Recursion Relations in Liouville Gravity coupled to Ising Model satisfying Fusion Rules

TL;DR

This paper reexamines recursion relations in 2D quantum gravity coupled to the Ising model with fusion rules, deriving a closed set of relations that match matrix model results despite divergences and structural differences.

Contribution

It derives and solves recursion relations for correlators in Liouville gravity with Ising matter satisfying fusion rules, addressing divergences and comparing with matrix model outcomes.

Findings

Recursion relations form a closed set among well-defined correlators.

Normalized correlator ratios agree with matrix model results.

Divergences cause some correlators to be ill-defined, affecting the recursion structure.

Abstract

The recursion relations of 2D quantum gravity coupled to the Ising model discussed by the author previously are reexamined. We study the case in which the matter sector satisfies the fusion rules and only the primary operators inside the Kac table contribute. The theory involves unregularized divergences in some of correlators. We obtain the recursion relations which form a closed set among well-defined correlators on sphere, but they do not have a beautiful structure that the bosonized theory has and also give an inconsistent result when they include an ill-defined correlator with the divergence. We solve them and compute the several normalization independent ratios of the well-defined correlators, which agree with the matrix model results.