String Theory on Lorentzian AdS_3 in Minisuperspace

TL;DR

This paper studies string theory on Lorentzian AdS_3 using a minisuperspace approximation, focusing on the Hilbert space, unitarity, and correlation functions, revealing differences from the Euclidean case and resolving ambiguities through wave function overlaps.

Contribution

It provides a detailed analysis of the Hilbert space, unitarity constraints, and correlation functions in Lorentzian AdS_3 string theory within the minisuperspace approximation, highlighting differences from Euclidean models.

Findings

Hilbert space consists of normalizable wave functions

Unitarity restricts wave functions and ensures probability conservation

Ambiguities in correlation functions are fixed by wave function overlaps

Abstract

We investigate string theory on Lorentzian AdS_3 in the minisuperspace approximation. The minisuperspace model reduces to the worldline theory of a scalar particle in the Lorentzian AdS_3. The Hilbert space consists of normalizable wave functions, and we see that the unitarity of the theory (or the self-adjointness of the Hamiltonian) restricts the possible sets of wave functions. The restricted wave functions have the property of probability conservation (or current conservation) across the horizons. Two and three point functions are also computed. In the Euclidean model functional forms of these quantities are restricted by the SL(2,R) symmetry almost uniquely, however, in the Lorentzian model there are several ambiguities left. The ambiguities are fixed by the direct computation of overlaps of wave functions.