2010/07/23 by A. Morozov, Morozov, A., Sh. Shakirov +1 · 1 citation
Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #hep-th
paper · pdf · doi:10.48550/arxiv.1007.4100
12 pages
arxiv created 2010/07/23 · arxiv updated 2010/07/26
Brezin-Hikami contour-integral representation of exponential multidensities in finite N Hermitian matrix model is a remarkable implication of the old Hermitian-Kontsevich duality. It is also a simplified version of Okounkov's formulas for the same multidensities in the cubic Kontsevich model and of Nekrasov calculus for LMNS integrals, a central piece of the modern studies of AGT relations. In this paper we use Brezin-Hikami representation to derive explicit expressions for the Harer-Zagier multidensities (from arXiv:0906.0036): the only known exhaustive generating functions of all-genera Gaussian correlators which are fully calculable and expressed in terms of elementary functions. Using the Brezin-Hikami contour integrals, we rederive the 1-point function of Harer and Zagier and the 2-point arctangent function of arXiv:0906.0036. We also present (without a proof) the explicit expression for the 3-point function in terms of arctangents. Derivation of the 3-point and higher Harer-Zagier functions remains a challenging problem.