Branes and Non-Invertible Symmetries

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.