Guido Carlet, Boris Dubrovin, Luca Philippe Mertens
We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/infinity respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
On a class of integrable deformations of the integrable hierarchy of topological type associated to a semisimple Frobenius manifold
本文针对半单弗罗贝尼乌斯流形构造了一类具有多项式 tau-结构的拓扑型可积层级形变，并猜想在一维情形下该构造为黎曼–霍普夫层级的可积形变提供普适对象。
We formulate some conjectures that relates semisimple Frobenius manifolds, their spectral curves and integrable hierarchies.
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 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…
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。
The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper a new theory of Frobenius-algebra valued integrable systems is developed. This is achieved for systems derived from Frobenius manifolds by utilizing the theory of tensor products for such manifolds, as developed by Kaufmann, Kontsevich and Manin \cite{Kaufmann,KMK}. By specializing this construction, using a fixed Frobenius algebra…
Motivated by the theory of Painlevé equations and associated hierarchies, we study non-autonomous Hamiltonian systems that are Frobenius integrable. We establish sufficient conditions under which a given finite-dimensional Lie algebra of Hamiltonian vector fields can be deformed to a time-dependent Lie algebra of Frobenius integrable vector fields spanning the same distribution as the original algebra. The results are applied to quasi-Stäckel systems.
We propose a new construction of an integrable hierarchy associated to any infinite series of Frobenius manifolds satisfying a certain stabilization condition. We study these hierarchies for Frobenius manifolds associated to $A_N$, $D_N$ and $B_N$ singularities. In the case of $A_N$ Frobenius manifolds our hierarchy turns out to coincide with the KP hierarchy; for $B_N$ Frobenius manifolds it coincides with the BKP hierarchy; and for $D_N$ hierarchy it is a certain reduction of the 2-component BKP hierarchy. As a side product to these results we illustrate the enumerative meaning of certain co…
We prove that the description of pencils of compatible (N x N)-metrics of constant Riemannian curvature is equivalent to a special class of integrable N-parametric deformations of quasi-Frobenius (in general, noncommutative) algebras.
We investigate the solutions to open WDVV equation, associated to type A and D Dubrovin-Frobenius manifolds. We show that these solutions satisfy some stabilization condition and associate to both of them the systems of commuting PDEs. In the type A we show that the system of PDEs constructed coincides with the dispersionless modifiled KP hierarchy written in the Fay form.
On B type family of Dubrovin-Frobenius manifolds and their integrable systems
本文研究B型Coxeter群对应的双指标Dubrovin-Frobenius流形族之间的关系，证明部分数据可由平移指标后的流形恢复，并构造出扩展无色散BKP族的无穷可交换偏微分方程组。
Following the approach of Carlet et al.(2011)\cite{CDM}, we construct a class of infinite-dimensional Frobenius manifolds underlying the Toda lattice hierarchy, which are defined on the space of pairs of meromorphic functions with possibly higher-order poles at the origin and at infinity. We also show a connection between these infinite-dimensional Frobenius manifolds and the finite-dimensional Frobenius manifolds on the orbit space of extended affine Weyl groups of type $A$ defined by Dubrovin and Zhang.
This article is the first one in a suite of three articles exploring connections between dynamical systems of Stäckel-type and of Painlevé- type. In this article we present a deformation of autonomous Stäckel-type systems to non-autonomous Frobenius integrable systems. First, we consider quasi-Stäckel systems with quadratic in momenta Hamiltonians containing separable potentials with time dependent coefficients and then we present a procedure of deforming these equations to non-autonomous Frobenius integrable systems. Then, we present a procedure of deforming quasi-Stäckel systems with so call…
We use the connection between the Frobrenius manifold and the Douglas string equation to further investigate Minimal Liouville gravity. We search a solution of the Douglas string equation and simultaneously a proper transformation from the KdV to the Liouville frame which ensure the fulfilment of the conformal and fusion selection rules. We find that the desired solution of the string equation has explicit and simple form in the flat coordinates on the Frobenious manifold in the general case of (p,q) Minimal Liouville gravity.
The Drinfeld - Sokolov construction associates a hierarchy of bihamiltonian integrable systems with every untwisted affine Lie algebra. We compute the complete set of invariants of the related bihamiltonian structures with respect to the group of Miura type transformations.
Duality of operator Frobenius algebras and solution of Eisenhart-Stäckel problem in the non-diagonal case
本文研究算子场的弗罗贝尼乌斯代数，引入其对偶性概念，证明在算子场互为对称性时，对偶代数中的算子场也互为对称性，从而构造新的无限维积分系统。主要应用是解决任意塞格雷特征和任意维度下的艾辛哈特-斯塔克尔问题。
We construct a class of infinite-dimensional Frobenius manifolds on the space of pairs of certain even functions meromorphic inside or outside the unit circle. Via a bi-Hamiltonian recursion relation, the principal hierarchies associated to such Frobenius manifolds are found to be certain extensions of the dispersionless two-component BKP hierarchy. Moreover, we show that these infinite-dimensional Frobenius manifolds contain finite-dimensional Frobenius submanifolds as defined on the orbit space of Coxeter groups of types B and D.
