Frobenius Manifolds Research Articles

Legendre transforms for type An and Bn V-systems

Abstract The Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations have a rich structure related to the theory of Frobenius manifolds, with many known families of solutions. A Legendre transformation is a symmetry of the WDVV equations, introduced by Dubrovin. We explicitly compute the results of a Legendre transformation applied to An - and Bn -type multi-parameter rational solutions, relating them to known and new trigonometric solutions.

Regular F-manifolds with eventual identities

Abstract Given an F-manifold one may construct a dual multiplication (generalizing the idea of an almost-dual Frobenius manifold introduced by Dubrovin) using a so-called eventual identity, the definition of which ensures that the dual object is also an F-manifold. In this paper we solve the equations for an eventual identity for a regular (so non-semi-simple) F-manifold and construct a dual coordinate system in which dual multiplication is preserved. As an application, families of Nijenhuis operators are constructed.

On B-type family of Dubrovin–Frobenius manifolds and their integrable systems

On B-type family of Dubrovin–Frobenius manifolds and their integrable systems

Commutativity equations and their trigonometric solutions

In the theory of Frobenius manifolds and Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, one normally assumes that Frobenius algebras associated with a solution F have an identity e . Equivalently, the corresponding flat metric can be expressed as a linear combination of the matrices of the third-order derivatives of the pre-potential function F . We show that under certain non-degeneracy conditions, this assumption can be omitted, that is, the identity field e exists automatically without further assumptions. We also study trigonometric solutions F determined by a finite collection of vectors with multiplicities, and we give an explicit formula for the field e for all the known such solutions. The corresponding collections of vectors are given by the non-simply laced root systems or are related to their projections to the intersection of mirrors.

Solutions of the Loop Equations of a Class of Generalized Frobenius Manifolds

Solutions of the Loop Equations of a Class of Generalized Frobenius Manifolds

Homotopy BV-algebras in Hermitian geometry

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative BV∞-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.

GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back

GUE via Frobenius Manifolds. I. From Matrix Gravity to Topological Gravity and Back

Dubrovin–Frobenius manifold structures on the orbit space of the symmetric group

By choosing different Sl-invariant metrics, we show the existence of (l − 1) different Dubrovin–Frobenius manifold structures on the orbit space of the symmetric group and also construct Landau–Ginzburg superpotentials for these Dubrovin–Frobenius manifold structures.

On the Quantum Deformations of Associative Sato Grassmannian Algebras and the Related Matrix Problems

We analyze the Lie algebraic structures related to the quantum deformation of the Sato Grassmannian, reducing the problem to studying co-adjoint orbits of the affine Lie subalgebra of the specially constructed loop diffeomorphism group of tori. The constructed countable hierarchy of linear matrix problems made it possible, in part, to describe some kinds of Frobenius manifolds within the Dubrovin-type reformulation of the well-known WDVV associativity equations, previously derived in topological field theory. In particular, we state that these equations are equivalent to some bi-Hamiltonian flows on a smooth functional submanifold with respect to two compatible Poisson structures, generating a countable hierarchy of commuting to each other’s hydrodynamic flows. We also studied the inverse problem aspects of the quantum Grassmannian deformation Lie algebraic structures, related with the well-known countable hierarchy of the higher nonlinear Schrödinger-type completely integrable evolution flows.

Hurwitz numbers for reflection groups II: Parabolic quasi-Coxeter elements

We define parabolic quasi-Coxeter elements in well generated complex reflection groups. We characterize them in multiple natural ways, and we study two combinatorial objects associated with them: the collections RedW(g) of reduced reflection factorizations of g and RGS(W,g) of relative generating sets of g. We compute the cardinalities of these sets for large families of parabolic quasi-Coxeter elements and, in particular, we relate the size #RedW(g) with geometric invariants of Frobenius manifolds. This paper is second in a series of three; we will rely on many of its results in part III to prove uniform formulas that enumerate full reflection factorizations of parabolic quasi-Coxeter elements, generalizing the genus-0 Hurwitz numbers.

Shifted Witten classes and topological recursion

The Witten r r -spin class defines a non-semisimple cohomological field theory. Pandharipande, Pixton and Zvonkine studied two special shifts of the Witten class along two semisimple directions of the associated Dubrovin–Frobenius manifold using the Givental–Teleman reconstruction theorem. We show that the R R -matrix and the translation of these two specific shifts can be constructed from the solutions of two differential equations that generalise the classical Airy differential equation. Using this, we prove that the descendant intersection theory of the shifted Witten classes satisfies topological recursion on two 1 1 -parameter families of spectral curves. By taking the limit as the parameter goes to zero, we prove that the descendant intersection theory of the Witten r r -spin class can be computed by topological recursion on the r r -Airy spectral curve. We finally show that this proof suffices to deduce Witten’s r r -spin conjecture, already proved by Faber, Shadrin and Zvonkine, which claims that the generating series of r r -spin intersection numbers is the tau function of the r r -KdV hierarchy that satisfies the string equation.

