# PHYSICS PHD DISSERTATION DEFENSE: Milind Shyani

Ph.D. Candidate:  Milind Shyani

Date: Wednesday, June 9, 2021

Time: 11:00 AM

Title: Dispersion relations in Conformal field theories

Abstract: Dispersion relations of analytic functions provide a powerful tool in understanding strongly coupled field theories. We discuss the Lorentzian Inversion formula of Caron-Huot, a dispersion relation that relates the OPE coefficients to the discontinuity of the four-point function in any conformal field theory, and elaborate upon the utility and importance of such inversion formulae in two different ways.

First, we argue that the Inversion formula takes a natural form in Mellin space and derive a Mellin space inversion formula. We then utilize the fact that Mellin space provides a natural representation for any Holographic correlator, and compute the anomalous dimensions and OPE coefficients of double-trace scalar primaries up to order $\frac{1}{N^4}$.

Second, we study the two-point function of a local operator on an $n$-sheeted replica manifold corresponding to the half-space in the vacuum state of a conformal field theory. Such two-point functions are used to compute the entanglement entropy of excited states. We argue that the two-point function is analytic in $n$ by deriving an inversion formula similar to Caron-Huot's. We discover an interesting identity between the discontinuity of a generating function and the matrix element of local operators in the eigenbasis of the modular Hamiltonian. We show that our formulae reproduce the recent results on the off-diagonal elements of the Eigenstate Thermalization Hypothesis in 2d CFTs.

June 9, 2021 - 11:00am