Geometry of Spectral Curves and All Order Dispersive Integrable System

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.

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Cited by in corpus (26)

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• Rational differential systems, loop equations, and application to the q-th reductions of KP
• Exact eigenfunctions and the open topological string
• Isomonodromic deformations of a rational differential system and reconstruction with the topological recursion: the ${\mathfrak{sl}}_2$ case
• Comment on integrability in Dijkgraaf-Vafa beta-ensembles
• Integrable differential systems of topological type and reconstruction by the topological recursion
• Partition functions and the continuum limit in Penner matrix models
• Asymptotic expansion of beta matrix models in the multi-cut regime
• The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition
• Whittaker vectors for $\mathcal{W}$-algebras from topological recursion
• Loop equations from differential systems
• Airy structures for semisimple Lie algebras
• Taking limits in topological recursion
• Painlevé equations, topological type property and reconstruction by the topological recursion
• Quantization of classical spectral curves via topological recursion
• Frobenius Manifolds, Spectral Curves, and Integrable Hierarchies
• Whittaker vectors at finite energy scale, topological recursion and Hurwitz numbers
• The factorial growth of topological recursion
• Szegő Kernel and Symplectic Aspects of Spectral Transform for Extended Spaces of Rational Matrices
• Blobbed topological recursion and KP integrability
• Any topological recursion on a rational spectral curve is KP integrable
• Fay Identities of Pfaffian Type for Hyperelliptic Curves
• Painlevé Kernels and Surface Defects at Strong Coupling