Non-Invertible Symmetries, Brane Dynamics, and Tachyon Condensation

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.


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 [1], 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 ∂X d+1 = M d [1][2][3]. Such TQFTs are referred to as Symmetry Topological Field Theories (SymTFTs) [4]; they naturally emerge via anomaly inflow methods for QFTs engineered from string theory and holography [4][5][6][7]. From this point of view, string theory provides an important avenue for studying topological symmetries and their role in the dynamics of QFTs.
In this paper, we will explore the SymTFT associated to 4d N = 1 Super Yang-Mills (SYM) theory with gauge algebra 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 [18] and Klebanov-Strassler [19] (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 [8], 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 [8]. 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 Symmetry Action Symmetry Generator Brane Configuration D3 wrapping Hopf base S 2 ⊂ S 3 Table 1: Symmetry generators/defects and their D-brane realizations. 1 operators which generate a bulk gauge symmetry from the SymTFT [8,20]. We then match each symmetry generator with a specific Dp-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.
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 Dp-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 Dp-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 noninvertibility of operators can be attributed to induced brane charges due to bulk-brane dynamics. For example, the operator O 1 (W 2 ) presented in Table 1 takes the explicit form, where λ 1 and ϕ 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 Z M × 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 Table 2: Fusion rules for the symmetry defects. The left column indicates the schematic fusion in terms of the associated Dp-branes, while the right column shows the explicit fusion rules. The coefficients A M,p , X M,q , and (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 [8][9][10][21][22][23]. We find that these defects are described naturally in string theory via tachyon condensation. When a Dp-and a Dp-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 (N , U) and (O, 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.

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. [4][5][6][7]24]). 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 [7]. 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 [8], 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, 2 where G 3 and H 3 are the field strengths for the RR 2-form and NSNS 2-form potentials respectively, and G 5 is defined such that G 5 = (1 + 10 )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, 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 ω 3 and Ω 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 ∧ G 3 , impose constraints on the external field strengths written above. These extra constraints reduce to the 5d Bianchi identities We may solve the leftmost equations via where H 3 , C 3 , F 2 ∈ H * (W 5 ; Z) and b 2 , c 2 , and A 1 are globally well-defined gauge potentials. 3 In the later computations 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 Z 2M and Z M gauge fields via the Stückelberg mechanism. As a result, we may solve the Bianchi identities as where A 1 ∈ H 1 (W 5 ; Z 2M ) and B 2 ∈ H 2 (W 5 ; Z M ), while c 0 and β 1 are globally well-defined gauge potentials. The coefficients of the cohomology groups in which A 1 and B 2 live can be understood as follows. The Bianchi identites in (2.3) put constraints on the basepoint fluxes F 2 and H 3 that we have written above, i.e. 2M F 2 and M H 3 must vanish identically for the right-hand side of the relations to be an exact cocycle. 4 This requirement implies that F 2 and H 3 are each contained in the image of a Bockstein homomorphism. Specifically, we may express In the above β and β are respectively the Bockstein maps 5 The fact that these are valued in integer cohomology follows from the flux quantization of the RR field strengths [25]. 4 In general one might demand 2M F2 to be simply cohomologically trivial rather than to vanish identically, but such an exact piece can be removed via a field redefinition of A1. 5 A brief review of Bockstein homomorphisms can be found in Appendix B.
With this notation we can now compute the 5d SymTFT by reducing (2.1) over the conifold T 1,1 . The topological action S top takes the following form,

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 W 5 as a time coordinate and performing a Hamiltonian analysis of the 5d bulk action [20,31,32]. 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 ,ĉ 2 , A 1 , and c 3 will impose a Gauss Law constraint, which were originally computed in [8] and we restate them here for convenience, The quantities 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 2πi M i P i acts as a symmetry generator supported on some submanifold M i ⊂ W 5 of the appropriate dimension. In contrast, if the symmetry is broken down to a finite group, e.g. Z k , then the proper generator to consider is the "k-th" root of the aforementioned one, i.e. D ∼ exp (2πi/k) M i P i , such that D ⊗k ∼ = 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 G A 1 . Using the prescription above, one would naïvely construct an operator of the form, 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 B 2 to a dynamical U (1) gauge field a 1 localized on M 3 [8,22], i.e.
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 Z 2M 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 N 1 (M 3 ) can be schematically decomposed into two pieces, i.e. 13) where 2M R-symmetry rotation, and A M,1 [B 2 ] is a 3d minimal TQFT associated with the Z (1) M symmetry [33]. Note that the simultaneous appearance of these two operators in N 1 (M 3 ) results from the presence of the anomaly term A 1 B 2 2 in (2.8). An analogous decomposition of the chiral symmetry defect can be found in [22] in the context of 4d massless QED (see also, e.g. [23,34] 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, Z M , and Z (1) M , using the Gauss Law constraints G c 3 , G b 2 , and Gĉ 2 . Explicitly, these defects take the respective forms, When constructing V 1 and U 1 , we have made use of the discussion surrounding (2.9).
Analogously to the previous case of N 1 , the operator 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 A 1 and B 2 respectively to dynamical U (1) gauge fields λ 1 and ϕ localized on W 2 , i.e.
For convenience, let us define the shorthand notation, assuming gcd(M, q) = 1. The anomaly label q ∼ q + M specifies the braiding statistics between λ 1 and ϕ. Note that in the absence of the coupling to A 1 and B 2 , the TQFT X M,1 is an untwisted 2d Z M × Z M BF theory. 9 Similarly to the defect N 1 (M 3 ), we observe that where O 1 (W 2 ) = exp 2πi W 2ĉ2 is an invertible operator generating a Z M transformation, and the appearance of the 2d TQFT X M,1 [A 1 , B 2 ] is a consequence of the interaction term b 2 A 1 B 2 in (2.8).

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 ≥ 2. The hallmark of such defects is two-fold: 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-symmetrygenerating 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 Dp-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 Dp-brane to the bulk RR and NSNS fields. To construct a topological operator out of a probe Dp-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 ∼ r p , causing the DBI term to decouple. This leaves us with the WZ action (see, e.g. [36,37]), given by where µ p is the charge of a single Dp-brane, and E is the Chan-Paton bundle over the worldvolume M p+1 . We sum over even q in Type IIA and odd q in Type IIB string theory. The quantities R T = 2πR T and R N = 2πR N denote respectively the (normalized) curvature 2forms of the tangent and normal bundles of the worldvolume M p+1 of the Dp-branes, whilê 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 Dp-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 ch B (E) = e B 2 +F 2 . Such couplings are physically interpreted as Dq-branes (q < p) being dissolved in M p+1 [38]. The numbers of these Dq-branes are determined by the worldvolume Chan-Paton flux on the Dp-brane. As a simple example, suppose M p+1 = M 2 × M p−1 , and the Chan-Paton field strength decomposes as F 2 =f 2 + f 2 withf 2 ∈ H 2 ( M 2 ; Z) and f 2 ∈ H 2 (M p−1 ; Z). The Chern character factorizes as and so the number, k, of D(p − 2)-branes within the Dp-brane is given by k = M 2f2 .
Let us now consider a single Dp-brane with zero net Dp charge, i.e. M p+1 C p+1 = 0, but with k units of worldvolume flux through 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 E on the brane stack has structure group U (k). Suppose we parametrize the u(k)-valued field strength F 2 = F 2 + f 2 1 k , with F 2 being su(k)-valued and f 2 ∈ H 2 (M p+1 ; Z), then the Chern character factorizes as We therefore recover the second term in (3.2) with k = M 2f2 . 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 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 Dp-branes.

