21 research outputs found
Frobenius submanifolds
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 so-called natural submanifold. These ideas are illustrated
using examples of Frobenius manifolds constructed from Coxeter groups, and for
the Frobenius manifolds governing the quantum cohomology of CP^2 and CP^1
\times CP^1.Comment: 23 pages. LaTeX 2
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
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 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
Transmogrifying Fuzzy Vortices
We show that the construction of vortex solitons of the noncommutative
Abelian-Higgs model can be extended to a critically coupled gauged linear sigma
model with Fayet-Illiopolous D-terms. Like its commutative counterpart, this
fuzzy linear sigma model has a rich spectrum of BPS solutions. We offer an
explicit construction of the degree static semilocal vortex and study in
some detail the infinite coupling limit in which it descends to a degree
\C\Pk^{N} instanton. This relation between the fuzzy vortex and
noncommutative lump is used to suggest an interpretation of the noncommutative
sigma model soliton as tilted D-strings stretched between an NS5-brane and a
stack of D3-branes in type IIB superstring theory.Comment: 21 pages, 4 figures, LaTeX(JHEP3
Coisotropic deformations of associative algebras and dispersionless integrable hierarchies
The paper is an inquiry of the algebraic foundations of the theory of
dispersionless integrable hierarchies, like the dispersionless KP and modified
KP hierarchies and the universal Whitham's hierarchy of genus zero. It stands
out for the idea of interpreting these hierarchies as equations of coisotropic
deformations for the structure constants of certain associative algebras. It
discusses the link between the structure constants and the Hirota's tau
function, and shows that the dispersionless Hirota's bilinear equations are,
within this approach, a way of writing the associativity conditions for the
structure constants in terms of the tau function. It also suggests a simple
interpretation of the algebro-geometric construction of the universal Whitham's
equations of genus zero due to Krichever.Comment: minor misprints correcte
Moduli space metrics for axially symmetric instantons
Under an axial symmetry the Yang–Mills self-duality equations for an arbitrary gauge group reduce to the Toda equation for that particular group, from which the finite action instantons (hyperbolic vortices) may be constructed. The space of such finite action instantons, with gauge equivalent solutions identified, is known as the moduli space, and carries a naturally defined Kähler metric. This metric is studied for the simply laced Lie algebras, and explicit examples are constructed for the 2-vortex system
Degenerate bi-Hamiltonian structures of the hydrodynamic type
Degenerate bi-Hamiltonian Poisson brackets of the hydrodynamic type are studied. They are bi-Hamiltonian structures of certain dispersionless rational Lax equations and are related to the notion of a degenerate Frobenius manifold
Low-velocity scattering of vortices in a modified Abelian Higgs model
The moduli space metric for hyperbolic vortices is constructed, and their slow motion scattering is calculated in terms of geodesics in this space