Polynomial hierarchies generated by Eynard-Orantin invariants

© 2011 Dr. Nicky Jason ScottThis thesis examines analytic properties of the Eynard-Orantin invariants of rational curves with two branch points. These are meromorphic multidifferential forms that live on a compact Riemann surface, often with complicated functional descriptions, attracting great interest from physicists and combinatorists. For each curve, coordinates can be chosen so that the corresponding Eynard-Orantin invariants have analytic Taylor expansions around the origin. The coefficients of these expansions are found to be quasi-polynomials; simple expressions containing equivalent information to the original invariants. These quasi-polynomials are a useful computational tool, and we will use them to provide insight into problems represented by this family of spectral curve. Their degree and asymptotics are determined from the definition and geometry of the Eynard-Orantin invariants, whilst other properties, such as the string and dilaton equations, translate to relationships between polynomials. A particular advantage of this interpretation is that residue and contour integration can be replaced by evaluation of expressions. One notable application is the Hermitian 1-matrix model, a typical problem in this family known to store combinatorial information as expectation values with respect to a matrix measure. We find simple expressions for these expectations, given by linear transformations of the quasi-polynomials. This approach is used to further understand the Gromov-Witten theory of P1. One difficulty with this theory has been the requirement of both stationary and non-stationary classes to recursively generate invariants. We begin by showing that the Eynard-Orantin invariants of the curve y = arccosh(x/2) generate stationary Gromov-Witten invariants. This is a rational curve with two branch points, enabling new expressions involving simple polynomials to be computed. Properties of the Eynard-Orantin invariants – now relationships amongst the polynomials – reduce the string, dilaton and Virasoro constraints to involve only stationary terms. This provides significant insight; the stationary theory is completely and independently determined by the Eynard-Orantin recursion. Another application involves integrals over Kontsevich cycles in the moduli space of curves, which can independently be troublesome to calculate. Kontsevich first stored them as the asymptotic expansion of a matrix model with specified polynomial potential, or equivalently, the symplectic invariants of some complicated spectral curve. Using a combinatorial trick, this information is viewed as the n point Eynard-Orantin invariants of the curve y^2 = xy − 1, radically simplifying the problem. This is a rational curve with two branch points, allowing us to express Kontsevich integrals in terms of simple polynomials