309 research outputs found
Conformally flat pencils of metrics, Frobenius structures and a modified Saito construction
No full textThe 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
Jordan manifolds and dispersionless KdV equations
No full textMulticomponent 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
Weyl groups and elliptic solutions of the WDVV equations
No full textA functional ansatz is developed which gives certain elliptic solutions of the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations. This ansatz 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 cases 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-Stickelberger relations
Logarithmic deformations of the rational superpotential/Landau-Ginzburg construction of solutions of the WDVV equations
No full textThe 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 solutions of the WDVV equations. Such solutions satisfy various pseudo-quasi-homogeneity conditions, on assigning a notional weight to the deformation parameters. These solutions originate in the so-called `water-bag' reductions of the dispersionless KP hierarchy. 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
Modular frobenius manifolds and their invariant flows
No full textThe space of Frobenius manifolds has a natural involutive symmetry on it: there exists a map I which sends 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
A note on the relationship between rational and trigonometric solutions of the WDVV equations
No full textLegendre 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
Frobenius submanifolds
No full textThe 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 Frobenius manifold — is studied. Two-dimensional submanifolds are particularly simple. More generally, sufficient conditions are given for a submanifold to be a so-called natural Frobenius submanifold. These ideas are illustrated using examples of Frobenius manifolds constructed from Coxeter groups, and for the Frobenius manifolds governing the quantum cohomology of CP2 and CP1 X CP1
Columbus\u27s Ghost: Tourism, Art and National Identity in the Bahamas
No full textIan Gregory Strachan (1969-), Bahamian writer, Chair of English Studies at College of the Bahamas, author of God\u27s Angry Babies (1997) and Paradise and Plantation (2002)
Frobenius manifolds: natural submanifolds and induced bi-Hamiltonian structures
No full textSubmanifolds 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 that these natural submanifolds carry a induced pencil of compatible metrics. It is then shown how one may constrain the bi-Hamiltonian hierarchies associated to a Frobenius manifold to live on these natural submanifolds whilst retaining their, now non-local, bi-Hamiltonian structure
El fantasma de Colón: El turismo, el arte y la identidad nacional en las Bahamas
No full textIan Gregory Strachan (1969-), Bahamian writer, Chair of English Studies at College of the Bahamas, author of God's Angry Babies (1997) and Paradise and Plantation (2002).
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