Branes and Non‐Invertible Symmetries

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


Introduction
The notion of symmetry is undergoing rapid evolution: during the last few years a number of works have convincingly argued that the classical textbook definition of symmetry as a group of transformations acting on local operators can (and should) be extended to include higher form symmetries acting on extended operators, [1] higher groups structures [2][3][4] and more generally higher categorical structures.
The importance of such higher categorical structures in two dimensions has been realised for a long time, where they often appear from discrete gauging. [5][6][7][8] A number of recent works have shown that symmetry operators without inverses (and which are therefore not elements of any group, but should rather be thought of in categorical terms) are also very common in higher dimensional theories. [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] In this paper we will focus on one class of theories where such non-invertible symmetries appear:  = 4 theories with gauge group 1 Pin + (4N), Sc(4N) and PO(4N). [14] The details are a little different in the three cases, so in this introduction we will focus on the Sc(4N) case for concreteness. This theory has three 2-surface symmetry generators, which we will call D ,e 2 (Σ 2 ), D ,m 2 (Σ 2 ) and their product D ,e 2 (Σ 2 )D ,m 2 (Σ 2 ). There is additionally a three-surface operator  ( 3 ). The terms in the fusion algebra involving  ( 3 ) are DOI: 10.1002/prop.202200154 The right hand side of (1a) is generically a sum of operators, and therefore  ( 3 ) is not invertible. All these theories can be obtained from the  = 4 SO(4N) theories by suitable gaugings of discrete symmetries. Whether we have performed the gauging or not is not visible for a local observer measuring processes on a topologically trivial (but arbitrarily large) neighbourhood of a point. This suggests that the holographic dual of all these theories is the same, which is indeed the case: the holographic dual is in all cases IIB on AdS 5 × ℝℙ 5 . The different theories arise from different choices of asymptotic behaviour for discrete gauge fields in the bulk, as discussed in related examples in [29,30].
Since all these theories share the same bulk description, it should be possible to describe the non-invertible symmetry generators (in the cases where they are present in the field theory) in terms of objects living on the holographic IIB dual. The goal of this note is to identify these objects, and to derive their fusion rules using IIB techniques. 2,3 Surprisingly, given the perhaps unfamiliar fusion relations (1), it will transpire that the symmetry generators are represented holographically by ordinary branes wrapping torsional cycles in the internal ℝℙ 5 .
In order to explain how this is possible, it is useful to review briefly how the fusion relations (1) are derived in [14] (see also [12]). We start with the SO(4N) theory, which has a ℤ 2 outer automorphism 0-form symmetry and a ℤ 2 × ℤ 2 1-form symmetry. We will denote the background for the 0-form symmetry A 1 , and the backgrounds for the two ℤ 2 factors B m 2 and C e 2 . There is a cubic 't Hooft anomaly represented by an anomaly theory with action We obtain the Sc(4N) theory by gauging both 1-form symmetries simultaneously. 4 (The Pin + and PO cases are obtained by gauging other pairs of symmetries involved in the cubic anomaly.) 2 The techniques we use in our analysis do not require knowledge of the Lagrangian of the boundary SCFT (although the choice of theories to study is certainly informed by the field theory results in [14], and we will chose our notation to dovetail the field theory analysis), so they apply equally well to the study of non-Lagrangian theories realised either holographically or via geometric engineering. See [21] for a recent study of non-invertible symmetries in non-Lagrangian theories using a different approach. 3 We refer the reader to [23] for a holographic study of a different class of non-invertible defects. 4 There is a discrete choice when gauging, related to the precise way in which we sum over B m 2 backgrounds. A slightly different choice (re-www.advancedsciencenews.com www.fp-journal.org Naively, we would say that the 0-form symmetry is broken due to the cubic anomaly (2). The more precise statement is that due to the anomaly the generator D (0) 3 ( 3 ) of the 0-form symmetry is not invariant under combined gauge transformations of B m 2 and C e 2 . But as argued in [12,14] it is possible to "dress" (or stack) D (0) 3 ( 3 ) with an anomalous TQFT  depending on B m 2 and C e 2 . The combined topological operator is gauge invariant, and survives as a topological operator of the gauged theory. The price to pay is that the fusion rules for  are more involved, and lead to non-invertibility of the dressed operator D (0) 3 ( 3 ) ×  (the details will be reviewed below).