D-brane Anomaly Polynomial
As for any Chern-Simons action, we can naturally construct an anomaly polynomial associated with the Dp-brane via descent. It can be defined locally as 10 The RR and NSNS field strengths locally take the forms, In the presence of backgroud D-branes, or when [H 3 ] ∈ H 3 (M p+1 ; Z) is nontrivial [39,40], the RR field strengths G q+2 can be topologically nontrivial. Globally, they should be defined by the Bianchi identities, 11 dG q+2 = −H 3 G q , (3.6) 10 Formally, I Dp p+2 is defined on a manifold M p+2 such that ∂M p+2 = M p+1 is the worldvolume of the Dp-brane. We also assume the gauge fields originally defined on M p+1 can be extended to M p+2 . 11 Equivalently, the Bianchi identities for the various RR fields Gq+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 [41,42].
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 (2.8) of the SymTFT is derived by reducing I 11 of Type IIB string theory on the conifold base T 1,1 of the KT-KS background. Similarly, one can reduce I Dp p+2 of a Dp-brane on some n-manifold X n ⊂ 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 W 5 .
We claim that each of the four symmetry generators, , and U 1 (Σ 2 ), derived in Section 2 with a Gauss Law constraint admits a D-brane origin. The first symmetry generator we are interested in is gauge symmetry. From the bulk string theory setup, this defect can be realized by reducing a D5-brane on S 3 ⊂ T 1,1 [8]. With this interpretation, the dynamical gauge field a 1 in A M,1 [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. Defect fusion. Let us study the parallel fusion between two copies of N 1 defined on the same 3-manifold M 3 . Here we would like to neglect the D5-brane origin of 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 N 1 fuse invertibly as N 1 ⊗ N 1 = N 2 = exp 4πi M 3 c 3 . 12 On the other hand, the minimal TQFT A M,1 [B 2 ] satisfies the fusion rule [22], assuming gcd(M, 2) = 1, i.e. M is odd. Note that the first copy of A M,2 is a decoupled TQFT [43], while the second one couples to the Z M gauge field B 2 . Combining the results, we conclude that the fusion of two R-symmetry defects is given by In general, one may construct an operator N p analogously to how we defined N 2 above, i.e. we take the naïve operator N p and stack it with a minimal TQFT A M,p [B 2 ]. The latter has an anomaly label p ∼ p + M dictating the braiding statistics of the anyons living on M 3 . This implies that A M,M ∼ = A M,0 is a trivial theory, so the operator N M ∼ = N M reduces to an invertible operator when we take p = M . The operator N M therefore generates the anomalyfree Z (0) 2 subgroup of the Z (0) 2M R-symmetry that is associated with the (−1) F operator [19,44]. It is important, however, to note that N M has a periodicity p ∼ p + 2M , and the same applies to the full operator N p .
We show in Appendix C that if gcd(M, p) = gcd(M, p ) = gcd(M, p + p ) = 1, 13 the fusion between general R-symmetry defects follows which reduces to (4.2) when p = p = 1. It should be emphasized again that the factor A M,pp (p+p ) does not couple to B 2 . Since A M,pp (p+p ) is symmetric in the labels p and p , the fusion rule (4.3) is manifestly commutative.
Dielectric branes. The above field-theoretic fusion for the N p operators can alternatively be understood geometrically through the dynamics of probe Dp-branes in the dual string theory. The N 1 ⊗ N 1 fusion is modeled by two coincident D5-branes on the M 3 × S 3 needed to generate the Z (0) 2M 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 Dp-branes will polarize and "puff up" into an S 2 transverse to the original stack [8,[45][46][47], 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 Dp-brane charge. In the case at hand, when two D5-branes wrapping M 3 × S 3 are brought close to one another, the configuration becomes indistinguishable from a single D7-brane wrapping M 3 × T 1,1 with two units of worldvolume flux through the S 2 ⊂ T 1,1 .
There is a subtlety arising from the description of two D5-branes polarizing into a D7brane. 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 ⊂ T 1,1 on which the D5-branes are wrapped. The reduction leads to exp 2πi where ∂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 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 where a i 1 denotes the Chan-Paton gauge field along S 3 , and f i 2 represents the worldvolume flux such that S 2 f i 2 = 1. We can then perform the reduction over T 1,1 using (2.2) to find where we have made the field redefinitions 2α 1 = a 1 1 + a 1 2 and α 1 = −a 1 1 + a 2 1 . Not only have we recovered the action for N 2 (M 3 ), but we have also gained an additional term. This term encodes the information of the minimal TQFT A M,2 −1 . 14 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 gcd(M, 2) = 1. Importantly, it is an integer so that (4.6) is properly quantized. We now further utilize the equivalence of minimal TQFTs [33], by rescaling the generating line a = e 2πi α 1 of a minimal TQFT as a → a r , for any r such that gcd(M, r) = 1. Setting r = 2 gives A M,2 −1 ∼ = 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 N p ⊗ N p . We can repeat the above computation for stacks of p and p coincident D5-branes, by viewing each stack as a single D7-brane with p and p units of D5-brane charge respectively, i.e. S 2 f 1 2 = p and S 2 f 2 2 = p . The reduction over T 1,1 now becomes where the required field redefinition here is (p + p )α 1 = pa 1 1 + p a 2 1 and α 1 = −a 1 1 + a 2 1 . This exactly reproduces the general fusion in (4.3), with the first term describing A M,pp (p+p ) −1 ∼ = A M,pp (p+p ) , and the terms in parentheses describing N p+p . Note that the latter can equivalently be described as a single D7-brane with p + p units of worldvolume flux. 14 We should be careful as there is an ambiguity in this description. As was discussed in [33], U (1)mn ∼ = A mn,1 ∼ = A m,n ⊗ A n,m are isomorphic as topological field theories. Furthermore, the effective Lagrangian descriptions for A m,n , 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.
In the former case, there is a center-of-mass mode α for the overall p + p + p stack, and two other relative modes α and α respectively for the p -(p + p ) and pp stacks. The same applies to the figure on the right.
Collective degrees of freedom. The conclusions above can be recasted from the perspective of the original stacks of p and p D5-branes. We see that the gauge field α 1 ∼ pa 1 1 + p a 2 1 defined on N p+p is the overall center-of-mass mode for the combined stack of p+p D5-branes. Likewise, a 1 1 and a 2 1 are the center-of-mass modes of the individual stacks. The other gauge field, α 1 , should be interpreted as a mode describing the relative motion of the two stacks of p and p 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) × U (1) CM . The same is true of course for the stack of p D5-branes as well as the final stack of p + p D5-branes. However, this final stack has another Abelian mode contained within it. Within SU (p + p ) there is a maximal subgroup given by where the U (1) rel is generated by the Cartan element diag(p 1 p , −p1 p ). It is this U (1) rel that is describing the relative motion of the D5-brane stacks, which manifests itself as the decoupled minimal TQFT A M,pp (p+p ) at the level of the fusion algebra (4.3). Note that if we were to start with stacks of, say, p + k and p − k D5-branes, then we would end up with a different U (1) rel , as is reflected by a different decoupled minimal TQFT A M,(p+k)(p −k)(p+p ) in the fusion.
We can carry out the same argument for more than two stacks of D5-branes. Consider, for instance, three stacks of p, p , and p 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 2 1 , and a 3 1 , but we can repackage these degrees of freedom in different combinations. We may first consider an overall center-of-mass mode, α 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 α 1 , as well as a mode describing the relative motion of the two rightmost D5-brane stacks given by α 1 . The explicit expressions for α 1 , α 1 , α 1 can be written as By regrouping our degrees of freedom in this way, we are effectively perfoming the fusion 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 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 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 Dp-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.

