Differential and Functional Identities for the Elliptic Trilogarithm

When written in terms of $\vartheta$-functions, the classical Frobenius-Stickelberger pseudo-addition formula takes a very simple form. Generalizations of this functional identity are studied, where the functions involved are derivatives (including derivatives with respect to the modular parameter) of the elliptic trilogarithm function introduced by Beilinson and Levin. A differential identity satisfied by this function is also derived. These generalized Frobenius-Stickelberger identities play a fundamental role in the development of elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde equations of associativity, with the simplest case reducing to the above mentioned differential identity

A construction of Multidimensional Dubrovin-Novikov Brackets

A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is based on an extension of the original construction of Gelfand and Dorfman which gave examples of Novikov algebras in terms of structures defined from commutative, associative algebras. Given such an algebra, the construction involves only linear algebra

Deformations of the Monge/Riemann hierarchy and approximately integrable systems

Dispersive deformations of the Monge equation u_u=uu_x are studied using ideas originating from topological quantum field theory and the deformation quantization programme. It is shown that, to a high-order, the symmetries of the Monge equation may also be appropriately deformed, and that, if they exist at all orders, they are uniquely determined by the original deformation. This leads to either a new class of integrable systems or to a rigorous notion of an approximate integrable system. Quasi-Miura transformations are also constructed for such deformed equations.Comment: 9 pages LaTe

Generalized Legendre transformations and symmetries of the WDVV equations

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 theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations.Comment: 23 page

Compatible metrics on a manifold and non-local bi-Hamiltonian structures

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 supporting such compatible metrics is studied and interpreted in terms of a multiplication on the cotangent bundle. With additional quasi-homogeneity assumptions one arrives at a so-called weak \F-manifold - a curved version of a Frobenius manifold (which is not, in general, an F-manifold). A submanifold theory is also developed.Comment: 17 page

Modular Frobenius manifolds and their invariant flows

The space of Frobenius manifolds has a natural involutive symmetry on it: there exists a map $I$ which send a Frobenius manifold to another Frobenius manifold. Also, from a Frobenius manifold one may construct a so-called almost dual Frobenius manifold which satisfies almost all of the axioms of a Frobenius manifold. The action of $I$ on the almost dual manifolds is studied, and the action of $I$ on objects such as periods, twisted periods and flows is studied. A distinguished class of Frobenius manifolds sit at the fixed point of this involutive symmetry, and this is made manifest in certain modular properties of the various structures. In particular, up to a simple reciprocal transformation, for this class of modular Frobenius manifolds, the flows are invariant under the action of $I\,.

Novikov algebras and a classification of multicomponent Camassa-Holm equations

A class of multi-component integrable systems associated to Novikov algebras, which interpolate between KdV and Camassa-Holm type equations, is obtained. The construction is based on the classification of low-dimensional Novikov algebras by Bai and Meng. These multi-component bi-Hamiltonian systems obtained by this construction may be interpreted as Euler equations on the centrally extended Lie algebras associated to the Novikov algebras. The related bilinear forms generating cocycles of first, second and third order are classified. Several examples, including known integrable equations, are presented.Comment: V2: some comments and references are adde

Einstein metrics on tangent bundles of spheres

We give an elementary treatment of the existence of complete Kahler-Einstein metrics with nonpositive Einstein constant and underlying manifold diffeomorphic to the tangent bundle of the (n+1)-sphere.Comment: 9 page