_{2}]$
which generates the $\mathbb{Z}_{2M}^{(0)}$ gauge symmetry. From the bulk
string theory setup, this defect can be realized by reducing a D5-brane on
$S^{3}\subset T^{1,1}$ Apruzzi:2022rei . With this interpretation, the
dynamical gauge field $a_{1}$ in $\mathcal{A}^{M,1}[\mathcal{B}_{2}]$ can be
understood as the Chan-Paton field living on the D5-brane. These localized
degrees of freedom thereby naturally encode the stacking of a minimal TQFT in
order to construct the symmetry defect.

##  4 $\mathbb{Z}_{2M}^{(0)}$ and $\widehat{\mathbb{Z}}_{M}^{(1)}$ Defect
Fusion

###  4.1 $\mathcal{N}_{p}\otimes\mathcal{N}_{p^{\prime}}$ Fusion and
Dielectric Branes

#### Defect fusion.

Let us study the parallel fusion between two copies of $\mathcal{N}_{1}$
defined on the same 3-manifold $M^{3}$. Here we would like to neglect the
D5-brane origin of $\mathcal{N}_{1}$ for a moment and treat them solely as
topological defects acting on the Hilbert space of the theory. It is
straightforward to deduce that $\widetilde{\mathcal{N}}_{1}$ fuse invertibly
as
$\widetilde{\mathcal{N}}_{1}\otimes\widetilde{\mathcal{N}}_{1}=\widetilde{\mathcal{N}}_{2}=\exp\big{(}4\pi
i\int_{M^{3}}c_{3}\big{)}$.141414To avoid notational clutter, we hereafter
often omit the dependency of the operators on manifolds. On the other hand,
the minimal TQFT $\mathcal{A}^{M,1}[\mathcal{B}_{2}]$ satisfies the fusion
rule Choi:2022jqy ,

$\mathcal{A}^{M,1}[\mathcal{B}_{2}]\otimes\mathcal{A}^{M,1}[\mathcal{B}_{2}]=\mathcal{A}^{M,2}\otimes\mathcal{A}^{M,2}[\mathcal{B}_{2}]\,,$
(4.1)

assuming $\mathop{\mathrm{gcd}}(M,2)=1$, i.e. $M$ is odd. Note that the first
copy of $\mathcal{A}^{M,2}$ is a decoupled TQFT Roumpedakis:2022aik , while
the second one couples to the $\mathbb{Z}_{M}$ gauge field $\mathcal{B}_{2}$.
Combining the results, we conclude that the fusion of two $R$-symmetry defects
is given by

$\mathcal{N}_{1}\otimes\mathcal{N}_{1}=\widetilde{\mathcal{N}}_{2}\otimes\mathcal{A}^{M,2}\otimes\mathcal{A}^{M,2}[\mathcal{B}_{2}]\coloneqq\mathcal{A}^{M,2}\otimes\mathcal{N}_{2}\,.$
(4.2)

In general, one may construct an operator $\mathcal{N}_{p}$ analogously to how
we defined $\mathcal{N}_{2}$ above, i.e. we take the naïve operator
$\widetilde{\mathcal{N}}_{p}$ and stack it with a minimal TQFT
$\mathcal{A}^{M,p}[\mathcal{B}_{2}]$. The latter has an anomaly label $p\sim
p+M$ dictating the braiding statistics of the anyons living on $M^{3}$. This
implies that $\mathcal{A}^{M,M}\cong\mathcal{A}^{M,0}$ is a trivial theory, so
the operator $\mathcal{N}_{M}\cong\widetilde{\mathcal{N}}_{M}$ reduces to an
invertible operator when we take $p=M$. The operator $\mathcal{N}_{M}$
therefore generates the anomaly-free $\mathbb{Z}_{2}^{(0)}$ subgroup of the
$\mathbb{Z}_{2M}^{(0)}$ $R$-symmetry that is associated with the $(-1)^{F}$
operator Klebanov:2000hb ; Klebanov:2002gr . It is important, however, to note
that $\widetilde{\mathcal{N}}_{M}$ has a periodicity $p\sim p+2M$, and the
same applies to the full operator $\mathcal{N}_{p}$.

We show in Appendix C that if
$\mathop{\mathrm{gcd}}(M,p)=\mathop{\mathrm{gcd}}(M,p^{\prime})=\mathop{\mathrm{gcd}}(M,p+p^{\prime})=1$,151515This
condition can be satisfied for all possible values of
$p,p^{\prime}=0,\dots,M-1$ if $M>2$ and is prime. the fusion between general
$R$-symmetry defects follows

$\mathcal{N}_{p}\otimes\mathcal{N}_{p^{\prime}}=\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}\otimes\mathcal{N}_{p+p^{\prime}}\,,$
(4.3)

which reduces to (4.2) when $p=p^{\prime}=1$. It should be emphasized again
that the factor $\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}$ does not couple to
$\mathcal{B}_{2}$. Since $\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}$ is
symmetric in the labels $p$ and $p^{\prime}$, the fusion rule (4.3) is
manifestly commutative.

#### Dielectric branes.

The above field-theoretic fusion for the $\mathcal{N}_{p}$ operators can
alternatively be understood geometrically through the dynamics of probe
D$p$-branes in the dual string theory. The
$\mathcal{N}_{1}\otimes\mathcal{N}_{1}$ fusion is modeled by two coincident
D5-branes on the $M^{3}\times S^{3}$ needed to generate the
$\mathbb{Z}_{2M}^{(0)}$ $R$-symmetry. As the two branes are brought together,
the worldvolume theory will undergo an enhancement of the gauge group from
$U(1)$ to $U(2)$ as can be seen from the new stringy modes stretching between
the two D5-branes. This enhancement will play a significant role in the
behavior of the stack of D5-branes.

It is known in the string theory literature that, in the presence of a
nontrivial $G_{p+4}$ or $B_{2}$ background, $k$ coincident D$p$-branes will
polarize and “puff up” into an $S^{2}$ transverse to the original stack
Myers:1999ps ; Polchinski:2000uf ; Apruzzi:2019ecr ; Apruzzi:2022rei , known
as the Myers Effect. The resulting configuration is equivalent to that of a
D$(p+2)$-brane with vanishing D$(p+2)$-brane charge and $k$ units of
D$p$-brane charge. In the case at hand, when two D5-branes wrapping
$M^{3}\times S^{3}$ are brought close to one another, the configuration
becomes indistinguishable from a single D7-brane wrapping $M^{3}\times
T^{1,1}$ with two units of worldvolume flux through the $S^{2}\subset
T^{1,1}$.

There is a subtlety arising from the description of two D5-branes polarizing
into a D7-brane. To illustrate the issue, let us reduce a D7-brane on
$T^{1,1}$, with two units of worldvolume flux threading the $S^{2}$ orthogonal
to the $S^{3}\subset T^{1,1}$ on which the D5-branes are wrapped. The
reduction leads to

$\exp\bigg{(}2\pi i\int_{M^{4}\times
T^{1,1}}\mathcal{I}^{\text{D7}}_{9}\bigg{)}=\exp\bigg{(}4\pi
i\int_{M^{3}}c_{3}+\frac{M}{2}a_{1}da_{1}+a_{1}\mathcal{B}_{2}\bigg{)}=\mathcal{N}_{2}(M^{3})\,,$
(4.4)

where $\partial M^{4}=M^{3}$. Note that here $a_{1}$ is the $U(1)$ Chan-Paton
field living on the D7-brane. A quick comparison with (4.2) reveals the issue:
the decoupled TQFT $\mathcal{A}^{M,2}$ seems to be absent from the naïve
computation above.

To recover this missing information, we can simply model each D5-brane as a
“neutral” D7-brane with one unit of worldvolume flux (see the discussion
following (3.3)). Specifically, the anomaly polynomial of each of these
neutral D7-branes can be expanded as

$\mathcal{I}_{9,i}^{\rm
D7}=G_{7}(B_{2}+da_{1}^{i}+f_{2}^{i})+\frac{1}{2}\,G_{5}(B_{2}+da_{1}^{i}+f_{2}^{i})^{2}+\frac{1}{6}\,G_{3}(B_{2}+da_{1}^{i}+f_{2}^{i})^{3}\,,$
(4.5)

where $a_{1}^{i}$ denotes the Chan-Paton gauge field along $S^{3}$, and
$f_{2}^{i}$ represents the worldvolume flux such that
$\int_{S^{2}}f_{2}^{i}=1$. We can then perform the reduction over $T^{1,1}$
using (2.2) to find

$\displaystyle\int_{M^{4}\times T^{1,1}}\mathcal{I}_{9,1}^{\rm
D7}+\mathcal{I}_{9,2}^{\rm D7}$
$\displaystyle=\int_{M^{4}}2dc_{3}+\frac{M}{2}\left(da_{1}^{1}da_{1}^{1}+da_{1}^{2}da_{1}^{2}\right)+d(a_{1}^{1}+a_{1}^{2})\mathcal{B}_{2}$
$\displaystyle=\int_{M^{3}}\frac{2^{-1}M}{2}\,\alpha_{1}^{\prime}d\alpha_{1}^{\prime}+2\left(c_{3}+\frac{M}{2}\,\alpha_{1}d\alpha_{1}+\alpha_{1}\mathcal{B}_{2}\right)\,,$
(4.6)

where we have made the field redefinitions $2\alpha_{1}=a_{1}^{1}+a_{2}^{1}$
and $\alpha_{1}^{\prime}=-a_{1}^{1}+a_{1}^{2}$. Not only have we recovered the
action for $\mathcal{N}_{2}(M^{3})$, but we have also gained an additional
term. This term encodes the information of the minimal TQFT
$\mathcal{A}^{M,2^{-1}}$.161616We should be careful as there is an ambiguity
in this description. As was discussed in Hsin:2018vcg ,
$U(1)_{mn}\cong\mathcal{A}^{mn,1}\cong\mathcal{A}^{m,n}\otimes\mathcal{A}^{n,m}$
are isomorphic as topological field theories. Furthermore, the effective
Lagrangian descriptions for $\mathcal{A}^{m,n}$, $\mathcal{A}^{n,m}$, and
$U(1)_{mn}$ are essentially identical. Technically, they can only be
distinguished by coupling to appropriate background fields. Our choice of the
identification is motivated by the field-theoretic results detailed in
Appendix C. Since the anomaly $p$ of the minimal TQFT is only defined modulo
$M$, the quantity $2^{-1}$ denotes the unique multiplicative inverse of $2$
modulo $M$ assuming $\mathop{\mathrm{gcd}}(M,2)=1$. Importantly, it is an
integer so that (4.1) is properly quantized. We now further utilize the
equivalence of minimal TQFTs Hsin:2018vcg ,

$\mathcal{A}^{M,p}\cong\mathcal{A}^{M,pr^{2}}\,,$ (4.7)

by rescaling the generating line $a=e^{2\pi i\oint\alpha_{1}^{\prime}}$ of a
minimal TQFT as $a\to a^{r}$, for any $r$ such that
$\mathop{\mathrm{gcd}}(M,r)=1$. Setting $r=2$ gives
$\mathcal{A}^{M,2^{-1}}\cong\mathcal{A}^{M,2}$, and so we have fully
reproduced the RHS of (4.2).

The question then arises of how to address the more general fusion
$\mathcal{N}_{p}\otimes\mathcal{N}_{p^{\prime}}$. We can repeat the above
computation for stacks of $p$ and $p^{\prime}$ coincident D5-branes, by
viewing each stack as a single D7-brane with $p$ and $p^{\prime}$ units of
D5-brane charge respectively, i.e. $\int_{S^{2}}f_{2}^{1}=p$ and
$\int_{S^{2}}f_{2}^{2}=p^{\prime}$. The reduction over $T^{1,1}$ now becomes

$\displaystyle\int_{M^{4}\times T^{1,1}}\mathcal{I}_{9,1}^{\rm
D7}+\mathcal{I}_{9,2}^{\rm D7}$
$\displaystyle=\int_{M^{4}}(p+p^{\prime})dc_{3}+\frac{M}{2}\left(pda_{1}^{1}da_{1}^{1}+p^{\prime}da_{1}^{2}da_{1}^{2}\right)+d(pa_{1}^{1}+p^{\prime}a_{1}^{2})\mathcal{B}_{2}$
$\displaystyle=\int_{M^{3}}\frac{pp^{\prime}(p+p^{\prime})^{-1}M}{2}\,\alpha_{1}^{\prime}d\alpha_{1}^{\prime}+(p+p^{\prime})\left(c_{3}+\frac{M}{2}\,\alpha_{1}d\alpha_{1}+\alpha_{1}\mathcal{B}_{2}\right)\,,$
(4.8)

where the required field redefinition here is
$(p+p^{\prime})\alpha_{1}=pa_{1}^{1}+p^{\prime}a_{1}^{2}$ and
$\alpha_{1}^{\prime}=-a_{1}^{1}+a_{1}^{2}$. This exactly reproduces the
general fusion in (4.3), with the first term describing
$\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})^{-1}}\cong\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}$,
and the terms in parentheses describing $\mathcal{N}_{p+p^{\prime}}$. Note
that the latter can equivalently be described as a single D7-brane with
$p+p^{\prime}$ units of worldvolume flux.

#### Collective degrees of freedom.

The conclusions above can be recasted from the perspective of the original
stacks of $p$ and $p^{\prime}$ D5-branes. We see that the gauge field
$\alpha_{1}\sim pa_{1}^{1}+p^{\prime}a_{1}^{2}$ defined on
$\mathcal{N}_{p+p^{\prime}}$ is the overall center-of-mass mode for the
combined stack of $p+p^{\prime}$ D5-branes. Likewise, $a_{1}^{1}$ and
$a_{1}^{2}$ are the center-of-mass modes of the individual stacks.