N p ⊗ N −p Fusion and Tachyon Condensation
Condensation defect. The case of p = −p mod M requires special treatment. As discussed above, the N p (M 3 ) component of the defect is invertible and thus becomes the identity for the N p ⊗ N −p fusion. Regarding the fusion between the minimal TQFTs, we have, in terms of explicit worldvolume actions, where a 1 and a 1 respectively denote the auxiliary fields living on A M,p and A M,−p , whilē a 1 = p(a 1 − a 1 ). In the third line, we used the fact that if M is odd, the Dijkgraaf-Witten (DW) term in (Z M ) K is trivial when K is an integer multiple of M , i.e. K ∼ K +M (assuming M 3 is a spin manifold) [48]. 15 Integrating out a 1 enforces We then rewrite (4.13) as a discrete sum of the Z where Σ 2 is the Poincaré dual of the torsional reduction ofā 1 with respect to M 3 . We will refer to 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 Z (1) M symmetry [43]. The symmetry operators condense on the submanifold M 3 of spacetime, hence the name. Note that the factor of |H 0 (M 3 ; Z M )| is a standard gauge-theoretic normalization coming from the normalization of the path integral measure [1,23].
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 [49,50]. Our starting point is a Dp -Dp-brane pair wrapped on a submanifold M p+1 of the total spacetime, where an anti-Dp-brane is the orientation reversal of a Dp-brane. Over the Dpbrane we denote the Chan-Paton bundle by E + , while over the anti-Dp-brane we denote the bundle E − . Without loss of generality, we may take E − = K and E + = L ⊗ K for L and K complex line bundles. 17 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 where (E − ) * denotes the dual bundle, reflecting that it is charged under the U (1) × 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 where s is a section of L which vanishes over a codimension-2 submanifold M p−1 ⊂ M p+1 , and c is a constant chosen such that the tachyon takes its vev away from M p−1 . By construction, this localizes a codimension-2 solution to exist only along M p−1 , and far from M p−1 the system is indistinguishable from the vacuum. In the above construction, we are only considering a single Dp-brane and Dp-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 [42]. More explicitly, note that the worldvolume bundle E has a natural 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 [53], where a ± 1 denote the Chan-Paton gauge fields over E ± , 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 Dp -Dp configuration to a stable D(p − 2)-brane. This action was proposed in [54], later elaborated on in [55][56][57][58], and can be written in the form where F = dA + A 2 is the supercurvature given by In the above, Tr s is the Z 2 -graded supertrace and the covariant derivatives DT and DT are defined as By writing a + 1 = a L + a K and a − 1 = a K to reflect the structure of the line bundles, we can see that the tachyon field T couples only to a L and thus is tied only to the bundle L. The action (4.17) may be rewritten as a WZ action for a D(p − 2)-brane living inside the Dp -Dp worldvolume, i.e. (4.20) where in the final equality we integrated out the two directions transverse to the D(p − 2)brane's worldvolume M p−1 . Due to our choice of bundle structure, this is equivalent to integrating out the dependence on the line bundle L, and thus the tachyon. 18 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 × S 3 in the KT-KS geometry W 5 × T 1,1 . When the branes are brought together there will be a solitonic D3-brane localized on a worldvolume M 4 ⊂ M 3 × S 3 , with a WZ action described by (4.20).
To reconstruct the condensation defect C ∼ U 1 in (4.14), we must first specify the submanifold M 4 that the solitonic D3-brane is wrapping. Equivalently, this amounts to specifying the bundle L defined along the two transverse directions of M 4 with respect to M 3 × 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 S 2 , and an application of Poincaré duality to land on Both of these are encoded by the Chan-Paton flux F L . Specifically, we can construct a U (1) R -equivariant ansatz for the Chan-Paton flux as where f 2 , η 1 , and a i 0 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 L = 0 leads us to identify a 1 0 = a 2 0 = a 0 and M η 1 = da 0 , so that where a L = a 0 (Dψ/2π) is a locally defined gauge field associated with L. The 2-form Λ 2 is closed, and it plays the role of a basepoint flux for F L , much like f i 2 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 Note that the expression for F L give two possible sources for a D3-brane in the D5-D5 worldvolume. The first is as a soliton arising from tachyon condensation as discussed above. 18 It is worth pointing out that the factor involving ch(L) is encoding the K-theoretic difference alluded to in the previous footnote. Had we not made the previous simplification and instead chosen E + = L1 ⊗ K and E − = L2 ⊗ K (we can always consider a common line bundle K), this term would have taken the form ch(L1) − ch(L2). For the purpose of illustration, we assume in (4.20) that the integral of ch(L) − 1 is equal to 2π. In general, the ratio of Dp-brane charges is given by µp/µ p = (2π √ α ) p −p [59], but we also adopt the convention that 2πα = 1, otherwise B2 + 2πα F2 should be the gauge-invariant combination instead.
The second is from any worldvolume flux present on the original D5-brane and anti-D5-brane. The D5-branes used to construct 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 I D3

5
for the D3-brane on the Hopf base S 2 , but end up finding an apparently vanishing result, i.e.
However, this is too quick. The operator U 1 (Σ 2 ) we are ultimately concerned with finding is a holonomy of a discrete gauge field B 2 ∈ H 2 (M 3 ; 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 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 Z M -valued operator, we shall follow our nose and consider reducing on Z M -valued torsional cycles. The correct topological operator is then not the standard holonomy of I 3 on an integervalued cycle, but a holonomy along a Z M -valued cycle. Namely, the proper operator to consider is 19 where Σ 3 is a formal extension of Σ 2 such that ∂ Σ 3 = M Σ 2 . In the final expression we have rewritten the integral as a pairing between Z M -valued cycles and cocycles, and should now consider everything to be valued in (co)homology with Z M coefficients. As we mentioned, I 3 should be viewed as a "field strength" for a discrete field, so we may express . This justifies our decision to consider the reduction on 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 B 2 ∈ H 2 (M 3 ; Z M ). 20 With this choice, we can evaluate the holonomy of I 3 as 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 β = ∂/M for the Bockstein homomorphism in homology (see Appendix B for details). This holonomy is nothing but the U 1 (Σ 2 ) operator we saw in the condensation defect (4.14). 19 We can alternatively phrase this in terms of the RR field strengths, by originally having a term in G5 of the form M η3 ∧ (V1 + V2) with η3 a ZM -valued field strength. Due to the torsion, this term would formally be zero and naïvely wouldn't be present in the expansion. 20 By exactness of the short exact sequence 0 → ZM → Z M 2 → ZM → 0, the quantity M B2 is trivial, and so is β (M B2). This means that UM (Σ 2 ) = U1(M Σ 2 ) = exp 2πi Σ 3 I3 = 0 as we saw in (4.23).
The arguments above regarding the D3-brane worldvolume action can be rephrased in the language of differential cohomology [60][61][62][63][64]. 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 Σ 2 × S 2 , there is one last thing to consider about the D3-brane soliton arising from the D5-D5 worldvolume. The Chan-Paton field on the D5-D5 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 L along the other two transverse directions in the original D5-D5 worldvolume. A sum over a 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 Σ 2 that the D3-brane is wrapping (via Poincaré duality), yields a sum over all possible 2-cycles Σ 2 ∈ H 2 (M 3 ; Z M ). Hence, taking into account the proper gauge theoretic normalization, our final defect is given by which is in exact agreement with (4.14) for p = 1. We have understood the condensation defect arising in the fusion N 1 ⊗ N −1 as being a sum of all possible D3-brane configurations in the original D5-and D5-brane system. It is worth noting that the a 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 N p ⊗ N p ) that the symmetry defects are those coupling to the center-of-mass modes for the stacks of Dp-branes. We can extrapolate the above argument to the more general fusion of N p ⊗ N −p . In this case, tachyon condensation will produce p D3-branes defined on the Σ 2 × 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 N p ⊗ N −p = 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-D5 worldvolume, the base 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 U q (Σ 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. Figure 2: Condensation defect arising from tachyon condensation on a Dp -Dp 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 Dp -Dp system.
For this choice the solitonic nature is more apparent as follows. When implementing tachyon condensation for a D5-D5 system, the tachyon profile is crucial in determining where the D3brane 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. 21 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.

