Motohico Mulase

A matrix model for simple Hurwitz numbers, and topological recursion

Abstract We introduce a new matrix model representation for the generating function of simple Hurwitz numbers. We calculate the spectral curve of the model and the associated symplectic invariants developed in Eynard and Orantin (2007) [3] . As an application, we prove the conjecture proposed by Bouchard and Marino (2008) [1] , relating Hurwitz numbers to the spectral invariants of the Lambert curve e x = y e − y .

Complete integrability of the Kadomtsev-Petviashvili equation

Abstract The total hierarchy of the Kadomtsev-Petviashvili (KP) equation is transformed to the system of linear partial differential equations with constant coefficients. The complete integrability of the KP equation is proved by using this linear system. The existence and uniqueness theorem of the Cauchy problem of the KP hierarchy is obtained.

The Laplace transform of the cut-and-join equation and the Bouchard-Marino conjecture on Hurwitz numbers

Author(s): Eynard, Bertrand; Mulase, Motohico; Safnuk, Brad | Abstract: We calculate the Laplace transform of the cut-and-join equation of Goulden, Jackson and Vakil. The result is a polynomial equation that has the topological structure identical to the Mirzakhani recursion formula for the Weil-Petersson volume of the moduli space of bordered hyperbolic surfaces. We find that the direct image of this Laplace transformed equation via the inverse of the Lambert W-function is the topological recursion formula for Hurwitz numbers conjectured by Bouchard and Marino using topological string theory.

Solvability of the super KP equation and a generalization of the Birkhoff decomposition

SummaryThe unique solvability of the initial value problem for the total hierarchy of the super Kadomtsev-Petviashvili system is established. To prove the existence we use a generalization of the Birkhoff decomposition which is obtained by replacing the loop variable and loop groups in the original setting by a super derivation operator and groups of infinite order super micro- (i.e. pseudo-) differential operators. To show the uniqueness we generalize the fact that every flat connection admits horizontal sections to the case of an infinite dimensional super algebra bundle defined over an infinite dimensional super space. The usual KP system with non-commutative coefficients is also studied. The KP system is obtained from the super KP system by reduction modulo odd variables. On the other hand, the first modified KP equation can be obtained from the super KP system by elimination of odd variables. Thus the super KP system is a natural unification of the KP system and the modified KP systems.

CATEGORY OF VECTOR BUNDLES ON ALGEBRAIC CURVES AND INFINITE DIMENSIONAL GRASSMANNIANS

Equivalence between the following categories is established: 1) A category of arbitrary vector bundles on algebraic curves defined over a field of arbitrary characteristic, and 2) a category of infinite dimensional vector spaces corresponding to certain points of Grassmannians together with their stabilizers. Our contravariant functor between these categories gives a full generalization of the well-known Krichever map, which assigns points of Grassmannians to the geometric data consisting of curves and line bundles. As an application, a solution to the classical problem of Wallenberg-Schur of classifying all commutative algebras consisting of ordinary differential operators is obtained. It is also shown that the KP flows produce all generic vector bundles on arbitrary algebraic curves of genus greater than one.

Spectral curves and the Schrödinger equations for the Eynard-Orantin recursion

It is predicted that the principal specialization of the partition function of a B-model topological string theory, that is mirror dual to an A-model enumerative geometry problem, satisfies a Schrodinger equation, and that the characteristic variety of the Schrodinger operator gives the spectral curve of the B-model theory, when an algebraic K -theory obstruction vanishes. In this paper we present two concrete mathematical A-model examples whose mirror dual partners exhibit these predicted features on the B-model side. The A-model examples we discuss are the generalized Catalan numbers of an arbitrary genus and the single Hurwitz numbers. In each case, we show that the Laplace transform of the counting functions satisfies the Eynard–Orantin topological recursion, that the B-model partition function satisfies the KP equations, and that the principal specialization of the partition function satisfies a Schrodinger equation whose total symbol is exactly the Lagrangian immersion of the spectral curve of the Eynard–Orantin theory.

Duality of orthogonal and Symplectic matrix integrals and quaternionic Feynman graphs

We present an asymptotic expansion for quaternionic self-adjoint matrix integrals. The Feynman diagrams appearing in the expansion are ordinary ribbon graphs and their non-orientable counterparts. We show that the 2N×2N Gaussian Orthogonal Ensemble (GOE) and N×N Gaussian Symplectic Ensemble (GSE) have exactly the same expansion term by term, except that the contributions from graphs on a non-orientable surface with odd Euler characteristic carry the opposite sign. As an application, we give a new topological proof of the known duality for correlations of characteristic polynomials, demonstrating that this duality is equivalent to Poincaré duality of graphs drawn on a compact surface. Another consequence of our graphical expansion formula is a simple and simultaneous (re)derivation of the Central Limit Theorem for GOE, GUE (Gaussian Unitary Ensemble) and GSE: The three cases have exactly the same graphical limiting formula except for an overall constant that represents the type of the ensemble.

Quantum spectral curve for the Gromov-Witten theory of the complex projective line

We construct the quantum curve for the Gromov-Witten theory of the complex projective line.

Quantum Curves for Hitchin Fibrations and the Eynard-Orantin Theory

We generalize the topological recursion of Eynard–Orantin (JHEP 0612:053, 2006; Commun Number Theory Phys 1:347–452, 2007) to the family of spectral curves of Hitchin fibrations. A spectral curve in the topological recursion, which is defined to be a complex plane curve, is replaced with a generic curve in the cotangent bundle T*C of an arbitrary smooth base curve C. We then prove that these spectral curves are quantizable, using the new formalism. More precisely, we construct the canonical generators of the formal