1,721,031 research outputs found
Simple Lie Algebras and Topological ODEs
No full textFor a simple Lie algebra g we define a system of linear ODEs with polynomial coeffi- cients, which we call the topological equation of g-type. The dimension of the space of solutions regular at infinity is equal to the rank of the Lie algebra. For the simplest example g = sl2(C) the regular solution can be expressed via products of Airy functions and their derivatives; this matrix-valued function was used in our previous work [4] for computing logarithmic derivatives of the Witten–Kontsevich tau-function. For an arbitrary simple Lie algebra we construct a basis in the space of regular solutions to the topological equation called generalized Airy resolvents. We also outline applica- tions of the generalized Airy resolvents for computing the Witten and Fan–Jarvis–Ruan invariants of the Deligne–Mumford moduli spaces of stable algebraic curves
Correlation functions of the KdV hierarchy and applications to intersection numbers over Mg,n
No full textWe derive an explicit generating function of correlation functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formulæ of a new type for the generating series of intersection numbers of ψ -classes as well as of mixed ψ - and κ -classes in full genera
Classical hurwitz numbers and related combinatorics
Full text linkWe give a polynomial-time algorithm of computing the classical Hurwitz numbers Hg,d, which were defined by Hurwitz 125 years ago. We show that the generating series of Hg,d for any fixed g > 2 lives in a certain subring of the ring of formal power series that we call the Lambert ring. We then define some analogous numbers appearing in enumerations of graphs, ribbon graphs, and in the intersection theory on moduli spaces of algebraic curves, such that their generating series belong to the same Lambert ring. Several asymptotics of these numbers (for large g or for large d) are obtained
Three-phase solutions of the Kadomtsev-Petviashvili equation
No full textThe Kadomtsev-Petviashvili (KP) equation is known to admit explicit periodic and quasiperiodic solutions with N independent phases, for any integer N, based on a Riemann theta-function of N variables. For N=1 and 2, these solutions have been used successfully in physical applications. This article addresses mathematical problems that arise in the computation of theta-functions of three variables and with the corresponding solutions of the KP equation. We identify a set of parameters and their corresponding ranges, such that every real-valued, smooth KP solution associated with a Riemann theta-function of three variables corresponds to exactly one choice of these parameters in the proper range. Our results are embodied in a program that computes these solutions efficiently and that is available to the reader. We also discuss some properties of three-phase solutions
On Critical Behaviour in Systems of Hamiltonian Partial Differential Equations
Full text linkWe study the critical behaviour of solutions to weakly dispersive Hamiltonian systems considered as perturbations of elliptic and hyperbolic systems of hydrodynamic type with two components. We argue that near the critical point of gradient catastrophe of the dispersionless system, the solutions to a suitable initial value problem for the perturbed equations are approximately described by particular solutions to the Painlevé-I (PI ) equation or its fourth-order analogue P2I . As concrete examples, we discuss nonlinear Schrödinger equations in the semiclassical limit. A numerical study of these cases provides strong evidence in support of the conjecture
Integrable systems of double ramification type
Full text linkIn this paper we study various aspects of the double ramification (DR) hierarchy, introduced by the 1st author, and its quantization. We extend the notion of tau- symmetry to quantum integrable hierarchies and prove that the quantum DR hierarchy enjoys this property. We determine explicitly the genus 1 quantum correction and, as an application, compute completely the quantization of the 3- and 4-KdV hierarchies (the DR hierarchies for Witten’s 3- and 4-spin theories). We then focus on the recursion relation satisfied by the DR Hamiltonian densities and, abstracting from its geometric origin, we use it to characterize and construct a new family of quantum and classical integrable systems that we call of DR type, as they satisfy all of the main properties of the DR hierarchy. In the 2nd part, we obtain new insight towards the Miura equivalence conjecture between the DR and Dubrovin-Zhang (DZ) hierarchies, via a geometric interpretation of the correlators forming the DR tau-function. We then show that the candidate Miura transformation between the DR and DZ hierarchies (which we uniquely identified in our previous paper) indeed turns the DZ Poisson structure into the standard form. Eventually, we focus on integrable hierarchies associated with rank-1 cohomological field theories and their deformations, and we prove the DR/DZ equivalence conjecture up to genus 5 in this context
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