One-Dimensional String Theory with Vortices as Upside-Down Matrix Oscillator

TL;DR

This paper explores matrix quantum mechanics at finite temperature, modeling vortex configurations in one-dimensional string theory, and develops a technique to compute partition functions involving vortex pairs, revealing a BKT phase transition.

Contribution

It introduces a method to calculate partition functions with vortex-anti-vortex pairs in matrix oscillators, extending to upside-down oscillators and discussing generalizations to higher dimensions.

Findings

Vortex configurations correspond to non-trivial U(N) representations.

Partition functions with vortex pairs are computed analytically in the double scaling limit.

The BKT phase transition occurs at the same temperature as in flat 2D space.

Abstract

We study matrix quantum mechanics at a finite temperature equivalent to one dimensional compactified string theory with vortex (winding) excitations. It is explicitly demonstrated that the states transforming under non-trivial U(N) representations describe various configurations vortices and anti-vortices. For example, for the adjoint representation the Feynman graphs (representing discretized world-sheets) contain two faces with the boundaries wrapping around the compactified target space which is equivalent to a vortex-anti-vortex pair. A technique is developed to calculate partition functions in a given representation for the standard matrix oscillator. It enables us to obtain the partition function in the presence of a vortex-anti-vortex pair in the double scaling limit using an analytical continuation to the upside-down oscillator. The Berezinski-Kosterlitz-Thouless phase transition occurs in a similar way and at the same temperature as in the flat 2D space. A possible generalization of our technique to any dimension of the embedding space is discussed.