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

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

2007-08-06

arxiv.org/abs/0708.0645v1

Physics

Mathematical Physics

17 pages, 2 figures, 1 table

Scientific paper

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.

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