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Probabilistic understanding of Liouville conformal blocks


Term:

Fall

Department:

18: Mathematics

Faculty Supervisor:

Promit Ghosal

Faculty email:

promit@mit.edu

Apply by:

Friday, August 28, 2020

Contact:

Promit Ghosal: promit@mit.edu

Project Description

Conformal field theory (CFT) is a quantum field theory invariant under conformal transformations. From its early days, CFT grown out to be one of the most prominent branches of modern theoretical physics, with applications to the critical behavior of the statistical mechanics model, string theory and, two dimensional quantum gravity as well as far reaching consequences in mathematics. The conformal boostrap, introduced by Belavin, Polyakov and Zamolodchikov for two dimensional CFT is an ubiquitous way of solving the CFTs by describing the infinitesimal conformal symmetries in terms of the Virasoro algebra and encoding them in the so called building blocks of CFT, namely, the conformal blocks. One of the earliest examples of CFT is Liouville CFT which arose in Polyakov's description of bosonic string theory. The conformal blocks of Liouville CFT is specially interesting since it paves the path for critical understanding of the interplay between the Liouville CFT and the supersymmetric gauge theory via Alday-Gaiotto-Tachikawa correspondence. There are recent advances in probabilistic understanding of the Liouville CFT and especially, the Liouville conformal block. In this project, you will explore the state-of-art construction of the probabilistic conformal blocks and its properties. This is one of the cutting-edge research directions and will require expertise on probability theory, special functions, complex analysis, representation theory and simulations.

Pre-requisites

One requires advanced level understanding of real analysis (18.100), differential equation (18.03) and probability theory (18.675). For simulation, prior knowledge of the softwares like mathematica, matlab and MaCaulay 2 would definitely be helpful.