Research
Research
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.
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 holomorphic 1-forms in the disc with values in the Lie algebra of the group. The construction and characterisation of these measures is the topic of a recent preprint.
In the future, I plan to leverage this result to construct natural measures on moduli spaces of G-bundles and study the coupling with random metrics (aka the coset WZW model). I also plan to expore the connections with the quantum analytic Langlands correspondence.
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...