Pantelis A. Damianou, Rui Loja Fernandes
We observe that the modular class of a Poisson-Nijhenhuis manifold has a canonical representative and that, under a cohomological assumption, this vector field is bi-hamiltonian. In many examples the associated hierarchy of flows reproduces classical integrable hierarchies.
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We prove that for any tau-symmetric bihamiltonian deformation of the tau-cover of the Principal Hierarchy associated with a semisimple Frobenius manifold, the deformed tau-cover admits an infinite set of Virasoro symmetries.
We consider a certain super extension, called the super tau-cover, of a bihamiltonian integrable hierarchy which contains the Hamiltonian structures including both the local and non-local ones as odd flows. In particular, we construct the super tau-cover of the principal hierarchy associated with an arbitrary Frobenius manifold, and the super tau-cover of the Korteweg-de Vries (KdV) hierarchy. We also show that the Virasoro symmetries of these bihamiltonan integrable hierarchies can be extended to symmetries of the associated super tau-covers.
Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations
本文提出变分双哈密顿上同调的定义并计算相关上同调群，作为研究半单Frobenius流形主层级Virasoro对称性形变的基础工具，并据此对具有半单流体力学极限的共形双哈密顿结构进行分类。
Legendre transformations of a class of generalized Frobenius manifolds and the associated integrable hierarchies
本文研究一类广义 Frobenius 流形在 Legendre 型变换下的可积层级关系，证明 Legendre 扩展主层级及其半单条件下的拓扑形变均由同一线性互反变换联系。
In this short note we show how Dubrovin's integrable hierarchies, defined using the Gromov-Witten theory of a closed symplectic manifold, generalizes to Hamiltonian Floer theory. In particular, we show how the required generalization of the PSS isomorphism, relating Gromov-Witten theory and Hamiltonian Floer theory, can be constructed in the framework of Eliashberg-Givental-Hofer's symplectic field theory.
In the present work, the integrable bi-Hamiltonian hierarchies related to compatible nonlocal Poisson brackets of hydrodynamic type are effectively constructed. For achieving this aim, first of all, the problem on the canonical form of a special type for compatible nonlocal Poisson brackets of hydrodynamic type is solved.
We solve the problem of reducing to the simplest and convenient for our purposes, canonical form for an arbitrary pair of compatible nonlocal Poisson brackets of hydrodynamic type generated by metrics of constant Riemannian curvature in order to get an effective construction of the integrable hierarchies related to all these compatible Poisson brackets.
Involutive scroll structures on solutions of 4D dispersionless integrable hierarchies
本文研究4维heavenly型无色散可积层级解上的有理正规卷轴结构，证明其与特征簇重合且具有对合性，并由共形自对偶方程层级支配。
We compute the central invariants of the bihamiltonian structures of the constrained KP hierarchies, and show that these integrable hierarchies are topological deformations of their hydrodynamic limits.
Generalized Frobenius Manifolds with Non-flat Unity and Integrable Hierarchies
本文为具有非平坦单位元的广义Frobenius流形构造了双哈密顿可积层次（主层次），证明其具有Virasoro对称性和tau结构，推导环方程并构造拓扑形变，且给出与Volterra、q-变形KdV及Ablowitz-Ladik层次相关的实例。
We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential $m$-form $Ω_m$ with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invarian…
We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov - Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative r…
Starting from a so-called flat exact semisimple bihamiltonian structures of hydrodynamic type, we arrive at a Frobenius manifold structure and a tau structure for the associated principal hierarchy. We then classify the deformations of the principal hierarchy which possess tau structures.
We obtain variational formulas for holomorphic objects on Riemann surfaces with respect to arbitrary local coordinates on the moduli space of complex structures. These formulas are written in terms of a canonical object on the moduli space which corresponds to the pairing between the space of quadratic differentials and the tangent space to the moduli space. This canonical object satisfies certain commutation relations which appear to be the same as the ones that emerged in the integrability theory of Whitham type hierarchies. Driven by this observation, we develop the theory of Whitham type h…
We present some general results on properties of the bihamiltonian cohomologies associated to bihamiltonian structures of hydrodynamic type, and compute the third cohomology for the bihamiltonian structure of the dispersionless KdV hierarchy. The result of the computation enables us to prove the existence of bihamiltonian deformations of the dispersionless KdV hierarchy starting from any of its infinitesimal deformations.
Super integrable hierarchies associated with color Lie algebra
本文通过考虑彩色李代数 $\mathfrak{sp}_{1}(6)$ 上的两个非等谱问题，构造了 (1+1) 维和 (2+1) 维超可积系统，并利用超迹恒等式建立了其超哈密顿结构。
We solve the problem of describing all nonlocal Hamiltonian operators of hydrodynamic type with flat metrics. This problem is also equivalent to the description of all flat submanifolds with flat normal bundle in a pseudo-Euclidean space. It is proved that every such Hamiltonian operator (or the submanifold corresponding to the operator) gives a pencil of compatible Poisson brackets, generates bi-Hamiltonian integrable hierarchies of hydrodynamic type, and also defines a family of integrals in involution. We prove that a natural special class of such Hamiltonian operators (submanifolds) is des…
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 generating function which yields a solution of the semiclassical limit of the KdV_r equations. We formulate axioms for a cohomology class on this moduli space which allow one to construct a cohomological field theory of rank $r-1$ in all genera. In genus zero it produces a Frobenius manifold which is isomorphic to the Frobenius manifold structure on the base o…
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