2D String Theory as Normal Matrix Model

TL;DR

This paper demonstrates that the $c=1$ bosonic string theory at finite temperature can be realized through two dual matrix models: the standard inverted oscillator quantum mechanics and a newly derived normal matrix model, both exhibiting Toda integrability.

Contribution

The paper introduces a novel normal matrix model realization of $c=1$ string theory, establishing its duality with the known quantum mechanics model and showing their equivalence under tachyon perturbations.

Findings

Both models exhibit Toda integrable structure.

The models are dual via a transformation related to complex curve cycles.

Equivalence holds for all tachyon perturbations and string coupling orders.

Abstract

We show that the $c=1$ bosonic string theory at finite temperature has two matrix-model realizations related by a kind of duality transformation. The first realization is the standard one given by the compactified matrix quantum mechanics in the inverted oscillator potential. The second realization, which we derive here, is given by the normal matrix model. Both matrix models exhibit the Toda integrable structure and are associated with two dual cycles (a compact and a non-compact one) of a complex curve with the topology of a sphere with two punctures. The equivalence of the two matrix models holds for an arbitrary tachyon perturbation and in all orders in the string coupling constant.