Topological Quantum Field Theory Seminar

Planned sessions

Reps of relative mapping class groups via conformal nets

Given a surface with boundary $\Sigma$, its relative mapping class group is the quotient of $\operatorname{Diff}(\Sigma)$ by the subgroup of maps which are isotopic to the identity via an isotopy that fixes the boundary pointwise. (If $\Sigma$ has no boundary, then that's the usual mapping class group; if $\Sigma$ is a disc, then that's the group $\operatorname{Diff}(S^1)$ of diffeomorphisms of $S^1$.)

Conformal nets are one of the existing axiomatizations of chiral conformal field theory (vertex operator algebras being another one). We will show that, given an arbitrary conformal net and a surface with boundary $\Sigma$, we get a continuous projective unitary representation of the relative mapping class group (orientation reversing elements act by anti-unitaries). When the conformal net is rational and $\Sigma$ is a closed surface (i.e. $\partial \Sigma = \emptyset$), then these representations are finite dimensional and well known. When the conformal net is not rational, then we must require $\partial \Sigma \neq \emptyset$ for these representations to be defined. We will try to explain what goes wrong when $\Sigma$ is a closed surface and the conformal net is not rational.

The material presented in this talk is partially based on my paper arXiv:1409.8672 with Arthur Bartels and Chris Douglas.

Counting Monster Potentials

The monster potentials were introduced by Bazhanov-Lukyanov-Zamolodchikov in the framework of the ODE/IM correspondence. They should in fact be in 1:1 correspondence with excited states of the Quantum KdV model (an Integrable Conformal Field Theory) since they are the most general potentials whose spectral determinant solves the Bethe Ansatz equations of such a theory. By studying the large momentum limit of the monster potentials, I retrieve that

1. The poles of the monster potentials asymptotically condensate about the complex equilibria of the ground state potential.
2. The leading correction to such asymptotics is described by the roots of Wronskians of Hermite polynomials.

This allows me to associate to each partition of N a unique monster potential with N roots, of which I compute the spectrum. As a consequence, I prove up to a few mathematical technicalities that, fixed an integer N, the number of monster potentials with N roots coincide with the number of integer partitions of N, which is the dimension of the level N subspace of the quantum KdV model. In striking accordance with the ODE/IM correspondence.

This is joint work with Riccardo Conti (Group of Mathematical Physics of Lisbon University).