This work continues the study of $F$--manifolds $(M,\circ)$, first defined by Hertling and Manin and investigated in [He]. The notion of a compatible flat structure $\nabla$ is introduced, and it is shown that many constructions known for Frobenius manifolds do not in fact require invariant metrics and can be developed for all such triples $(M,\circ ,\nabla).$ In particular, we extend and generalize recent Dubrovin's duality.
In this note we introduce the concept of F-algebroid, and give its elementary properties and some examples. We provide a description of the almost duality for Frobenius manifolds, introduced by Dubrovin, in terms of a composition of two anchor maps of a unique cotangent F-algebroid.
A vector field E on an F-manifold (M, o, e) is an eventual identity if it is invertible and the multiplication X*Y := X o Y o E^{-1} defines a new F-manifold structure on M. We give a characterization of such eventual identities, this being a problem raised by Manin. We develop a duality between F-manifolds with eventual identities and we show that is compatible with the local irreducible decomposition of F-manifolds and preserves the class of Riemannian F-manifolds. We find necessary and sufficient conditions on the eventual identity which insure that harmonic Higgs bundles and DChk-structure…
We study regular non-semisimple Dubrovin-Frobenius manifolds in dimensions 2,3,4. We focus on the case where the Jordan canonical form of the operator of multiplication by the Euler vector field has a single Jordan block. Our results rely on the existence of special local coordinates introduced in [4] for regular flat F-manifolds with Euler vector field. In such coordinates the invariant metric of the Dubrovin-Frobenius manifold takes a special form which is the starting point of our construction.
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…
First order Hamiltonian operators of differential-geometric type were introduced by Dubrovin and Novikov in 1983, and thoroughly investigated by Mokhov. In 2D, they are generated by a pair of compatible flat metrics $g$ and $\tilde g$ which satisfy a set of additional constraints coming from the skew-symmetry condition and the Jacobi identity. We demonstrate that these constraints are equivalent to the requirement that $\tilde g$ is a linear Killing tensor of $g$ with zero Nijenhuis torsion. This allowed us to obtain a complete classification of $n$-component operators with $n\leq 4$ (for $n=1…
We show that the small quantum product of the generalized flag manifold $G/B$ is a product operation on $H^*(G/B)\otimes \bR[q_1,..., q_l]$ uniquely determined by the fact that it is a deformation of the cup product on $H^*(G/B)$, it is commutative, associative, graded with respect to $°(q_i)=4$, it satisfies a certain relation (of degree two), and the corresponding Dubrovin connection is flat. We deduce that it is again the flatness of the Dubrovin connection which characterizes essentially the solutions of the "quantum Giambelli problem" for $G/B$. This result gives new proofs of the quantum…
This is a survey of the current state of the theory of $F$--(super)manifolds $(M,\circ)$, first defined in [HeMa] and further developed in [He], [Ma2], [Me1]. Here $\circ$ is an $\Cal{O}_M$--bilinear multiplication on the tangent sheaf $\Cal{T}_M$, satisfying an integrability condition. $F$--manifolds and compatible flat structures on them furnish a useful weakening of Dubrovin's Frobenius structure which naturally arises in the quantum $K$--theory, theory of extended moduli spaces, and unfolding spaces of singularities.
We discuss some aspects of index and secondary index theory for flat bundles with duality. This theory was first developed by J. Lott. Our main purpose in the present paper is to provide a modification with better functorial properties.
In a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem of wave fronts in space forms. As an application, we show that the following four objects are essentially same; * conformally flat n-manifolds (n>=3) with admissible singular points (i.e. admissible GCF-manifolds), * frontals as hypersurfaces in the lightcone Q^{n+1}_+, * frontals as hypersurfaces in the hyperbolic space H^{n+1}, * sapacelike frontals as hypersurfaces in the de Sitter space S^{n+1}_1. Recently, the duality of conformally flat Riemannian mani…
We show that bi-flat $F$-manifolds can be interpreted as natural geometrical structures encoding the almost duality for Frobenius manifolds without metric. Using this framework, we extend Dubrovin's duality between orbit spaces of Coxeter groups and Veselov's $\vee$-systems, to the orbit spaces of exceptional well-generated complex reflection groups of rank $2$ and $3$. On the Veselov's $\vee$-systems side, we provide a generalization of the notion of $\vee$-systems that gives rise to a dual connection which coincides with a Dunkl-Kohno-type connection associated with such groups. In particula…
We solve the problem on flat extensions of a generic surface with boundary in Euclidean 3-space, relating it to the singularity theory of the envelope generated by the boundary. We give related results on Legendre surfaces with boundaries via projective duality and observe the duality on boundary singularities. Moreover we give formulae related to remote singularities of the boundary-envelope.
In this paper, we consider an equivalence problem of second order partially differential equations (PDE) and a duality of the flat differential equation. For the equivalence problem, explicit form of invariants (curvatures) are given. We also investigate a duality associated with the flat equation using double fibrations. These double fibrations are described in terms of transformation groups.
As was shown by a part of the authors, for a given $(2, 3, 5)$-distribution $D$ on a $5$-dimensional manifold $Y$, there is, locally, a Lagrangian cone structure $C$ on another $5$-dimensional manifold $X$ which consists of abnormal or singular paths of $(Y, D)$. We give a characterization of the class of Lagrangian cone structures corresponding to $(2, 3, 5)$-distributions. Thus we complete the duality between $(2, 3, 5)$-distributions and Lagrangian cone structures via pseudo-product structures of type $G_2$. A local example of non-flat perturbations of the global model of flat Lagrangian co…
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 solutions to the Painlevé-VI equation with coefficients $(1/8,-1/8,1/8,3/8).$ Analogous deformations of the real doubles of the Hurwitz Frobenius manifolds are also found; these deformations depend on $g(g+1)/2$ real parameters.
Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local bi-Hamiltonian structure, and with additional quasi-homogeneity conditions one obtains the structure of a Frobenius manifold. With appropriate curvature conditions one may define a curved pencil of compatible metrics and these give rise to an associated non-local bi-Hamiltonian structure. Specific examples include the F-manifolds of Hertling and Manin equipped with an invariant metric. In this paper the geometry su…
We study the moduli spaces of flat SL(r)- and PGL(r)-connections, or equivalently, Higgs bundles, on an algebraic curve. These spaces are noncompact Calabi-Yau orbifolds; we show that they can be regarded as mirror partners in two different senses. First, they satisfy the requirements laid down by Strominger-Yau-Zaslow (SYZ), in a suitably general sense involving a B-field or flat unitary gerbe. To show this, we use their hyperkahler structures and Hitchin's integrable systems. Second, their Hodge numbers, again in a suitably general sense, are equal. These spaces provide significant evidence …
We describe two extensions of the notion of a self-dual connection in a vector bundle over a manifold M from dim M=4 to higher dimensions. The first extension, Omega-self-duality, is based on the existence of an appropriate 4-form Omega on the Riemannian manifold M and yields solutions of the Yang-Mills equations. The second is the notion of half-flatness, which is defined for manifolds with certain Grassmann structure T^C M \cong E \otimes H. In some cases, for example for hyper-Kaehler manifolds M, half-flatness implies Omega-self-duality. A construction of half-flat connections inspired by …
We prove a duality principle for a special class of submanifolds in pseudo-Euclidean spaces. This class of submanifolds with potential of normals is introduced in this paper. We prove also, for example, that an arbitrary Frobenius manifold can be realized as a certain flat submanifold of this very natural class.