Geometry and arithmetic of integrable hierarchies of KdV type. I. Integrality

For each of the simple Lie algebras g=Al, Dl or E6, we show that the all-genera one-point FJRW (Fan–Jarvis–Ruan, Witten) invariants of g-type, after multiplication by suitable products of Pochhammer symbols, are the coefficients of an algebraic generating function and hence are integral. Moreover, we find that the all-genera invariants themselves coincide with the coefficients of the unique calibration of the Frobenius manifold of g-type evaluated at a special point. For the A4 (5-spin) case we also find two other normalizations of the sequence that are again integral and of at most exponential growth, and hence conjecturally are the Taylor coefficients of some period functions.

The sixth Painlevé equation as isomonodromy deformation of an irregular system: monodromy data, coalescing eigenvalues, locally holomorphic transcendents and Frobenius manifolds

The sixth Painlevé equation PVI is both the isomonodromy deformation condition of a 2-dimensional isomonodromic Fuchsian system and of a 3-dimensional irregular system. Only the former has been used in the literature to solve the nonlinear connection problem for PVI, through the computation of invariant quantities . We prove a new simple formula expressing the invariants p jk in terms of the Stokes matrices of the irregular system, making the irregular system a concrete alternative for the nonlinear connection problem. We classify the transcendents such that the Stokes matrices and the p jk can be computed in terms of special functions, providing a full non-trivial class of 3-dim. examples such that the theory of non-generic isomonodromy deformations of Cotti et al (2019 Duke Math. J. 168 967–1108) applies. A sub-class of these transcendents realises the local structure of all the 3-dim Dubrovin–Frobenius manifolds with semisimple coalescence points of the type studied in Cotti et al (2020 SIGMA 16 105). We compute all the monodromy data for these manifolds (Stokes matrix, Levelt exponents and central connection matrix).

On real forms of Dubrovin-Ugaglia Poisson brackets

Dubrovin and Ugaglia introduced a holomorphic Poisson structure on the space of semisimple Frobenius manifolds, which is identified with the entries of an upper triangular matrix with ones on the diagonal. In this paper, we compute explicitly the real forms of the Dubrovin-Ugaglia Poisson structures, arising from the real forms of the Lie algebra son(C).

Dubrovin–Frobenius manifolds associated with Bn and the constrained KP hierarchy

In this paper, we will show that the Dubrovin–Frobenius prepotentials on the orbit space of the Coxeter group Bn constructed by Arsie et al. [Sel. Math. New Ser. 29, 1 (2023)] coincide with the solutions of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations associated with the constrained KP hierarchy.

On a Connection Formula of a Higher Rank Analog of Painlevé VI

Abstract In this paper, we derive a connection formula for a higher rank analog of Painlevé VI arising from the isomonodromic deformation theory and the study of Frobenius manifolds.

Low-dimensional bihamiltonian structures of topological type

We construct local bihamiltonian structures from classical W-algebras associated with non-regular nilpotent elements of regular semisimple type in Lie algebras of types A2 and A3. They form exact Poisson pencils and admit a dispersionless limit, and their leading terms define logarithmic or trivial Dubrovin–Frobenius manifolds. We calculate the corresponding central invariants, which are expected to be constants. In particular, we get Dubrovin–Frobenius manifolds associated with the focused Schrödinger equation and Hurwitz space M0;1,0 and the corresponding bihamiltonian structures of topological type.

Variational Bihamiltonian Cohomologies and Integrable Hierarchies I: Foundations

This series of papers is devoted to the study of Virasoro symmetries of deformations of the Principal Hierarchy associated with a semisimple Frobenius manifold. The main tool we use is a generalization of the bihamiltonian cohomology called the variational bihamiltonian cohomology. In the present paper, we give its definition and compute the associated cohomology groups that will be used in our study of Virasoro symmetries. As an application of the variational bihamiltonian cohomology theory, we classify conformal bihamiltonian structures with semisimple hydrodynamic limits.

Frobenius Manifolds and a New Class of Extended Affine Weyl Groups of A-type (II)

Frobenius Manifolds and a New Class of Extended Affine Weyl Groups of A-type (II)

THE GENERAL KOVALEVSKAYA THEOREM FOR A CAUCHY PROBLEM

The Kovalevskaya theorem is a well-known mathematical result that provides a powerful tool for solving systems of ordinary differential equations (ODEs). However, there has been a longstanding question regarding whether the theorem can be extended to more general systems of partial differential equations (PDEs). In this paper, we address this question by presenting a discussion of the only general Kovalevskaya theorem that can be extended from the theory of ODEs to systems of PDEs. Our discussion begins by reviewing the classical Kovalevskaya theorem for ODEs and its various applications in mathematics and physics. We then introduce the concept of Frobenius manifold, which provides a framework for extending the Kovalevskaya theorem to PDEs. We show how the Frobenius manifold structure can be used to transform a system of PDEs into a system of ODEs, which can then be solved using the classical Kovalevskaya theorem. Overall, our discussion provides a novel perspective on the Kovalevskaya theorem and its potential applications in PDE theory. By extending the theorem to this broader class of equations, we open up new possibilities for solving complex problems in mathematics, physics, and engineering.