Gauge Theory Formulation of the $c=1$ Matrix Model: Symmetries and Discrete States

TL;DR

This paper formulates the $c=1$ matrix model as a gauge theory with symmetries described by the $W^{(+)}_{1+initeinfty}$ algebra, exploring residual symmetries, classical solutions, and connections to string theory and gravity.

Contribution

It introduces a gauge theory framework for the $c=1$ matrix model, revealing its symmetry structure and classical solutions, and discusses its relation to string theory and gravity.

Findings

Residual gauge symmetries are realized through an analytic continuation.

Classical solutions and actions are explicitly constructed.

The $SL(2)$ structure emerges within the $W^{(+)}_{1+initeinfty}$ algebra.

Abstract

We present a non-relativistic fermionic field theory in 2-dimensions coupled to external gauge fields. The singlet sector of the $c=1$ matrix model corresponds to a specific external gauge field. The gauge theory is one-dimensional (time) and the space coordinate is treated as a group index. The generators of the gauge algebra are polynomials in the single particle momentum and position operators and they form the group $W^{(+)}_{1+\infty}$. There are corresponding Ward identities and residual gauge transformations that leave the external gauge fields invariant. We discuss the realization of these residual symmetries in the Minkowski time theory and conclude that the symmetries generated by the polynomial basis are not realized. We motivate and present an analytic continuation of the model which realises the group of residual symmetries. We consider the classical limit of this theory and make the correspondence with the discrete states of the $c=1$ (Euclidean time) Liouville theory. We explain the appearance of the $SL(2)$ structure in $W^{(+)}_{1+\infty}$. We also present all the Euclidean classical solutions and the classical action in the classical phase space. A possible relation of this theory to the $N=2$ string theory and also self-dual Einstein gravity in 4-dimensions is pointed out.