Abstract
We study Matrix Quantum Mechanics on the Euclidean time orbifold S1/Z2. 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 τ-function of an integrable hierarchy.
Highlights
Introduction and MotivationLittle is known in Quantum Gravity about how the space-like singularities in general, and cosmological singularities in particular, can be resolved—if they can be resolved at all
1 16 log μ0 matches precisely the world-sheet result. This provides a non-trivial check of the duality we propose between the n = N/2 representation of the orbifold matrix quantum mechanics and the 2D non-critical string theory on S1/Z2
In this paper we considered the quantum mechanics of an N × N dimensional Hermitean matrix M compactified on Euclidean time τ and orbifolded by a Z2 action that contains the reflection τ → −τ, which we embedded into the gauge group
Summary
Little is known in Quantum Gravity about how the space-like singularities in general, and cosmological singularities in particular, can be resolved—if they can be resolved at all. Using the matrix model techniques we calculate the torus partition function in the large R limit for the n = 0 and n = N/2 representations, especially the twisted state contribution to it, and show that the n = N/2 representation matches precisely the result obtained from the world-sheet CFT This provides a non-trivial check of the equivalence we propose between the orbifold MQM and the orbifold 2D non-critical string theory. In addition we find that this partition function admits a natural continuation into Lorentzian signature, provides a possible connection to the cosmological toy universe In particular it has a nice structure from which the initial and final wavefunctions and the transition amplitude of the toy cosmological space-time can be read off.
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