M Defect
We have so far seen that the existence of the Z (0) 2M symmetry defect N 1 (M 3 ) in the 5d bulk W 5 can be attributed to a D5-brane wrapping M 3 × S 3 . Furthermore, the fusion of such a D5-brane with its orientation reversal produces a condensate of the Z M defects U 1 (Σ 2 ) as solitonic D3-branes localized on M 3 . A natural question to ask then is whether a U 1 (Σ 2 ) defect defined on a generic Σ 2 ⊂ 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-D5 worldvolume, we can consider a D3-brane on its own wrapping the Hopf base S 2 ⊂ S 3 inside the internal T 1,1 , and an arbitrary external Z M -valued cycle Σ 2 ∈ H 2 (W 5 ; Z M ). Following the identical arguments given previously, the D3-brane reproduces the Z (1) M magnetic symmetry defect, It is interesting to note that this symmetry defect can also be constructed by a closed fundamental (F1) string sitting in the external spacetime W 5 . As F1-strings do not couple to the Ramond-Ramond sector, the anomaly polynomial for this F1-string is simply given by 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 ∼ exp 2πi Σ 2 B 2 /M (up to a sign convention). We have thus showed that these F1-strings can also give rise to the symmetry defect U 1 (Σ 2 ). An alternative understanding of this comes from a modification of the "baryon vertex" introduced in [65]. 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 ⊂ T 1,1 [66]. From the Bianchi identities, 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 gcd(N, M ) = M F1-strings. This implies that the defect produced by an F1-string will generate a Z M (rather than Z N ) symmetry. Due to the baryon vertex, a collection of k F1-strings also generates the full Z M symmetry as long as gcd(M, k) = 1 (otherwise it will only generate a 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 U 1 to be a bound state of one D3-brane and k F1-strings. However, this is indistinguishable from a lone D3-brane as the latter may simply absorb the F1-strings into its worldvolume. We will therefore use the D3-brane description in the discussion that follows.
Recall that earlier we constructed the Z 2M defects N p (M 3 ) by stacking p D5-branes on M 3 × 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 ⊂ T 1,1 . It is straightforward to apply the same argument here to construct the general Z from a stack of D3-branes wrapping Σ 2 × S 2 . Again equivalently, this can also be realized by taking a D5-brane with q units of worldvolume flux threading S 2 ⊂ T 1,1 .
Defect fusion. Unlike the D5-brane operator N p (M 3 ) on which the Chan-Paton fields form a nontrivial TQFT, the D3-brane operator U q (Σ 2 ) has a relatively simple structure as seen in (4.30). For two such defects wrapping the same external 2-cycle (mod M ) Σ 2 ⊂ W 5 , their fusion obeys where the label q of U q is only defined modulo M . In other words, the U q operators realize a group-like fusion algebra corresponding to a Z M group.

Z
M and Z

2M Defect Fusion
Let us proceed to discuss the remaining two symmetry defects resulting from the Gauss Law constraints presented in Section 2, namely, O q and V p , which respectively generate the Z 2M symmetries. As we will show, these operators can also be realized by certain Dbrane 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 Z M symmetry defect 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 Dp-brane. In fact, we can construct the desired operator through a bound state of a single D3-brane and k D1-branes. 22 Let us first consider a D3-brane wrapping the (homologically) nontrivial S 2 ⊂ 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 + 10 )G 5 is given by where g 5 is an external gauge field. By imposing the IIB Bianchi identities, we find that 5 g 2 can be solved as 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 3) 22 We previously defined k via N = kM . As long as M 2 N M , there is no significant backreaction from the stack of k probe D1-branes. We assume this limit for the above discussion.
where we have made use of the expression for g 2 in (2.5). We also hereafter absorb β 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) A 1 of g 1 as in (2.5). 23 The topological operator associated to the D3-brane wrapping S 2 ⊂ T 1,1 is thus given by exp 2πi where In addition to the D3-brane configuration described above, we also consider a D1-brane in the external W 5 , whose characteristic class is simply To reproduce the Z M symmetry generator, we need k such D1-branes wrapping the W 2 of the D3-brane worldvolume, combining to give Alternatively, we can derive the above expression for O 1 (W 2 , W 3 ) by considering a D3-brane with k units of worldvolume flux threading the S 2 ⊂ T 1,1 . This combination of branes pairs nicely with the discussion of the U 1 (Σ 2 ) operator surrounding (4.29). There, a bound state of a single D3-brane and k F1-strings gave rise to a Z (1) M symmetry generator. It is natural to expect that the symmetry generator for the Pontryagin dual symmetry Z (1) M 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 O 1 (W 2 , W 3 ).
As mentioned in Section 2, this operator is only well-defined when W 2 = ∂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 λ 1 and ϕ, We emphasize that, similarly to the worldvolume Chan-Paton field a 1 in A M,p [B 2 ] (see (2.12)), the presence of the worldvolume gauge fields λ 1 and ϕ can be understood in terms of anomaly inflow. The A 1 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 Z M global symmetries on the D3-D1 bound state. It is known in general 23 The potential c0 may be turned off via a gauge transformation.
M generator 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. that a theory with a global symmetry G (k) can be equivalently viewed as a gauge theory for the Pontryagin-dual symmetry G (d−k−2) [1]. In our case, this corresponds to a Z Before proceeding, let us pause to examine the field content of the O 1 (W 2 ) operator. The first term withĉ 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 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 ϕ ∈ H 0 (W 2 ; Z M ) and λ 1 ∈ H 1 (W 2 ; Z M ) by virtue of the discussion in footnote 9. The coupling to B 2 in (5.7) then implies that the D3-D1 worldvolume has a ϕ number of induced F1-string charge, where ϕ is summed from 0 to M − 1 in order to create a gauge-invariant operator. Similarly, the coupling to A 1 in (5.7) describes induced D1-brane charge localized on 1-cycles γ 1 which is Poincaré-dual to λ 1 (with respect to W 2 ). We will soon explain why A 1 is associated with D1-branes when we discuss the V 1 (γ 1 ) operator. See Figure 3 for a schematic picture of these induced charges on O 1 (W 2 ).

