String Beta Function Equations From c=1 Matrix Model

TL;DR

This paper derives the tachyon beta-function equations of 2D string theory within the c=1 matrix model, highlighting the nonlocal nonlinear relations needed to connect matrix model variables to spacetime physics.

Contribution

It provides a derivation of the tachyon beta-function equations directly from the c=1 matrix model, emphasizing the nonlocal nonlinear transformations involved.

Findings

The tachyon beta-function is satisfied by a nonlocal nonlinear combination of matrix model fields.

The work discusses the representation of discrete states and gravitational backgrounds within the matrix model.

It comments on the realization of W-infinity symmetry in the string theory context.

Abstract

We derive the $\sigma$-model tachyon $\beta$-function equation of 2-dimensional string theory, in the background of flat space and linear dilaton, working entirely within the $c=1$ matrix model. The tachyon $\beta$-function equation is satisfied by a \underbar{nonlocal} and \underbar{nonlinear} combination of the (massless) scalar field of the matrix model. We discuss the possibility of describing the `discrete states' as well as other possible gravitational and higher tensor backgrounds of 2-dimensional string theory within the $c=1$ matrix model. We also comment on the realization of the $W$-infinity symmetry of the matrix model in the string theory. The present work reinforces the viewpoint that a nonlocal (and nonlinear) transform is required to extract the space-time physics of 2-dimensional string theory from the $c=1$ matrix model.