S. V. Shadrin
In this note we prove that the enlarged Witten's conjecture is true in the case of the Boussinesq hierarchy for correlators in genus 3 with descendants only at one point.
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We prove that the genus-0 invariants in K-theoretic Gromov--Witten theory are governed by an integrable hierarchy of hydrodynamic type. If the K-theoretic quantum product is semisimple, then we also prove the completeness of the hierarchy.
Isoperiodic deformations of Abelian differentials of the second kind over elliptic curves and the Boussinesq equation
研究保持周期不变的黎曼曲面和第二类阿贝尔微分的形变，导出控制极点位置的有理系数二阶常微分方程，并应用于Boussinesq方程的亏格一解。
Buchstaber and Mikhailov introduced the polynomial dynamical systems in $\mathbb{C}^4$ with two polynomial integrals on the basis of commuting vector fields on the symmetric square of hyperelliptic curves. In our previous paper, we constructed the field of meromorphic functions on the sigma divisor of hyperelliptic curves of genus 3 and solutions of the systems for $g=3$ by these functions. In this paper, as an application of our previous results, we construct two parametric deformation of the KdV-hierarchy. This new system is integrated in the meromorphic functions on the sigma divisor of hyp…
We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $τ$-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter $\hbar$ are $τ$-functions of the $\hbar$-KP hierarchy and the expansion in $\hbar$ for the $\hbar$-KP coincides with the genus expansion for these models. Furthermore, …
Genus expansion of matrix models and $\hbar$ expansion of $B$KP hierarchy
本文研究矩阵模型亏格展开与$B$KP层级$\hbar$展开的联系，证明BGW模型、Kontsevich模型及spin Hurwitz数生成函数在插入$\hbar$后均为具有良好准经典行为的$\hbar$-$B$KP层级解，并揭示了模型间形变规则的相似性与差异。
We study the singular iso-level manifold $M_g(0)$ of the genus $g$ Mumford system associated to the spectral curve $y^2=x^{2g+1}$. We show that $M_g(0)$ is stratified by $g+1$ open subvarieties of additive algebraic groups of dimension $0,1,...,g$ and we give an explicit description of $M_g(0)$ in terms of the compactification of the generalized Jacobian. As a consequence, we obtain an effective algorithm to compute rational solutions to the genus $g$ Mumford system, which is closely related to rational solutions of the KdV hierarchy.
In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's $r$-spin classes. They are parameterized by a phase space which has one extra dimension and in genus $0$ they correspond to the extended $r$-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the $r$-spin class. They do not form a cohomological field theory, but a more general object which we call F-Coh…
Among solutions of n-Gelfand-Dikii's hierarchy there exists a remarkable solution W, which satisfies the string equation. We call it Witten's solution because according to the Witten conjecture the function F(x_1, x_2, x_3,...) = W(x_1,(x_2)/2, (x_3)/3,...) is the generating function for intersection nambers of Mumford-Morita-Muller cohomological classes of the moduli space of n-spin Riemann surfaces. This conjecture was proved by Kontsevich for n=2 and by Witten himself for surfaces of genus 0. In this paper we find recurrence relations between coefficients of Taylor series of W. This reduces…
KdV integrability in GUE correlators
本文基于GUE相关子项与Witten交集数的关系，通过Toda晶格的tau函数证明了Witten-Kontsevich定理，揭示了KdV可积结构与GUE相关子项的联系。
In our previous two papers, we constructed an $r$-spin theory in genus zero for Riemann surfaces with boundary and fully determined the corresponding intersection numbers, providing an analogue of Witten's $r$-spin conjecture in genus zero in the open setting. In particular, we proved that the generating series of open $r$-spin intersection numbers is determined by the genus-zero part of a special solution of a certain extension of the Gelfand-Dickey hierarchy, and we conjectured that the whole solution controls the open $r$-spin intersection numbers in all genera, which do not yet have a geom…
In this paper we continue the study of the double ramification hierarchy of [Bur15]. After showing that the DR hierarchy satisfies tau-symmetry we define its partition function as the (logarithm of the) tau-function of the string solution and show that it satisfies various properties (string, dilaton and divisor equations plus some important degree constraints). We then formulate a stronger version of the conjecture from [Bur15]: for any semisimple cohomological field theory, the Dubrovin-Zhang and double ramification hierarchies are related by a normal (i.e. preserving the tau-structure [DLYZ…
Combinatorics and large genus asymptotics of the Brézin--Gross--Witten numbers
本文研究 Brézin--Gross--Witten 数的组合与渐近性质，证明其归一化形式的一致大亏格首项渐近及多项式现象，并提出单调性与整性等新猜想，同时给出在 Painlevé II 层级和 BGW-kappa 数上的应用。
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 number of cases, the Hamiltonian dispersionless hierarchies that govern the full descendent genus zero theory. Our main tool is the application of Dubrovin's formalism, based on associativity equations, to the known results on the genus zero theory from local mirror symmetry and localization. The hierarchies we find are apparently new, with the exception of t…
The main goal of this paper is to prove the following two conjectures for genus up to two: 1. Witten's conjecture on the relations between higher spin curves and Gelfand--Dickey hierarchy. 2. Virasoro conjecture for target manifolds with conformal semisimple Frobenius manifolds. The main technique used in the proof is the invariance of tautological equations under loop group action.
We prove that the logarithm of an arbitrary tau-function of the KdV hierarchy can be approximated, in the topology of graded formal series by the logarithmic expansions of hyperelliptic theta-functions of finite genus, up to at most quadratic terms. As an example we consider theta-functional approximations of the Witten--Kontsevich tau-function.
A polynomial bracket for the Dubrovin--Zhang hierarchies
本研究定义了与Givental群半单轨道中任意tau函数关联的哈密顿PDE层级，证明其方程、哈密顿量和括号为加权齐次多项式，且共形Frobenius结构下与Dubrovin-Zhang层级一致，从而证明了两人提出的多项式性猜想。
Mapping partition functions
本文引入形式幂级数群对 Witten-Kontsevich 型配分函数的群作用，构造 WK 映射配分函数与相应可积方程族，证明其作为 Riemann-Hopf 方程族双哈密顿微扰的普适性，并建立广义 Hodge-WK 对应。
We derive an effective recursion for Witten's r-spin intersection numbers, using Witten's conjecture relating r-spin numbers to the Gel'fand-Dikii hierarchy (Theorem 4.1). Consequences include closed-form descriptions of the intersection numbers (for example, in terms of gamma functions: Propositions 5.2 and 5.4, Corollary 5.5). We use these closed-form descriptions to prove Harer-Zagier's formula for the Euler characteristic of M_{g,1}. Finally in Section 6, we extend Witten's series expansion formula for the Landau-Ginzburg potential to study r-spin numbers in the small phase space in genus …
On Vanishing of Gromov--Witten Invariants
本文研究部分旗流形上三点亏格零 Gromov–Witten 不变量是否为零的判定问题，在广义黎曼假设下证明其属于复杂度类 AM，并借助显式多项式方程组和参数化 Hilbert 零点定理的推广完成归约。
We conclude the construction of $r$-spin theory in genus zero for Riemann surfaces with boundary. In particular, we define open $r$-spin intersection numbers, and we prove that their generating function is closely related to the wave function of the $r$th Gelfand-Dickey integrable hierarchy. This provides an analogue of Witten's $r$-spin conjecture in the open setting and a first step toward the construction of an open version of Fan-Jarvis-Ruan-Witten theory. As an unexpected consequence, we establish a mysterious relationship between open $r$-spin theory and an extension of Witten's closed t…
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