Riemann Hypothesis and Master Matrix for FZZT Brane Partition Functions

TL;DR

This paper explores the connection between the Riemann zeta function and FZZT brane partition functions using matrix models, deriving a master matrix and analyzing zeros of the Airy function, with implications for understanding the Riemann Hypothesis.

Contribution

It introduces a master matrix for minimal models related to FZZT branes and proposes an iterative method to represent the Riemann Xi function via deformed minimal models.

Findings

Zeros of the FZZT partition function lie on the real axis.

Derived the master matrix for specific minimal models.

Connected the Riemann Xi function to deformed minimal models.

Abstract

We continue to investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence begun in arxiv:0708.0645. We derive the master matrix of the $(2,1)$ minimal and $(3,1)$ minimal matrix model. We use it's characteristic polynomial to understand why the zeros of the FZZT partition function, which is the Airy function, lie on the real axis. We also introduce an iterative procedure that can describe the Riemann $\Xi$ function as a deformed minimal model whose deformation parameters are related to a Konsevich integrand. Finally we discuss the relation of our work to other approaches to the Riemann $\Xi$ function including expansion in terms of Meixner-Pollaczek polynomials and Riemann-Hilbert problems.