The other gauge field, $\alpha_{1}^{\prime}$, should be interpreted as a mode
describing the relative motion of the two stacks of $p$ and $p^{\prime}$
D5-branes. This can be understood in the following way. The Chan-Paton bundle
for the stack of $p$ D5-branes has a gauge group $U(p)$. It is known that the
center-of-mass mode and the non-Abelian modes are decoupled from each other,
so that the gauge group can be schematically considered as $SU(p)\times
U(1)_{\rm CM}$. The same is true of course for the stack of $p^{\prime}$
D5-branes as well as the final stack of $p+p^{\prime}$ D5-branes. However,
this final stack has another Abelian mode contained within it. Within
$SU(p+p^{\prime})$ there is a maximal subgroup given by

$SU(p+p^{\prime})\supset SU(p)\times SU(p^{\prime})\times U(1)_{\rm rel}\,,$
(4.9)

where the $U(1)_{\rm rel}$ is generated by the Cartan element
$\mathrm{diag}(p^{\prime}\mathds{1}_{p},-p\mathds{1}_{p^{\prime}})$. It is
this $U(1)_{\rm rel}$ that is describing the relative motion of the D5-brane
stacks, which manifests itself as the decoupled minimal TQFT
$\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}$ at the level of the fusion algebra
(4.3). Note that if we were to start with stacks of, say, $p+k$ and
$p^{\prime}-k$ D5-branes, then we would end up with a different $U(1)_{\rm
rel}$, as is reflected by a different decoupled minimal TQFT
$\mathcal{A}^{M,(p+k)(p^{\prime}-k)(p+p^{\prime})}$ in the fusion.

Figure 1: Fusion of three stacks of D5-branes. The left figure depicts the
fusion
$\mathcal{N}_{p}\otimes(\mathcal{N}_{p^{\prime}}\otimes\mathcal{N}_{p^{\prime\prime}})$,
while the right depicts the fusion
$(\mathcal{N}_{p}\otimes\mathcal{N}_{p^{\prime}})\otimes\mathcal{N}_{p^{\prime\prime}}$.
In the former case, there is a center-of-mass mode $\alpha$ for the overall
$p+p^{\prime}+p^{\prime\prime}$ stack, and two other relative modes
$\alpha^{\prime}$ and $\alpha^{\prime\prime}$ respectively for the $p$ –
$(p^{\prime}+p^{\prime\prime})$ and $p^{\prime}$ – $p^{\prime\prime}$ stacks.
The same applies to the figure on the right.

We can carry out the same argument for more than two stacks of D5-branes.
Consider, for instance, three stacks of $p$, $p^{\prime}$, and
$p^{\prime\prime}$ D5-branes as in Figure 1. Each stack of D5-branes will
itself have a center-of-mass mode describing the collective motion, i.e.
$a_{1}^{1},a_{1}^{2},$ and $a_{1}^{3}$, but we can repackage these degrees of
freedom in different combinations. We may first consider an overall center-of-
mass mode, $\alpha_{1}$, describing the collective motion of the three stacks.
We can then consider a mode describing the relative motion of the leftmost
D5-brane stack compared to the collective motion of the rightmost two D5-brane
stacks given by $\alpha_{1}^{\prime}$, as well as a mode describing the
relative motion of the two rightmost D5-brane stacks given by
$\alpha_{1}^{\prime\prime}$. The explicit expressions for
$\alpha_{1},\alpha_{1}^{\prime},\alpha_{1}^{\prime\prime}$ can be written as

$\displaystyle\alpha_{1}$
$\displaystyle=\frac{1}{p+p^{\prime}+p^{\prime\prime}}\left(pa_{1}^{1}+p^{\prime}a_{1}^{2}+p^{\prime\prime}a_{1}^{3}\right)\,,$
$\displaystyle\alpha_{1}^{\prime}$
$\displaystyle=\frac{1}{p(p^{\prime}+p^{\prime\prime})}\left((p^{\prime}+p^{\prime\prime})(pa_{1}^{1})-p(p^{\prime}a_{1}^{2})-p(p^{\prime\prime}a_{1}^{3})\right)\,,$
(4.10) $\displaystyle\alpha_{1}^{\prime\prime}$
$\displaystyle=\frac{1}{p^{\prime}p^{\prime\prime}}\left(p^{\prime\prime}(p^{\prime}a_{1}^{2})-p^{\prime}(p^{\prime\prime}a_{1}^{3})\right)\,.$

By regrouping our degrees of freedom in this way, we are effectively perfoming
the fusion

$\mathcal{N}_{p}\otimes(\mathcal{N}_{p^{\prime}}\otimes\mathcal{N}_{p^{\prime\prime}})\cong\mathcal{A}^{M,p^{\prime}p^{\prime\prime}(p^{\prime}+p^{\prime\prime})}\otimes\mathcal{A}^{M,p(p^{\prime}+p^{\prime\prime})(p+p^{\prime}+p^{\prime\prime})}\otimes\mathcal{N}_{p+p^{\prime}+p^{\prime\prime}}.$
(4.11)

The above describes the left side of Figure 1. If we grouped the stacks of
D5-branes in the other way as in the right side, we could define similar modes
and would be performing the fusion

$(\mathcal{N}_{p}\otimes\mathcal{N}_{p^{\prime}})\otimes\mathcal{N}_{p^{\prime\prime}}\cong\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}\otimes\mathcal{A}^{M,p^{\prime\prime}(p+p^{\prime})(p+p^{\prime}+p^{\prime\prime})}\otimes\mathcal{N}_{p+p^{\prime}+p^{\prime\prime}}.$
(4.12)

Since the only difference between the two processes is the order in which we
fuse the stacks of D5-branes, it is natural to argue that locality of the
theory demands the fusion to be associative, i.e. (4.11) and (4.12) should be
physically equivalent. In Appendix C, we show that these two triple fusion
rules are indeed isomorphic.

It is worth taking a moment to reassess the previous results. We are able to
model the symmetry defects $\mathcal{N}_{p}$ as stacks of $p$ D5-branes, which
themselves can equivalently be described as a single D7-brane with $p$ units
of worldvolume flux. The fusion (4.3) of the defects themselves does not seem
to favor a particular approach, as it extracts only the Abelian information
from the D$p$-branes. All the brane information captured by the fusion rules
comes from a single center-of-mass mode as well as a single mode describing
relative motion, or multiple relative modes in the case of more instances of
fusion. In this sense, the dynamics of the non-Abelian modes in the
worldvolume theory are “washed out” in the symmetry defects.

###  4.2 $\mathcal{N}_{p}\otimes\mathcal{N}_{-p}$ Fusion and Tachyon
Condensation

#### Condensation defect.

The case of $p^{\prime}=-p\mod M$ requires special treatment. As discussed
above, the $\widetilde{\mathcal{N}}_{p}(M^{3})$ component of the defect is
invertible and thus becomes the identity for the
$\mathcal{N}_{p}\otimes\mathcal{N}_{-p}$ fusion. Regarding the fusion between
the minimal TQFTs, we have, in terms of explicit worldvolume actions,

$\displaystyle\mathcal{A}^{M,p}[\mathcal{B}_{2}]\otimes\mathcal{A}^{M,-p}[\mathcal{B}_{2}]$
$\displaystyle=\int\mathcal{D}a_{1}\mathcal{D}a^{\prime}_{1}\exp\bigg{(}2\pi
i\int_{M^{3}}\frac{pM}{2}a_{1}da_{1}-\frac{pM}{2}a^{\prime}_{1}da^{\prime}_{1}+p(a_{1}-a^{\prime}_{1})\mathcal{B}_{2}\bigg{)}$
$\displaystyle=\int\mathcal{D}\bar{a}_{1}\mathcal{D}a^{\prime}_{1}\exp\bigg{(}2\pi
i\int_{M^{3}}M\bar{a}_{1}da^{\prime}_{1}+\frac{p^{-1}M}{2}\bar{a}_{1}d\bar{a}_{1}+\bar{a}_{1}\mathcal{B}_{2}\bigg{)}$
$\displaystyle=(\mathcal{Z}_{M})_{0}[\mathcal{B}_{2}]\,,$ (4.13)

where $a_{1}$ and $a^{\prime}_{1}$ respectively denote the auxiliary fields
living on $\mathcal{A}^{M,p}$ and $\mathcal{A}^{M,-p}$, while
$\bar{a}_{1}=p(a_{1}-a^{\prime}_{1})$. In the third line, we used the fact
that if $M$ is odd, the Dijkgraaf-Witten (DW) term in $(\mathcal{Z}_{M})_{K}$
is trivial when $K$ is an integer multiple of $M$, i.e. $K\sim K+M$ (assuming
$M^{3}$ is a spin manifold) cmp/1104180750 .171717The continuum action for a
3d DW theory $(\mathcal{Z}_{M})_{K}$ is given by $\int
Ma_{1}da_{1}+(k/2)a_{1}dc_{1}$. Integrating out $a^{\prime}_{1}$ enforces
$M\bar{a}_{1}\in H^{1}(M^{3};\mathbb{Z})$ to be an integral lift of a
1-cocycle in $H^{1}(M^{3};\mathbb{Z}_{M})$. We then rewrite (4.13) as a
discrete sum of the $\widehat{\mathbb{Z}}_{M}^{(1)}$ symmetry defects
$\mathcal{U}_{1}(\Sigma^{2})$ in (2.16),181818For general $M$, there is an
additional factor in the condensation defect coming from a choice of discrete
torsion, or the choice to stack with an SPT phase when gauging. For $M$ odd,
which we are assuming throughout this work, this discrete torsion factor will
be trivial and thus not present in the condensation defect.

$\mathcal{N}_{p}\otimes\mathcal{N}_{-p}=\frac{1}{|H^{0}(M^{3};\mathbb{Z}_{M})|}\sum_{\Sigma^{2}\in
H_{2}(M^{3};\mathbb{Z}_{M})}\mathcal{U}_{1}(\Sigma^{2})\coloneqq\mathcal{C}\,,$
(4.14)

where $\Sigma^{2}$ is the Poincaré dual of the torsional reduction of
$\bar{a}_{1}$ with respect to $M^{3}$. We will refer to $\mathcal{C}$ as the
“condensation defect” as is common in the literature. This nomenclature stems
from the interpretation that the sum of operators signifies a “higher gauging”
of the $\widehat{\mathbb{Z}}_{M}^{(1)}$ symmetry Roumpedakis:2022aik . The
symmetry operators condense on the submanifold $M^{3}$ of spacetime, hence the
name. Note that the factor of $|H^{0}(M^{3};\mathbb{Z}_{M})|$ is a standard
gauge-theoretic normalization coming from the normalization of the path
integral measure Gaiotto:2014kfa ; Kaidi:2021xfk .

#### Tachyon condensation.

The presence of the condensation defect can be understood completely in terms
of the dynamical process of tachyon condensation, suggesting an interesting
interpretation of non-invertible fusion in holographic setups. Here we review
some relevant generic features of tachyon condensation Sen:1998sm ; Sen:1998ii
; Sen:1999mg .

Our starting point is a D$p$ -$\overline{{\rm D}p}$-brane pair wrapped on a
submanifold $\mathcal{M}^{p+1}$ of the total spacetime, where an
anti-D$p$-brane is the orientation reversal of a D$p$-brane. Over the
D$p$-brane we denote the Chan-Paton bundle by $\mathcal{E}^{+}$, while over
the anti-D$p$-brane we denote the bundle $\mathcal{E}^{-}$. Without loss of
generality, we may take $\mathcal{E}^{-}=\mathcal{K}$ and
$\mathcal{E}^{+}=\mathcal{L}\otimes\mathcal{K}$ for $\mathcal{L}$ and
$\mathcal{K}$ complex line bundles.191919The proper classification of D-branes
is well-known to be formalized not by ordinary cohomology but by K-theory
Minasian:1997mm ; Witten:1998cd ; Olsen:1999xx ; Witten:2000cn . In this
formalism, the relevant quantity which classifies the D-branes is the
K-theoretic “difference” between the bundles $\mathcal{E}^{\pm}$. Hence, the
choice of parametrization in the main text is for convenience. It is known
that the worldvolume theory for a brane-antibrane pair suffers from a
tachyonic mode. The tachyon in this case should be viewed as a section of
$\mathcal{E}^{+}\otimes(\mathcal{E}^{-})^{*}\cong\mathcal{L}$, where
$(\mathcal{E}^{-})^{*}$ denotes the dual bundle, reflecting that it is charged
under the $U(1)\times U(1)$ gauge symmetry of the worldvolume theory. We may
consider vortex-like solutions correpsonding to a choice of tachyon profile
that vanishes in a specified region of spacetime. To be precise, let us take
the tachyon profile to be

$T=c\cdot s\,,$ (4.15)

where $s$ is a section of $\mathcal{L}$ which vanishes over a codimension-2
submanifold $\mathcal{M}^{p-1}\subset\mathcal{M}^{p+1}$, and $c$ is a constant
chosen such that the tachyon takes its vev away from $\mathcal{M}^{p-1}$. By
construction, this localizes a codimension-2 solution to exist only along
$\mathcal{M}^{p-1}$, and far from $\mathcal{M}^{p-1}$ the system is
indistinguishable from the vacuum. In the above construction, we are only
considering a single D$p$-brane and $\overline{{\rm D}p}$-brane producing a
single D$(p-2)$-brane, but in general one may consider $2^{k-1}$ each of
branes and antibranes to produce a D$(p-2k)$-brane Witten:1998cd .

