Research

Random Jordan curves

This has been an active topic in probability theory since the early years of the 21st century, with four different approaches:

Stochastic Loewner evolution (now Schramm-Loewner evolution or SLE), describing scaling limits in statistical systems at the critical temperature.
Virasoro unitarising measures: hypothetical measures whose L2-space carry a unitary representation of the Virasoro algebra.
Restriction covariant measures: transforming covariantly upon restriction to subdomains.
Random conformal weldings: solutions to conformal welding problems involving Gaussian multiplicative chaos.

That SLE satisfies conformal restriction was known since the work of Lawler-Schramm-Werner, and the equivalence of 1 and 4 was proved by Sheffield. The more elusive parts were that SLE loops are the unique restriction covariant measures, and their equivalence with Virasoro unitarising measures, which is the main conjecture of Kontsevich & Suhov.

This conjecture has been proved in a series of joint works with Jego, where we also clarify the different realisations of the Virasoro algebra, give a new proof of Sheffield's theorem, as well as a mathematical interpretation of the Faddeev-Popov ghost in Polyakov's path integral. We are currently exploring the link with the Belavin-Knizhnik measure on the moduli space of Riemann surfaces.

Kac-Moody unitarising measures (KMU)

The questions above have natural analogues if we replace the Virasoro algebra by a Kac-Moody algebra, and the relevant measure now lives on the space of flat connections in the trivial G-bundle over the disc. The construction and characterisation of these measures is the topic of a recent preprint.

One motivation is the (quantum) analytic Langlands correspondence, which uses the natural inner-product on (twisted) half-densities to introduce a functional analytic approach to the moduli spaces of G-bundles. The KMUs provide an alternative Hilbert space in a disjoint region of the κ-plane (=level).

In the future, I will explore the links with coset WZW models and the quantisation of moduli spaces of flat connections on Riemann surfaces with boundary (an infinite dimensional version of the Hitchin connection).

Critical level -h^\vee: Analytic Langlands correspondence (ALC)Vertical line: quantum ALCBold half-line: KMU region

Liouville conformal field theory (LCFT)

In the last two decades, a major achievement of probability theory was the rigorous construction of the LCFT path integral and the proofs of spectacular formulas within the probabilistic framework (structure constants and conformal bootstrap).

My contribution to this topic was to help constructing and giving a probabilistic interpretation of the algebraic and geometric structures of the theory: Virasoro algebra, semigroup of annuli, degenerate modules, conformal blocks...

Page updated

Google Sites

Report abuse