Related papers: Structure of Dubrovin-Zhang free energy functions …
In [41] Okuyama and Sakai gave a conjectural equality for the higher genus generalized Br\'ezin--Gross--Witten (BGW) free energies. In a recent work [46] we established the Hodge-BGW correspondence on the relationship between certain…
In this paper we prove that the generating series of the Hodge integrals over the moduli space of stable curves is a solution of a certain deformation of the KdV hierarchy. This hierarchy is constructed in the framework of the…
We provide an integral formula for the free energy of the two-matrix model with polynomial potentials of arbitrary degree (or formal power series). This is known to coincide with the tau-function of the dispersionless two--dimensional Toda…
We consider the Dubrovin--Frobenius manifold of rank $2$ whose genus expansion at a special point controls the enumeration of a higher genera generalization of the Catalan numbers, or, equivalently, the enumeration of maps on surfaces,…
We define a hierarchy of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the equations, the Hamiltonians, and the…
We introduce a natural generalization of Maya diagrams -- the space of infinite Fibonacci configurations, which are specified functions on $\mathbb{Z}$ with values $1$ and $0$. Infinite Fibonacci configurations are particularly interesting…
Free energy functionals of Ginzburg-Landau type lie at the heart of a broad class of continuum dynamical models, such as the Cahn-Hilliard and Swift-Hohenberg equations. Despite the wide use of such models, the assumptions embodied in the…
Following Natanzon-Zabrodin, we explore the Kadomtsev-Petviashvili hierarchy as an infinite system of mutually consistent relations on the second derivatives of the free energy with some universal coefficients. From this point of view,…
B. Dubrovin introduced the structure of a Dubrovin--Frobenius manifold on a space of ramified coverings of a sphere by a genus $g$ Riemann surface with the prescribed ramification profile. This is now known as a genus $g$ Hurwitz--Frobenius…
We use genus zero free energy functions of Hermitian matrix models to define spectral curves and their special deformations. They are special plane curves defined by formal power series with integral coefficients generalizing the Catalan…
The bi-Hamiltonian structure of the Benney hierarchy is revisited. We show that the compatibility condition of the Poisson brackets provides the genus zero free energy of a topological field theory coupled to 2d gravity. We calculate the…
We consider the Hurwitz spaces of ramified coverings of $\mathbb{P}^1$ with prescribed ramification profile over the point at infinity. By means of a particular symmetric bidifferential on a compact Riemann surface, we introduce…
We investigate symmetries of Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) equations proposed by Dubrovin from bi-hamiltonian point of view. These symmetries can be viewed as canonical Miura transformations between genus-zero bi-hamiltonian…
We consider the Hurwitz Dubrovin--Frobenius manifold structure on the space of meromorphic functions on the Riemann sphere with exactly two poles, one simple and one of arbitrary order. We prove that the all genera partition function (also…
We prove a formula for the genus one free energy $\mathcal{F}^{(1)}$ of the quartic Kontsevich model for arbitrary ramification by working out a boundary creation operator for blobbed topological recursion. We thus investigate the…
In this paper we begin the study of the relationship between the local Gromov-Witten theory of Calabi-Yau rank two bundles over the projective line and the theory of integrable hierarchies. We first of all construct explicitly, in a large…
We prove the genus zero part of the generalized Witten conjecture relating moduli spaces of spin curves to Gelfand-Dickey hierarchies. That is, we show that intersection numbers on the moduli space of stable r-spin curves assemble into a…
We prove the existence of a power structure over the Grothendieck ring of geometric dg categories. We show that a conjecture by Galkin and Shinder (proved recently by Bergh, Gorchinskiy, Larsen, and Lunts) relating the motivic and…
We construct Frobenius structures of "dual type" on the moduli space of ramified coverings of $\mathbb{P}^1$ with given ramification type over two points, generalizing a construction of Dubrovin. A complete hierarchy of hydrodynamic type is…
Okounkov [36] proved a remarkable formula relating $n$-point GUE (Gaussian unitary ensemble) correlators of a fixed genus to Witten's intersection numbers of the same genus. The partition function of GUE correlators is a tau-function for…