Coming back to the IIB holographic setup, the main observation of this paper is that D (0) 3 ( 3 ) ×  is precisely the IR limit of the theory on branes wrapping suitable torsional cycles in the holographic dual background. For instance, we will see that in the Sc(4N) case the non-invertible operator arises from a D3 brane wrapping  3 × ℝℙ 1 , where ℝℙ 1 ⊂ ℝℙ 5 . Reducing on the ℝℙ 5 leaves an effective 3-dimensional brane wrapping  3 inside the five dimensional bulk, which becomes D A pleasing consequence of the identification in this note is that anomaly cancellation of the dressed operator follows automatically: the background fields for the symmetries of the theory are given by asymptotic values for the supergravity fields in the IIB dual, and the D3 brane action is necessarily gauge invariant under all possible gauge transformations of these (although the precise way in which this happens is often subtle). Since anomaly cancellation is automatic once we start talking about branes, it is illuminating to understand why non-invertible symmetries appear in the holographic dual without referring to anomalous operators. This is also desirable since the split between the bare D( 3 ) and its "dressing"  is unnatural in terms of the brane theory, particularly once we try to formulate things in the language of K-theory. We provide such an explanation below in terms of incomplete cancellation of induced brane charges due to quantum effects.

Note Added
I thank the authors of [31] for informing me of their related upcoming work, where they give complementary evidence for the identification of non-invertible symmetries with branes in holographic settings, and for agreeing to coordinate submissions.

4d  = 4 (4N) SYM and Non-Invertibles
The Spin(4N) SYM theory has a 2-group structure, with one-form symmetry group 5 lated by the outer automorphism) gives the Ss(4N) global form instead, which also has non-invertible symmetries. The analysis of both cases is essentially identical, so we will focus on the Sc(4N) case. 5 Our conventions are as follows: Spin(4n) has two spinor irreps unrelated by complex conjugation, which we denote by " " and " ". ℤ 2 acts on , and leaves invariant, while ℤ 2 acts on and leaves invariant. This choice of notation is motivated by consistency with the fact that the diagonal ℤ 2 combination, traditionally denoted ℤ V 2 , does not act on the vector. We define Sc(4N) := Spin(4N)∕ℤ 2 .
and a 0-form symmetry part ℤ (0) 2 which is an outer automorphism that acts on the 1-form symmetry by exchanging the two factors: ℤ 2 ↔ ℤ 2 . We will now construct the topological defects that generate these symmetries in the holographic dual.
This holographic dual is obtained as the near horizon limit of a stack of D3-branes on top of an O3 − orientifold, and is given by IIB string theory on AdS 5 × ℝℙ 5 . [29] In general we want to put the field theory on some spin 6 manifold  4 different from S 4 , so we will replace AdS 5 by a non-compact manifold X 5 which asymptotically becomes ℝ ×  4 . [32] There is a non-trivial SL(2, ℤ) duality fibration over ℝℙ 5 , which acts with the −1 ∈ SL(2, ℤ) element as we go around the non-trivial generator of 1 (ℝℙ 5 ) = ℤ 2 . (This element can be represented alternatively as ΩF L in worldsheet terms, but with future generalisations in mind we will describe it as an SL(2, ℤ) bundle instead.) The 2-form supergravity fields B 2 and C 2 get a sign under this action, and project down to ℤ 2 fields on AdS 5 , while C 4 does not get a sign and survives as a continuous field. We will find it useful to work in a democratic formulation, where we also include the B 6 and C 6 fields magnetic dual to B 2 and C 2 . SL(2, ℤ) is a gauge symmetry of the theory on the (orientable) space AdS 5 × ℝℙ 5 , so in order for the action to be well defined we need B 6 and C 6 to also transform with a minus sign under −1 ∈ SL (2, ℤ).
