arXiv research

The notion of a Frobenius submanifold - a submanifold of a Frobenius manifold which is itself a Frobenius manifold with respect to structures induced from the original manifold - is studied. Two dimensional submanifolds are particularly simple. More generally, sufficient conditions are given for a submanifold to be a s…

Symplectic and Poisson structures proved for information geometry's Frobenius manifold.

Flat coordinates found for algebraic Frobenius manifolds in low dimensions.

A new algebra for Frobenius manifolds solves PDEs and constraints.

The study classifies Kähler-Frobenius manifolds and their properties.

Paper proves algebraic structure of a specific Frobenius manifold.

Curvature interpretation for WDVV equation in Frobenius manifolds.

Curved Frobenius manifolds link to Hessian metrics in geometry.

New integrable deformations for topological hierarchies from Frobenius manifolds.

Study on Frobenius manifold structures and inversion symmetry.

Proves existence and uniqueness of loop equation solutions for semisimple Frobenius manifolds.

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 dispersionle…

Let π: V \rightarrow M be a (real or holomorphic) vector bundle whose base has an almost Frobenius structure (\circ_{M},e_{M}, g_{M}) and typical fiber has the structure of a Frobenius algebra (\circ_{V},e_{V},g_{V}). Using a connection D on the bundle V and a morphism α: V \rightarrow TM, we construct an almost Froben…

Introduces a new canonical connection for Riemannian manifolds and proves Frobenius theorem geometrically.

The article gives a necessary and sufficient condition for a Frobenius manifold to be a CDV-structure. We show that there exists a positive definite CDV-structure on any semi-simple Frobenius manifold. We also compare three natural connections on a CDV-structure and conclude that the underlying Hermitian manifold of a …

Researchers find Frobenius manifold structures on orbits spaces of finite groups.

Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.

New Frobenius manifold structures found on Dicyclic group orbits.

The paper geometrically characterizes graded manifolds and proves the Frobenius theorem.

Linearizes Virasoro symmetries for semisimple Frobenius manifolds.

Study on special coordinates for Dubrovin-Frobenius manifolds in low dimensions.

Submanifolds of Frobenius manifolds are studied. In particular, so-called natural submanifolds are defined and, for semi-simple Frobenius manifolds, classified. These carry the structure of a Frobenius algebra on each tangent space, but will, in general, be curved. The induced curvature is studied, a main result being …

The paper studies transformations of Frobenius manifolds and their properties.

Study bihamiltonian structures and Frobenius manifolds for specific Toda hierarchies.

Constructs integrable hierarchies for generalized Frobenius manifolds with non-flat unity.

New Poisson bracket connects to logarithmic manifolds.

New findings on Frobenius structures on Kodaira manifolds.

New geometric properties discovered in a specific Frobenius manifold.

I.A.B. Strachan introduced the notion of a natural Frobenius submanifold of a Frobenius manifold and gave a sufficient but not necessary condition for a submanifold to be a natural Frobenius submanifold. This paper will give a necessary and sufficient condition and classify the natural Frobenius hypersurfaces.

Constructs generalized Frobenius manifolds for specific Weyl groups.

In this paper, we show that the quotient space of the domain by the reflection group for an elliptic root system has a structure of Frobenius manifold for the case of codimension 1. We also give a characterization of this Frobenius manifold structure under some suitable condition.

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 betw…

We introduce and study a superversion of Dubrovin's notion of semisimple Frobenius manifolds. We establish a correspondence between semisimple Frobenius (super)manifolds and special solutions to the (supersymmetric) Schlesinger equations. Finally, we calculate the Schlesinger initial conditions for solutions describing…

We obtain algebraic Frobenius manifolds from classical $W$-algebras associated to subregular nilpotent elements in simple Lie algebras of type $D_r$ where $r$ is even and $E_r$. The resulting Frobenius manifolds are certain hypersurfaces in the total spaces of semiuniversal deformation of simple hypersurface singularit…

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 Frobeniu…

We present a universal construction of almost duality for Frobenius manifolds. The analytic setup of this construction is described in details for the case of semisimple Frobenius manifolds. We illustrate the general considerations by examples from the singularity theory, mirror symmetry, the theory of Coxeter groups a…

Main mathematical applications of Frobenius manifolds are in the theory of Gromov - Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes rema…

New manifold structures on Weyl group orbit spaces proven.

We construct a differential Gerstenhaber-Batalin-Vilkovisky algebra from Dolbeault complex of any close Kaehler manifold, and a Frobenius manifold structure on Dolbeault cohomology.

Deformations of Dubrovin's Hurwitz Frobenius manifolds are constructed. The deformations depend on $g(g+1)/2$ complex parameters where $g$ is the genus of the corresponding Riemann surface. In genus one, the flat metric of the deformed Frobenius manifold coincides with a metric associated with a one-parameter family of…

We obtain polynomial Frobenius manifolds from classical $W$-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan subalgebras.

Constructs tri-Hamiltonian structure and Frobenius manifold for asymmetric gAL hierarchy

The paper proves convergence of WDVV potentials and semisimplicity of Frobenius manifolds.

Paper constructs solutions to WDVV equations for Frobenius manifolds.

One of the methods to obtain Frobenius manifold structures is via DGBV (differential Gerstenhaber-Batalin-Vilkovisky) algebra construction. An important problem is how to identify Frobenius manifold structures constructed from two different DGBV algebras. For DGBV algebras with suitable conditions, we show the functori…

Quantum Frobenius map for $SL_3$ skein modules constructed and described.

We construct a functor from the derived category of homotopy Gerstenhaber algebras with finite-dimensional cohomology to the purely geometric category of so-called $F_{\infty}$-manifolds. The latter contains Frobenius manifolds as a subcategory (so that a pointed Frobenius manifold is itself a homotopy Gerstenhaber alg…

New insights into mirror symmetry via Monge-Ampère domains and pre-Frobenius manifolds.