Bulk-boundary correlators in the hermitian matrix model and minimal Liouville gravity

TL;DR

This paper establishes a correspondence between bulk-boundary correlators in minimal Liouville gravity and the hermitian matrix model, analyzing resonance transformations and boundary couplings to ensure agreement between discrete and continuous approaches.

Contribution

It introduces a detailed method to relate matrix model correlators to minimal Liouville gravity, including explicit transformations and resonance encoding for boundary couplings.

Findings

Confirmed agreement between matrix model and Liouville gravity correlators in multiple cases.

Derived explicit boundary resonance encoding polynomials.

Developed an alternative method for boundary resonance analysis.

Abstract

We construct the one matrix model (MM) correlators corresponding to the general bulk-boundary correlation numbers of the minimal Liouville gravity (LG) on the disc. To find agreement between both discrete and continuous approach, we investigate the resonance transformation mixing boundary and bulk couplings. It leads to consider two sectors, depending on whether the matter part of the LG correlator is vanishing due to the fusion rules. In the vanishing case, we determine the explicit transformation of the boundary couplings at the first order in bulk couplings. In the non-vanishing case, no bulk-boundary resonance is involved and only the first order of pure boundary resonances have to be considered. Those are encoded in the matrix polynomials determined in our previous paper. We checked the agreement for the bulk-boundary correlators of MM and LG in several non-trivial cases. In this process, we developed an alternative method to derive the boundary resonance encoding polynomials.