What this means is that H 3 and F 3 are elements of the cohomology group with local coefficients H 3 (X 5 × ℝℙ 5 ;Z) (we refer the reader to appendix 3.H of [33] for details), and similarly their magnetic duals H 7 and F 7 are elements of H 7 (X 5 × ℝℙ 5 ;Z). On the other hand F 5 is classified by H 5 (X 5 × ℝℙ 5 ; ℤ). In what follows we will focus on the structure on ℝℙ 5 , as the SL(2, ℤ) bundle is trivial on X 5 . The untwisted cohomology groups of ℝℙ 5 are standard, and the twisted ones can be derived easily from the results in [34]: Similar considerations hold for homology: (p, q) 1-branes (such as fundamental strings and D1 branes) are elements of H 2 (X 5 × ℝℙ 5 ;Z), (p, q) 5-branes are elements of H 6 (X 5 × ℝℙ 5 ;Z), and D3 branes are elements of H 4 (X 5 × ℝℙ 5 ; ℤ). The relevant homology groups are (by Poincaré duality, which holds since ℝℙ 5 is orientable) With an understanding of the cycles that the branes can wrap, it is straightforward to identify the charged operators of the Spin(4N) theory [29] : the vector Wilson line W V is a fundamental string on a point of ℝℙ 5 , the -spinor Wilson line W is a D5-brane on ℝℙ 4 , and finally the -spinor Wilson line W is the www.advancedsciencenews.com www.fp-journal.org combination of both previous lines: a D5-brane/F1 bound state, again wrapped on ℝℙ 4 . In all cases the branes wrap a surface on X 5 extending to the boundary, where they end on a line.
We can go to the SO(4N) theory by gauging the diagonal factor The vector line W V is unaffected by the gauging, so it survives, but the W and W lines are no longer gauge invariant, and become non-genuine (that is, boundaries of surface operators). A non-genuine line H V of the Spin(4N) theory, with w 2 = w 2 flux around it, now becomes a genuine line operator in the SO(4N) theory. Holographically this operator corresponds to a D1 brane wrapping a point in H 0 (ℝℙ 5 ;Z) = ℤ 2 . 7 We denote the generators for these two symmetries D B,e 2 ( 2 ) (acting on fundamental strings) and D C,m 2 () (acting on D1 branes), and the corresponding background fields B e 2 and C m 2 . Starting from the SO(4N) theory we can gauge various pairs of global symmetries, an operation that, due to the cubic anomaly (2), results in theories with non-invertible symmetries [12,14]: In these expressions D We are thus led to the crucial question in this paper: having identified the charged operators in the field theory in terms of the holographic dual, what is the holographic description of the charge operators implementing the global symmetries in the SO(4N) theory?
For concreteness, let us specialise to the holographic dual of the symmetry generator D C,m 2 ( 2 ) of the SO(4N) theory, measuring how many 't Hooft lines (mod 2) H V are linked by  2 , without taking into account the Wilson lines W V . Given our identification of lines above, a natural guess would be where  2 lives on  4 , and becomes the symmetry operator when pushed to the boundary. This holonomy certainly measures the number of D1 branes linked by  2 (the basic argument is given below in case of the outer automorphism 0-form symmetry), but it cannot be the right answer for a number of reasons. First, we know that in IIB string theory fluxes are not measured by cohomology, but rather K-theory. [35][36][37] A way of capturing the right K-theoretic formula is to phrase the answer in terms of the Wess-Zumino coupling in the D5 brane action: 7 The simplest derivation of this fact follows from recalling that the SO(4N) field theory is invariant under SL(2, ℤ), which maps to an SL(2, ℤ) action on the holographic dual. We refer the reader to [28] for a systematic analysis.
where [37][38][39] A second reason why we expect neither (7) nor (8) to be the full answer is that in string theory there are no local operators, only dynamical objects. So we should aim to represent the charge generator by a dynamical object, and not simply a defect. The dynamical objects that are electrically charged under C 6 , and would arise when fixing the insertion of the defect as a boundary condition, are D5 branes. While neither argument is conclusive, they both suggest that the holographic description of the symmetry generator is a full D5, pushed to the boundary: 8 This ansatz has the additional virtue of restoring the common origin between lines and charge generators, familiar from the formulation of symmetries in terms of relative field theories. [40] An objection one might raise about (10) is that branes are not topological, while charge operators should be. As we will see in a moment, the worldvolume theory on the branes, when reduced to  2 ⊂ X 5 , is a discrete ℤ 2 gauge theory. Therefore the potential lack of deformation-invariance coming from the gauge fields on the brane is not an issue. There is still an overall factor of the volume, but it does not couple to the dynamical fields of the field theory on the boundary, so it can be absorbed into a counterterm.
