Frobenius manifolds and Frobenius algebra-valued integrable systems

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 (Int Math Res Not 19:929–952, 1996), Kontsevich and Manin (Inv Math 124: 313–339, 1996). By specializing this construction, using a fixed Frobenius algebra \({\mathcal {A}},\) one can arrive at such a theory. More generally, one can apply the same idea to construct an \({\mathcal {A}}\)-valued topological quantum field theory. The Hamiltonian properties of two classes of integrable evolution equations are then studied: dispersionless and dispersive evolution equations. Application of these ideas are discussed, and as an example, an \({\mathcal {A}}\)-valued modified Camassa–Holm equation is constructed.     

Polynomial modular Frobenius manifolds

The moduli space of Frobenius manifolds carries a natural involutive symmetry, and a distinguished class–so-called modular Frobenius manifolds–lie at the fixed points of this symmetry. In this paper a classification of semi-simple modular Frobenius manifolds which are polynomial in all but one of the variables is begun, and completed for three and four dimensional manifolds. The resulting examples may also be obtained from higher dimensional manifolds by a process of folding. The relationship of these results with orbifold quantum cohomology is also discussed.     

Proof of character-valued index theorems

A character-valued index is a generalization of the ordinary Dirac index to manifolds with nontrivial automorphism groups. A simple proof of the corresponding fixed-point theorem is presented which uses the techniques of supersymmetric quantum mechanics. This theorem relates the character-valued index to a topological integral of curvature forms on the fixed-point space of the automorphism in question.     

“Real Doubles” of Hurwitz Frobenius Manifolds

New Frobenius structures on Hurwitz spaces are found. A Hurwitz space is considered as a real manifold; therefore the number of coordinates is twice as large as the number of coordinates on Hurwitz Frobenius manifolds of Dubrovin. Simple branch points of a ramified covering and their complex conjugates play the role of canonical coordinates on the constructed Frobenius manifolds. Corresponding solutions to WDVV equations and G-functions are obtained.     

Proof of a conjecture on the genus two free energy associated to the An singularity

In a recent paper Dubrovin et al. (1998), it is proved that the genus two free energy of an arbitrary semisimple Frobenius manifold can be represented as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so called genus two $G$-function, and for a certain class of Frobenius manifolds it is conjectured that the associated genus two $G$-functions vanish. In this paper, we prove this conjecture for the Frobenius manifolds associated with simple singularities of type A.     

Derived Bracket Construction and Manin Products

We will extend the classical derived bracket construction to any algebra over a binary quadratic operad. We will show that the derived product construction is a functor given by the Manin white product with the operad of permutation algebras. As an application, we will show that the operad of prePoisson algebras is isomorphic to the Manin black product of the Poisson operad with the preLie operad. We will show that differential operators and Rota–Baxter operators are, in a sense, Koszul-dual to each other.     

Givental Graphs and Inversion Symmetry

Inversion symmetry is a very non-trivial discrete symmetry of Frobenius manifolds. It was obtained by Dubrovin from one of the elementary Schlesinger transformations of a special ODE associated to a Frobenius manifold. In this paper, we review the Givental group action on Frobenius manifolds in terms of Feynman graphs and obtain an interpretation of the inversion symmetry in terms of the action of the Givental group. We also consider the implication of this interpretation of the inversion symmetry for the Schlesinger transformations and for the Hamiltonians of the associated principle hierarchy.     

Stability for manifolds of equilibrium states of generalized Birkhoff system

<正>Stability for the manifolds of equilibrium states of a generalized Birkhoff system is studied.A theorem for the stability of the manifolds of equilibrium states of the general autonomous system is used to the generalized Birkhoffian system and two propositions on the stability of the manifolds of equilibrium states of the system are obtained.An example is given to illustrate the application of the results.     

A Remark on Deformations of Hurwitz Frobenius Manifolds

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.     

On precise center stable manifold theorems for certain reaction–diffusion and Klein–Gordon equations

We consider positive, radial and exponentially decaying steady state solutions of the general reaction–diffusion and Klein–Gordon type equations and present an explicit construction of infinite-dimensional invariant manifolds in the vicinity of these solutions. The result is a precise stable manifold theorem for the reaction–diffusion equation and a precise center-stable manifold theorem for the Klein–Gordon equation, which include the co-dimension of the manifolds and the decay rates for even perturbations.     

