Riemann Hypothesis and Master Matrix for FZZT Brane Partition Functions

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.

17 pages, 8 figures

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