A subtle feature of (10) is that the worldvolume theories on the brane are quantum field theories, so we should sum over them. As we will argue, the sum over worldvolume degrees of freedom provides precisely the minimal anomalous TQFT "dressing" the bare symmetry generator identified in [14]. This is a very nontrivial test of the identification (10).
Clearly, if the ansatz (10) is correct, the holographic dual of the operator counting 't Hooft lines H V is the S-dual of (10): Additionally, the SO(4M) theory has the 0-form parity symmetry discussed above. The point operator charged under this symmetry is known as the Pfaffian operator. As discussed in [29] the Pfaffian operator is represented holographically by a D3 brane wrapping the ℝℙ 3 cycle inside ℝℙ 5 , and extending to a point on the boundary. We will refer to this brane as the "Pfaffian brane". We now argue that the holographic dual of the generator of this symmetry is www.advancedsciencenews.com www.fp-journal.org that the endpoint of the Pfaffian operator is at t = 0. Now we wrap the putative symmetry generator D3 on  3 × ℝℙ 1 , where  3 is at the boundary at t = 0. Because the F 5 RR flux is self-dual, the two D3 branes that we have introduced do not commute [41][42][43][44] : where Pf (pt) is the D3 brane representing the Pfaffian operator, (ℝℙ 1 , ℝℙ 3 ) = 1 2 is the linking pairing between the given cycles of ℝℙ 5 , and we have used that the given branes intersect at a point on the t = 0 spatial slice on X 5 . Equation (13) is the Hamiltonian version of the statement that the Pfaffian operator is charged under D3( 3 × ℝℙ 1 ), as claimed. This discussion can be generalised straightforwardly to show that the branes (10) and (11) do indeed give the expected charges to the W V and H V lines of the SO (4N) theory, as claimed.

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

Fluxes and Twisted Differential Cohomology
Our basic tool will be differential cohomology. We refer the reader to [45] for a review of the basic techniques and notation that we use. The analysis in this paper has some novelties with respect to the discussion in [45], which we now discuss.
The main difference is that we will be working with twisted differential cohomology. The twisted and untwisted cohomology groups of ℝℙ 5 were given in (4) above. The ring structure induced by the cup product for ℝℙ 5 can be obtained by adapting the discussion in Lemma 1 of [46] (see also [34]). It is most easily described by adjoining the twisted and untwisted cohomology groups * (ℝℙ 5 That is, we have free components of degree 0 and 5, and ℤ 2 torsional components of degrees 1 to 5, generated by t n 1 . In particular, taking an even number of powers of t 1 gives an untwisted class, while taking an odd number of powers gives a twisted one. In what follows we will use the notation t 2n+1 = t 2n+1 1 for twisted classes and u 2n = t 2n 1 together with u 5 for untwisted ones. We denote byt k a flat differential cohomology class with characteristic class t k , which we denote by I(t k ) = t k , and similarly foȓ u k . We note that I(t 2 1 ) = u 2 , and similarly I(t 4 1 ) = u 4 , so perfectness of the linking pairing on H 2 (ℝℙ 5 ; ℤ) × H 4 (ℝℙ 5 ; ℤ) = ℤ 2 × ℤ 2 implies that This equation together with the ring structure (14) will be our workhorses in what follows. Finally, before moving on to the examples, we need to know how to represent background fluxes in terms of differential cohomology. To lighten notation, in this section we introduce b 2 := B e 2 and c 2 := C m 2 . Recall that the objects charged under these backgrounds are F1 and D1 branes, respectively, so the holographic fluxes encoding these backgrounds areH 3 andF 3 , which are asymptotically of the formH 3 = b 2 ⋆t 1 andF 3 = c 2 ⋆t 1 . By imposing this asymptotic form we ensure that the charged lines in the field theory acquire the right holonomies, see [45,47] for analysis of similar examples. We could also include terms proportional tot 3 in these expansions, but they would correspond to a change of the gauge algebra to (4N + 1) (forF 3 ) of (4N) (forH 3 ) [29] so we will not consider these terms further. 9 Finally, a field theory 0-form symmetry background a 1 for D (0) 2 is represented byF 5 = a 1 ⋆ȗ 4 . There are additional terms possible in the expansion forF 5 , we will discuss these below.