The extended Ablowitz-Ladik hierarchy and a generalized Frobenius manifold
本研究定义了Ablowitz-Ladik层级的一种扩展形式，证明其与带非平坦单位元的广义Frobenius流形主层级的拓扑形变等价，丰富了可积系统与Frobenius流形的关联理论。
Infinite-dimensional Frobenius Manifolds and Extensions of Genus-Zero Whitham Hierarchies
该研究在黎曼球面特定区域的亚纯函数对空间中构造一类无穷维Frobenius流形，推导其主层级并证明是零亏格Whitham层级的延拓，丰富了相关数学结构研究。
We give a short review of Frobenius manifolds and algebraic integrability and study their intersection. The simplest case is the relation between the Frobenius manifold of simple singularities, which is almost dual to the integrable open Toda chain. New types of manifolds called extra special Kaehler and special F-manifolds are introduced which capture the intersection.
We construct a class of infinite-dimensional Frobenius manifolds on the spaces of pairs of meromorphic functions with a pole at infinity and a movable pole. Such Frobenius manifolds are shown to be underlying the universal Whitham hierarchy, which is an extension of the dispersionless Kadomtsev-Petviashvili hierarchy.
Principal hierarchy of Infinite-dimensional Frobenius Manifold Underlying the Extended Kadomtsev-Petviashvili Hierarchy
本文构造了扩展KP层级对应的无穷维Frobenius流形的主层级，并证明它是带一个可移动极点的零亏格Whitham层级的扩展。
We investigate the correspondence between two dimensional topological gauge theories and quantum integrable systems discovered by Moore, Nekrasov, Shatashvili. This correspondence means that the hidden quantum integrable structure exists in the topological gauge theories. We showed the correspondence between the G/G gauged WZW model and the phase model in JHEP 11 (2012) 146 (arXiv:1209.3800). In this paper, we study a one-parameter deformation for this correspondence and show that the G/G gauged WZW model coupled to additional matters corresponds to the q-boson model. Furthermore, we investiga…
The Stability Limit of Prepotentials for Hurwitz-Frobenius Manifolds: An Infinite-Dimensional Approach
本文通过无限维Frobenius流形的方法，证明了预势稳定性与Whitham层次tau结构的内在联系，并扩展到由开放WDVV方程解稳定性产生的层次。
We study the Dubrovin equation of the infinite-dimensional 2D Toda Dubrovin-Frobenius manifold at its irregular singularity. We first revisit the definition of the canonical coordinates, proving that they emerge naturally as generalized eigenvalues of the operator of multiplication by the Euler vector field. We then show that the formal solutions to the Dubrovin equation with exponential type behaviour at the irregular singular point are not uniquely determined by their leading order, but instead depend on an infinite number of parameters, contrary to what happens in the finite-dimensional cas…
We define a dispersionless tau-symmetric bihamiltonian integrable hierarchy on the space of pairs of functions analytic inside/outside the unit circle with simple poles at $0$/$\infty$ respectively, which extends the dispersionless 2D Toda hierarchy of Takasaki and Takebe. Then we construct the deformed flat connection of the infinite-dimen\-sional Frobenius manifold $M_0$ introduced by Carlet, Dubrovin and Mertens in Math. Ann. 349 (2011) 75--115 and, by explicitly solving the deformed flatness equations, we prove that the extended 2D Toda hierarchy coincides with principal hierarchy of $M_0$…
For any semisimple Frobenius manifold, we prove that a tau-symmetric bihamiltonian deformation of its Principal Hierarchy admits an infinite family of linearizable Virasoro symmetries if and only if all the central invariants of the corresponding deformation of the bihamiltonian structure are equal to $\frac{1}{24}$. As an important application of this result, we prove that the Dubrovin-Zhang hierarchy associated with the semisimple Frobenius manifold possesses a bihamiltonian structure which can be represented in terms of differential polynomials.
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 obtained from the corresponding deformed flat connection. This provides a suitable framework for the Whitham theory of an enlarged class of integrable hierarchies; we treat as examples the q-deformed Gelfand-Dickey hierarchy and the sine-Gordon equation, and compute the corresponding solutions of the WDVV equations.
In this paper, we introduce the notion of cyclic Frobenius algebras (CF-algebras). Canonical structures of CF-algebras exist on associative and Poisson algebras. It turns out that the modern theory of integrable systems yields non-trivial examples of CF-algebras. In the theory of the KdV hierarchy there is a structure of CF-algebra which leads to explicit expressions for the first integrals of the N-th Novikov hierarchy, suitable for classical, non-Abelian and quantum cases.
