$W$-Infinity Ward Identities and Correlation Functions in the $C=1$ Matrix Model

TL;DR

This paper investigates the $W$-infinity symmetry in the $c=1$ matrix model, deriving exact Ward identities that relate correlation functions and provide non-perturbative insights, including calculations of two-point functions and constraints on the partition function.

Contribution

It introduces exact Ward identities for the $W$-infinity symmetry in the $c=1$ matrix model, linking correlation functions to an effective three-dimensional theory and deriving new constraints on the partition function.

Findings

Derived exact Ward identities relating correlation functions.

Calculated the two-point function in the double scaling limit.

Rewrote $W$-infinity charges to impose constraints on the partition function.

Abstract

We explore consequences of $W$-infinity symmetry in the fermionic field theory of the $c=1$ matrix model. We derive exact Ward identities relating correlation functions of the bilocal operator. These identities can be expressed as equations satisfied by the effective action of a {\it three} dimensional theory and contain non-perturbative information about the model. We use these identities to calculate the two point function of the bilocal operator in the double scaling limit. We extract the operator whose two point correlator has a {\it single} pole at an (imaginary) integer value of the energy. We then rewrite the \winf~ charges in terms of operators in the matrix model and use this derive constraints satisfied by the partition function of the matrix model with a general time dependent potential.