More explicitly, note that the worldvolume bundle $\mathcal{E}$ has a natural
$\mathbb{Z}_{2}$ grading induced by the splitting between the brane and
antibrane degrees of freedom, thereby giving the structure of a superbundle.
Taking advantage of the superbundle structure, we may collect the worldvolume
degrees of freedom into a single superconnection Quillen:1985vya ,

$\mathbb{A}=\begin{pmatrix}{}\;a_{1}^{+}\;&\;iT\\\
\;i\overline{T}\;&\;a_{1}^{-}\end{pmatrix},$ (4.16)

where $a_{1}^{\pm}$ denote the Chan-Paton gauge fields over
$\mathcal{E}^{\pm}$, and $T$ is the tachyon field as a section. Using this
superconnection, we may write an action that allows us to go directly from a
D$p$ -$\overline{{\rm D}p}$ configuration to a stable D$(p-2)$-brane. This
action was proposed in Kennedy:1999nn , later elaborated on in
Alishahiha:2000du ; Takayanagi:2000rz ; Szabo:2001yd ; Garousi:2007fk , and
can be written in the form

$S_{{\rm
WZ}}^{\text{D$p$\,-$\overline{\text{D}p}$}}=\mu_{p}\int_{\mathcal{M}^{p+1}}C\wedge
e^{B}\wedge{\rm
Tr_{s}}e^{\mathbb{F}}\wedge\sqrt{\frac{\hat{\mathcal{A}}(\mathcal{R}_{T})}{\hat{\mathcal{A}}(\mathcal{R}_{N})}}\Bigg{|}_{(p+1)-{\rm
form}}\,,$ (4.17)

where $\mathbb{F}=d\mathbb{A}+\mathbb{A}^{2}$ is the supercurvature given by

$\mathbb{F}=\begin{pmatrix}{}da_{1}^{+}-{T}\overline{T}&iDT\\\
i\overline{DT}&da_{1}^{-}-{T}\overline{T}\end{pmatrix}\,.$ (4.18)

In the above, ${\rm Tr_{s}}$ is the $\mathbb{Z}_{2}$-graded supertrace and the
covariant derivatives $DT$ and $\overline{DT}$ are defined as

$DT:=d{T}+(a_{1}^{+}-a_{1}^{-}){T},\qquad\overline{DT}:=d\overline{T}-(a_{1}^{+}-a_{1}^{-})\overline{T}\,.$
(4.19)

By writing $a_{1}^{+}=a_{\mathcal{L}}+a_{\mathcal{K}}$ and
$a_{1}^{-}=a_{\mathcal{K}}$ to reflect the structure of the line bundles, we
can see that the tachyon field $T$ couples only to $a_{\mathcal{L}}$ and thus
is tied only to the bundle $\mathcal{L}$. The action (4.17) may be rewritten
as a WZ action for a D$(p-2)$-brane living inside the D$p$ -$\overline{{\rm
D}p}$ worldvolume, i.e.

$S_{{\rm
WZ}}^{\text{D$p$\,-$\overline{\text{D}p}$}}=\mu_{p}\int_{\mathcal{M}^{p+1}}C\,e^{B}\,{\rm
ch}(\mathcal{K})\,({\rm
ch}(\mathcal{L})-1)=\mu_{p-2}\int_{\mathcal{M}^{p-1}}C\,e^{B+F_{\mathcal{K}}}=S^{{\rm
D}(p-2)}_{\rm WZ}\,,$ (4.20)

where in the final equality we integrated out the two directions transverse to
the D$(p-2)$-brane’s worldvolume $\mathcal{M}^{p-1}$. Due to our choice of
bundle structure, this is equivalent to integrating out the dependence on the
line bundle $\mathcal{L}$, and thus the tachyon.202020It is worth pointing out
that the factor involving ${\rm ch}(\mathcal{L})$ is encoding the K-theoretic
difference alluded to in the previous footnote. Had we not made the previous
simplification and instead chosen
$\mathcal{E}^{+}=\mathcal{L}_{1}\otimes\mathcal{K}$ and
$\mathcal{E}^{-}=\mathcal{L}_{2}\otimes\mathcal{K}$ (we can always consider a
common line bundle $\mathcal{K}$), this term would have taken the form ${\rm
ch}(\mathcal{L}_{1})-{\rm ch}(\mathcal{L}_{2})$. For the purpose of
illustration, we assume in (4.20) that the integral of ${\rm
ch}(\mathcal{L})-1$ is equal to $2\pi$. In general, the ratio of D$p$-brane
charges is given by
$\mu_{p}/\mu_{p^{\prime}}=(2\pi\sqrt{\alpha^{\prime}})^{p^{\prime}-p}$
Johnson:2003glb , but we also adopt the convention that
$2\pi\alpha^{\prime}=1$, otherwise $B_{2}+2\pi\alpha^{\prime}F_{2}$ should be
the gauge-invariant combination instead.

#### Condensation defect from tachyon condensation.

Let us apply the general discussion above to our case where a D5-brane and an
anti-D5-brane wrap the submanifold $M^{3}\times S^{3}$ in the KT-KS geometry
$\mathcal{W}^{5}\times T^{1,1}$. When the branes are brought together there
will be a solitonic D3-brane localized on a worldvolume
$\mathcal{M}^{4}\subset M^{3}\times S^{3}$, with a WZ action described by
(4.20).

To reconstruct the condensation defect $\mathcal{C}\sim\sum\mathcal{U}_{1}$ in
(4.14), we must first specify the submanifold $\mathcal{M}^{4}$ that the
solitonic D3-brane is wrapping. Equivalently, this amounts to specifying the
bundle $\mathcal{L}$ defined along the two transverse directions of
$\mathcal{M}^{4}$ with respect to $M^{3}\times S^{3}$. The integration in
(4.20) can be viewed as a two-step process: a fiber integration from the
$S^{3}$ down to the Hopf base $\widetilde{S^{2}}$, and an application of
Poincaré duality to land on $\Sigma^{2}\in H_{2}(M^{3};\mathbb{Z}_{M})$. Both
of these are encoded by the Chan-Paton flux $F_{\mathcal{L}}$. Specifically,
we can construct a $U(1)_{R}$ -equivariant ansatz for the Chan-Paton flux as

$F_{\mathcal{L}}=f_{2}+M\eta_{1}\,\frac{D\psi}{2\pi}+a_{0}^{i}V_{i}\,,$ (4.21)

where $f_{2},\eta_{1},$ and $a_{0}^{i}$ are fields defined over the external
spacetime, and $V_{i}$ are the volume forms of the spheres present in the
$T^{1,1}$. Enforcing the Bianchi identity $dF_{\mathcal{L}}=0$ leads us to
identify $a_{0}^{1}=a_{0}^{2}=a_{0}$ and $M\eta_{1}=da_{0}$, so that

$F_{\mathcal{L}}=M\eta_{1}\,\frac{D\psi}{2\pi}-a_{0}\left(V_{1}+V_{2}-\mathsf{F}_{2}\right)+\Lambda_{2}=da_{\mathcal{L}}+\Lambda_{2}\,,$
(4.22)

where $a_{\mathcal{L}}=a_{0}(D\psi/2\pi)$ is a locally defined gauge field
associated with $\mathcal{L}$. The 2-form $\Lambda_{2}$ is closed, and it
plays the role of a basepoint flux for $F_{\mathcal{L}}$, much like
$f_{2}^{i}$ in the discussion surrounding (4.5). Performing the integration in
(4.20) using this field strength yields the desired WZ action for a solitonic
D3-brane localized on $\mathcal{M}^{4}=\Sigma^{2}\times\widetilde{S^{2}}$.

Note that the expression for $F_{\mathcal{L}}$ give two possible sources for a
D3-brane in the D5-$\overline{{\rm D}5}$ worldvolume. The first is as a
soliton arising from tachyon condensation as discussed above. The second is
from any worldvolume flux present on the original D5-brane and anti-D5-brane.
The D5-branes used to construct $\mathcal{N}_{1}(M^{3})$ did not carry any
worldvolume flux, so we may take the corresponding term to be zero if desired.

At this point, one may try to reduce the anomaly polynomial
$\mathcal{I}^{\text{D3}}_{5}$ for the D3-brane on the Hopf base
$\widetilde{S^{2}}$, but end up finding an apparently vanishing result, i.e.

$\int_{\widetilde{S^{2}}\subset S^{3}}\mathcal{I}_{5}^{\rm
D3}=\mathcal{I}_{3}\ \text{``}\\!=\\!\text{''}\ 0\,.$ (4.23)

However, this is too quick. The operator $\mathcal{U}_{1}(\Sigma^{2})$ we are
ultimately concerned with finding is a holonomy of a discrete gauge field
$\mathcal{B}_{2}\in H^{2}(M^{3};\mathbb{Z}_{M})$. The standard descent
procedure falls short as the reduction above merely indicates that there is no
nontrivial holonomy for continuous gauge fields; we may still have discrete
holonomies. Roughly speaking, we can again think of $\mathcal{I}_{3}$ as a
“field strength” for torsional flux and thus would vanish when evaluated on
integer-valued cycles. If not integer-valued, then what cycles should we
evaluate this on? Since we aim to recover a $\mathbb{Z}_{M}$-valued operator,
we shall follow our nose and consider reducing on $\mathbb{Z}_{M}$-valued
torsional cycles.

The correct topological operator is then not the standard holonomy of
$\mathcal{I}_{3}$ on an integer-valued cycle, but a holonomy along a
$\mathbb{Z}_{M}$-valued cycle. Namely, the proper operator to consider
is212121We can alternatively phrase this in terms of the RR field strengths,
by originally having a term in $G_{5}$ of the form
$M\eta_{3}\wedge(V_{1}+V_{2})$ with $\eta_{3}$ a $\mathbb{Z}_{M}$-valued field
strength. Due to the torsion, this term would formally be zero and naïvely
wouldn’t be present in the expansion.

$\mathcal{U}_{1}(\Sigma^{2})=\exp\left(\frac{2\pi
i}{M}\int_{\widetilde{\Sigma}^{3}}\mathcal{I}_{3}\right)=\exp\left(\frac{2\pi
i}{M}\,\big{\langle}\widetilde{\Sigma}^{3},\mathcal{I}_{3}\big{\rangle}\right)\,,$
(4.24)

where $\widetilde{\Sigma}^{3}$ is a formal extension of $\Sigma^{2}$ such that
$\partial\widetilde{\Sigma}^{3}=M\Sigma^{2}$. In the final expression we have
rewritten the integral as a pairing between $\mathbb{Z}_{M}$-valued cycles and
cocycles, and should now consider everything to be valued in (co)homology with
$\mathbb{Z}_{M}$ coefficients. As we mentioned, $\mathcal{I}_{3}$ should be
viewed as a “field strength” for a discrete field, so we may express
$\mathcal{I}_{3}=\beta^{\prime}(\mathcal{B}_{2})\in
H^{3}(M^{3};\mathbb{Z}_{M})$. This justifies our decision to consider the
reduction on $\mathbb{Z}_{M}$-valued cycles, as opposed to cycles valued in
any other coefficients, as the only discrete 2-form gauge field present in the
supergravity setup is $\mathcal{B}_{2}\in
H^{2}(M^{3};\mathbb{Z}_{M})$.222222By exactness of the short exact sequence
$0\to\mathbb{Z}_{M}\to\mathbb{Z}_{M^{2}}\to\mathbb{Z}_{M}\to 0$, the quantity
$M\mathcal{B}_{2}$ is trivial, and so is $\beta^{\prime}(M\mathcal{B}_{2})$.
This means that
$\mathcal{U}_{M}(\Sigma^{2})=\mathcal{U}_{1}(M\Sigma^{2})=\exp\big{(}2\pi
i\int_{\widetilde{\Sigma}^{3}}\mathcal{I}_{3}\big{)}=0$ as we saw in (4.23).

With this choice, we can evaluate the holonomy of $\mathcal{I}_{3}$ as

$\exp\left(\frac{2\pi
i}{M}\,\big{\langle}\widetilde{\Sigma}^{3},\beta^{\prime}\left(\mathcal{B}_{2}\right)\big{\rangle}\right)=\exp\left(\frac{2\pi
i}{M}\,\big{\langle}\beta^{\prime}\big{(}\widetilde{\Sigma}^{3}\big{)},\mathcal{B}_{2}\big{\rangle}\right)=\exp\left(\frac{2\pi
i}{M}\,\big{\langle}\Sigma^{2},\mathcal{B}_{2}\big{\rangle}\right)\,.$ (4.25)

In the first equality, we switched the Bockstein homomorphism in cohomology to
its dual in homology, and in the second equality, we used the property that
$\beta^{\prime}=\partial/M$ for the Bockstein homomorphism in homology (see
Appendix B for details). This holonomy is nothing but the
$\mathcal{U}_{1}(\Sigma^{2})$ operator we saw in the condensation defect
(4.14).

The arguments above regarding the D3-brane worldvolume action can be rephrased
in the language of differential cohomology 10.1007/BFb0075216 ; Hopkins:2002rd
; Freed:2006yc ; Cordova:2019jnf ; Hsieh:2020jpj . The standard characteristic
class construction via the descent procedure constructs the curvature form for
a (higher) anomaly bundle over the base spacetime. Naïvely, this curvature
form only captures the continuous data of a Cheeger-Simons differential
cocycle; the torsional data is overlooked. What we see above is that the
torsional data is not completely lost, but can be reconstructed through the
presence of torsional flux.

