Bornological Methods in TQFTs

Topological Quantum Field Theories (TQFTs), axiomatised by Atiyah in the 80s, lie at the frontier of research in Physics and Mathematics. Most QFTs are sensitive to changes in the local structure of spacetime, but TQFTs are sensitive to its global structure, its topology. Roughly, a TQFT is a symmetric monoidal functor from a certain category of cobordisms to the category of vector spaces. Although their origin was in Physics, TQFTs are related to many areas of Mathematics, such as Knot Theory, moduli spaces, and quantum groups.

A semisimple Lie algebra g can be associated with a non-commutative algebra Uq(g), its quantised universal enveloping algebra, which is a Hopf algebra dependent on a parameter q. These algebras are members of a family called algebraic quantum groups, the Hopf algebra structure emulating that of a group. When q is a root of unity, these quantum groups give rise to invariants of 3-manifolds.

These 3D TQFTs are well understood but constructing 4D TQFTs is more difficult. Motivated by the idea that replacing algebraic structures with similar categorical ones lifts the dimension of the corresponding TQFT by one, Crane and Frenkel developed the concept of categorification. Recent work by Kremnitzer et al. focuses on categorifying the bialgebra structure on Borel subgroups of quantum groups using a geometric approach to Hall algebras.

Although quantum groups are usually approached from an algebraic perspective, several analytic analogues of quantum groups have recently arisen. Soibelman's quantum deformations of algebras of locally analytic functions on p-adic Lie groups paved the way for this new area of analytic geometry, but recent work by Smith on deformations of analytic Nichols algebras is of particular interest to us. Smith creates analytic analogues of quantised universal enveloping algebras using Majid's double-bosonisation construction to glue together analytic Nichols algebras over IndBanach spaces.

If topological spaces are the natural settings to discuss continuity, then bornological spaces are the correct setting to talk about boundedness. The category of IndBanach spaces is closely related to the category of complete bornological spaces. Translating our analytic quantum groups, which are IndBanach Hopf algebras, to the category of complete bornological spaces has many advantages. In particular, following Prosmans and Schneiders, many concepts from homological algebra can be extended to this quasi-abelian category.

As in the algebraic case, we expect our analytic quantum groups to give rise to TQFTs and that categorification will produce four-dimensional TQFTs. The main focus of my research will be to explore analytic quantum groups, themselves an underexplored area of mathematics, and to develop and use bornological methods to study their invariants.

The starting point for my research will be developing my previous work on Koszul duality in quasi-abelian categories. The quasi-abelian categories we are interested in working in, namely the category of complete bornological spaces, should contain many interesting types of Koszul duality. A further avenue of research would be to develop 'analytic homological algebra'. By replacing complexes of graded bornological spaces by their analytically graded analogues, we could provide a framework for extending Koszul duality to analytic quantum groups. Additionally, Nichols algebras could connect the theory I have developed for symmetric and exterior monoids with quantum groups.

TQFTs appear across numerous disciplines, with potential applications in particle accelerators, superconductors, and quantum computers. In addition, the category of bornological spaces is thought to be a natural setting for doing analysis, so developments in bornological methods would be of interest to many mathematicians.

This project falls within the EPSRC Algebra, Geometry and Topology, and Mathematical Physics