String Field Theory for d \leq 0 Matrix Models via Marinari-Parisi

TL;DR

This paper extends the Marinari-Parisi approach to all non-unitary multicritical points in 2D quantum gravity, analyzing the resulting fermionic matrix models using collective field theory and exploring their nonperturbative properties.

Contribution

It generalizes the Marinari-Parisi formulation to higher multicritical points and studies the collective field theory in the planar limit, including nonperturbative aspects and integrability.

Findings

Eigenvalue distribution matches original multicritical point scaling

Collective field theory exhibits appropriate scaling behavior

Comments on nonperturbative properties and integrability

Abstract

We generalize the Marinari-Parisi definition for pure two dimensional quantum gravity $(k = 2)$ to all non unitary minimal multicritical points $(k \geq 3)$. The resulting interacting Fermi gas theory is treated in the collective field framework. Making use of the fact that the matrices evolve in Langevin time, the Jacobian from matrix coordinates to collective modes is similar to the corresponding Jacobian in $d = 1$ matrix models. The collective field theory is analyzed in the planar limit. The saddle point eigenvalue distribution is the one that defines the original multicritical point and therefore exhibits the appropriate scaling behaviour. Some comments on the nonperturbative properties of the collective field theory as well as comments on the Virasoro constraints associated with the loop equations are made at the end of this letter. There we also make some remarks on the fermionic formulation of the model and its integrability, as a nonlocal version of the non linear Schr\"{o}dinger model.