Now that we have constructed an operator for the D3-brane wrapping
$\Sigma^{2}\times\widetilde{S^{2}}$, there is one last thing to consider about
the ${\rm D}3$-brane soliton arising from the D5-$\overline{{\rm D}5}$
worldvolume. The Chan-Paton field on the D5-$\overline{{\rm D}5}$ brane pair
is dynamical and therefore should be summed over in a quantum theory. This
includes a summing over the gauge field not only from the solitonic D3-brane
itself, but also from $a_{\mathcal{L}}$ along the other two transverse
directions in the original D5-$\overline{{\rm D}5}$ worldvolume. A sum over
$a_{\mathcal{L}}$ configurations is equivalent to a sum over the
configurations of $a_{0}$ defined in (4.22). This in turn, as $a_{0}$ dictates
the torsional 2-cycle $\Sigma^{2}$ that the D3-brane is wrapping (via Poincaré
duality), yields a sum over all possible 2-cycles $\Sigma^{2}\in
H_{2}(M^{3};\mathbb{Z}_{M})$. Hence, taking into account the proper gauge
theoretic normalization, our final defect is given by

$\mathcal{C}=\frac{1}{|H^{0}(M^{3};\mathbb{Z}_{M})|}\sum_{\Sigma^{2}\in
H_{2}(M^{3};\mathbb{Z}_{M})}\exp\left(\frac{2\pi
i}{M}\int_{\Sigma^{2}}\mathcal{B}_{2}\right)\,,$ (4.26)

which is in exact agreement with (4.14) for $p=1$. We have understood the
condensation defect arising in the fusion
$\mathcal{N}_{1}\otimes\mathcal{N}_{-1}$ as being a sum of all possible
D3-brane configurations in the original ${\rm D}5$\- and $\overline{{\rm
D}5}$-brane system. It is worth noting that the $a_{\mathcal{L}}$ that the
condensation defect couples to is the center-of-mass mode for the brane-
antibrane configuration. Even though it arises as a difference of gauge
fields, this “difference” is really a sum with an orientation reversal coming
from the anti-D5-brane. The condensation defect aligns with the earlier
observation (in the case of $\mathcal{N}_{p}\otimes\mathcal{N}_{p^{\prime}}$)
that the symmetry defects are those coupling to the center-of-mass modes for
the stacks of D$p$-branes.

Figure 2: Condensation defect arising from tachyon condensation on a D$p$
-$\overline{\text{D}p}$ system wrapping some internal manifold. We
schematically decomposed the worldvolume of the branes as a direct product of
internal and external manifolds. Chan-Paton modes exist both for open strings
ending on the same brane or on different branes. The latter includes tachyonic
modes, which, upon integrating out, lead to a condensate of lower branes
within the (both internal and external) worldvolume of the original D$p$
-$\overline{\text{D}p}$ system.

We can extrapolate the above argument to the more general fusion of
$\mathcal{N}_{p}\otimes\mathcal{N}_{-p}$. In this case, tachyon condensation
will produce $p$ D3-branes defined on the $\Sigma^{2}\times\widetilde{S^{2}}$
submanifold. As we will elaborate upon later, this will cause each term in the
sum over 2-cycles to have charge $p$ rather than charge one. As the sum is
over all 2-cycles in $M^{3}$, this rescaling will simply reshuffle the terms
in the sum and has no effect otherwise. We can then immediately see the brane
realization of the more general fusion
$\mathcal{N}_{p}\otimes\mathcal{N}_{-p}=\mathcal{C}$ as derived in (4.14).

There is one last comment to make regarding the computation of the
condensation defect. When constructing the solitonic D3-brane on the
D5-$\overline{{\rm D}5}$ worldvolume, the base $\widetilde{S^{2}}$ of the Hopf
fibration was really one of the many possible choices on which to wrap the
D3-brane. We could have chosen any possible linear combination of the two
spheres in the base of the $T^{1,1}$ fibration and would have produced the
same defect $\mathcal{U}_{q}(\Sigma^{2})$. From the point of view of the
external spacetime, the defects are identical. In fact, one might argue that a
more natural choice for the worldvolume of the D3-brane is one wrapping the
equatorial $S^{2}$ of the $S^{3}$ cycle. For this choice the solitonic nature
is more apparent as follows. When implementing tachyon condensation for a
D5-$\overline{{\rm D}5}$ system, the tachyon profile is crucial in determining
where the D3-brane is located. We could have picked the tachyon profile such
that it approaches $T_{0}$, its minimum, at one pole of the $S^{3}$, and
$-T_{0}$ at the other pole.232323We would need to pick similar configurations
in the external spacetime within $M^{3}$. The tachyon will then have a profile
such that it vanishes exactly along the equatorial $S^{2}$ specified above.
The fact that this choice of internal directions for the D3-brane has no
effect on the produced operator is an interesting phenomenon that we will
encounter again later in this paper.

###  4.3 Nature of the $\widehat{\mathbb{Z}}_{M}^{(1)}$ Defect

We have so far seen that the existence of the $\mathbb{Z}_{2M}^{(0)}$ symmetry
defect $\mathcal{N}_{1}(M^{3})$ in the 5d bulk $\mathcal{W}^{5}$ can be
attributed to a D5-brane wrapping $M^{3}\times S^{3}$. Furthermore, the fusion
of such a D5-brane with its orientation reversal produces a condensate of the
$\widehat{\mathbb{Z}}_{M}^{(1)}$ defects $\mathcal{U}_{1}(\Sigma^{2})$ as
solitonic D3-branes localized on $M^{3}$. A natural question to ask then is
whether a $\mathcal{U}_{1}(\Sigma^{2})$ defect defined on a generic
$\Sigma^{2}\not\subset M^{3}$ can be realized as a bona fide D3-brane existing
on its own right.

The answer is actually baked in already in the presentation of (4.20). Instead
of considering solitonic D3-brane excitations on a D5-$\overline{{\rm D}5}$
worldvolume, we can consider a D3-brane on its own wrapping the Hopf base
$\widetilde{S^{2}}\subset S^{3}$ inside the internal $T^{1,1}$, and an
arbitrary external $\mathbb{Z}_{M}$-valued cycle $\Sigma^{2}\in
H_{2}(\mathcal{W}^{5};\mathbb{Z}_{M})$. Following the identical arguments
given previously, the D3-brane reproduces the $\widehat{\mathbb{Z}}_{M}^{(1)}$
magnetic symmetry defect,

$\mathcal{U}_{1}(\Sigma^{2})=\exp\left(\frac{2\pi
i}{M}\int_{\Sigma^{2}}\mathcal{B}_{2}\right)\,.$ (4.27)

It is interesting to note that this symmetry defect can also be constructed by
a closed fundamental (F1) string sitting in the external spacetime
$\mathcal{W}^{5}$. As F1-strings do not couple to the Ramond-Ramond sector,
the anomaly polynomial for this F1-string is simply given by

$\mathcal{I}_{3}^{\rm
F1}=h_{3}=db_{2}-\widetilde{\beta}^{\prime}(\mathcal{B}_{2})\,,$ (4.28)

where we used (2.5) and (2.6). The fluctuating field $b_{2}$ is suppressed as
we push the F1-string towards the conformal boundary, so what is left behind
is a holonomy $\sim\exp\big{(}2\pi i\int_{\Sigma^{2}}\mathcal{B}_{2}/M\big{)}$
(up to a sign convention). We have thus showed that these F1-strings can also
give rise to the symmetry defect $\mathcal{U}_{1}(\Sigma^{2})$.

An alternative understanding of this comes from a modification of the “baryon
vertex” introduced in Witten:1998xy . This utilizes the fact that F1-strings
act as Wilson operators in the boundary field theory, which can be screened by
operators carrying the appropriate charge. In the background we are concerned
with, there are two possible causes for the screening of F1-strings: a
D5-brane wrapping the entirety of the $T^{1,1}$, or a D3-brane wrapping only
the $S^{3}\subset T^{1,1}$ Apruzzi:2021phx . From the Bianchi identities,

$\int_{T^{1,1}}dG_{7}=-\int_{T^{1,1}}H_{3}G_{5}=-Ndb_{2}\,,\qquad\int_{S^{3}}dG_{5}=-\int_{S^{3}}H_{3}G_{3}=-Mdb_{2}\,,$
(4.29)

we can see that the former will screen $N$ F1-strings while the latter will
screen $M$ F1-strings. The combined effect will be to screen
$\mathop{\mathrm{gcd}}(N,M)=M$ F1-strings. This implies that the defect
produced by an F1-string will generate a $\mathbb{Z}_{M}$ (rather than
$\mathbb{Z}_{N}$) symmetry.

Due to the baryon vertex, a collection of $k$ F1-strings also generates the
full $\widehat{\mathbb{Z}}_{M}^{(1)}$ symmetry as long as
$\mathop{\mathrm{gcd}}(M,k)\neq 1$ (otherwise it will only generate a
$\mathbb{Z}_{M/k}$ subgroup). With an eye towards the construction of the dual
symmetry defect in Section 5, it is convenient for us to take $k=N/M$. One may
then interpret the defect $\mathcal{U}_{1}(\Sigma^{2})$ to be either one
D3-brane or $k$ coincident F1-strings. However, these descriptions are
equivalent when considering the Myers Effect discussed earlier242424We thank
Oren Bergman for pointing this out to us.. To see this, it is best to first
consider the construction of Maldacena:2000yy  which describes an S-dual
description to the KS/KT background. In that analysis, there exists nontrivial
$H_{3}$ flux threading the $S^{3}$ of the internal geometry. If one imagines
$k$ coincident D1-branes placed in the external spacetime of this background,
the D1-branes will “puff up” into a single D3-brane with $k$ units of
worldvolume flux through an $S^{2}\subset S^{3}$ via the standard Myers
Effect. Transforming to the KS/KT background252525This was described in
Herzog:2001fq , and a similar effect was discussed in Kachru:2002gs ., this
tells us that a stack of $k$ F1-strings in the presence of a nontrivial
$G_{3}$ background is equivalent to a D3-brane with $k$ units of worldvolume
flux threading a transverse $S^{2}\subset S^{3}$, which we identify as the
Hopf base $\widetilde{S^{2}}$ above. We thus see that the two descriptions
presented above for $\mathcal{U}_{1}(\Sigma^{2})$ converge. We will use the
D3-brane description, as opposed to the one involving F1-strings, in the
discussion that follows.

Recall that earlier we constructed the $\mathbb{Z}_{2M}^{(0)}$ defects
$\mathcal{N}_{p}(M^{3})$ by stacking $p$ D5-branes on $M^{3}\times S^{3}$,
thanks to the fact that only the (Abelian) center-of-mass mode, but not the
non-Abelian modes, of the stack contributes to this topological operator.
Equivalently, we may take a D7-brane with $p$ units of worldvolume flux
threading $S^{2}\subset T^{1,1}$. It is straightforward to apply the same
argument here to construct the general $\widehat{\mathbb{Z}}_{M}^{(1)}$
defects,

$\mathcal{U}_{q}(\Sigma^{2})=\exp\bigg{(}\frac{2\pi
iq}{M}\int_{\Sigma^{2}}\mathcal{B}_{2}\bigg{)}\,,$ (4.30)

from a stack of D3-branes wrapping $\Sigma^{2}\times\widetilde{S^{2}}$. Again
equivalently, this can also be realized by taking a D5-brane with $q$ units of
worldvolume flux threading $S^{2}\subset T^{1,1}$.

#### Defect fusion.

Unlike the D5-brane operator $\mathcal{N}_{p}(M^{3})$ on which the Chan-Paton
fields form a nontrivial TQFT, the D3-brane operator
$\mathcal{U}_{q}(\Sigma^{2})$ has a relatively simple structure as seen in
(4.30). For two such defects wrapping the same external 2-cycle (mod $M$)
$\Sigma^{2}\subset\mathcal{W}^{5}$, their fusion obeys

$\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{U}_{q^{\prime}}(\Sigma^{2})=\mathcal{U}_{q+q^{\prime}}(\Sigma^{2})=\exp\bigg{(}\frac{2\pi
i(q+q^{\prime})}{M}\int_{\Sigma^{2}}\mathcal{B}_{2}\bigg{)}\,,$ (4.31)

where the label $q$ of $\mathcal{U}_{q}$ is only defined modulo $M$. In other
words, the $\mathcal{U}_{q}$ operators realize a group-like fusion algebra
corresponding to a $\mathbb{Z}_{M}$ group.

##  5 $\mathbb{Z}_{M}^{(1)}$ and $\widehat{\mathbb{Z}}_{2M}^{(2)}$ Defect
Fusion

Let us proceed to discuss the remaining two symmetry defects resulting from
the Gauss Law constraints presented in Section 2, namely, $\mathcal{O}_{q}$
and $\mathcal{V}_{p}$, which respectively generate the $\mathbb{Z}_{M}^{(1)}$
and $\widehat{\mathbb{Z}}_{2M}^{(2)}$ symmetries. As we will show, these
operators can also be realized by certain D-brane configurations in the KT-KS
background. Remarkably, their fusion rules share some qualitative features
with those we studied in the previous section, thus shedding light on the
underlying brane kinematics and dynamics.