Non-Invertibles in Sc(4N) from D3 Branes
We start with the case of the Sc(4N) theory, where following the analysis in [12,14], we expect to get the topological defect that is non-invertible from the generator of the 0-form symmetry in the SO(4N) theory. We identified this generator above with the D3-brane wrapped on  3 × ℝℙ 1 =  3 × S 1 . The worldvolume U(1) field on the D3 brane is odd under −1 ∈ SL(2, ℤ), because it is a trivialisation of B 2 , which is odd. Therefore it takes values in the twisted cohomology group H 2 ( 3 × ℝℙ 1 ;Z). We have H * (ℝℙ 1 ;Z) = {0, ℤ 2 }.
When computing the path integral on the D3, the field strength F 2 on the brane will induce D1 charge due to the Wess-Zumino term (9), while the magnetic field strength F D 2 will induce F1 charge. The Wess-Zumino action written in terms of the electric variable F 2 is withF 2 =F 2 −B 2 ,F 1 a differential cohomology uplift of C 0 , and e is a differential cohomology uplift of the Euler class of  3 × ℝℙ 1 . The term proportional toF 1 will be relevant only for analysis of anomalies in the space of coupling constants, which we do not analyse in this note (although this is certainly an interesting direction to explore further).
We also need to consider couplings of the formF 3 ⋆B 2 . As argued in [37] (elaborating on results of [48,49]), these couplings are not present when measuring the actual K-theory charges, which is what we are ultimately interested in, so we will simply setB 2 to 0. A more careful treatment of this issue would be desirable, but given that inclusion of these background fields would not change our conclusions (since they would provide overall invertible prefactors on the brane action in any case, even if we included them), we will postpone a more careful treatment of this point to future work.
With these simplifications taken into account, the relevant part of the Wess-Zumino action for the D3 becomes The first term is the differential cohomology avatar of the naive guess ∫  3 ×ℝℙ 1 C 4 for the flux operator in the field theory.F 5 is even under the −1 ∈ SL(2, ℤ) action, so its general decomposition is of the formF 5 = N ⋆ȗ 5 + a 1 ⋆ȗ 4 + a 3 ⋆ȗ 2 + N ⋆1. Here N are the number of units of RR 5-form flux on the ℝℙ 5 , and we have used that F 5 is self-dual to relate the components of degrees 5 and 0. In terms of this decomposition we have an effective operator in AdS 5 of the form where in the second equality we have used Poincaré duality on ℝℙ 5 to relate ℝℙ 1 to u 4 , and in the third used that the only non-trivial pairing in ℝℙ 5 appearing after the expansion ofF 5 is (15). This is the expected formula for the operator measuring discrete electric flux for the outer automorphism symmetry in the SO(4N) theory. The second term is the more interesting one for our purposes. As explained above, field theory backgrounds for the symmetry D C,m 2 are described holographically by fluxes with asymptotic formF 3 = c 2 ⋆t 1 . Similarly, we can expandF 2 = 1 ⋆t 1 . We then have (using the formulas for integration on products reviewed in [45]) where in the last step we have again used the fact that the linking pairing is perfect, so So far we have considered the charge induced on a D3 due to the gauge field strength F 2 . The computation above shows that it induces an effective coupling on  3 to the background for D C,m 2 . By IIB S-duality, this implies that a dual field strength F D 2 = 1 ⋆̆t 1 induces a coupling of the form on the effective operator on AdS 5 . The same result can be obtained from the effective action presented in the magnetic variables obtained in [50]. In elementary terms, the two couplings (19) and (21) that we have just derived can be understood as encoding the well known facts that worldvolume flux on the D3 induces D1 charge, and magnetic worldvolume flux F1 charge. Recall that the D1 and F1 are the charged objects in the SO(4N) theory before gauging their corresponding symmetries. After gauging, they will become the symmetry generators for the dual magnetic symmetries in the Sc(4N) theory (at least if our general philosophy of identifying branes with symmetries is correct). So what we have just shown, is that when doing the path integral on the D3 we will have to sum over insertions of the symmetry generators for the 1-forms of the theory. This is certainly suggestive that condensations [9,13,16] are going to enter the picture after gauging.
