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

Sergei Gukov, Du Pei, Charles Reid, Ali Shehper

TL;DR

The paper develops a unified framework for generalized symmetries in 2d semisimple TQFTs and shows that 0-form symmetries permute the idempotent basis while 1-form symmetries act by phases, enabling a detailed gauging program. It extends the Verlinde formalism to equivariant settings for arbitrary simple groups by interpreting Hitchin moduli space quantization as the partition function of an appropriate TQFT in symmetry backgrounds, including non-simply-connected groups via center and higher-form symmetries. The authors provide explicit gauging rules for 0- and 1-form symmetries, derive the resulting partition functions and Frobenius algebras, and illustrate these with concrete computations for PSL(2, C) and related groups, including parabolic and once-punctured torus cases. They connect these results to geometry through the equivariant Verlinde algebra, compute Hitchin characters for moduli spaces with nontrivial fundamental groups, and discuss categorification via line-operator categories in partially topological 3d theories. Overall, the work unifies several strands of Verlinde-type formulas, Hitchin moduli quantization, and 2d TQFT symmetries, offering a scalable method to obtain equivariant dimensions and their geometric implications across 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 SO(3) gauge group.

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

TL;DR

The paper develops a unified framework for generalized symmetries in 2d semisimple TQFTs and shows that 0-form symmetries permute the idempotent basis while 1-form symmetries act by phases, enabling a detailed gauging program. It extends the Verlinde formalism to equivariant settings for arbitrary simple groups by interpreting Hitchin moduli space quantization as the partition function of an appropriate TQFT in symmetry backgrounds, including non-simply-connected groups via center and higher-form symmetries. The authors provide explicit gauging rules for 0- and 1-form symmetries, derive the resulting partition functions and Frobenius algebras, and illustrate these with concrete computations for PSL(2, C) and related groups, including parabolic and once-punctured torus cases. They connect these results to geometry through the equivariant Verlinde algebra, compute Hitchin characters for moduli spaces with nontrivial fundamental groups, and discuss categorification via line-operator categories in partially topological 3d theories. Overall, the work unifies several strands of Verlinde-type formulas, Hitchin moduli quantization, and 2d TQFT symmetries, offering a scalable method to obtain equivariant dimensions and their geometric implications across 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 SO(3) gauge group.
Paper Structure (29 sections, 180 equations, 6 figures, 1 table)

This paper contains 29 sections, 180 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Generalized symmetries of 2D semisimple TQFTs
  3. Quantization of moduli space of Higgs Bundles
  4. Quantization of moduli space of parabolic Higgs bundles
  5. Finite gauging of 2d TQFTs
  6. Symmetries of two-dimensional TQFTs
  7. Gauging of 0-form and 1-form symmetries
  8. Equivariant Verlinde formula for any simple group
  9. Review: the simply connected case
  10. 0-form symmetry and its gauging
  11. Examples
  12. 1-form symmetry and its gauging
  13. Examples
  14. Quantization of $\mathcal{M} (G)$
  15. Examples
  16. ...and 14 more sections

Figures (6)

  • Figure 1: Symmetry of a 2D TQFT is reflected in its partition function as well as its algebra of local operators.
  • Figure 2: The action of 0-form and 1-form symmetry transformations on a state in Hilbert space.
  • Figure 3: The action of 0-form and 1-form symmetry transformations on surfaces in a two-dimensional TQFT.
  • Figure 4: Once-punctured torus
  • Figure 5: Decomposition of $\Sigma$ into caps and handles
  • ...and 1 more figures