### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 8 (2012), 100, 53 pages      arXiv:1110.4936      https://doi.org/10.3842/SIGMA.2012.100

### Geometry of Spectral Curves and All Order Dispersive Integrable System

Gaëtan Borot a and Bertrand Eynard b, c
a) Section de Mathématiques, Université de Genève, 2-4 rue du Lièvre, 1211 Genève 4, Switzerland
b) Institut de Physique Théorique, CEA Saclay, Orme des Merisiers, 91191 Gif-sur-Yvette, France
c) Centre de Recherche Mathématiques de Montréal, Université de Montréal, P.O. Box 6128, Montréal (Québec) H3C 3J7, Canada

Received November 14, 2011, in final form December 11, 2012; Published online December 18, 2012

Abstract
We propose a definition for a Tau function and a spinor kernel (closely related to Baker-Akhiezer functions), where times parametrize slow (of order 1/N) deformations of an algebraic plane curve. This definition consists of a formal asymptotic series in powers of 1/N, where the coefficients involve theta functions whose phase is linear in N and therefore features generically fast oscillations when N is large. The large N limit of this construction coincides with the algebro-geometric solutions of the multi-KP equation, but where the underlying algebraic curve evolves according to Whitham equations. We check that our conjectural Tau function satisfies Hirota equations to the first two orders, and we conjecture that they hold to all orders. The Hirota equations are equivalent to a self-replication property for the spinor kernel. We analyze its consequences, namely the possibility of reconstructing order by order in 1/N an isomonodromic problem given by a Lax pair, and the relation between ''correlators'', the tau function and the spinor kernel. This construction is one more step towards a unified framework relating integrable hierarchies, topological recursion and enumerative geometry.

Key words: topological recursion; Tau function; Sato formula; Hirota equations; Whitham equations.

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