O q ⊗ O q 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 O 1 (W 2 , W 3 ) where W 2 is a boundary such that W 2 = ∂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 × S 2 × S 2 and one unit of worldvolume flux threading the S 2 . Each D5-brane will have a characteristic class I 7,i similar to what we used before in (4.5), that can be written as where S 2 f i 2 = 1 and S 2 f i 2 = 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 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, The same can be shown to hold for general O q (W 2 , W 3 ) and O q (W 2 , W 3 ). We emphasize that this fusion, and by extension this method, will only be valid in the case W 2 = ∂W 3 . If W 2 is a non-contractible closed manifold, we will need to take more care in handling the discrete worldvolume fields ϕ and λ 1 discussed previously.
To account for these additional worldvolume fields, we can compute the fusion directly at the level of O 1 (W 2 ) or, in general, O q (W 2 ). We find, for q = −q, 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 ϕ and λ 1 are localized worldvolume degrees of freedom coupled respectively to the 5d bulk fields B 2 and 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.
(q + q ) ϕ = qϕ + q ϕ , (q + q ) λ 1 = qλ 1 + q λ 1 , Ultimately, the ϕ and λ 1 fields are associated to the O q+q operator while the ϕ and λ 1 fields comprise the decoupled TQFT coefficient. This lends itself to an interpretation reminiscent of that given for the fusion of the 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 X M,qq (q+q ) −1 arising as the fusion coefficient. Despite the similarity, the mechanisms from which the 3d TQFT A M,pp (p+p ) and the 2d TQFT X M,qq (q+q ) −1 arise are slightly different. In the former case, since the Chan-Paton field a 1 is what couples to B 2 in the 3d worldvolume action of N p , it captures both the centerof-mass and relative modes of the brane stacks. However, in the latter case, these modes are instead captured by the pair of gauge fields ϕ and λ 1 , which respectively couple to B 2 and A 1 in the 2d worldvolume action of O q .

O q ⊗ O −q Fusion
Due to the existence of the nontrivial 2d TQFT X M,q [A 1 , B 2 ] within the O q (W 2 ) operator, one naturally expects it to be non-invertible. The fusion of O q with its orientation reversal produces another condensation defect C through 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 W 5 ) will annihilate completely, but the induced charges will not. This logic is the same as that of the non-invertible fusion of [9]. 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)-(D3-D1) 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 λ 1 in the definition of O q (W 2 ). The reason why the sum includes terms of the form V 2 ⊗ 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 Z (2) 2M 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 U 1 (Σ 2 ) factors in the condensation defect 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 γ 1 ∈ H 1 (W 2 ; 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-D5 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 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 U 1 (Σ 2 ), and a solitonic D1-brane operator V 2 (γ 1 ).
We shall emphasize that the D1-brane operator V 2 (γ 1 ) resulting from tachyon condensation is stable only because of its solitonic nature. To construct a generic Z 2M 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 C in Section 4, namely, when constructing symmetry operators from Dp-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) × SU (2) isometries, corresponding to the S 2 × S 2 base of T 1,1 in the KT-KS background.

Nature of the Z
(2)

2M Defect
As alluded to earlier, one can construct a stand-alone Z (2) 2M symmetry operator, with γ 1 ∈ H 1 (W 5 ; 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 S 2 , such that the brane is protected from shrinking to zero. The explicit construction of V 1 (γ 1 ) follows closely to that of U q (Σ 2 ) in Section 4, i.e. if we naïvely reduce the characteristic class over the Hopf fiber, we find Much akin to the construction of the Z M 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, Z 2M R-symmetry. Following the same arguments as presented around (4.24), we may express I 2 = β(A 1 ) ∈ H 2 (W 5 ; Z 2M ), and by wrapping the D1-brane on a cycle γ 1 such that ∂ γ 2 = 2M γ 1 , one can reproduce (5.14). More generally, the operator where p ∼ 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.
Similarly to before, the fusion rule can be understood by considering the D1-brane used to construct V 1 (γ 1 ) as being instead a neutral D3-brane wrapping γ 1 × S 1 × 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, . . . , 2M − 1 ∈ Z 2M .
6 Cross-symmetry Defect Fusion and Braiding

Dissolved Branes
We discussed in Sections 4 and 5 the brane kinematics and dynamics underlying the parallel fusion rules for N p ⊗ N p , O q ⊗ O q , U q ⊗ U q , and V p ⊗ V p . 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, The D-branes associated with U q (Σ 2 ) and N p (M 3 ) are respectively q D3-branes wrapping Σ 2 × S 2 and p D5-branes wrapping M 3 × S 3 , where Σ 2 ⊂ M 3 and S 2 ⊂ 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. 24 Moreover, adding D3-branes to the condensation defect C(M 3 ), which is itself a sum of all D3-branes localized on M 3 (×S 3 ), has the identical effect of reshuffling the summands in the condensation defect. This naturally explains the fusion rule, Alternatively, locality of the fusion allows us to interpret (6.2) from a different but equivalent perspective, i.e. 3) The expression above tells us that U q ⊗ 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-D5 system. Analogous arguments can be used to provide a brane interpretation of the following fusion rules, We now turn our focus to the fusion between the Z (0) 2M defect N p (M 3 ) with its associated condensation defect 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, assuming gcd(M, p + p ) = 1. As per the discussion in Section 4, there is a center-of-mass mode associated with each stack of D5-branes (or D5-branes). Suppose we first fuse p and p D5-branes. This results in a stack of p + p D5-branes, in addition to the relative mode A M,pp (p+p ) between these two stacks. Next, we fuse this new stack of p + p D5-branes with p D5-branes, which results in a stack of p D5-branes, in addition to the relative mode A M,−pp (p+p ) between the two original stacks. It turns out that these two minimal TQFTs, each of which is associated with an independent Z M symmetry, combines to form a Z M × Z M 3d DW theory (Z M ) 0 [33] (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 N p ⊗C = (Z M ) 0 ⊗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. (6.6)

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 Importantly, we see that (b 2 ,ĉ 2 ) and (A 1 , c 3 ) are conjugate pairs. Note that at low energies, the fluctuating fields M b 2 and 2M A 1 are respectively Higgsed to the discrete gauge fields −B 2 ∈ H 2 (W 5 ; Z M ) and A 1 ∈ H 1 (W 5 ; Z 2M ) via the Stückelberg mechanism. It follows that the commutation relation between the Z where (Σ 2 , W 2 ) denotes the linking number of Σ 2 and W 2 with respect to W 5 . 25 As discussed in [12,62,68,69], the nontriviality of the commutation relation is very natural when considering the Dp-branes associated with these operators. In Type II string theory, Dp-branes are (electromagnetically) dual to D(6 − p)-branes, and particularly, D3-branes are self-dual. 26 Therefore one does expect the commutation relation between U q (Σ 2 ) and O q (W 2 ), both of which are constructed from D3-branes, to be nontrivial. By the same token, the Z 2M defect V p (γ 1 ) (from D1-branes) and the Z (0) 2M defect N p (M 3 ) (from D5-branes) do not commute. More explicitly, we find where (γ 1 , M 3 ) again denotes the linking number of γ 1 and M 3 with respect to W 5 . For completeness, let us provide a broad-brush argument for the commutation relation between N p (M 3 ) and O q (W 2 ), as was also studied by [8]. The corresponding manifolds typically do not link in W 5 , in which case these two operators commute. However, if we consider a refined case where W 2 ∼ = S 1 ×L 1 , with S 1 ⊂ M 3 , then heuristically the commutation relation takes the following form, (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 25 The linking number we are using in this context is defined in the sense of [67]. By construction, . 26 This duality is not to be confused with the SL(2, Z) duality of Type IIB string theory, under which the pair (B2, C2) transforms as a doublet while C4 transforms as a singlet. 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 O q (S 1 × L 1 ) is dragged through the D5-brane operator 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 → −B 2 /M at low energies), so we may schematically express the action of where I is the interval along the transverse direction between N p (M 3 ) and O q (S 1 × L 1 ). Such a phenomenon is akin to the Hanany-Witten transition [70], 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. (NS5-D3-D5) we arrive at the desired combination of branes in our construction.