The precise details are nevertheless somewhat subtle. In general, when performing the path integral the standard prescription is that we choose whether we formulate the theory in terms of electric or magnetic variables, and then sum over the specified variables only. From this point of view the two couplings (19) and (21) seem somewhat at odds, and it is not clear which one we should choose. What saves the day is that this standard prescription has to be subtly modified whenever the cohomology groups where the electric and magnetic fluxes live contain torsional components. In this case, as originally pointed out by [42][43][44], the electric and magnetic flux operators do not commute. As shown in [45] (see also [51] for a different derivation of the same result) this flux non-commutativity leads to the existence of a discrete gauge theory when the theory is compactified on the space with torsion. The argument, adapted to the system at hand, goes as follows.
Our initial theory is four dimensional U(1) Maxwell theory on the D3, compactified on  3 × ℝℙ 1 . We will present a Hamiltonian quantisation analysis, so we assume that  3 =  2 × ℝ, and we identify the last component with the time direction. 10 The spatial slice is of the form  2 × ℝℙ 1 . There is a non-trivial SL(2, ℤ) duality bundle along the ℝℙ 1 = S 1 direction inherited from the ℝℙ 5 background, with holonomy −1, which induces a (F 2 , F D 2 ) → (−F 2 , −F D 2 ) transformation of the worldvolume gauge field. Therefore, just as in the IIB background itself, the worldvolume gauge fields on the D3 are valued in twisted cohomology. In particular H 1 (ℝℙ 1 ;Z) = ℤ 2 , which justifies the statement above that there is torsion in this problem.
Consider the operators Φ e (a ⊗ t 1 ), Φ m (b ⊗ t 1 ) that measure electric and magnetic fluxes on the torsional sector. They are associated with flat, topologically non-trivial elements of Tor H 2 ( 2 × ℝℙ 1 ;Z), [43,44] which in our case are all of the form a ⊗ t 1 , where a ∈ H 1 ( 2 ; ℤ) and t 1 is the generator of H 1 (ℝℙ 1 ;Z). Alternatively, using Poincaré duality, we can view these operators as the holonomy of the twisted fluxesF 2 andF D 2 on cycles ×pt and ×pt, where and are Poincaré dual to a and b in  2 , andpt 10 A Lagrangian derivation will appear in [52]. www.advancedsciencenews.com www.fp-journal.org the (twisted) point in H 0 ( 2 ;Z), which is Poincaré dual on ℝℙ 1 to t 1 . So we have Φ e (a ⊗ t 1 ) = exp and similarly Now, it follows from the general analysis of [42][43][44] that or equivalently, formulating everything in terms of homology on  2 (and abusing notation slightly): These commutation relations are precisely those of a ℤ 2 theory. We can represent this theory by a gauge theory on the two fields 1 , 1 with action [53] S ℤ 2 = i ∫  3 1 1 .
We have identified the fields appearing in the Lagrangian with 1 and 1 since these are precisely the fields whole holonomies are measured by the operators in the theory, by construction. Assembling all the pieces together, we find that the effective partition function on the D3, seen as an 3-surface dynamical object on X 5 , is (up to an overall normalisation) The path integral over 1 , 1 is the remnant of the U(1) YM path integral in this torsional setting. This is precisely the noninvertible operator found in [14].

