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

TL;DR

We study a finite-temperature matrix quantum mechanics model that maps to one-dimensional bosonic string theory with vortex excitations. By classifying states into $U(N)$ representations and performing an analytic continuation to the upside-down oscillator in the double scaling limit, we compute representation-specific partition functions, showing the adjoint sector corresponds to a vortex–anti-vortex pair and identifying a Berezinskii-Kosterlitz-Thouless transition at $β_{KT}=4π$ with a representation-dependent spectral gap. The work introduces a twisted partition function formalism, derives exact and approximate partition functions in the singlet and adjoint sectors, and discusses generalizations to $D+1$ dimensions, while noting ambiguities and limitations of the continuation approach. The results illuminate how angular (non-eigenvalue) degrees of freedom encode vortex dynamics on fluctuating world sheets and suggest a path toward understanding higher-dimensional bosonic strings through matrix models.

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.