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 [1,2,6,20,71]. Importantly, these choices also prescribe the spectrum of topological operators existing on the boundary.

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 where the first two terms are BF couplings respectively pairing b 2 andĉ 2 , and c 3 and 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 where we have made a choice in having the variation onĉ 2 . By adding a boundary term we may equivalently view the variation as δS BF = 2π ∂W 5 Mĉ 2 δb 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. 27 The difference between these two choices corresponds to different global structures for the gauge group of the 4d N = 1 SYM field theory living on the boundary W 4 = ∂W 5 . Specifically, an SU (M ) gauge theory has a Z  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 A 1 B 2 2 term is a mixed 't Hooft anomaly between the Z 2M symmetry to become non-invertible [8,23,66]. Meanwhile, the self-anomaly term A 1 β(A 1 )β(A 1 ) indicates A 1 (and A 1 ) cannot be given Neumann boundary conditions, reflecting the fact that one cannot gauge the anomalous Z

Boundary Conditions and Branes
The choice of boundary conditions for the bulk supergravity fields determines the set of allowed configurations of the probe Dp-branes in W 5 . A priori, there are 2 4 = 16 choices of boundary conditions on the set of fields (b 2 ,ĉ 2 , c 3 , A 1 ). However, 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.
(D,N,N,D) boundary conditions. To see the effect of boundary conditions concretely, let us again focus on the (b 2 ,ĉ 2 ) BF pair discussed in the previous subsection, keeping in mind that c 3 and A 1 must be given Neumann and Dirichlet boundary conditions respectively. 27 These 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 [72].  Table 3: The allowed boundary conditions for the four bulk gauge fields (b 2 ,ĉ 2 , c 3 , 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.
We first consider the case where we impose Neumann boundary condition onĉ 2 . By the variational principle, we must require that M b 2 , and by extension B 2 , vanish on the conformal boundary 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 P SU (M ).
With this choice of boundary conditions it is straightforward to see from (4.27) that U q (Σ 2 ) becomes a trivial operator when pushed to the boundary. This is expected for a theory with gauge group SU (M ), as the defects U q would generate the magnetic 1-form symmetry not present in this theory. Instead of having the D3-branes being pushed parallel to W 4 , we can consider transverse D3-branes such that one of their external dimensions is aligned with the radial coordinate of W 5 . Using (7.3), we can write b 2 in terms of a Z M -valued flat connection B 1 defined only on the conformal boundary, i.e. M b 2 W 4 = dB 1 = 0 , (7.4) in which case the resultant operator becomes with γ 1 closed. The commutation relation (6.8), when localized to W 4 , tells us that such a line operator living in the boundary field theory is charged under the Z M electric symmetry generated by the O q defects. Following the discussion below (4.27), this could have been equivalently argued from an alternative construction of the U q defects using F1-strings. It is known in the literature that F1-strings ending on the conformal boundary trace out Wilson lines [73]. The line operators U(γ 1 ) should therefore be identified with these Wilson lines in the SU (M ) gauge theory. 28 Similarly to before, imposing Neumann boundary condition on c 3 leads to where A 0 ∈ {0, . . . , 2M − 1} is a flat Z 2M -valued 0-form field. This corresponds to a local operator V(x ∈ W 4 ), where x = ∂γ 1 , that is charged under the Z (7.7) Note that the b 2 A 1 B 2 anomaly trivializes as both fields A 1 and B 2 are turned off on W 4 , so the D3-D1 operator no longer couples to the TQFT X M,q on which the auxiliary fields λ 1 and ϕ live. Importantly, O q is an invertible operator, which agrees with our expectation that this defect generates the (invertible) Z M center symmetry in the SU (M ) gauge theory. By the same token, the A 1 B 2 2 anomaly trivializes on W 4 , so the Z 2M defect also becomes an invertible operator [8], i.e. (N,D,N,D) boundary conditions. Let us now consider the case whereĉ 2 is chosen to have Dirichlet boundary conditions instead of b 2 . This is equivalent to having a P SU (M ) gauge theory on the boundary. As has been discussed previously, this causes the U q operators to generate an invertible Z (1) M symmetry and the N p operators to generate a non-invertible Z (0) 2M symmetry on the boundary. Furthermore, the bulk line operator V p ends on the boundary to become the local operator V(x ∈ W 4 ). What remains to be discussed is the fate of the O q operators. Whenĉ 2 is chosen to vanish on W 4 , the defect O q takes the form, where we have kept in mind that A 1 W 4 = 0 due to the self-anomaly term. In the above, λ 1 acts as a Lagrange multiplier causing ϕ to take on integer values, such that the remaining operator turns into a (normalized) sum over trivial operators. 29 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 W 4 , i.e. we take one of the directions in W 2 to align with the radial coordinate of W 5 . From the variation (7.3), we can write on the boundary,

