Quantum invariants of maps from 3-manifolds to homotopy 2-types

Abstract: Topological quantum field theories (TQFTs) provide a powerful framework for constructing invariants of manifolds. Their extension to homotopy quantum field theories (HQFTs) refines these invariants by incorporating maps to a target space. In this talk, I recall Dijkgraaf–Witten invariants and their state-sum description, and explain their extension to HQFTs with target BG. I then discuss the passage from groups to crossed modules as algebraic models for homotopy 2-types, and outline the associated tensor-categorical and Hopf-algebraic structures, including Hopf crossed module coalgebras. Finally, I present joint work with Alexis Virelizier, where we construct Kuperberg-type invariants for pairs (M,g), with g∈[M,B\chi] where \chi is a crossed module. The construction uses \chi-labeled Heegaard diagrams and involutory Hopf \chi-coalgebras, extending the classical Kuperberg invariant and admitting an interpretation in terms of invariant of flat principal 2-bundles.

Mathematics

Audience: researchers in the topic

European Quantum Algebra Lectures (EQuAL)

Series comments: EQuAL is an online seminar series on quantum algebra and related topics such as topological and conformal field theory, operator algebra, representation theory, quantum topology, etc. sites.google.com/view/equalseminar/home