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date: 13 December 2018

Abstract and Keywords

This article discusses the connection between the matrix models and algebraic geometry. In particular, it considers three specific applications of matrix models to algebraic geometry, namely: the Kontsevich matrix model that describes intersection indices on moduli spaces of curves with marked points; the Hermitian matrix model free energy at the leading expansion order as the prepotential of the Seiberg-Witten-Whitham-Krichever hierarchy; and the other orders of free energy and resolvent expansions as symplectic invariants and possibly amplitudes of open/closed strings. The article first describes the moduli space of algebraic curves and its parameterization via the Jenkins-Strebel differentials before analysing the relation between the so-called formal matrix models (solutions of the loop equation) and algebraic hierarchies of Dijkgraaf-Witten-Whitham-Krichever type. It also presents the WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations, along with higher expansion terms and symplectic invariants.

Keywords: matrix model, algebraic geometry, Kontsevich matrix model, moduli space, Hermitian matrix model, free energy, Seiberg-Witten-Whitham-Krichever hierarchy, symplectic invariants, formal matrix model, WDVV (Witten-Dijkgraaf-Verlinde-Verlinde) equations

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