The $\mathbb{Z}_{M}^{(1)}$ symmetry defect $\mathcal{O}_{q}(W^{2})$, as
written in (2.15), is composed of multiple 5d bulk gauge fields of different
origins. Therefore, it is natural to expect that the string-theoretic
construction is not given by a single type of D$p$-brane. In fact, we can
construct the desired operator through a bound state of a single D3-brane and
$k$ D1-branes.262626We previously defined $k$ via $N=kM$. As long as $M^{2}\gg
N\gg M$, there is no significant backreaction from the stack of $k$ probe
D1-branes. We assume this limit for the above discussion. Let us first
consider a D3-brane wrapping the (homologically) nontrivial $S^{2}\subset
T^{1,1}$. To reduce its WZ action, we have to carefully isolate the components
of $G_{5}$ along the directions of the $S^{2}$. With (2.2), the full
expression for $G_{5}=(1+\star_{10})\mathcal{G}_{5}$ is given by

$G_{5}=N\Omega_{5}+g_{2}\omega_{3}+Ng_{5}-\frac{N}{2}(V_{1}+V_{2})\star_{5}\mathsf{F}_{2}+\star_{5}g_{2}\omega_{2}\,,$
(5.1)

where $g_{5}$ is an external gauge field. By imposing the IIB Bianchi
identities, we find that $\star_{5}g_{2}$ can be solved as

$\star g_{2}=da_{2}+\frac{1}{M}\,g_{1}g_{2}\,,$ (5.2)

for some globally well-defined $a_{2}$. Now that we have this expansion, the
reduction of the D3-brane characteristic class over $S^{2}$ becomes

$\int_{S^{2}}\mathcal{I}_{5}^{\rm
D3}=\int_{S^{2}}G_{5}+G_{3}(B_{2}+da_{1})=da_{2}+\frac{1}{M}\,g_{1}\mathcal{B}_{2}+g_{1}\left(da_{1}+\frac{d\beta_{1}}{M}\right)\,,$
(5.3)

where we have made use of the expression for $g_{2}$ in (2.5). We also
hereafter absorb $\beta_{1}$ into the worldvolume Chan-Paton gauge field
$a_{1}$ via a simple field redefinition.

As argued previously in Section 4, the Chan-Paton field is dynamical and hence
should be summed over in the worldvolume gauge theory. For the D3-brane
operator at hand, performing the path integral enforces the condition
$dg_{1}=0$, so we retain only the cohomological class (representative)
$\mathcal{A}_{1}$ of $g_{1}$ as in (2.5).272727The potential $c_{0}$ may be
turned off via a gauge transformation. The topological operator associated to
the D3-brane wrapping $S^{2}\subset T^{1,1}$ is thus given by

$\exp\bigg{(}2\pi i\int_{W^{2}}a_{2}+\frac{2\pi
i}{M}\int_{W^{3}}\mathcal{A}_{1}\mathcal{B}_{2}\bigg{)}\,,$ (5.4)

where $\partial W^{3}=W^{2}\subset\mathcal{W}^{5}$.

In addition to the D3-brane configuration described above, we also consider a
D1-brane in the external $\mathcal{W}^{5}$, whose characteristic class is
simply

$\mathcal{I}_{3}^{\rm D1}=G_{3}\big{|}_{\mathcal{W}^{5}}=dc_{2}\,.$ (5.5)

To reproduce the $\mathbb{Z}_{M}^{(1)}$ symmetry generator, we need $k$ such
D1-branes wrapping the $W^{2}$ of the D3-brane worldvolume, combining to give

$\mathcal{O}_{1}(W^{2},W^{3})=\exp\bigg{(}2\pi
i\int_{W^{2}}(kc_{2}+a_{2})+\frac{2\pi
i}{M}\int_{W^{3}}\mathcal{A}_{1}\mathcal{B}_{2}\bigg{)}\,.$ (5.6)

Alternatively, we can derive the above expression for
$\mathcal{O}_{1}(W^{2},W^{3})$ by considering a D3-brane with $k$ units of
worldvolume flux threading the $S^{2}\subset T^{1,1}$.

This combination of branes pairs nicely with the discussion of the
$\mathcal{U}_{1}(\Sigma^{2})$ operator surrounding (4.29). There, a bound
state of a single D3-brane and $k$ F1-strings gave rise to a
$\widehat{\mathbb{Z}}_{M}^{(1)}$ symmetry generator. It is natural to expect
that the symmetry generator for the Pontryagin dual symmetry
$\mathbb{Z}_{M}^{(1)}$ should be generated by the S-dual brane configuration,
i.e. a bound state of a single D3-brane and $k$ D1-branes, which is precisely
what was used to construct $\mathcal{O}_{1}(W^{2},W^{3})$.

As mentioned in Section 2, this operator is only well-defined when
$W^{2}=\partial W^{3}$. In order to find an operator that is well-defined even
when $W^{2}$ is closed but is not a boundary, we may rewrite it in terms of
auxiliary fields which we denote as $\lambda_{1}$ and $\varphi$,

$\mathcal{O}_{1}(W^{2})=\int\mathcal{D}\lambda_{1}\mathcal{D}\varphi\exp\left(2\pi
i\int_{W^{2}}\hat{c}_{2}+\lambda_{1}\mathcal{A}_{1}+\varphi\mathcal{B}_{2}-M\varphi
d\lambda_{1}\right)\,.$ (5.7)

We emphasize that, similarly to the worldvolume Chan-Paton field $a_{1}$ in
$\mathcal{A}^{M,p}[\mathcal{B}_{2}]$ (see (2.12)), the presence of the
worldvolume gauge fields $\lambda_{1}$ and $\varphi$ can be understood in
terms of anomaly inflow. The $\mathcal{A}_{1}\mathcal{B}_{2}/M$ term in (5.6)
should be interpreted as an anomaly defined in one dimension higher than the
D3-D1 worldvolume. Specifically, it describes a mixed anomaly between the
$\mathbb{Z}_{M}^{(0)}\subset\mathbb{Z}_{2M}^{(0)}$ and $\mathbb{Z}_{M}^{(1)}$
global symmetries on the D3-D1 bound state. It is known in general that a
theory with a global symmetry $G^{(k)}$ can be equivalently viewed as a gauge
theory for the Pontryagin-dual symmetry $\widehat{G}^{(d-k-2)}$
Gaiotto:2014kfa . In our case, this corresponds to a
$\widehat{\mathbb{Z}}_{M}^{(0)}\times\widehat{\mathbb{Z}}_{M}^{(-1)}$ gauge
theory on the D3-D1 worldvolume, which is exactly the TQFT $\mathcal{X}^{M,1}$
within $\mathcal{O}_{1}(W^{2})$.

Before proceeding, let us pause to examine the field content of the
$\mathcal{O}_{1}(W^{2})$ operator. The first term with $\hat{c}_{2}$ describes
the D3- and D1-brane charges present in the bound state, while the final term
is simply a BF term that Higgses the two gauge fields from $U(1)$ down to
$\mathbb{Z}_{M}$. On the other hand, the second and third terms can be
interpreted as induced charges from F1-strings and D1-branes. To see this
explicitly, note that $\varphi\in H^{0}(W^{2};\mathbb{Z}_{M})$ and
$\lambda_{1}\in H^{1}(W^{2};\mathbb{Z}_{M})$ by virtue of the discussion in
footnote 11. The coupling to $\mathcal{B}_{2}$ in (5.7) then implies that the
D3-D1 worldvolume has a $\varphi$ number of induced F1-string charge, where
$\varphi$ is summed from $0$ to $M-1$ in order to create a gauge-invariant
operator. Similarly, the coupling to $\mathcal{A}_{1}$ in (5.7) describes
induced D1-brane charge localized on 1-cycles $\gamma^{1}$ which is Poincaré-
dual to $\lambda_{1}$ (with respect to $W^{2}$). We will soon explain why
$\mathcal{A}_{1}$ is associated with D1-branes when we discuss the
$\mathcal{V}_{1}(\gamma^{1})$ operator. See Figure 3 for a schematic picture
of these induced charges on $\mathcal{O}_{1}(W^{2})$.

Figure 3: A schematic picture for the induced charges on the
$\mathbb{Z}_{M}^{(1)}$ generator $\mathcal{O}_{q}(W^{2})$. In addition to the
D3- and D1-brane charges from the constituent branes, there are additional
induced charges describing F1-strings and D1-branes. The F1-charge is spread
throughout the manifold $W^{2}$, while the D1-charge is localized to
codimension-1 submanifolds.

###  5.1 $\mathcal{O}_{q}\otimes\mathcal{O}_{q^{\prime}}$ Fusion

We are now interested in the fusion of these operators, which can be computed
using methods similar to those utilized in Section 4. First we can consider
the self-fusion of $\mathcal{O}_{1}(W^{2},W^{3})$ where $W^{2}$ is a boundary
such that $W^{2}=\partial W^{3}$. In this case, we may compute the fusion by
taking the D3-D1 configuration to be a neutral D5-brane with the appropriate
units of worldvolume flux. Specifically, we take two D5-branes each with
worldvolume $W^{2}\times S^{2}\times\widetilde{S^{2}}$ and one unit of
worldvolume flux threading the $\widetilde{S^{2}}$. Each D5-brane will have a
characteristic class $\mathcal{I}_{7,i}$ similar to what we used before in
(4.5), that can be written as

$\mathcal{I}_{7,i}^{{\rm
D}5}=G_{5}(B_{2}+da_{1}^{i}+f_{2}^{i})+\frac{1}{2}\,G_{3}(B_{2}+da_{1}^{i}+f_{2}^{i})^{2}\,,$
(5.8)

where $\int_{\widetilde{S^{2}}}f_{2}^{i}=1$ and $\int_{S^{2}}f_{2}^{i}=k$ to
account for the $k$ D1-branes in the bound state. Summing the two
characteristic classes and performing the reduction over the internal geometry
yields

$\int_{S^{2}\times\widetilde{S^{2}}}\mathcal{I}_{7,1}^{{\rm
D}5}+\mathcal{I}_{7,2}^{{\rm
D}5}=2\int_{S^{2}}G_{5}+G_{3}(B_{2}+da_{1}+f_{2}^{1}+f_{2}^{2})\,,$ (5.9)

which is twice that of the reduction for a single D3-D1 bound state. We
therefore find that the operators obey a group-like fusion,

$\mathcal{O}_{1}(W^{2},W^{3})\otimes\mathcal{O}_{1}(W^{2},W^{3})=\mathcal{O}_{2}(W^{2},W^{3})\,.$
(5.10)

The same can be shown to hold for general $\mathcal{O}_{q}(W^{2},W^{3})$ and
$\mathcal{O}_{q^{\prime}}(W^{2},W^{3})$. We emphasize that this fusion, and by
extension this method, will only be valid in the case $W^{2}=\partial W^{3}$.
If $W^{2}$ is a non-contractible closed manifold, we will need to take more
care in handling the discrete worldvolume fields $\varphi$ and $\lambda_{1}$
discussed previously.

To account for these additional worldvolume fields, we can compute the fusion
directly at the level of $\mathcal{O}_{1}(W^{2})$ or, in general,
$\mathcal{O}_{q}(W^{2})$. We find, for $q^{\prime}\neq-q$,

$\mathcal{O}_{q}(W^{2})\otimes\mathcal{O}_{q^{\prime}}(W^{2})=\mathcal{X}^{M,qq^{\prime}(q+q^{\prime})^{-1}}(W^{2})\otimes\mathcal{O}_{q+q^{\prime}}(W^{2})\,,$
(5.11)

where the coefficient is a decoupled 2d TQFT using the notation introduced in
(2.19). The full details of this fusion can be found in Appendix C, but we
highlight some important features here. It is first worth noting that the
gauge fields $\varphi$ and $\lambda_{1}$ are localized worldvolume degrees of
freedom coupled respectively to the 5d bulk fields $\mathcal{B}_{2}$ and
$\mathcal{A}_{1}$. To arrive at the fusion rule (5.11), we make a field
redefinition analogously to what we did in (4.8), i.e.

$\displaystyle(q+q^{\prime})\widetilde{\varphi}$
$\displaystyle=q\varphi+q^{\prime}\varphi^{\prime}\,,$
$\displaystyle(q+q^{\prime})\widetilde{\lambda}_{1}$
$\displaystyle=q\lambda_{1}+q^{\prime}\lambda^{\prime}_{1}\,,$
$\displaystyle\widetilde{\varphi}^{\prime}$
$\displaystyle=-\varphi+\varphi^{\prime}\,,$
$\displaystyle\widetilde{\lambda}_{1}^{\prime}$
$\displaystyle=-\lambda_{1}+\lambda^{\prime}_{1}\,.$ (5.12)

Ultimately, the $\widetilde{\varphi}$ and $\widetilde{\lambda}_{1}$ fields are
associated to the $\mathcal{O}_{q+q^{\prime}}$ operator while the
$\widetilde{\varphi}^{\prime}$ and $\widetilde{\lambda}_{1}^{\prime}$ fields
comprise the decoupled TQFT coefficient. This lends itself to an
interpretation reminiscent of that given for the fusion of the
$\mathcal{N}_{p}$ operators; the symmetry defects themselves couple to the
center-of-mass modes of their corresponding brane stacks, whereas the
topological sector that decouples from the system is encoded by the relative
modes. In this case, the relative modes yield the 2d TQFT
$\mathcal{X}^{M,qq^{\prime}(q+q^{\prime})^{-1}}$ arising as the fusion
coefficient.

Despite the similarity, the mechanisms from which the 3d TQFT
$\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}$ and the 2d TQFT
$\mathcal{X}^{M,qq^{\prime}(q+q^{\prime})^{-1}}$ arise are slightly different.
In the former case, since the Chan-Paton field $a_{1}$ is what couples to
$\mathcal{B}_{2}$ in the 3d worldvolume action of $\mathcal{N}_{p}$, it
captures both the center-of-mass and relative modes of the brane stacks.
However, in the latter case, these modes are instead captured by the pair of
gauge fields $\varphi$ and $\lambda_{1}$, which respectively couple to
$\mathcal{B}_{2}$ and $\mathcal{A}_{1}$ in the 2d worldvolume action of
$\mathcal{O}_{q}$.

