Matrix integrals $\&$ finite holography

TL;DR

This paper investigates the duality between multicritical matrix integrals and non-unitary minimal models, matching critical exponents through diagrammatic and saddle point methods, and discusses the finiteness of the continuum theory's Hilbert space.

Contribution

It provides a detailed analysis of the duality, including novel combinatorial expressions and a systematic saddle point evaluation of multicritical matrix integrals.

Findings

Matching of critical exponents between matrix integrals and continuum models.

Development of a diagrammatic expansion revealing new combinatorial structures.

Evidence supporting the finiteness of the continuum theory's Hilbert space on different topologies.

Abstract

We explore the conjectured duality between a class of large $N$ matrix integrals, known as multicritical matrix integrals (MMI), and the series $(2m-1,2)$ of non-unitary minimal models on a fluctuating background. We match the critical exponents of the leading order planar expansion of MMI, to those of the continuum theory on an $S^2$ topology. From the MMI perspective this is done both through a multi-vertex diagrammatic expansion, thereby revealing novel combinatorial expressions, as well as through a systematic saddle point evaluation of the matrix integral as a function of its parameters. From the continuum point of view the corresponding critical exponents are obtained upon computing the partition function in the presence of a given conformal primary. Further to this, we elaborate on a Hilbert space of the continuum theory, and the putative finiteness thereof, on both an $S^2$ and a $T^2$ topology using BRST cohomology considerations. Matrix integrals support this finiteness.