In this short survey, we describe our approach for constructing hierarchies of Poisson brackets for classical integrable systems using its' spectral curves.
Discrete $C^*$-Frobenius Algebras and Infinite-Index Extensions in Algebraic Quantum Field Theories
本文引入离散 $C^*$-Frobenius 代数作为范畴数据，构造 Möbius 协变网的无限指标扩张，并给出简单电流扩张和离散 Longo-Rehren 构造两类例子。
We study structural aspects of the Ablowitz-Ladik (AL) hierarchy in the light of its realization as a two-component reduction of the two-dimensional Toda hierarchy, and establish new results on its connection to the Gromov-Witten theory of local CP1. We first of all elaborate on the relation to the Toeplitz lattice and obtain a neat description of the Lax formulation of the AL system. We then study the dispersionless limit and rephrase it in terms of a conformal semisimple Frobenius manifold with non-constant unit, whose properties we thoroughly analyze. We build on this connection along two m…
In this note we use the formalism of multi-KP hierarchies in order to give some general formulas for infinitesimal deformations of solutions of the Darboux-Egoroff system. As an application, we explain how Shramchenko's deformations of Frobenius manifold structures on Hurwitz spaces fit into the general formalism of Givental-van de Leur twisted loop group action on the space of semi-simple Frobenius manifolds.
For an arbitrary calibrated Frobenius manifold, we construct an infinite dimensional Lie algebra, called the Virasoro-like algebra, which is a deformation of the Virasoro algebra of the Frobenius manifold. By using the Virasoro-like algebra we give a family of quadratic PDEs that are satisfied by the genus-zero free energy of the Frobenius manifold. We also derive, under the semisimplicity assumption, the Virasoro constraints for the corresponding abstract Hodge partition function.
We apply Frobenius integrability theorem in the search of invariants for one-dimensional Hamiltonian systems with a time-dependent potential. We obtain several classes of potential functions for which Frobenius theorem assures the existence of a two-dimensional foliation to which the motion is constrained. In particular, we derive a new infinite class of potentials for which the motion is assurately restricted to a two-dimensional foliation. In some cases, Frobenius theorem allows the explicit construction of an associated invariant. It is proven the inverse result that, if an invariant is kno…
Motivated by the algebraic open-closed string models, we introduce and discuss an infinite-dimensional counterpart of the open-closed Hurwitz theory describing branching coverings generated both by the compact oriented surfaces and by the foam surfaces. We manifestly construct the corresponding infinite-dimensional equipped Cardy-Frobenius algebra, with the closed and open sectors are represented by conjugation classes of permutations and the pairs of permutations, i.e. by the algebra of Young diagrams and bipartite graphes respectively.
We consider the deformations of Whitham systems including the "dispersion terms" and having the form of Dubrovin-Zhang deformations of Frobenius manifolds. The procedure is connected with B.A. Dubrovin problem of deformations of Frobenius manifolds corresponding to the Whitham systems of integrable hierarchies. Under some non-degeneracy requirements we suggest a general scheme of the deformation of the hyperbolic Whitham systems using the initial non-linear system. The general form of the deformed Whitham system coincides with the form of the "low-dispersion" asymptotic expansions used by B.A.…
We consider a Frobenius structure associated with the dispersionless Kadomtsev-Petviashvili equation. This is done, essentially, by applying a continuous analogue of the finite dimensional theory in the space of Schwartz functions on the line. The potential of the Frobenius manifold is found to be a logarithmic potential with quadratic external field. Following the construction of the principal hierarchy, we construct a set of infinitely many commuting flows, which extends the classical dKP hierarchy.
