Correlation function with the insertion of zero modes of modular Hamiltonians

TL;DR

This paper investigates zero modes of modular Hamiltonians in 2D massless free scalar theory, deriving finite correlators and establishing their relation to conformal blocks, with explicit fixing of certain correlators using 2D CFT data.

Contribution

It identifies zero modes of modular Hamiltonians in momentum space and connects correlators involving these modes to conformal blocks, providing explicit calculations in 2D CFTs.

Findings

Zero modes are found in momentum space for 2D massless scalar theory.

Finite correlators are extracted from connected correlation functions with zero mode insertions.

Correlators of (n,1)-type are shown to be conformal blocks, with explicit fixing of (2,1)-type correlators.

Abstract

Zero modes of modular Hamiltonian of one interval are found in momentum space for two dimensional massless free scalar theory. Finite correlators are extracted from separate region connected correlation functions with the insertion of zero modes. Correlators of $(n,1)$-type are claimed to be conformal block up to a set of theory dependent constants. We fix the correlators of $(2,1)$-type with the coefficients of three point function in 2d CFTs.