Searcharxiv⌕ Search

arXiv subjects

Ian A. B. Strachan

Publications and source records attributed to Ian A. B. Strachan.

30 records · Page 2Linked to original sources
arXiv2010

Dubrovin's duality for $F$-manifolds with eventual identities

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-structures are preserved by our duality. We use eventual identities to construct compatible pair of metrics.

math.DG↗
arXiv2010

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 $I\,.$

nlin.SI↗
arXiv2009

Weyl groups and Elliptic Solutions of the WDVV equations

A functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equation. This is based on the elliptic trilogarithm function introduced by Beilinson and Levin. For this to be a solution results in a number of purely algebraic conditions on the set of vectors that appear in the ansatz, this providing an elliptic version of the idea, introduced by Veselov, of a V-system. Rational and trigonometric limits are studied together with examples of elliptic V-systems based on various Weyl groups. Jacobi group orbit spaces are studied: these carry the structure of a Frobenius manifold. The corresponding almost dual structure is shown, in the A_N and B_N and conjecturally for an arbitrary Weyl group, to correspond to the elliptic solutions of the WDVV equations. Transformation properties, under the Jacobi group, of the elliptic trilogarithm are derived together with various functional identities which generalize the classical Frobenius-Stickelburger relations.

math-ph↗
arXiv2009

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.

math-ph↗
arXiv2006

Logarithmic deformations of the rational superpotential/Landau-Ginzburg construction of solutions of the WDVV equations

The superpotential in the Landau-Ginzburg construction of solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations is modified to include logarithmic terms. This results in deformations - quadratic in the deformation parameters - of the normal prepotential solution of the WDVV equations. Such solution satisfy various pseudo-quasi-homogeneity conditions, on assigning a notional weight to the deformation parameters. This construction includes, as a special case, deformations which are polynomial in the flat coordinates, resulting in a new class of polynomial solutions of the WDVV equations.

math-ph↗
arXiv2006

A note on the relationship between rational and trigonometric solutions of the WDVV equations

Legendre transformations provide a natural symmetry on the space of solutions to the WDVV equations, and more specifically, between different Frobenius manifolds. In this paper a twisted Legendre transformation is constructed between solutions which define the corresponding dual Frobenius manifolds. As an application it is shown that certain trigonometric and rational solutions of the WDVV equations are related by such a twisted Legendre transform.

nlin.SI↗
arXiv2005

Duality for Jacobi group orbit spaces and elliptic solutions of the WDVV equations

From any given Frobenius manifold one may construct a so-called dual structure which, while not satisfying the full axioms of a Frobenius manifold, shares many of its essential features, such as the existence of a prepotential satisfying the WDVV equations of associativity. Jacobi group orbit spaces naturally carry the structures of a Frobenius manifold and hence there exists a dual prepotential. In this paper this dual prepotential is constructed and expressed in terms of the elliptic polylogarithm function of Beilinson and Levin.

math-ph↗
arXiv2005

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, much of the structure comes from the compatibility properties 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 studied.

math.DG↗
arXiv2004

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.

math.DG↗
arXiv2002

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.

math.DG↗
arXiv2000

Jordan manifolds and dispersionless KdV equations

Multicomponent KdV-systems are defined in terms of a set of structure constants and, as shown by Svinolupov, if these define a Jordan algebra the corresponding equations may be said to be integrable, at least in the sense of having higher-order symmetries, recursion operators and hierarchies of conservation laws. In this paper the dispersionless limits of these Jordan KdV equations are studied, under the assumptions that the Jordan algebra has a unity element and a compatible non-degenerate inner product. Much of this structure may be encoded in a so-called Jordan manifold, akin to a Frobenius manifold. In particular the Hamiltonian properties of these systems are investigated.

nlin.SI↗