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On unitary 2-representations of finite groups and topological quantum field theory

Bruce Bartlett

TL;DR

The work investigates unitary 2-representations of finite groups within extended TQFT, introducing the 2-character and proving its unitarily faithful embedding into equivariant vector bundles, while computing the 2-categorical dimension as the category of conjugation-equivariant bundles with fusion. It develops the framework of even-handed structures to unify dualities across 2-categories, and applies it to fusion categories to characterize pivotal structures as twisted monoidal natural transformations, with sphericality governed by a vanishing pivotal cohomology class. The geometry of ordinary representations is connected to holomorphic line bundles and Bergman kernels, establishing a categorical bridge to 2-representations via equivariant gerbes and geometric characters. Collectively, the thesis links higher-categorical representations, geometric quantization, and fusion-category dualities to provide a cohesive picture of how extended TQFT principles arise frompoint-like data and how dualities govern pivotal and spherical structures.

Abstract

This thesis contains various results on unitary 2-representations of finite groups and their 2-characters, as well as on pivotal structures for fusion categories. The motivation is extended topological quantum field theory (TQFT), where the 2-category of unitary 2-representations of a finite group is thought of as the `2-category assigned to the point' in the untwisted finite group model. The first result is that the braided monoidal category of transformations of the identity on the 2-category of unitary 2-representations of a finite group computes as the category of conjugation equivariant vector bundles over the group equipped with the fusion tensor product. This result is consistent with the extended TQFT hypotheses of Baez and Dolan, since it establishes that the category assigned to the circle can be obtained as the `higher trace of the identity' of the 2-category assigned to the point. The second result is about 2-characters of 2-representations, a concept which has been introduced independently by Ganter and Kapranov. It is shown that the 2-character of a unitary 2-representation can be made functorial with respect to morphisms of 2-representations, and that in fact the 2-character is a unitarily fully faithful functor from the complexified Grothendieck category of unitary 2-representations to the category of unitary equivariant vector bundles over the group. The final result is about pivotal structures on fusion categories, with a view towards a conjecture made by Etingof, Nikshych and Ostrik. It is shown that a pivotal structure on a fusion category cannot exist unless certain involutions on the hom-sets are plus or minus the identity map, in which case a pivotal structure is the same thing as a twisted monoidal natural transformation of the identity functor on the category. Moreover the pivotal structure can be made spherical if and only if these signs can be removed.

On unitary 2-representations of finite groups and topological quantum field theory

TL;DR

The work investigates unitary 2-representations of finite groups within extended TQFT, introducing the 2-character and proving its unitarily faithful embedding into equivariant vector bundles, while computing the 2-categorical dimension as the category of conjugation-equivariant bundles with fusion. It develops the framework of even-handed structures to unify dualities across 2-categories, and applies it to fusion categories to characterize pivotal structures as twisted monoidal natural transformations, with sphericality governed by a vanishing pivotal cohomology class. The geometry of ordinary representations is connected to holomorphic line bundles and Bergman kernels, establishing a categorical bridge to 2-representations via equivariant gerbes and geometric characters. Collectively, the thesis links higher-categorical representations, geometric quantization, and fusion-category dualities to provide a cohesive picture of how extended TQFT principles arise frompoint-like data and how dualities govern pivotal and spherical structures.

Abstract

This thesis contains various results on unitary 2-representations of finite groups and their 2-characters, as well as on pivotal structures for fusion categories. The motivation is extended topological quantum field theory (TQFT), where the 2-category of unitary 2-representations of a finite group is thought of as the `2-category assigned to the point' in the untwisted finite group model. The first result is that the braided monoidal category of transformations of the identity on the 2-category of unitary 2-representations of a finite group computes as the category of conjugation equivariant vector bundles over the group equipped with the fusion tensor product. This result is consistent with the extended TQFT hypotheses of Baez and Dolan, since it establishes that the category assigned to the circle can be obtained as the `higher trace of the identity' of the 2-category assigned to the point. The second result is about 2-characters of 2-representations, a concept which has been introduced independently by Ganter and Kapranov. It is shown that the 2-character of a unitary 2-representation can be made functorial with respect to morphisms of 2-representations, and that in fact the 2-character is a unitarily fully faithful functor from the complexified Grothendieck category of unitary 2-representations to the category of unitary equivariant vector bundles over the group. The final result is about pivotal structures on fusion categories, with a view towards a conjecture made by Etingof, Nikshych and Ostrik. It is shown that a pivotal structure on a fusion category cannot exist unless certain involutions on the hom-sets are plus or minus the identity map, in which case a pivotal structure is the same thing as a twisted monoidal natural transformation of the identity functor on the category. Moreover the pivotal structure can be made spherical if and only if these signs can be removed.

Paper Structure

This paper contains 104 sections, 92 theorems, 641 equations, 19 figures, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Verifying $Z(S^1) \simeq \text{Dim } Z(\text{pt})$ in the finite group model
  4. The 2-character functor is unitarily fully faithful
  5. Characterizing pivotal structures on fusion categories
  6. Comparison with previous work
  7. Overview of thesis
  8. The geometry of ordinary representations of groups
  9. Holomorphic hermitian line bundles and kernels
  10. The Bergman kernel
  11. The Bergman kernel via modes of propagation
  12. The Bergman kernel as the large time limit of the heat kernel
  13. The Bergman kernel as a path integral
  14. The Bergman kernel and coherent states
  15. The category of line bundles and kernels
  16. ...and 89 more sections

Key Result

Theorem 1

The higher-categorical dimension of the 2-category of unitary 2-representations of a finite group $G$ is equivalent, as a braided monoidal category, to the category of conjugation-equivariant hermitian vector bundles over $G$ equipped with the fusion tensor product. In symbols,

Figures (19)

  • Figure 1: A holomorphic kernel from a line bundle $L \rightarrow X$ to another line bundle $Q \rightarrow Y$.
  • Figure 2: The Bergman kernel $\langle y | x \rangle$ sums over the fundamental modes of propagation from $L_x$ to $L_y$.
  • Figure 3: The section $s_v$ of the geometric line bundle over projective space associated to a vector $v \in V$.
  • Figure 4: The kernel $\langle l' | A | l \rangle$ is defined by applying $A$ and then projecting onto the line $l'$.
  • Figure 5: A morphism between finite spaces is a Hilbert bundle over their product.
  • ...and 14 more figures

Theorems & Definitions (191)

  • Theorem
  • Theorem
  • Theorem
  • Lemma 2.1
  • proof
  • Theorem 2.2: refer Bär and Pfäffle ref:bar_pfaffle
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • ...and 181 more