###  5.2 $\mathcal{O}_{q}\otimes\mathcal{O}_{-q}$ Fusion

Due to the existence of the nontrivial 2d TQFT
$\mathcal{X}^{M,q}[\mathcal{A}_{1},\mathcal{B}_{2}]$ within the
$\mathcal{O}_{q}(W^{2})$ operator, one naturally expects it to be non-
invertible. The fusion of $\mathcal{O}_{q}$ with its orientation reversal
produces another condensation defect $\widetilde{\mathcal{C}}$ through

$\mathcal{O}_{q}(W^{2})\otimes\mathcal{O}_{-q}(W^{2})=\frac{1}{|H^{0}(W^{2};\mathbb{Z}_{M})|}\bigoplus_{\begin{subarray}{c}\gamma^{1}\in
H_{1}(W^{2};\mathbb{Z}_{2M})\\\ \Sigma^{2}\in
H_{2}(W^{2};\mathbb{Z}_{M})\end{subarray}}\mathcal{V}_{2}(\gamma^{1})\otimes\mathcal{U}_{1}(\Sigma^{2})\coloneq\widetilde{\mathcal{C}}(W^{2})\,.$
(5.13)

More technical details are provided in Appendix C. The presence of this sum
over operators can be understood as a consequence of the discussion following
(5.7), as it is simply enumerating the additional induced charges on the
worldvolume $W^{2}$ of the D3-D1 operator.

We can interpret this condensation defect analogously to what we did to that
in Section 4: through tachyon condensation between the D3-D1 bound state and
its orientation reversal. In this case, the overall D3-brane charge and the
overall D1-brane charge (from the $k$ D1-branes in $\mathcal{W}^{5}$) will
annihilate completely, but the induced charges will not. This logic is the
same as that of the non-invertible fusion of GarciaEtxebarria:2022vzq . As the
initial configuration is a product of sums of the induced states, the final
result will be a sum over charges that have not been completely annihilated.
The charges remaining in the (D3-D1)-($\overline{{\rm D}3\text{-}{\rm D}1}$)
configuration are those of the F1-strings induced in the worldvolume, and some
remaining D1-brane charge that can be seen coming from the coupling to
$\lambda_{1}$ in the definition of $\mathcal{O}_{q}(W^{2})$. The reason why
the sum includes terms of the form $\mathcal{V}_{2}\otimes\mathcal{U}_{1}$ can
be understood from the BF term in (5.7). It acts as a constraint that the
induced charges are equal for the F1-strings and D1-branes, so that we do not
sum over all insertions of the $\widehat{\mathbb{Z}}_{2M}^{(2)}$ symmetry
generator but only those which satisfy the imposed charge constraint.

It is clear from the discussion following (4.27) that the F1-strings in the
external spacetime represent the $\mathcal{U}_{1}(\Sigma^{2})$ factors in the
condensation defect $\widetilde{\mathcal{C}}(W^{2})$. The question remains of
where the remaining D1-brane charge is localized. As stated previously, this
D1-brane charge is induced on the original D3-D1 worldvolume and should not be
thought of as a stand-alone D1-brane. We should interpret the induced charge
as a D1-brane soliton on the worldvolume of the bound state, which is picked
out by the tachyon condensation process. Specifically, we see that the net
D1-brane charge is localized to a torsional 1-cycle $\gamma^{1}\in
H_{1}(W^{2};\mathbb{Z}_{M})$ in the external spacetime. In order for the
soliton D1-brane to be stable, there must be a locus in the internal geometry
where the tachyon profile vanishes. Analogously to the discussion in Section 4
for the D5-$\overline{{\rm D}5}$ condensation, we can pick the tachyon profile
such that it reaches its minimum $T_{0}$ at one pole of the internal $S^{2}$,
and takes the value $-T_{0}$ at the other. This leaves a codimension-1 locus
where the tachyon vanishes, and thus where a solitonic D1-brane can live. For
simplicity, we take this to be the $\widetilde{S^{1}}$ equator of the $S^{2}$
cycle wrapped by the original D3-D1 bound states. The end result is then a
collection of F1-string operators $\mathcal{U}_{1}(\Sigma^{2})$, and a
solitonic D1-brane operator $\mathcal{V}_{2}(\gamma^{1})$.

We shall emphasize that the D1-brane operator $\mathcal{V}_{2}(\gamma^{1})$
resulting from tachyon condensation is stable only because of its solitonic
nature. To construct a generic $\widehat{\mathbb{Z}}_{2M}^{(2)}$ defect from a
stable D1-brane configuration, as we will do in the next subsection, we have
to consider a different $S^{1}$ to wrap in the internal geometry of the KT-KS
background. This is the same phenomenon we witnessed in constructing the
condensation defect $\mathcal{C}$ in Section 4, namely, when constructing
symmetry operators from D$p$-branes, there is an ambiguity arising from which
internal manifolds the branes wrap. In fact, multiple distinct configurations
may lead to the same 5d bulk operator that cannot otherwise be distinguished.
For the setup of interest, it is plausible that such a degeneracy can be
lifted if we also consider the charges of the operators under the $SU(2)\times
SU(2)$ isometries, corresponding to the $S^{2}\times S^{2}$ base of $T^{1,1}$
in the KT-KS background.

###  5.3 Nature of the $\widehat{\mathbb{Z}}_{2M}^{(2)}$ Defect

As alluded to earlier, one can construct a stand-alone
$\widehat{\mathbb{Z}}_{2M}^{(2)}$ symmetry operator,

$\mathcal{V}_{1}(\gamma^{1})=\exp\left(\frac{2\pi
i}{2M}\int_{\gamma^{1}}\mathcal{A}_{1}\right)\,,$ (5.14)

with $\gamma^{1}\in H_{1}(\mathcal{W}^{5};\mathbb{Z}_{2M})$, by wrapping a
D1-brane on the $S^{1}$ Hopf fiber of the $T^{1,1}$. The stability of this
D1-brane can be attributed to the fact that the $S^{1}$ fiber is non-vanishing
over the base $\widetilde{S^{2}}$, such that the brane is protected from
shrinking to zero.

The explicit construction of $\mathcal{V}_{1}(\gamma^{1})$ follows closely to
that of $\mathcal{U}_{q}(\Sigma^{2})$ in Section 4, i.e. if we naïvely reduce
the characteristic class over the Hopf fiber, we find

$\int_{S^{1}\subset S^{3}}\mathcal{I}_{3}^{{\rm D}1}=\mathcal{I}_{2}\
\text{``}\\!=\\!\text{''}\ 0$ (5.15)

Much akin to the construction of the $\widehat{\mathbb{Z}}_{M}^{(1)}$
operator, this is simply a side effect of the descent formalism overlooking
information regarding discrete holonomies in our theory. The appropriate
torsional flux that should appear this time can be anticipated, using the
simple fact that the $S^{1}$ Hopf fiber is associated with the $U(1)$, or more
precisely, $\mathbb{Z}_{2M}$ $R$-symmetry. Following the same arguments as
presented around (4.24), we may express
$\mathcal{I}_{2}=\widetilde{\beta}(\mathcal{A}_{1})\in
H^{2}(\mathcal{W}^{5};\mathbb{Z}_{2M})$, and by wrapping the D1-brane on a
cycle $\gamma^{1}$ such that $\partial\widetilde{\gamma}^{2}=2M\gamma^{1}$,
one can reproduce (5.14). More generally, the operator

$\mathcal{V}_{p}(\gamma^{1})=\exp\left(\frac{2\pi
ip}{2M}\int_{\gamma^{1}}\mathcal{A}_{1}\right)\,,$ (5.16)

where $p\sim p+2M$, can be constructed from a stack of $p$ D1-branes.

It is straightforward to show that these operators obey a group-like fusion
rule, i.e.

$\mathcal{V}_{p}(\gamma^{1})\otimes\mathcal{V}_{p^{\prime}}(\gamma^{1})=\mathcal{V}_{p+p^{\prime}}(\gamma^{1})\,.$
(5.17)

Similarly to before, the fusion rule can be understood by considering the
D1-brane used to construct $\mathcal{V}_{1}(\gamma^{1})$ as being instead a
neutral D3-brane wrapping $\gamma^{1}\times S^{1}\times S^{2}$ with one unit
of worldvolume flux through the sphere $S^{2}$. Bringing two such D3-branes
together is equivalent to a single D3-brane with two units of worldvolume
flux, which we can then extrapolate to arbitrary
$p=1,2,\dots,2M-1\in\mathbb{Z}_{2M}$.

##  6 Cross-symmetry Defect Fusion and Braiding

###  6.1 Dissolved Branes

We discussed in Sections 4 and 5 the brane kinematics and dynamics underlying
the parallel fusion rules for
$\mathcal{N}_{p}\otimes\mathcal{N}_{p^{\prime}}$,
$\mathcal{O}_{q}\otimes\mathcal{O}_{q^{\prime}}$,
$\mathcal{U}_{q}\otimes\mathcal{U}_{q^{\prime}}$, and
$\mathcal{V}_{p}\otimes\mathcal{V}_{p^{\prime}}$. These fusion rules alone,
however, only form part of the full-fledged story involving the interplay
between defects associated with different bulk gauge symmetries, as summarized
in Table 2 and derived in Appendix C. In the following, we will discuss the
brane origin of the rest of these fusion rules.

Let us begin with the fusion rule,

$\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{N}_{p}(M^{3})=\mathcal{N}_{p}(M^{3})\,.$
(6.1)

The D-branes associated with $\mathcal{U}_{q}(\Sigma^{2})$ and
$\mathcal{N}_{p}(M^{3})$ are respectively $q$ D3-branes wrapping
$\Sigma^{2}\times\widetilde{S^{2}}$ and $p$ D5-branes wrapping $M^{3}\times
S^{3}$, where $\Sigma^{2}\subset M^{3}$ and $\widetilde{S^{2}}\subset S^{3}$.
It now becomes clear why (6.1) should be the way it is. Since the entire
worldvolume of the D3-branes is a submanifold of the D5-branes, the former can
be absorbed by the latter without a cost. This can be understood by the fact
that D3-branes are dissolved in the D5-branes through the WZ coupling
$da_{1}C_{4}$, where we integrate over all configurations of the (dynamical)
Chan-Paton field $a_{1}$. The effect of embedding extra D3-branes into the
D5-branes then is merely to reshuffle the summands in the said
integration.282828If we instead interpret the defect
$\mathcal{U}_{q}(\Sigma^{2})$ to be coming from F1-strings (or more generally,
a D3-F1 bound state), we can understand the fusion rule as the simple
statement that a D5-brane freely absorbs F1-strings.

Moreover, adding D3-branes to the condensation defect $\mathcal{C}(M^{3})$,
which is itself a sum of all D3-branes localized on $M^{3}(\times S^{3})$, has
the identical effect of reshuffling the summands in the condensation defect.
This naturally explains the fusion rule,

$\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{C}(M^{3})=\mathcal{C}(M^{3})\,.$
(6.2)

Alternatively, locality of the fusion allows us to interpret (6.2) from a
different but equivalent perspective, i.e.

$\mathcal{U}_{q}\otimes\mathcal{C}=\mathcal{U}_{q}\otimes(\mathcal{N}_{p}\otimes\mathcal{N}_{-p})=(\mathcal{U}_{q}\otimes\mathcal{N}_{p})\otimes\mathcal{N}_{-p}=\mathcal{N}_{p}\otimes\mathcal{N}_{-p}=\mathcal{C}\,.$
(6.3)

The expression above tells us that $\mathcal{U}_{q}\otimes\mathcal{C}$ can
equivalently be thought of as a two-step process, namely, absorption of
D3-branes by D5-branes, followed by tachyon condensation on the
D5-$\overline{\text{D5}}$ system. Analogous arguments can be used to provide a
brane interpretation of the following fusion rules,

$\displaystyle\mathcal{V}_{p}(\gamma^{1})\otimes\mathcal{O}_{q}(W^{2})$
$\displaystyle=\mathcal{O}_{q}(W^{2})\,,$
$\displaystyle\qquad\mathcal{V}_{p}(\gamma^{1})\otimes\widetilde{\mathcal{C}}(W^{2})$
$\displaystyle=\widetilde{\mathcal{C}}(W^{2})\,,$ (6.4)
$\displaystyle\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{O}_{q^{\prime}}(W^{2})$
$\displaystyle=\mathcal{O}_{q^{\prime}}(W^{2})\,,$
$\displaystyle\qquad\mathcal{U}_{q}(\Sigma^{2})\otimes\widetilde{\mathcal{C}}(W^{2})$
$\displaystyle=\widetilde{\mathcal{C}}(W^{2})\,.$

We now turn our focus to the fusion between the $\mathbb{Z}_{2M}^{(0)}$ defect
$\mathcal{N}_{p}(M^{3})$ with its associated condensation defect
$\mathcal{C}(M^{3})$. To motivate the subsequent explanation, we utilize the
associativity of the fusion algebra again to rewrite the fusion in the
following manner,

$\displaystyle\mathcal{N}_{p}\otimes\mathcal{C}$
$\displaystyle=\mathcal{N}_{p}\otimes(\mathcal{N}_{p^{\prime}}\otimes\mathcal{N}_{-p^{\prime}})$
$\displaystyle=(\mathcal{N}_{p}\otimes\mathcal{N}_{p^{\prime}})\otimes\mathcal{N}_{-p^{\prime}}$
$\displaystyle=\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}\otimes\mathcal{N}_{p+p^{\prime}}\otimes\mathcal{N}_{-p^{\prime}}$
$\displaystyle=\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}\otimes\mathcal{A}^{M,-pp^{\prime}(p+p^{\prime})}\otimes\mathcal{N}_{p}\,,$
(6.5)

assuming $\mathop{\mathrm{gcd}}(M,p+p^{\prime})=1$. As per the discussion in
Section 4, there is a center-of-mass mode associated with each stack of
D5-branes (or $\overline{\text{D5}}$-branes). Suppose we first fuse $p$ and
$p^{\prime}$ D5-branes. This results in a stack of $p+p^{\prime}$ D5-branes,
in addition to the relative mode $\mathcal{A}^{M,pp^{\prime}(p+p^{\prime})}$
between these two stacks. Next, we fuse this new stack of $p+p^{\prime}$
D5-branes with $p^{\prime}$ $\overline{\text{D5}}$-branes, which results in a
stack of $p$ D5-branes, in addition to the relative mode
$\mathcal{A}^{M,-pp^{\prime}(p+p^{\prime})}$ between the two original stacks.
It turns out that these two minimal TQFTs, each of which is associated with an
independent $\mathbb{Z}_{M}$ symmetry, combines to form a
$\mathbb{Z}_{M}\times\mathbb{Z}_{M}$ 3d DW theory $(\mathcal{Z}_{M})_{0}$
Hsin:2018vcg  (see Appendix C for details).