Discussion and Outlook
In this paper, we have studied in detail the complete fusion algebra of gauge symmetry generators in the 5d bulk 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 W 4 ⊂ W 5 . Via the Gauss Law constraints, the SymTFT indicates the existence of topological extended operators (near 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 I 11 and the WZ action of the D-branes. 30 We leave a careful analysis of this to future work.
The consistency between the supergravity bulk and the D-branes teaches us three important lessons. Firstly, the construction of, say, non-invertible global symmetry generators in the dual field theory typically requires a manual stacking of individually non-topological operators (e.g. [22]), whereas we can obtain the full operator at once from a straightforward 29 Alternatively, we can see that the A1B2 anomaly again vanishes on the conformal boundary due to the Dirichlet boundary condition on A1. The associated TQFT involving ϕ and λ1 will then vanish and the operator Oq will still be trivial. 30 We expect in this case that the mixed chiral-gravitational anomaly of 4d N = 1 SU (M ) SYM theory disguises as some nontrivial TQFT attached to the appropriate D-brane operator. The resultant defect should be similar to that constructed by [74].
dimensional reduction of the D-brane WZ action. For example, the 3d TQFT A M,1 within the R-symmetry defect N 1 comes for free when we reduce a D5-brane over S 3 . The fact that the WZ action is compatible with the bulk supergravity action enables the former to capture the necessary anomaly data from the latter. Secondly, the explicit fusion algebra of symmetry generators in the boundary field theory can be understood holographically as brane interactions in the bulk. For concreteness, in the case of the 4d N = 1 P SU (M ) SYM theory, charge-p and charge-p R-symmetry defects fuse not only into a charge-(p + p ) defect, but also an extra decoupled TQFT. We argued in Section 4 that this fusion rule is fully characterized by brane kinematics, i.e. the former comes from the overall center-of-mass mode of the combined stack of branes, and the latter comes from the relative contributions from the center-of-mass modes of the individual stacks.
On the contrary, a condensation defect arises from the fusion between a charge-p defect with its orientation reversal. We suggest that this can instead be interpreted as a dynamical process known as tachyon condensation. In fact, the dressing of an otherwise invertible operator with a nontrivial TQFT (as in (2.13)) is indicative of lower-dimensional D3-branes being dissolved within the worldvolume of the D5-brane. As a proof of concept, we showed that by engineering a certain tachyon profile, the D5-D5 fusion indeed results in a sum of (codimension-2) solitonic D3-brane operators that matches the field-theoretic fusion rule.
That being said, it is not entirely clear to us how unique the specific tachyon profile is. To illustrate this, recall that the worldvolume of the D5-branes is M 3 × S 3 , so there are at least three naïve ways to construct a codimension-2 soliton, namely, where γ 1 , Σ 2 ⊂ M 3 , and S 2 , S 1 are respectively the Hopf base and fiber of S 3 . The first option amounts to integrating out a 2-cycle in M 3 , and the second option amounts to integrating out a one-cycle in M 3 plus a fiber integration in S 3 , whereas the third option does not seem to be viable because we cannot perform a "base integration" in S 3 . Ruling out the first option requires some technical details, but essentially one can show that the resultant operator is incompatible with the fluxes present in the KT-KS background. This then singles out the second option, as far as the condensation defect is concerned. Nevertheless, there is still a degeneracy in the choice of the internal manifold on which the tachyon profile is defined. We may for example replace the Hopf base S 2 with the equatorial S 2 of S 3 , and one cannot tell the distinction between these two D3-brane configurations from the perspective of the 5d bulk operator. This degeneracy may be lifted by considering the isometries of the S 2 ×S 2 base in the T 1,1 fibration. Including these isometries would introduce an additional SU (2) L × SU (2) R gauge symmetry in the bulk; in the dual field theory there are two sets of chiral superfields A r , B u in bifundamental representations of the UV gauge group SU (N + M ) × SU (N ). Each set forms a doublet under an SU (2) global symmetry, which gives a combined SU (2) L × SU (2) R global symmetry in the boundary field theory. In the supergravity theory, the effect of this symmetry can be seen in the following way. A D3-brane wrapping a submanifold of the S 3 cycle will act as a point particle on S 2 in the presence of a nonzero "magnetic field" arising from the G 5 flux threading T 1,1 . This will cause the correpsonding operator to have a Landau-level label as discussed in [75,76]. It is possible that this additional label will remove the degeneracy in the choice of internal manifold for the tachyon condensation.
Moreover, as we saw in Section 4, the D3-brane operators U q obey a group-like fusion rule, and notably, it annihilates with its orientation reversal. In other words, they do not undergo further tachyon condensation to form D1-branes. One may heuristically argue that there is a qualitative difference between the N p ⊗ N −p and U q ⊗ U −q fusions. The operator N p consists of a 3d TQFT A M,p on which Chan-Paton fields reside, whereas all the Chan-Paton modes are already integrated out in U q . Since the presence of worldvolume Chan-Paton modes are necessary (but not sufficient) for tachyon condensation to happen, as effective 5d bulk operators the former is non-invertible while the latter is not. It is likely that we can shed some light on these open questions by examining more constructions of condensation defects in other holographic setups.
The third lesson that we can learn is the bulk SymTFT contains a rich symmetry structure that encodes the data of multiple possible boundary field theories. As explained in Section 7, the global symmetries present in the boundary field theory is not only specified by the SymTFT itself, but also by a choice of boundary conditions for the various supergravity gauge fields. Consequently, this determines the field content of a given D-brane operator when placed at the conformal boundary (transverse to the radial direction of W 5 ). For instance, the electric can only be invertible if W 2 ⊂ W 4 , but once we move slightly into the bulk (but still near W 4 ), the 2d TQFT X M,q attached to O q (W 2 ⊂ W 5 ) renders the defect non-invertible. This illustrates a crucial point: a non-invertible gauge symmetry can be trivialized to an invertible global symmetry depending on the choice of boundary conditions.
There is a further subtlety that one should consider when using the Gauss Laws to derive the symmetry structure from the SymTFT. The analysis performed in this paper was only concerned with the classical Gauss Laws of the associated symmetries. In general there may be nontrivial and important effects from the quantum Gauss Laws as considered in [20]. These effects may contribute extra torsional phases to the symmetry generators. It would be interesting to study the effect of these phases as well as their origin in holographic constructions.
Recently in the literature, the study of symmetry-generating D-brane operators has provided us with fruitful insights into the nature of non-invertible and/or higher-form global symmetries in quantum field theories. Multiple examples of non-invertible symmetries have indeed been uncovered in different string theory solutions using D-branes. On top of that, we have shown in this paper brane interactions in the bulk play a significant role in characterizing the fusion algebra of the boundary field theory. It is widely believed a fusion algebra involving defects of various dimensions furnishes part of the data of some higher fusion category (e.g. [77][78][79][80]). The data contained in the bulk theory should then be interpreted as describing the Drinfeld center of this higher fusion category [81][82][83][84][85]. We therefore hope our work will offer a complementary understanding of the intricacies of such higher categorical structures.

Acknowledgments
We are grateful to Federico Bonetti, T. Daniel Brennan, Pierre Heidmann, Ho Tat Lam, Ruben Minasian, Salvatore Pace, Sebastian Schulz, Hannah Tillim, and Peter Weck for interesting conversations and correspondence. We also thank the authors of [13] for communicating about a related work and coordinating submission. IB, EL, and TW are supported by NSF grant PHY-2112699, and also in part by the Simons Collaboration on Global Categorical Symmetries.

A SymTFT Computation
We start with a brief review of the Klebanov-Tseylin (KT) [18] and Klebanov-Strassler (KS) setups [19] dual to the cascading field theory. There are two relevant features. The first is a stack of N D3-branes extending along flat space R 1,3 and placed at the tip of the conifold background as was studied in [86]. The conifold is a non-compact Calabi-Yau cone over a Sasaki-Einstein 5-manifold denoted T 1,1 . The space T 1,1 is constructed as a Hopf fibration over S 2 × S 2 with Einstein metric given by [8,87,88] where in the above {θ i , φ i } denote the coordinates of the two S 2 geometries each with periodicity 2π, and ψ is the coordinate of the S 1 fiber with 4π periodic. The T 1,1 space admits the topology of S 2 × S 3 and cohomologically nontrivial one 2-cycle, one 3-cycle, and 5-cycle that we denote by ω 2 , ω 3 , and Ω 5 respectively. The near-horizon geometry of the stack of D3-branes in type IIB string theory is correspond to AdS 5 × T 1,1 , with N units of G 5 flux. We will be interested in gauging the U (1) isometry of the fiber coordinate ψ through the introduction of a connection dψ → dψ + 2πA 1 , where the normalization is chosen such that the flux of A 1 has integral periods. In general one may consider gauging the SU (2) × SU (2) isometries of the S 2 × S 2 base of T 1,1 , in which case one would then have to incorporate global angular forms to account for the equivariant completion [24,[89][90][91]. The effect of these forms do not play a role in this paper, so we refrain from gauging these isometries. We can construct a U (1)-equivariant completion of the volume forms mentioned earlier as where V i denote the normalized volume forms of the 2-spheres in the base of the T 1,1 fibration. The normalization of these forms comes from demanding that T 1,1 ω 2 ω 3 = 1. The U (1)equivariant completion of Ω 5 is written where F 2 is the field strength of A 1 . The second main ingredient of the KT and KS setup is a stack of M D5-branes wrapping the topological S 2 of the conifold and extending along the flat space R 1,3 . The backreaction of this stack of branes has two effects on the supergravity setup, the first of which being that the conifold metric is replaced with that of the deformed conifold 31 where the S 3 cycle is blown up at the tip of the cone. The change in metric is felt very close to the original stack of N D3-branes and similarly in the deep IR of the dual field theory, but far away from the origin the space is indistinguishable from the original conifold. In the near horizon limit, the effect of the stack of M D5-branes is that the G 5 flux runs along the AdS 5 radius. The asymptotic region of the KS background corresponds to the KT [18]. This latter geometry is enough to determined the symmetry structure of interest in this paper.
The dual field theory of the setup is N = 1 Super Yang-Mills with gauge group SU (N + M ) × SU (N ) coupled to bifundamental matter [18]. This is a non-conformal background where the gauge couplings run. As this theory flows to the IR a sequence of Seiberg duality transformations occurs, often referred to as the "duality cascade". For N a multiple of M , this cascade ends with pure N = 1 Super Yang-Mills theory with gauge group SU (M ). In this paper we will study the symmetry structure of the N = 1 SU (M ) theory from the SymTFT derived from the dual gravitational background. In particular we will use various brane probes of the supergravity to construct the possible symmetry generators and their fusion rules.
To construct the SymTFT in this background, we first write down the 5d topological action using the nontrivial fluxes for h 3 and F 2 as discussed in Section 2, where the A 1 cubic coefficient is given by The subleading corrections in N may be computed directly from the UV quiver of the KT-KS setup. The full SymTFT is given by the above action augmented with standard kinetic terms for g 1 , g 2 , g 3 , h 3 , and F 2 with associated couplings κ 1 , κ 2 , κ 3 , κ h , and κ F . To elucidate the symmetry structure of the boundary field theory, we may dualize some of the fields in the above SymTFT. We first dualize g 1 by introducing a 3-form c 3 as a Lagrange multiplier to enforce the Bianchi identity.
There is another construction which considers the resolved conifold rather than the deformed conifold [92]. The difference is in which cycle is blown up, but the physical theory is still that of 4d N = 1 Super Yang-Mills. It would be interesting to explore the connections between the two theories with regard to the operators constructed out of branes.
The equation of motion for g 1 yields After integrating out g 1 by replacing it with its equation of motion, we may also dualize g 2 through an additional field a 2 acting as a Lagrange multiplier The equation of motion for g 2 can then be found as After performing both dualizations, we may write our final action S top as thus producing (2.8). In the low energy limit, the kinetic terms present in the full SymTFT are suppressed, so we often refer to S top as "the SymTFT" unless otherwise specified.

