[Paper Review] Matrix Models of 2D String Theory in Non--trivial Backgrounds

Arnaud Koetsier|arXiv (Cornell University)|Sep 2, 2005

Black Holes and Theoretical Physics48 references3 citations

TL;DR

This thesis constructs a matrix quantum mechanics (MQM) model for 2D string theory in non-trivial backgrounds, focusing on tachyon perturbations in a linear dilaton background. It generalizes the sine-Liouville CFT to two non-vanishing tachyon couplings, deriving the Fermi sea profile and free energy via the Toda lattice hierarchy, though an explicit free energy expression remains elusive beyond the one-coupling case.

ABSTRACT

After a brief review of critical string theory in trivial backgrounds we begin with introduction to strings in non--trivial backgrounds and noncritical string theory. In particular, we relate the latter to critical string theory in a linear dilaton background. We then show how a black hole background arises from 2D string theory and discuss some of its properties. A time--dependant tachyon background is constructed by perturbing the CFT describing string theory in a linear dilaton background. It is then explained that the T--dual of this theory with one non--vanishing tachyon coupling, which is a sine-Liouville CFT, is seemingly equivalent to the exact CFT describing the Euclidean black hole background. Subsequently, we launch into a review of some important facts concerning random matrix models and matrix quantum mechanics (MQM), culminating in an MQM model of 2D string theory in a dynamic tachyon background. We then solve this theory explicitly in the tree level approximation for the case of two non--vanishing tachyon couplings, which generalises the case of sine-Liouville CFT previously considered in the literature.

Motivation & Objective

• To extend the matrix quantum mechanics (MQM) framework to describe 2D string theory with time-dependent tachyon backgrounds beyond the one-coupling case.
• To investigate the T-dual of a perturbed CFT with one non-zero tachyon coupling, relating it to the exact Euclidean black hole CFT via the FZZ conjecture.
• To generalize the sine-Liouville CFT model to include second-order tachyon couplings using the Toda lattice hierarchy.
• To compute the phase space profiles of the Fermi sea in MQM for two non-zero tachyon couplings.
• To explore critical behavior and potential phase transitions in the free energy under two-coupling perturbations.

Proposed method

• Formulates matrix quantum mechanics (MQM) as a collective field theory for 2D string theory, mapping the system to non-interacting fermions in a Fermi sea.
• Applies the chiral coordinate representation of MQM to compute the S-matrix and tachyon vertex operators in a perturbed CFT.
• Uses the Toda lattice hierarchy to describe tachyon perturbations, with the string equation and τ-function encoding the dynamics.
• Derives the Fermi sea profiles for two non-zero tachyon couplings by solving the Toda lattice equations in the double scaling limit.
• Analyzes the free energy via the functional K[y₁, y₂, X] and investigates critical points where dμ/dX = 0 to detect phase transitions.
• Extends the one-coupling sine-Liouville model to two couplings, but cannot solve the resulting polynomial with irrational powers explicitly.

Experimental results

Research questions

• RQ1 How can matrix quantum mechanics be used to model 2D string theory with two non-vanishing tachyon couplings in a time-dependent background?
• RQ2 What is the structure of the Fermi sea profile in MQM when perturbed by two tachyon operators?
• RQ3 Can the free energy be explicitly computed in the two-coupling case, or is it only analyzable near critical points?
• RQ4 What is the role of the Toda lattice hierarchy in describing tachyon perturbations beyond the sine-Liouville case?
• RQ5 Are there phase transitions signaled by discontinuities in the μ(X) curve when two couplings are non-zero?

Key findings

• The Fermi sea profiles for two non-zero tachyon couplings are derived using the Toda lattice hierarchy, extending the one-coupling case.
• The free energy cannot be solved explicitly in the two-coupling case due to a polynomial with irrational powers in the variables y₁ and y₃.
• Critical behavior is signaled by a vanishing total derivative dμ/dX = 0, indicating possible phase transitions or emergence of c=0 quantum gravity.
• The model suggests that double-critical points may correspond to the Ising model or Majorana fermions coupled to 2D quantum gravity.
• The T-dual of the two-coupling perturbed CFT is conjectured to be equivalent to the exact black hole CFT, generalizing the FZZ duality.
• The system remains solvable in principle via the Toda hierarchy, but explicit analytical solutions for the free energy are obstructed by the complexity of the resulting equations.