This result suggests that conceptually, rather than thinking of the
condensation defect as a collection of D3-branes, it is more appropriate to
think of the defect as a “zero-charge” D5-brane with solitonic D3-branes
dissolved within, with an emphasis on “D5-brane” such that we can associate it
with an appropriate notion of a center-of-mass mode. Accordingly, the
decoupled DW theory in the fusion
$\mathcal{N}_{p}\otimes\mathcal{C}=(\mathcal{Z}_{M})_{0}\otimes\mathcal{N}_{p}$
can be interpreted as the relative mode between the $p$ D5-branes and such a
neutral D5-brane. Similar remarks apply to the remaining nontrivial fusion
rules in Table 2, i.e.

$\mathcal{C}\otimes\mathcal{C}=(\mathcal{Z}_{M})_{0}\otimes\mathcal{C}\,,\quad\mathcal{O}_{q}\otimes\widetilde{\mathcal{C}}=(\mathcal{X}^{M,-1})^{\otimes
2}\otimes\mathcal{O}_{q}\,,\quad\widetilde{\mathcal{C}}\otimes\widetilde{\mathcal{C}}=(\mathcal{X}^{M,-1})^{\otimes
2}\otimes\widetilde{\mathcal{C}}\,.$ (6.6)

###  6.2 Commutation Relations

What we have worked out earlier in this paper is the full set of parallel
fusion rules for the symmetry operators we constructed from D-branes. While
the fusion algebra plays a crucial role in the interplay between the symmetry
defects in the bulk, an equally important piece of information is the braiding
rules of these defects. Here we study the case where there is a nontrivial
linking between the manifolds on which the defects are defined. Depending on
the field content of these defects, there may be nontrivial commutation
relations, meaning that they are charged under each other.

To determine which pairs of operators are mutually charged, we first recall
from the SymTFT (2.8) that the conjugate momenta of the various 5d gauge
fields are given by

$\displaystyle\Pi_{b_{2}}$ $\displaystyle=-2\pi M\hat{c}_{2}\,,$
$\displaystyle\qquad\Pi_{\hat{c}_{2}}$ $\displaystyle=2\pi
Mb_{2}\to-2\pi\mathcal{B}_{2}\,,$ (6.7) $\displaystyle\Pi_{\mathsf{A}_{1}}$
$\displaystyle=4\pi Mc_{3}\,,$ $\displaystyle\qquad\Pi_{c_{3}}$
$\displaystyle=-4\pi M\mathsf{A}_{1}\to-2\pi\mathcal{A}_{1}\,.$

Importantly, we see that $(b_{2},\hat{c}_{2})$ and $(\mathsf{A}_{1},c_{3})$
are conjugate pairs. Note that at low energies, the fluctuating fields
$Mb_{2}$ and $2M\mathsf{A}_{1}$ are respectively Higgsed to the discrete gauge
fields $-\mathcal{B}_{2}\in H^{2}(\mathcal{W}^{5};\mathbb{Z}_{M})$ and
$\mathcal{A}_{1}\in H^{1}(\mathcal{W}^{5};\mathbb{Z}_{2M})$ via the
Stückelberg mechanism.

It follows that the commutation relation between the
$\widehat{\mathbb{Z}}_{M}^{(1)}$ defect $\mathcal{U}_{q}(\Sigma^{2})$ and the
$\mathbb{Z}_{M}^{(1)}$ defect $\mathcal{O}_{q^{\prime}}(W^{2})$ is

$\mathcal{U}_{q}(\Sigma^{2})\,\mathcal{O}_{q^{\prime}}(W^{2})=\exp\bigg{(}\frac{2\pi
iqq^{\prime}}{M}\,\ell(\Sigma^{2},W^{2})\bigg{)}\,\mathcal{O}_{q^{\prime}}(W^{2})\,\mathcal{U}_{q}(\Sigma^{2})\,,$
(6.8)

where $\ell(\Sigma^{2},W^{2})$ denotes the linking number of $\Sigma^{2}$ and
$W^{2}$ with respect to $\mathcal{W}^{5}$.292929The linking number we are
using in this context is defined in the sense of Bott1982 . By construction,
$\ell(\Sigma^{2},W^{2})=0$ if $\Sigma^{2}\subset W^{2}$, such that it is
compatible with the parallel fusion rule
$\mathcal{U}_{q}(\Sigma^{2})\otimes\mathcal{O}_{q^{\prime}}(W^{2})=\mathcal{O}_{q^{\prime}}(W^{2})$.
As discussed in Moore:2004jv ; Freed:2006yc ; Freed:2006ya ; Etheredge:2023ler
, the nontriviality of the commutation relation is very natural when
considering the D$p$-branes associated with these operators. In Type II string
theory, D$p$-branes are (electromagnetically) dual to D$(6-p)$-branes, and
particularly, D3-branes are self-dual.303030This duality is not to be confused
with the $SL(2,\mathbb{Z})$ duality of Type IIB string theory, under which the
pair $(B_{2},C_{2})$ transforms as a doublet while $C_{4}$ transforms as a
singlet. Therefore one does expect the commutation relation between
$\mathcal{U}_{q}(\Sigma^{2})$ and $\mathcal{O}_{q^{\prime}}(W^{2})$, both of
which are constructed from D3-branes, to be nontrivial. By the same token, the
$\widehat{\mathbb{Z}}_{2M}^{(2)}$ defect $\mathcal{V}_{p}(\gamma^{1})$ (from
D1-branes) and the $\mathbb{Z}_{2M}^{(0)}$ defect
$\mathcal{N}_{p^{\prime}}(M^{3})$ (from D5-branes) do not commute. More
explicitly, we find

$\mathcal{V}_{p}(\gamma^{1})\,\mathcal{N}_{p^{\prime}}(M^{3})=\exp\bigg{(}\\!-\frac{2\pi
ipp^{\prime}}{2M}\,\ell(\gamma^{1},M^{3})\bigg{)}\,\mathcal{N}_{p^{\prime}}(M^{3})\,\mathcal{V}_{p}(\gamma^{1})\,.$
(6.9)

where $\ell(\gamma^{1},M^{3})$ again denotes the linking number of
$\gamma^{1}$ and $M^{3}$ with respect to $\mathcal{W}^{5}$.

For completeness, let us provide a broad-brush argument for the commutation
relation between $\mathcal{N}_{p}(M^{3})$ and $\mathcal{O}_{q}(W^{2})$, as was
also studied by Apruzzi:2022rei . The corresponding manifolds typically do not
link in $\mathcal{W}^{5}$, in which case these two operators commute. However,
if we consider a refined case where $W^{2}\cong S^{1}\times L^{1}$, with
$S^{1}\subset M^{3}$, then heuristically the commutation relation takes the
following form,

$\mathcal{N}_{p}(M^{3})\,\mathcal{O}_{q}(S^{1}\times
L^{1})\sim\exp\bigg{(}\\!-2\pi
ipq\,\ell(L^{1},M^{3})\int_{S^{1}}a_{1}\bigg{)}\,\mathcal{O}_{q}(S^{1}\times
L^{1})\,\mathcal{N}_{p}(M^{3})\,.$ (6.10)

The “charge” here is not a c-number phase, but rather a holonomy of the Chan-
Paton field of (multiple) open F1-strings, whose endpoints trace out the two
copies of $S^{1}$ respectively embedded in $M^{3}$ and $W^{2}$. Via the open-
closed string duality, one may equivalently interpret this process as the
creation of a 2d worldsheet of closed F1-strings when the D3-brane operator
$\mathcal{O}_{q}(S^{1}\times L^{1})$ is dragged through the D5-brane operator
$\mathcal{N}_{p}(M^{3})$. Note that the Chan-Paton field $a_{1}$ is a
trivialization of the NSNS field $b_{2}$ on the boundary of this worldsheet
(with $b_{2}\to-\mathcal{B}_{2}/M$ at low energies), so we may schematically
express the action of $\mathcal{N}_{p}(M^{3})$ on $\mathcal{O}_{q}(S^{1}\times
L^{1})$ as

$\mathcal{N}_{p}(M^{3}):\quad\mathcal{O}_{q}(S^{1}\times
L^{1})\to\exp\bigg{(}\frac{2\pi ipq}{M}\,\ell(L^{1},M^{3})\int_{S^{1}\times
I}\mathcal{B}_{2}\bigg{)}\,\mathcal{O}_{q}(S^{1}\times L^{1})\,,$ (6.11)

where $I$ is the interval along the transverse direction between
$\mathcal{N}_{p}(M^{3})$ and $\mathcal{O}_{q}(S^{1}\times L^{1})$. Such a
phenomenon is akin to the Hanany-Witten transition Hanany:1996ie , where a
D3-brane is created when a D5-brane crosses an NS5-brane (each wrapping
manifolds along appropriate directions). By successively applying T and S
dualities on this brane system, i.e.

$(\text{NS5--D3--D5})\xlongrightarrow{\text{T}^{2}}(\text{NS5--D1--
D3})\xlongrightarrow{\text{S}}(\text{D5--F1--D3})\,,$

we arrive at the desired combination of branes in our construction.

##  7 Boundary Conditions and Field Theory

In the previous sections, we have been primarily concerned with the presence
and action of the symmetry defects in the 5d bulk gauge theory, where all the
discussed operators are realized and interact in nontrivial ways. While the
bulk alone has an involved structure, there is another half of the story,
namely, the symmetry structure of the 4d boundary field theory is determined
by choices of (Neumann or Dirichlet) boundary conditions for the bulk gauge
fields Maldacena:2001ss ; Belov:2004ht ; Gaiotto:2014kfa ; Gaiotto:2020iye ;
Bah:2020uev . Importantly, these choices also prescribe the spectrum of
topological operators existing on the boundary.

###  7.1 Allowed Choices from the SymTFT

Owing to the canonical quantization imposed on conjugate gauge fields, one
cannot simultaneously pick Neumann (or Dirichlet) boundary conditions on the
pair of fields. In addition, there are potential obstructions from the anomaly
terms in the SymTFT that forbids us from imposing Neumann boundary condition
on a given gauge field. In the following, we are going to systematically study
all the allowed choices of boundary conditions for the 5d bulk gauge fields in
the KT-KS setup.

Recall that the SymTFT is given by

$S_{\rm
top}=2\pi\int_{\mathcal{W}^{5}}\bigg{(}Mb_{2}d\hat{c}_{2}-2Mc_{3}d\mathsf{A}_{1}+b_{2}\mathcal{A}_{1}\mathcal{B}_{2}-\mathsf{A}_{1}\mathcal{B}_{2}^{2}-\frac{\mathcal{K}}{M}\,\mathcal{A}_{1}\beta(\mathcal{A}_{1})\beta(\mathcal{A}_{1})\bigg{)}\,,$
(7.1)

where the first two terms are BF couplings respectively pairing $b_{2}$ and
$\hat{c}_{2}$, and $c_{3}$ and $\mathsf{A}_{1}$. For concreteness, let us
consider the former pair; the story for the latter pair is completely
analogous. Varying the topological action, we find

$\delta S_{\rm
BF}=2\pi\int_{\partial\mathcal{W}^{5}}Mb_{2}\delta\hat{c}_{2}\,,$ (7.2)

where we have made a choice in having the variation on $\hat{c}_{2}$. By
adding a boundary term we may equivalently view the variation as $\delta
S_{\rm BF}^{\prime}=2\pi\int_{\partial\mathcal{W}^{5}}M\hat{c}_{2}\delta
b_{2}$. Because the variation is first-order in derivatives, we must impose
Dirichlet boundary condition on one of the gauge fields, and Neumann boundary
condition on the other to have a well-defined variational problem.313131These
boundary conditions are topological boundary conditions as they do not require
a metric on the boundary manifold, and are indeed invariant under
diffeomorphisms of the boundary manifold. One can also discuss topological
boundary conditions in theories other than BF theories. One such example is 3d
Abelian Chern-Simons theory, where the existence of topological boundary
conditions depends on the choice of the coefficient matrix for the theory
Kapustin:2010hk . The difference between these two choices corresponds to
different global structures for the gauge group of the 4d $\mathcal{N}=1$ SYM
field theory living on the boundary $\mathcal{W}^{4}=\partial\mathcal{W}^{5}$.
Specifically, an $SU(M)$ gauge theory has a $\mathbb{Z}_{M}^{(1)}$ (electric)
global symmetry, whereas a $PSU(M)$ gauge theory has a
$\widehat{\mathbb{Z}}_{M}^{(1)}$ (magnetic) global symmetry. In general, one
may consider a larger family of boundary conditions when $M=kk^{\prime}$ is
not prime. This realizes a $\mathbb{Z}_{k}\times\mathbb{Z}_{k^{\prime}}$
global symmetry on the boundary 4d $\mathcal{N}=1$ $SU(M)/\mathbb{Z}_{k}$ SYM
theory. We leave the study of such cases to future work.

