Riemann Hypothesis, Matrix/Gravity Correspondence and FZZT Brane Partition Functions

TL;DR

This paper explores the Riemann zeta function through the lens of matrix models and FZZT branes, proposing a new physical interpretation that links zeros of the zeta function to matrix/gravity correspondence and develops a Kontsevich matrix model.

Contribution

It introduces a novel physical interpretation of the Riemann zeta function as an FZZT brane partition function within a matrix/gravity framework and constructs a related Kontsevich matrix model.

Findings

Zeros of the zeta function are related to a critical line via an Airy function analogy.

A Kontsevich matrix model for multiple FZZT branes is developed.

The model connects matrix instantons with Liouville-type matrix models.

Abstract

We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large N matrix model. Using a related function $\Xi(z)$ we develop an analogy between this function and the Airy function Ai(z) of the Gaussian matrix model. The analogy gives an intuitive physical reason why the zeros lie on a critical line. Using a Fourier transform of the $\Xi(z)$ function we identify a Kontsevich integrand. Generalizing this integrand to $n \times n$ matrices we develop a Kontsevich matrix model which describes n FZZT branes. The Kontsevich model associated with the $\Xi(z)$ function is given by a superposition of Liouville type matrix models that have been used to describe matrix model instantons.