Fusion Rules
Now that we have a full description of the symmetry defect, including its TQFT sector, we can derive the fusion rules for the extended operators in the Sc(4N) theory, in particular showing that  ( 3 ) is a non-invertible operator of the Sc(4N) theory. Since the TQFT that comes out of the brane dynamics is identical to the one conjectured in [14], the rest of our derivation of the fusion rules can proceed exactly as in that paper (and the similar analysis in [12]). We include the details of the argument for completeness and convenience for the reader, and then offer some comments reinterpreting some of the features of the computation from a brane perspective. Consider first the fusion of two copies of  ( 3 ). Each defect comes with its own ℤ 2 TQFT, so we have two sets of dynamical fields: ) . (28) Switching to new variables 1 ,̂1 := 1 + ′ 1 , 1 ,̂1 := 1 + ′ 1 , the action becomes . (29) We can integrate 1 and 1 out, which imposeŝ1 =̂1 = 0, sô 1̂1 = 0. We then have Poincaré dualisinĝ1 and̂1 to Γ, Φ ∈ H 2 ( 3 ; ℤ), this can be rewritten as where D ,e 2 and D ,m 2 are the 1-form symmetry generators of the Sc(4N) theory. (The notation is explained below.) So  is indeed a non-invertible defect in the Sc(4N) theory, since the right hand side is a sum of operators.
This was the derivation in [14]. Holographically, the physical meaning of the computation can be understood as follows. We have argued that the defect  ( 3 ) corresponds to a D3 wrapping  3 × ℝℙ 1 , including its quantum dynamics. The effect of the quantum dynamics is to sum over induced charges, which in this case means summing over D3/F1 and D3/F1 bound states. (The precise way in which this sum happens involves, as shown above, a ℤ 2 gauge theory.) If there was no sum, but only a fixed induced charge (the trivial one, say), then taking the square would lead to a complete annihilation of the ℤ 2 charges, and therefore a trivial operator. Since there is a sum involved some of the crossterms in the square of the sum will lead to incomplete annihilations, leaving a sum over F1 and D1 insertions along the worldvolume of the D3. The D3 charge is always there no matter the induced charge, and disappears, so only the sum over D1 and F1 insertions remains.
In order to show that this physical process does indeed produce (31), all we need to verify is that the symmetry generators of the Sc(4N) theory are the F1 and D1. This is immediate, since www.advancedsciencenews.com www.fp-journal.org they are the genuine lines in the SO(4N) theory, and we are gauging the symmetry they are charged under, so they become the magnetic symmetry generators in the gauged theory. It is also instructive to derive it from the Spin(4N) starting point. Thespinor Wilson line W , given by a D5 brane wrapped on ℝℙ 4 , is neutral under ℤ 2 (recall our conventions from footnote ), which is the symmetry that we gauge to go to Sc (4N). So the corresponding charge operator, the D1 onpt, survives as a charge operator on the Sc(4N) theory. We have denoted it above by D ,e 2 . On the other hand the -spinor and vector Wilson lines are not invariant, due to the presence of fundamental strings in them, which are not invariant under ℤ 2 . So after gauging ℤ 2 the fundamental string on a twisted point in ℝℙ 5 becomes the second (magnetic) symmetry generator in the Sc(4N) theory, which we have denoted above by D ,m 2 . We are finally left with the task of determining the fusion of  ( 3 ) with the one-form symmetry generators D ,e 2 (Γ) and D ,m 2 (Φ). Consider for example D ,e 2 (Γ). We have just argued that it corresponds to a D1 brane on Γ ×pt. Fusing it with  ( 3 ), which involves a sum over induced D1 branes wrapping the Poincaré dual PD[ 1 ] ×pt to 1 amounts to shifting 1 → 1 + PD[Γ] in (27). But this can clearly be reabsorbed in a change of variables, giving back  ( 3 ). So An identical argument shows We have shown that the  ( 3 ) operators of the Sc (4N) theory are non-invertible, and are represented holographically by D3 branes. A small puzzle remains: our starting point was that the bulk of the holographic dual was the same for all global forms, so the same D3 brane appears in the bulk of all theories with the same local dynamics, including theories that are not expected to have non-invertible symmetries. The reason that the D3 does not lead to non-invertible symmetries in some cases has to do with boundary conditions (as it should, as this is the only thing that is different in the various cases). Consider for instance the SO(4N) theory, where the D3 on ℝℙ 1 is also a symmetry operator, implementing the outer automorphism. As we push to the boundary, we obtain an operator of the form (27), but with a crucial difference: the IIB B 2 and C 2 fields have a Dirichlet boundary condition in this case, so they are not dynamical but instead they provide backgrounds for the global 1-form symmetries for the SO(4N) theory. So the term in (27) (where we have capitalised B 2 and C 2 to indicate that now they are fixed background fields) does not depend on any dynamical field in the SO(4N) theory, so it is essentially trivial as an operator of the SO(4N) theory (it can be taken out of the path integral). In this case it is consistent to split it off from the invertible part D 3 ( 3 ), which can meaningfully be considered in isolation.

