Matrix Quantum Mechanics on $S^{1}/{\mathbb Z}_{2}$

TL;DR

This paper investigates Matrix Quantum Mechanics on an orbifolded circle, exploring its connection to non-critical string cosmology, and confirms the duality between matrix model and world-sheet results through explicit calculations of twisted state contributions.

Contribution

It introduces a detailed analysis of MQM on $S^{1}/\mathbb{Z}_2$, deriving partition functions in different ensembles, and explicitly matches twisted state contributions with world-sheet calculations, testing the duality.

Findings

Partition functions derived in canonical and grand canonical ensembles agree.

Twisted state contributions are explicitly calculated and match world-sheet results.

The partition function may be expressed as a $ au$-function of an integrable hierarchy.

Abstract

We study Matrix Quantum Mechanics on the Euclidean time orbifold $S_1/\mathbb{Z}_2$. Upon Wick rotation to Lorentzian time and taking the double-scaling limit this theory provides a toy model for a big-bang/big crunch universe in two dimensional non-critical string theory where the orbifold fixed points become cosmological singularities. We derive the MQM partition function both in the canonical and grand canonical ensemble in two different formulations and demonstrate agreement between them. We pinpoint the contribution of twisted states in both of these formulations either in terms of bi-local operators acting at the end-points of time or branch-cuts on the complex plane. We calculate, in the matrix model, the contribution of the twisted states to the torus level partition function explicitly and show that it precisely matches the world-sheet result, providing a non-trivial test of the proposed duality. Finally we discuss some interesting features of the partition function and the possibility of realising it as a $\tau$-function of an integrable hierarchy.