The remaining cubic terms in (7.1) place additional constraints on the allowed
choices of boundary conditions for the bulk gauge fields. From the point of
view of the boundary field theory, these interaction terms describe ’t Hooft
anomalies between different global symmetries. For example, the
$\mathsf{A}_{1}\mathcal{B}_{2}^{2}$ term is a mixed ’t Hooft anomaly between
the $\mathbb{Z}_{2M}^{(0)}$ and $\mathbb{Z}_{M}^{(1)}$ symmetries. Imposing
Neumann boundary conditions on $b_{2}$ (and $\mathcal{B}_{2}$ by extension)
amounts to gauging the $\mathbb{Z}_{M}^{(1)}$ symmetry on the boundary, which
causes the $\mathbb{Z}_{2M}^{(0)}$ symmetry to become non-invertible
Kaidi:2021xfk ; Apruzzi:2021phx ; Apruzzi:2022rei . Meanwhile, the self-
anomaly term $\mathcal{A}_{1}\beta(\mathcal{A}_{1})\beta(\mathcal{A}_{1})$
indicates $\mathsf{A}_{1}$ (and $\mathcal{A}_{1}$) cannot be given Neumann
boundary conditions, reflecting the fact that one cannot gauge the anomalous
$\mathbb{Z}_{2M}^{(0)}$ R-symmetry in the 4d $\mathcal{N}=1$ SYM field theory.

###  7.2 Boundary Conditions and Branes

The choice of boundary conditions for the bulk supergravity fields determines
the set of allowed configurations of the probe D$p$-branes in
$\mathcal{W}^{5}$. A priori, there are $2^{4}=16$ choices of boundary
conditions on the set of fields $(b_{2},\hat{c}_{2},c_{3},\mathsf{A}_{1})$.
However, $\mathsf{A}_{1}$ only admits Dirichlet boundary condition due to its
self-anomaly, so its conjugate, $c_{3}$, only admits Neumann boundary
condition. This narrows the number of combinations down to 2. Table 3
summarizes the consequences of each of these two choices in the corresponding
boundary field theories.

Boundary Conditions | (D,N,N,D) | (N,D,N,D)
---|---|---
Gauge Group | $SU(M)$ | $PSU(M)$
Global Symmetries | $\mathbb{Z}_{2M}^{(0)},\;{\mathbb{Z}}_{M}^{(1)}$ | $\mathbb{Z}_{2M}^{(0)},\;\widehat{\mathbb{Z}}_{M}^{(1)}$
Gauged Symmetries | $\widehat{\mathbb{Z}}_{2M}^{(2)},\;\widehat{\mathbb{Z}}_{M}^{(1)}$ | $\widehat{\mathbb{Z}}_{2M}^{(2)},\;{\mathbb{Z}}_{M}^{(1)}$
Symmetry Generators | $\widetilde{\mathcal{N}}_{p}(M^{3}),\;\widetilde{\mathcal{O}}_{q}(W^{2})$ | ${\mathcal{N}}_{p}(M^{3}),\;\mathcal{U}_{q}(\Sigma^{2})$
Charged Operators | $\overline{\mathcal{V}}(\overline{x}),\;\overline{\mathcal{U}}(\overline{\gamma}^{1})$ | $\overline{\mathcal{V}}(\overline{x}),\;\overline{\mathcal{O}}(\overline{W}^{1})$
Table 3:  The allowed boundary conditions for the four bulk gauge fields
$(b_{2},\hat{c}_{2},c_{3},\mathsf{A}_{1})$, and their effect on the boundary
field theory. Here and in the main text “N” (“D”) denotes Neumann (Dirichlet)
boundary condition imposed on a given gauge field.

#### (D,N,N,D) boundary conditions.

To see the effect of boundary conditions concretely, let us again focus on the
$(b_{2},\hat{c}_{2})$ BF pair discussed in the previous subsection, keeping in
mind that $c_{3}$ and $\mathsf{A}_{1}$ must be given Neumann and Dirichlet
boundary conditions respectively. We first consider the case where we impose
Neumann boundary condition on $\hat{c}_{2}$. By the variational principle,

$Mb_{2}\delta\hat{c}_{2}\big{|}_{\mathcal{W}^{4}}=0\,,$ (7.3)

we must require that $Mb_{2}$, and by extension $\mathcal{B}_{2}$, vanish on
the conformal boundary $\mathcal{W}^{4}$. In terms of the symmetry structure
of the boundary field theory, this is equivalent to choosing a gauge group of
$SU(M)$ rather than $PSU(M)$.

With this choice of boundary conditions it is straightforward to see from
(4.27) that $\mathcal{U}_{q}(\Sigma^{2})$ becomes a trivial operator when
pushed to the boundary. This is expected for a theory with gauge group
$SU(M)$, as the defects $\mathcal{U}_{q}$ would generate the magnetic 1-form
symmetry not present in this theory. Instead of having the D3-branes being
pushed parallel to $\mathcal{W}^{4}$, we can consider transverse D3-branes
such that one of their external dimensions is aligned with the radial
coordinate of $\mathcal{W}^{5}$. Using (7.3), we can write $b_{2}$ in terms of
a $\mathbb{Z}_{M}$-valued flat connection $\mathcal{B}_{1}$ defined only on
the conformal boundary, i.e.

$Mb_{2}\big{|}_{\mathcal{W}^{4}}=d\mathcal{B}_{1}=0\,,$ (7.4)

in which case the resultant operator becomes

$\overline{\mathcal{U}}(\overline{\gamma}^{1}\subset\mathcal{W}^{4})=\exp\left(\frac{2\pi
i}{M}\int_{\overline{\gamma}^{1}}\mathcal{B}_{1}\right)\,,$ (7.5)

with $\gamma^{1}$ closed. The commutation relation (6.8), when localized to
$\mathcal{W}^{4}$, tells us that such a line operator living in the boundary
field theory is charged under the $\mathbb{Z}_{M}^{(1)}$ electric symmetry
generated by the $\mathcal{O}_{q}$ defects. Following the discussion below
(4.27), this could have been equivalently argued from an alternative
construction of the $\mathcal{U}_{q}$ defects using F1-strings. It is known in
the literature that F1-strings ending on the conformal boundary trace out
Wilson lines Maldacena:1998im . The line operators
$\overline{\mathcal{U}}(\overline{\gamma}^{1})$ should therefore be identified
with these Wilson lines in the $SU(M)$ gauge theory.323232More accurately, the
$\overline{\mathcal{U}}$ operators describe the topological data of a Wilson
line in the picture of Gaiotto:2020iye ; Freed:2022qnc .

Similarly to before, imposing Neumann boundary condition on $c_{3}$ leads to

$2M\mathsf{A}_{1}\big{|}_{\mathcal{W}^{4}}=d\mathcal{A}_{0}=0\,,$ (7.6)

where $\mathcal{A}_{0}\in\\{0,\dots,2M-1\\}$ is a flat
$\mathbb{Z}_{2M}$-valued 0-form field. This corresponds to a local operator
$\overline{\mathcal{V}}(\overline{x}\in\mathcal{W}^{4})$, where
$\overline{x}=\partial\gamma^{1}$, that is charged under the
$\mathbb{Z}_{2M}^{(0)}$ symmetry generated by the $\mathcal{N}_{p}$ defects
according to (6.9).

The (D,N,N,D) boundary conditions also affect the non-invertibility of the
$\mathbb{Z}_{M}^{(1)}$ symmetry generator. From either (5.6) or (5.7), we see
that $\mathcal{O}_{q}$ simplifies dramatically when $\mathcal{A}_{1}$ and
$\mathcal{B}_{2}$ have Dirichlet boundary conditions, i.e.

$\widetilde{\mathcal{O}}_{q}\left(W^{2}\subset\mathcal{W}^{4}\right)=\exp\left(2\pi
iq\int_{W^{2}}\hat{c}_{2}\right)\,.$ (7.7)

Note that the $b_{2}\mathcal{A}_{1}\mathcal{B}_{2}$ anomaly trivializes as
both fields $\mathcal{A}_{1}$ and $\mathcal{B}_{2}$ are turned off on
$\mathcal{W}^{4}$, so the D3-D1 operator no longer couples to the TQFT
$\mathcal{X}^{M,q}$ on which the auxiliary fields $\lambda_{1}$ and $\varphi$
live. Importantly, $\widetilde{\mathcal{O}}_{q}$ is an invertible operator,
which agrees with our expectation that this defect generates the (invertible)
$\mathbb{Z}_{M}^{(1)}$ center symmetry in the $SU(M)$ gauge theory. By the
same token, the $\mathsf{A}_{1}\mathcal{B}_{2}^{2}$ anomaly trivializes on
$\mathcal{W}^{4}$, so the $\mathbb{Z}_{2M}^{(0)}$ defect also becomes an
invertible operator Apruzzi:2022rei , i.e.

$\widetilde{\mathcal{N}}_{p}\left(M^{3}\subset\mathcal{W}^{4}\right)=\exp\left(2\pi
ip\int_{M^{3}}c_{3}\right)\,.$ (7.8)

This conforms with the fact that the $\mathbb{Z}_{2M}^{(0)}$ $R$-symmetry is
invertible in the $SU(M)$ gauge theory.

#### (N,D,N,D) boundary conditions.

Let us now consider the case where $\hat{c}_{2}$ is chosen to have Dirichlet
boundary conditions instead of $b_{2}$. This is equivalent to having a
$PSU(M)$ gauge theory on the boundary. As has been discussed previously, this
causes the $\mathcal{U}_{q}$ operators to generate an invertible
$\widehat{\mathbb{Z}}_{M}^{(1)}$ symmetry and the $\mathcal{N}_{p}$ operators
to generate a non-invertible $\mathbb{Z}_{2M}^{(0)}$ symmetry on the boundary.
Furthermore, the bulk line operator $\mathcal{V}_{p}$ ends on the boundary to
become the local operator
$\overline{\mathcal{V}}(\overline{x}\in\mathcal{W}^{4})$. What remains to be
discussed is the fate of the $\mathcal{O}_{q}$ operators. When $\hat{c}_{2}$
is chosen to vanish on $\mathcal{W}^{4}$, the defect $\mathcal{O}_{q}$ takes
the form,

$\mathcal{O}_{q}\left(W^{2}\subset\mathcal{W}^{4}\right)=\int\mathcal{D}\varphi\mathcal{D}\lambda_{1}\exp\left(2\pi
i\int_{W^{2}}\varphi\mathcal{B}_{2}-M\varphi d\lambda_{1}\right)\,,$ (7.9)

where we have kept in mind that $\mathcal{A}_{1}\big{|}_{\mathcal{W}^{4}}=0$
due to the self-anomaly term. In the above, $\lambda_{1}$ acts as a Lagrange
multiplier causing $\varphi$ to take on integer values, such that the
remaining operator turns into a (normalized) sum over trivial
operators.333333Alternatively, we can see that the
$\mathcal{A}_{1}\mathcal{B}_{2}$ anomaly again vanishes on the conformal
boundary due to the Dirichlet boundary condition on $\mathcal{A}_{1}$. The
associated TQFT involving $\varphi$ and $\lambda_{1}$ will then vanish and the
operator $\mathcal{O}_{q}$ will still be trivial. This implies that the D3-D1
operator trivializes when placed parallel to the conformal boundary.

Nonetheless, we may consider D3-D1 bound states that instead end transversely
to $\mathcal{W}^{4}$, i.e. we take one of the directions in $W^{2}$ to align
with the radial coordinate of $\mathcal{W}^{5}$. From the variation (7.3), we
can write on the boundary,

$M\hat{c}_{2}\big{|}_{\mathcal{W}^{4}}=d\hat{\mathcal{C}}_{1}=0\,,$ (7.10)

where $\hat{\mathcal{C}}_{1}$ is a $\mathbb{Z}_{M}$-valued flat connection.
The associated operator on $\mathcal{W}^{4}$ is then given by

$\overline{\mathcal{O}}\big{(}\overline{W}^{1}\subset\mathcal{W}^{4}\big{)}=\exp\left(\frac{2\pi
i}{M}\int_{\overline{W}^{1}}\hat{\mathcal{C}}_{1}\right)\,,$ (7.11)

which is charged under the $\widehat{\mathbb{Z}}_{M}^{(1)}$ magnetic symmetry
generated by the $\mathcal{U}_{q}$ defects. Hence, the line operators
$\overline{\mathcal{O}}(\overline{W}^{1})$ should be identified with ’t Hooft
lines in the $PSU(M)$ gauge theory.

##  8 Discussion and Outlook

In this paper, we have studied in detail the complete fusion algebra of gauge
symmetry generators in the 5d bulk $\mathcal{W}^{5}$ of the KT-KS background
(see Table 2 for a summary). More precisely, the gauge symmetries of the bulk
are encoded by the SymTFT (2.8), whose kinetic terms are suppressed near the
conformal boundary $\mathcal{W}^{4}\subset\mathcal{W}^{5}$. Via the Gauss Law
constraints, the SymTFT indicates the existence of topological extended
operators (near $\mathcal{W}^{4}$) which generate the gauge symmetries.

Intuitively, the candidates for these operators are nothing but the solitonic
objects in Type II string theory, i.e. D-branes, wrapping suitable internal
submanifolds. We find that this is indeed the case, and particularly, the WZ
coupling of D-branes to lower-degree RR fields naturally reproduces the mixed
anomaly terms in the SymTFT. The agreement between the symmetry generators and
the D-branes therefore serves as a nontrivial check for the consistency
between the Type IIB supergravity action and the WZ action of D-branes. An
even more stringent test can be carried out by further including the
gravitational terms respectively in the 11-form Type IIB anomaly polynomial
$\mathcal{I}_{11}$ and the WZ action of the D-branes.343434We expect in this
case that the mixed chiral-gravitational anomaly of 4d $\mathcal{N}=1$ $SU(M)$