4d PO(4N) and Pin + Non-Invertibles
The other two theories with non-invertible symmetries in (6) can be analysed in a very similar way. Let us start with the PO (4N) case. Here we gauge D (0) 3 and D B,e 2 , so we expect the non-invertible 2-surface operator to be associated with D C,m 2 , which we argued above is given by a D5 brane wrapping ℝℙ 4 ⊂ ℝℙ 5 . We will need the twisted and untwisted cohomology groups of ℝℙ 4 , these are (The second line follows from analysing the twisted Gysin sequence in [34].) As above,F 2 is in the twisted sector, so it expands as 1 ⊗t 1 , but its magnetic dualF D 4 is now untwisted: this is needed to be able to write a kinetic term on the twisted ℝℙ 4 . It therefore has an expansion of the formF D 4 = 4 ⋆ 1 + 2 ⋆ȗ 2 + 0 ⋆ȗ 4 . In the electric frame the action on the D5 is of the form where the missing terms are proportional toF 1 , so we will ignore them. The term proportional toF 2 2 ⋆F 3 does not contribute for degree reasons, as it goes ast 3 1 . The curvature termȇ ⋆F 3 could in principle contribute, but it does not depend on the electric field so it will not enter our considerations. We are left with the first two terms. The first one does clearly contribute, and leads to the expected "naive" 2-surface holonomy operator on  4 , entirely analogously to the discussion around (18). The second term is also interesting. Given our expansion ofF 5 above, there is a single non-vanishing contribution of the form 2 i ∫  2 ×ℝℙ 4 ( 1 ⋆t 1 ) ⋆ (N ⋆ȗ 5 + a 1 ⋆ȗ 4 + a 3 ⋆ȗ 2 + N ⋆1) where we have used that ℝℙ 4 is Poincaré dual to t 1 in ℝℙ 5 . This is the statement that worldvolume flux F 2 induces D3 charge. The magnetic flux F D 4 will induce F1 charge (by a generalisation of the analysis in [50]), via a coupling of the form where we have used the expansionH 3 = b 2 ⋆t 1 as above.
All that remains is to obtain the prescription for how to sum over electric and magnetic fluxes. As above, flux noncommutativity can be used to argue that there is an effective ℤ 2 gauge theory on  2 with action i ∫  2 1 0 .
www.advancedsciencenews.com www.fp-journal.org The only new subtlety in this derivation comes from the fact that on ℝℙ 4 , being non-orientable, the perfect torsional pairing is between a twisted classt 1 and an untwisted oneȗ 4 . An easy way to verify the existence of such a coupling is to use Poincaré duality on ℝℙ 5 : Putting all these terms together we obtain the topological action S PO(4N) TFT = i ∫  2 1 0 + 0 b 2 + 1 a 1 (41) which is precisely the action proposed in [14]. The fusion algebra can be derived as above.
Finally, in the 4d Pin + (4N) SYM theory we gauge D 3 and D C,m 2 , so the non-invertible surface defects are realised as NS5-branes on ℝℙ 4 ×  2 . The worldvolume theory on the NS5 is just as on the D5, but the gauge fields couple to the S-dual supergravity fields. We can therefore write down the answer immediately from (41):