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

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.

arXiv commentPhD thesis, University of Sheffield, submitted Oct 2008. 243 pages, many figures

References in corpus (21)

• Mirror Manifolds And Topological Field Theory
• Physics, Topology, Logic and Computation: A Rosetta Stone
• Higher-Dimensional Algebra V: 2-Groups
• Chern-Simons Theory with Finite Gauge Group
• Spherical Categories
• Higher Algebraic Structures and Quantization
• Categorical formulation of quantum algebras
• Categorical Aspects of Topological Quantum Field Theories
• Introduction to the language of stacks and gerbes
• Categorical representations of categorical groups
• On the Rozansky-Witten weight systems
• Infinite-Dimensional Representations of 2-Groups
• 2-categorical Poincare Representations and State Sum Applications
• Path integrals on manifolds by finite dimensional approximation
• Smooth Functors vs. Differential Forms
• Quantum Groups from Path Integrals
• Some non-braided fusion categories of rank 3
• Locality and Integration in Topological Field Theory
• The periodic table of $n$-categories for low dimensions II: degenerate tricategories
• A strict totally coordinatized version of Kapranov and Voevodsky's 2-category {\bf 2Vect}
• Small linearly equivalent $G$-sets and a construction of Beaulieu

Cited by in corpus (13)

• Diagrammatics for Soergel categories
• Higher categorical symmetries and gauging in two-dimensional spin systems
• Traces on Module Categories over Fusion Categories
• A Prehistory of n-Categorical Physics
• Categorifications from planar diagrammatics
• Gray categories with duals and their diagrams
• Extended 3-dimensional bordism as the theory of modular objects
• Explicit Formulas for 2-Characters
• A covariant Stinespring theorem
• Crossing with the circle in Dijkgraaf-Witten theory and applications to topological phases of matter
• The Symmetric Tensor Product on the Drinfeld Centre of a Symmetric Fusion Category
• Coherence for adjunctions in a $3$-category via string diagrams
• On the $3$-representations of groups and the $2$-categorical Traces