Algebras of local observables on a manifold

We propose a generalization of the Haag-Kastler axioms for local observables to Lorentzian manifolds. The framework is intended to resolve ambiguities in the construction of quantum field theories on manifolds. As an example we study linear scalar fields for globally hyperbolic manifolds.Supported by National Science Foundation PHY 77-21740.On leave from Department of Mathematics, SUNY at Buffalo, Buffalo, NY 14214, USA     

Mackey Theorem and Two-Electron Wave Function of a Multi-Centre System

The two-electron wave function in a system of many equivalent atoms is investigated group-theoretically. It is shown that the classification of different types of two-electron (two-hole) localizations can be made by the double-coset decomposition of the symmetry group with respect to the local subgroup, and that the group appearing in the Mackey theorem can be used for the additional classification of states. The Mackey theorem on symmetrized squares and the generalized Frobenius reciprocity theorem are applied to the construction of two-electron states in octahedral symmetry. Received October 23, 1995; revised June 21, 1996; accepted for publication July 1, 1996     

Hamiltonian structures and their reciprocal transformations for the -KdV–CH hierarchy

The r-KdV–CH hierarchy is a generalization of the Korteweg–de Vries and Camassa–Holm hierarchies parameterized by r+1 constants. In this paper we clarify some properties of its multi-Hamiltonian structures including the explicit expressions of the Hamiltonians, the formulae of the central invariants of the associated bihamiltonian structures and the relationship of these bihamiltonian structures with Frobenius manifolds. By introducing a class of generalized Hamiltonian structures, we present in a natural way the transformation formulae of the Hamiltonian structures of the hierarchy under certain reciprocal transformations, and prove the validity of the formulae at the level of dispersionless limit.     

Conformally flat pencils of metrics,Frobenius structures and a modified Saito construction

The structure of a Frobenius manifold encodes the geometry associated with a flat pencil of metrics. However, as shown in the authors’ earlier work [L. David, I.A.B. Strachan, Compatible metrics on manifolds and non-local bi-Hamiltoninan structures, Int. Math. Res. Notices 66 (2004) 3533–3557], much of the structure comes from the compatibility property of the pencil rather than from the flatness of the pencil itself. In this paper conformally flat pencils of metrics are studied and examples, based on a modification of the Saito construction, are developed.     

The Construction of Frobenius Manifolds¶from KP tau-Functions

Frobenius manifolds (solutions of WDVV equations) in canonical coordinates are determined by the system of Darboux–Egoroff equations. This system of partial differential equations appears as a specific subset of the n-component KP hierarchy. KP representation theory and the related Sato infinite Grassmannian are used to construct solutions of this Darboux–Egoroff system and the related Frobenius manifolds. Finally we show that for these solutions Dubrovin's isomonodromy tau-function can be expressed in the KP tau-function. Received: 1 September 1998 / Accepted: 7 March 1999     

On the AKSZ Formulation of the Poisson Sigma Model

We review and extend the Alexandrov–Kontsevich–Schwarz–Zaboronsky construction of solutions of the Batalin–Vilkovisky classical master equation. In particular, we study the case of sigma models on manifolds with boundary. We show that a special case of this construction yields the Batalin–Vilkovisky action functional of the Poisson sigma model on a disk. As we have shown in a previous paper, the perturbative quantization of this model is related to Kontsevich's deformation quantization of Poisson manifolds and to his formality theorem. We also discuss the action of diffeomorphisms of the target manifolds.     

A Symplectic Generalization of the Peradzyński Helicity Theorem and Some Applications

Symplectic and symmetry analysis for studying MHD superfluid flows is devised, a new version of the Z. Peradzyński (Int. J. Theor. Phys. 29(11):1277–1284, [1990]) helicity theorem based on differential-geometric and group-theoretical methods is derived. Having reanalyzed the Peradzyński helicity theorem within the modern symplectic theory of differential-geometric structures on manifolds, a new unified proof and a new generalization of this theorem for the case of compressible MHD superfluid flow are proposed. As a by-product, a sequence of nontrivial helicity type local and global conservation laws for the case of incompressible superfluid flow, playing a crucial role for studying the stability problem under suitable boundary conditions, is constructed.     

On the genus two free energies for semisimple Frobenius manifolds

We represent the genus two free energy of an arbitrary semisimple Frobenius manifold as the sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called ??genus two G-function.?? Conjecturally, the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities, as well as for ?¹-orbifolds with positive Euler characteristics. We explain the reasons for the conjecture and prove it in particular cases.     

Le théorème de réduction de Marsden-Weinstein en géométrie cosymplectique et de contact

We show that the classical Marsden-Weinstein Reduction theorem for Hamiltonian systems with symmetries is still true for contact manifolds and cosympletic manifolds (i.e. canonical manifolds in the sense of A. Lichnerowicz).

In fact, we precise the notion of transitive almost contact structure, which enables us to consider the cosymplectic geometry as a limit of the contact geometry when a certain parameter goes to zero. This point of view unifies both theories.

However, we have to give two distinct proofs for the contact Reduction theorem and the cosymplectic one.     

Inverse Problem and Monodromy Data for Three-Dimensional Frobenius Manifolds

We study the inverse problem for semi-simple Frobenius manifolds of dimension 3 and we explicitly compute a parametric form of the solutions of the WDVV equations in terms of Painlevé VI transcendents. We show that the solutions are labeled by a set of monodromy data. We use our parametric form to explicitly construct polynomial and algebraic solutions and to derive the generating function of Gromov–Witten invariants of the quantum cohomology of the two-dimensional projective space. The procedure is a relevant application of the theory of isomonodromic deformations.