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

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.