The Branes Behind Generalized Symmetry Operators

Abstract

The modern approach to m-form global symmetries in a d-dimensional quantum field theory (QFT) entails specifying dimension d − m − 1 topological generalized symmetry operators which non-trivially link with m-dimensional defect operators. In QFTs engineered via string constructions on a non-compact geometry X, these defects descend from branes wrapped on non-compact cycles which extend from a localized source / singularity to the boundary 𝝏X. The generalized symmetry operators which link with these defects arise from magnetic dual branes wrapped on cycles in 𝝏X. This provides a systematic way to read off various properties of such topological operators, including their worldvolume topological field theories, and the resulting fusion rules. We illustrate these general features in the context of 6D superconformal field theories, where we use the F-theory realization of these theories to read off the worldvolume theory on the generalized symmetry operators. Defects of dimension 3 which are charged under a suitable 3-form symmetry detect a non-invertible fusion rule for these operators. We also sketch how similar considerations hold for related systems.

Exploring non-invertible symmetries in free theories

“…Non-invertible symmetries are long known to exist in two-dimensional rational CFTs since the work of Verlinde [2] (see also [3][4][5][6][7][8][9][10][11][12][13][14]). It was recently realized that non-invertible symmetries also exist in non-conformal theories in two dimensions [15], as well as in QFTs in higher dimensions [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30], and have turned out to be useful in various contexts [31][32][33][34][35][36][37][38][39][40]. However, so far in the literature the existence of non-invertible symmetries has been shown either by using indirect methods or by using methods that are particular to certain classes JHEP03(2023)005 of theories.…”

Branes and symmetries for $\mathcal N=3$ S-folds

“…Gauging the 1-form symmetries leads to the 0-form symmetries becoming non-invertible [20,21]. A detailed field theory analysis of the non-invertible symmetries for the cases discussed in this paper has been presented in [22] and has been given a holographic interpretation in [23] (see also [29,30] for related work in other cases).…”

“…The non-commutativity comes from couplings of the branes to the RR fields in the Wess-Zumino terms in the action. We note that we are interpreting the asymptotic D1 and D5 as the symmetry operators themselves (associated to the integrals of F3 ∼ dC 2 and * F3 ∼ * dC 2 respectively) [23,29,30]. Choosing C 2 as our basic variable, as above, and still modeling IIB by generalized Maxwell (we will refine this momentarily), our previous discussion implies that…”

A Panorama Of Physical Mathematics c. 2022

“…Higher-group symmetries can be computed from the geometry, but anomalies for these should also fall into the framework of symmetry TFTs and should arise from string theory. Non-invertible symmetry in higher dimensions have recently been constructed within string theory [198,199,1110,1111], and play a role in the context of the swampland program [1112].…”