B Brief Review of Bockstein Homomorphisms
For a detailed introduction to Bockstein homomorphisms, we refer the reader to standard references in the literature, e.g. [93,94]. Here we briefly review and explain the notation adopted in the main text. Let us first consider the following short exact sequence with k prime, 32 where the "mod k" map means reduction modulo k. Associated with it is a long exact sequence in homology, where the Bockstein homomorphism is defined to be a map Explicitly, it acts on cycles (mod k), C n ∈ H n (M ; Z k ), as A closely related construction starts with the short exact sequence which leads to the long exact sequence The codomain of this Bockstein homomorphism is different from that in the previous case, namely, On a cycle C n ∈ H n (M ; Z k ), the map similarly acts as where the two connecting homomorphisms are related via a reduction map, Just like how the Bockstein homomorphisms act on cycles as "boundary operators," they act on cocycles as "coboundary operators." More explicitly, β(c n ) = 1 k δc n (B.14) for any c n ∈ H n (M ; Z k ). The Bockstein homomorphisms have important applications in physics, especially when torsional manifolds or fluxes are involved. One of them being the definition of the integral Stiefel-Whitney classes. For a degree-n (generalized) Stiefel-Whitney class w n ∈ H n (M ; Z k ), the corresponding integral Stiefel-Whitney class is defined as If n = k = 2, the degree-3 integral class W 3 = β(w 2 ) measures the obstruction to having a spin c structure on the manifold M . Another example, used in our contruction, is the flat but topologically nontrivial part of the NSNS field B 2 ∈ H 2 (W 5 ; Z M ), which corresponds to the cohomological class of the 't Hooft magnetic flux via H 3 = β (B 2 ) as in (2.6). In fact, these two examples are closely related through the Freed-Witten anomaly [39,40]. Moreover, the Bockstein homomorphism for k = 2 coincides with the first Steenrod square (see, e.g. [95]), where w 1 (M ) ∈ H 1 (M ; Z 2 ) is the first Stiefel-Whitney class of M (or more precisely, its normal bundle). As an aside, we clarify that the Bockstein homomorphisms invoked in this paper shall be distinguished from that commonly used in differential cohomology, which is associated with another short exact sequence, 0 → Z → R → R/Z → 0.

C Derivation of Fusion Rules
In this appendix, we are going to derive the nontrivial fusion rules between the symmetry defects. The reader may find similar derivations in [96]. We assume all manifolds are spin throughout the appendix. ]. An efficient way to compute the fusion rule is as follows. 33 The minimal TQFT A M,p is a 3d theory with a Z M 1-form symmetry generated by Wilson lines denoted by a s where s = 0, . . . , M − 1 [33], (C.7) One may see that p + p corresponds to the anomaly of A M,p+p as desired.
To impose the condition that A M,n and A M,p+p are decoupled from each other, we demand the line b s to braid trivially with b r for any (s, r), i.e. , (C.9) 33 We thank Ho Tat Lam for discussions about this point. 34 In a braided fusion category with fusion rules a ⊗ b = c N c ab c, the S-matrix is given by S ab = c N c ab (θc/θaθ b )(dc/D), where θa = exp(−2πih) is the topological twist of the simple object (or anyon) a. If we represent a finite group G with a fusion category, then the quantum dimension of any anyon is da = 1, and the total quantum dimension of the category is D = a d 2 a = |G|. whose spins are given by In this case, one can check that the following redefinition of lines, b s = (a p p a −p(p+p +p ) ) s , b r = (aa ) r , (C.21) gives rise to (C.14) as well. Since the fusion is also commutative, this triple fusion rule indeed holds for arbitrary permutations of (p, p , p ) with an appropriate redefinition of lines.
Holographically, this implies that we can freely reshuffle the three stacks of D5-branes without altering the end result, although the decoupled modes are sensitive to the number of branes in each individual stack. where (Z M ) 0 is decoupled from B 2 and acts as a coefficient. We can immediately verify that associativity (and commutativity) holds for the following triple fusion, N p ⊗ N p ⊗ C . (C. 25) More explicitly, performing the fusion in different orders yield the same answer, C ⊗ C fusion. The most efficient way to compute this fusion rule is to observe that Alternatively, one may derive this using a Lagrangian description, i.e. (C.28) If we defineΛ 1 = Λ 1 + Λ 1 andΛ 1 = a 1 − a 1 , then we will obtain M 3 which can be identified as (Z M ) 0 ⊗ C. It can be readily checked that N p ⊗ C ⊗ C and C ⊗ C ⊗ C are both commutative and associative. then we may express U q (Σ 2 ) ⊗ N p = N p ⊗ U q (Σ 2 ) = (−1) q 2 p −1 Q(Σ 2 ) ⊗ N p .
(C. 33) Similarly to before, the fusions U q (Σ 2 ) ⊗ U q (Σ 2 ) ⊗ N p and U q (Σ 2 ) ⊗ N p ⊗ N p are both commutative and associative. If M is odd, the phase (−1) Q(Σ 2 ) is always trivial [96], so we recover the fusion rule presented in Section 4. Since the condensation defect C is essentially a sum of magnetic symmetry defects, i.e.
This result is consistent with the previously derived fusion rule for C ⊗ N p .
U q (Σ 2 ) ⊗ C fusion. We can again use (C.34) to deduce that where Σ 2 = Σ 2 + qΣ 2 ∈ H 2 (M 3 ; Z M ). In other words, the effect of such a fusion is merely to relabel the 2-cycles (mod M ) that are being summed over in the condensation defect. We can again easily verify that this fusion rule is compatible with the previous ones that we have derived.