Algebraic presentation of cobordisms and quantum invariants in dimensions 3 and 4

Abstract: The category 2Cob of 2-dimensional cobordisms is freely generated by a commutative Frobenius algebra: the circle. This yields a complete classification of 2-dimensional TQFTs (Topological Quantum Field Theories). In this talk, I will discuss some consequences of analogous algebraic presentations in dimensions 3 and 4, due to Bobtcheva and Piergallini. In both cases, the fundamental algebraic structures are provided by certain Hopf algebras called BPH algebras. In dimension 3, I will consider the category 3Cob of connected cobordisms between connected surfaces with connected boundary. I will explain that an algebraic presentation conjectured (or rather announced without proof) by Habiro is in fact equivalent to the one established by Bobtcheva and Piergallini. In dimension 4, I will focus on a category denoted 4HB, whose morphisms are 2-deformation classes of 4-dimensional 2-handlebodies. I will show that any unimodular ribbon category contains a BPH algebra, which can be characterized very explicitly. This result proves the existence of a very large family of TQFT functors on 4HB. Finally, I will explain that a unimodular ribbon category has the potential to detect exotic phenomena in dimension 4 only if it is neither semisimple nor factorizable. This is a joint work with A. Beliakova, I. Bobtcheva, and R. Piergallini.

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