Integrable Differential Systems of Topological Type and Reconstruction by the Topological Recursion (original) (raw)
Painlevé equations, topological type property and reconstruction by the topological recursion
Journal of Geometry and Physics, 2018
In this article, we prove that we can introduce a small parameter in the six Painlevé equations through their corresponding Lax pairs and Hamiltonian formulations. Moreover, we prove that these-deformed Lax pairs satisfy the Topological Type property proposed by Bergère, Borot and Eynard for any generic choice of the monodromy parameters. Consequently we show that one can reconstruct the formal series expansion of the tau-function and of the determinantal formulas by applying the so-called topological recursion on the spectral curve attached to the Lax pair in all six Painlevé cases. Eventually we illustrate the former results with the explicit computations of the first orders of the six tau-functions.
Rational Differential Systems, Loop Equations, and Application to the qth Reductions of KP
Annales Henri Poincaré, 2015
To any solution of a linear system of differential equations, we associate a matrix kernel, correlators satisfying a set of loop equations, and in the presence of isomonodromic parameters, a Tau function. We then study their semiclassical expansion (WKB type expansion in powers of the weight per derivative) of these quantities. When this expansion is of topological type (TT), the coefficients of expansions are computed by the topological recursion with initial data given by the semiclassical spectral curve of the linear system. This provides an efficient algorithm to compute them at least when the semiclassical spectral curve is of genus 0. TT is a non-trivial property, and it is an open problem to find a criterion which guarantees it is satisfied. We prove TT and illustrate our construction for the linear systems associated to the qth reductions of KP-which contain the (p, q) models as a specialization. We thank the referee for helpful comments.
Journal of Mathematical Physics, 2020
In this paper, we show that it is always possible to deform a differential equation ∂ x Ψ(x) = L(x)Ψ(x) with L(x) ∈ sl 2 (C)(x) by introducing a small formal parameter in such a way that it satisfies the Topological Type properties of Bergère, Borot and Eynard. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce. The topological recursion is then proved to provide a formal series expansion of the corresponding tau-function whose coefficients can thus be expressed in terms of intersections of tautological classes in the Deligne-Mumford compactification of the moduli space of surfaces. We present a few examples including any Fuchsian system of sl 2 (C)(x) as well as some elements of Painlevé hierarchies.
New approaches to integrable hierarchies of topological type
Russian Mathematical Surveys
This survey is devoted to a large class of systems of partial differential equations which on the one hand appear in classical problems of mathematical physics and on the other hand provide an efficient tool for the description of enumerative invariants in algebraic geometry. Particular attention is paid to new approaches to these systems, in particular, to the approach proposed in a recent paper of the author. Bibliography: 57 titles.
WKB solutions of difference equations and reconstruction by the topological recursion
Nonlinearity, 2017
The purpose of this article is to analyze the connection between Eynard-Orantin topological recursion and formal WKB solutions of a-difference equation: Ψ(x +) = e d dx Ψ(x) = L(x;)Ψ(x) with L(x;) ∈ GL 2 ((C(x))[ ]). In particular, we extend the notion of determinantal formulas and topological type property proposed for formal WKB solutions of-differential systems to this setting. We apply our results to a specific-difference system associated to the quantum curve of the Gromov-Witten invariants of P 1 for which we are able to prove that the correlation functions are reconstructed from the Eynard-Orantin differentials computed from the topological recursion applied to the spectral curve y = cosh −1 x 2. Finally, identifying the large x expansion of the correlation functions, proves a recent conjecture made by B. Dubrovin and D. Yang regarding a new generating series for Gromov-Witten invariants of P 1 .
2017
We present a new spectral problem, a generalization of the D-Kaup-Newell spectral problem, associated with the Lie algebra sl(2,R). Zero curvature equations furnish the soliton hierarchy. The trace identity produces the Hamiltonian structure for the hierarchy. Lastly, a reduction of the spectral problem is shown to have a different soliton hierarchy with a bi-Hamiltonian structure. The first major motivation of this dissertation is to present spectral problems that generate two soliton hierarchies with infinitely many commuting conservation laws and high-order symmetries, i.e., they are Liouville integrable. We use the soliton hierarchies and a non-seimisimple matrix loop Lie algebra in order to construct integrable couplings. An enlarged spectral problem is presented starting from a generalization of the D-Kaup-Newell spectral problem. Then the enlarged zero curvature equations are solved from a series of Lax pairs producing the desired integrable couplings. A reduction is made of ...
International Journal of Modern Physics A, 1997
This paper provides a systematic description of the interplay between a specific class of reductions denoted as cKP r,m(r,m ≥ 1) of the primary continuum integrable system — the Kadomtsev–Petviashvili (KP) hierarchy and discrete multi-matrix models. The relevant integrable cKP r,m structure is a generalization of the familiar r-reduction of the full KP hierarchy to the SL (r) generalized KdV hierarchy cKP r,0. The important feature of cKP r,m hierarchies is the presence of a discrete symmetry structure generated by successive Darboux–Bäcklund (DB) transformations. This symmetry allows for expressing the relevant tau-functions as Wronskians within a formalism which realizes the tau-functions as DB orbits of simple initial solutions. In particular, it is shown that any DB orbit of a cKP r,1 defines a generalized two-dimensional Toda lattice structure. Furthermore, we consider the class of truncated KP hierarchies (i.e. those defined via Wilson–Sato dressing operator with a finite trun...
q-Painlevé Systems Arising from q-KP Hierarchy
Letters in Mathematical Physics
A system of q-Painlevé type equations with multi-time variables t1,. .. , tM is obtained as a similarity reduction of the N-reduced q-KP hierarchy. This system has affine Weyl group symmetry of type A (1) M −1 × A (1) N−1. Its rational solutions are constructed in terms of q-Schur functions.