Symmetries of 2d TQFTs and Equivariant Verlinde Formulae for General Groups

Abstract

We study (generalized) discrete symmetries of 2d semisimple TQFTs. These are 2d TQFTs whose "fusion rules" can be diagonalized. We show that, in this special basis, the 0-form symmetries always act as permutations while 1-form symmetries act by phases. This leads to an explicit description of the gauging of these symmetries. One application of our results is a generalization of the equivariant Verlinde formula to the case of general Lie groups. The generalized formula leads to many predictions for the geometry of Hitchin moduli spaces, which we explicitly check in several cases with low genus and SOp3q gauge group.

Generalized symmetries of topological field theories

“…These results are suspiciously close to those of [5], which showed that 0-form symmetries of unextended oriented topological field theories whose corresponding commutative Frobenius algebras are semisimple act on a basis of idempotents by permutations preserving the trace map, while 1-form symmetries act by multiplication of idempotents by elements of C * . In fact this is no coincidence, because the semisimple commutative Frobenius algebras correspond precisely to oriented topological field theories which are extendable: by extending, one can make a proper definition of a 1-form symmetry, and show that the conditions obtained in [5] are not only necessary, but also sufficient. Our results thus not only generalize those of [5], but also place them in their proper context.…”

“…In fact this is no coincidence, because the semisimple commutative Frobenius algebras correspond precisely to oriented topological field theories which are extendable: by extending, one can make a proper definition of a 1-form symmetry, and show that the conditions obtained in [5] are not only necessary, but also sufficient. Our results thus not only generalize those of [5], but also place them in their proper context.…”

“…The same trick of extending theories can be applied to understanding the results of [5], where it is shown that 0-and 1-form symmetries of a sub-class of oriented nonmaximally extended TFTs in d = 2 (i.e. SO(2) TFT 2,1 ) valued in Vect C , to wit those corresponding to commutative Frobenius algebras that are additionally semisimple, are given by permutations and phasings, respectively.…”

Snowmass White Paper: Generalized Symmetries in Quantum Field Theory and Beyond

“…Similar to the anomaly constraints from an ordinary symmetry, the non-invertible symmetries also have dramatic consequences on renormalization group flows. In two-dimensional QFT, invertible and noninvertible topological lines have been explored extensively in [39,91,343,97,246,101,94,[344][345][346][347][348][349][350][351]229,98,352,96,353] in recent years with various dynamical applications. In particular, using a modular invariance argument, it was shown in [91] (see also [97] for generalizations) that the existence of certain non-invertible topological lines is incompatible with a trivially gapped phase.…”

Reading between the rational sections: Global structures of 4d $\mathcal{N}=2$ KK theories

“…We will thus learn how to 'read between the rational sections' of the Seiberg-Witten geometry, in close parallel with how one can 'read between the defect lines' of 4d gauge theories more generally [3]. Our approach builds on many previous works on this and other closely related subjects [2,[10][11][12][13][14][15] -in particular, a proposed relation between global structures and isogenies already appeared (somewhat obliquely) in [2]. We will further comment on the wider picture, and on perspectives for future work, in the final section.…”