Genus zero Whitham hierarchy via Hurwitz--Frobenius manifolds
本文研究了零 genus Hurwitz-Frobenius 流形与 Whitham 级数的关系，证明了其微分方程组等价于零 genus Whitham 级数，并展示了其与 KP 级数的扩展方法。
We prove that a local Hamiltonian operator of hydrodynamic type K_1 is compatible with a nondegenerate local Hamiltonian operator of hydrodynamic type K_2 if and only if the operator K_1 is locally the Lie derivative of the operator K_2 along a vector field in the corresponding domain of local coordinates. This result gives a natural invariant definition of the class of special flat manifolds corresponding to all the class of compatible Dubrovin--Novikov Hamiltonian operators (the Frobenius--Dubrovin manifolds naturally belong to this class). There is an integrable bi-Hamiltonian hierarchy cor…
We prove that the associativity equations of two-dimensional topological quantum field theories are very natural reductions of the fundamental nonlinear equations of the theory of submanifolds in pseudo-Euclidean spaces and give a natural class of potential flat torsionless submanifolds. We show that all potential flat torsionless submanifolds in pseudo-Euclidean spaces bear natural structures of Frobenius algebras on their tangent spaces. These Frobenius structures are generated by the corresponding flat first fundamental form and the set of the second fundamental forms of the submanifolds (i…
Transmission matrices for two types of integrable defect are calculated explicitly, first by solving directly the nonlinear transmission Yang-Baxter equations, and second by solving a linear intertwining relation between a finite dimensional representation of the relevant Borel subalgebra of the quantum group underpinning the integrable quantum field theory and a particular infinite dimensional representation expressed in terms of sets of generalized `quantum' annihilation and creation operators. The principal examples analysed are based on the $a_2^{(2)}$ and $a_n^{(1)}$ affine Toda models bu…
Cartan connections for an infinite family of integrable vortices
研究无穷族可积涡旋方程与其底层黎曼面的Cartan几何关系，揭示涡旋方程为非阿贝尔连接的平坦性，且解可产生特定Dirac算子的磁零模。
The relation between the infinite-dimensional 3-algebras and the dispersionless KdV hierarchy is investigated. Based on the infinite-dimensional 3-algebras, we derive two compatible Nambu Hamiltonian structures. Then the dispersionless KdV hierarchy follows from the Nambu-Poisson evolution equation given the suitable Hamiltonians. We find that the dispersionless KdV system is not only a bi-Hamiltonian system, but also a bi-Nambu-Hamiltonian system. Due to the Nambu-Poisson evolution equation involving two Hamiltonians, more intriguing relationships between these Hamiltonians are revealed. As a…
The Darboux-KP system as an integrable Chern-Simons multiform theory in infinite dimensional space
本研究在已有广义Darboux系统拉格朗日3-形式结构基础上，为Darboux-KP系统建立Miwa变量无穷维空间中的Chern-Simons作用量拉格朗日多形式族，给出嵌入该空间的通用3维可积系统的变分形式。
The Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class - the so-called Legendre transformations - were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theo…
Infinite Family of Integrable Sigma Models Using Auxiliary Fields
本文引入一类由单变量函数参数化的二维sigma模型，含辅助场，证明其经典可积性，涵盖主手征模型及其能量-动量张量形变。
The exchange operator formalism in polar coordinates, previously considered for the Calogero-Marchioro-Wolfes problem, is generalized to a recently introduced, infinite family of exactly solvable and integrable Hamiltonians $H_k$, $k=1$, 2, 3,..., on a plane. The elements of the dihedral group $D_{2k}$ are realized as operators on this plane and used to define some differential-difference operators $D_r$ and $D_φ$. The latter serve to construct $D_{2k}$-extended and invariant Hamiltonians $\chh_k$, from which the starting Hamiltonians $H_k$ can be retrieved by projection in the $D_{2k}$ identi…
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…
A general unifying framework for integrable soliton-like systems on time scales is introduced. The $R$-matrix formalism is applied to the algebra of $δ$-differential operators in terms of which one can construct infinite hierarchy of commuting vector fields. The theory is illustrated by two infinite-field integrable hierarchies on time scales which are difference counterparts of KP and mKP. The difference counterparts of AKNS and Kaup-Broer soliton systems are constructed as related finite-field restrictions.
An infinite family of exactly-solvable and integrable potentials on a plane is introduced. It is shown that all already known rational potentials with the above properties allowing separation of variables in polar coordinates are particular cases of this family. The underlying algebraic structure of the new potentials is revealed as well as its hidden algebra. We conjecture that all members of the family are also superintegrable and demonstrate this for the first few cases. A quasi-exactly-solvable and integrable generalization of the family is found.
We describe an infinite-dimensional algebra of hidden symmetries of N=4 supersymmetric Yang-Mills (SYM) theory. Our derivation is based on a generalization of the supertwistor correspondence. Using the latter, we construct an infinite sequence of flows on the solution space of the N=4 SYM equations. The dependence of the SYM fields on the parameters along the flows can be recovered by solving the equations of the hierarchy. We embed the N=4 SYM equations in the infinite system of the hierarchy equations and show that this SYM hierarchy is associated with an infinite set of graded symmetries re…
In the present paper we prove the integrability (in the sense of existence of formal symmetry of infinite rank) for a class of block-triangular inhomogeneous extensions of (1+1)-dimensional integrable evolution systems. An important consequence of this result is the existence of formal symmetry of infinite rank for "almost integrable" systems, recently discovered by Sanders and van der Kamp.
We carry out the explicit computations that are used to write down the integrable hierarchy associated with the quintic Calabi-Yau threefold. We also do the calculations for the geometric structures emerging in the Gromov-Witten theory of the quintic, such as the Frobenius manifold structure and the special Kähler structure.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。