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

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.

💡 Research Summary

This thesis investigates unitary 2‑representations of finite groups and their connections to extended topological quantum field theory (ETQFT). The work is organized around three main results, each addressing a different aspect of the categorical structure that underlies the “point‑circle‑surface” assignment in ETQFT.

First, the author defines the 2‑category 2Repᵤ(G) of unitary 2‑representations of a finite group G. Objects are Hilbert‑space‑valued 2‑vector spaces equipped with a unitary G‑action, 1‑morphisms are G‑linear functors, and 2‑morphisms are natural transformations. The identity 2‑functor Id on this 2‑category has a transformation 2‑category End(Id). By analysing the components of a transformation, the author shows that End(Id) is equivalent to the braided monoidal category of conjugation‑equivariant complex vector bundles over G, equipped with the fusion tensor product. This equivalence provides a concrete realization of the “higher trace of the identity” that Baez and Dolan proposed as the categorical object assigned to the circle in an extended TQFT. In other words, the circle‑level state space obtained from the point‑level 2‑category is precisely the category of equivariant bundles with fusion, confirming the expected compatibility between the point and circle assignments.

Second, the thesis turns to 2‑characters, a categorified analogue of ordinary group characters introduced independently by Ganter and Kapranov. For a unitary 2‑representation ρ, the 2‑character χ_ρ assigns to each group element g a finite‑dimensional vector space (the trace of the action of g on ρ) together with natural isomorphisms encoding conjugation invariance. The author proves that this construction extends to a functor from the complexified Grothendieck category K₀(2Repᵤ(G))⊗ℂ to the category of unitary G‑equivariant vector bundles. Moreover, the functor is fully faithful: distinct 2‑representations yield non‑isomorphic 2‑characters, and morphisms between 2‑representations correspond bijectively to bundle maps between their characters. This result elevates the 2‑character from a numerical invariant to a complete, functorial invariant that captures the entire linear‑categorical data of a unitary 2‑representation. In the language of ETQFT, the 2‑character provides the precise boundary‑to‑bulk map for the circle, encoding how point‑level data propagate to one‑dimensional manifolds.

Third, the thesis addresses pivotal structures on fusion categories, motivated by a conjecture of Etingof, Nikshych, and Ostrik concerning the existence of spherical structures. The author shows that a pivotal structure can exist only if certain involutive endomorphisms on Hom‑spaces are either the identity or minus the identity. When all such involutions are +id, the pivotal structure coincides with a twisted monoidal natural transformation of the identity functor, and it can be made spherical by eliminating the signs. Conversely, the presence of a –id involution obstructs sphericality. This characterization translates the abstract existence problem for pivotal (and spherical) structures into a concrete algebraic condition on the Hom‑spaces, offering a new perspective on the classification of fusion categories suitable for TQFT constructions.

In the concluding discussion, the author synthesizes these three strands. The equivalence between End(Id) and equivariant bundles validates the higher‑trace picture for the circle, the fully faithful 2‑character functor demonstrates that unitary 2‑representations are completely captured by equivariant bundle data, and the pivotal‑structure analysis clarifies which fusion categories can serve as the surface‑level data in an ETQFT. Together, the results provide a coherent categorical framework that bridges unitary 2‑representation theory, higher character theory, and the algebraic underpinnings of extended topological quantum field theories, and they lay groundwork for future investigations into higher‑dimensional quantum symmetries and their physical realizations.