151 research outputs found
A remark on deformations of Hurwitz Frobenius manifolds
In this note we use the formalism of multi-KP hierarchies in order to give
some general formulas for infinitesimal deformations of solutions of the
Darboux-Egoroff system. As an application, we explain how Shramchenko's
deformations of Frobenius manifold structures on Hurwitz spaces fit into the
general formalism of Givental-van de Leur twisted loop group action on the
space of semi-simple Frobenius manifolds.Comment: 10 page
On deformations of quasi-Miura transformations and the Dubrovin-Zhang bracket
In our recent paper we proved the polynomiality of a Poisson bracket for a
class of infinite-dimensional Hamiltonian systems of PDE's associated to
semi-simple Frobenius structures. In the conformal (homogeneous) case, these
systems are exactly the hierarchies of Dubrovin-Zhang, and the bracket is the
first Poisson structure of their hierarchy. Our approach was based on a very
involved computation of a deformation formula for the bracket with respect to
the Givental-Y.-P. Lee Lie algebra action. In this paper, we discuss the
structure of that deformation formula. In particular, we reprove it using a
deformation formula for weak quasi-Miura transformation that relates our
hierarchy of PDE's with its dispersionless limit.Comment: 21 page
On parabolic Whittaker functions
We derive a Mellin-Barnes integral representation for solution to generalized
(parabolic) quantum Toda lattice introduced in \cite{GLO}, which presumably
describes the -equivariant Gromov-Witten invariants of
Grassmann variety.Comment: 14 page
Coordinate Change of Gauss-Manin System and Generalized Mirror Transformation
In this paper, we explicitly derive the generalized mirror transformation of
quantum cohomology of general type projective hypersurfaces, proposed in our
previous article, as an effect of coordinate change of the virtual Gauss-Manin
system.Comment: 19 pages, latex, minor errors are corrected, discussions in Section 4
are refine
Quantum cohomology of flag manifolds and Toda lattices
We discuss relations of Vafa's quantum cohomology with Floer's homology
theory, introduce equivariant quantum cohomology, formulate some conjectures
about its general properties and, on the basis of these conjectures, compute
quantum cohomology algebras of the flag manifolds. The answer turns out to
coincide with the algebra of regular functions on an invariant lagrangian
variety of a Toda lattice.Comment: 35 page
On a Conjecture of Givental
These brief notes record our puzzles and findings surrounding Givental's
recent conjecture which expresses higher genus Gromov-Witten invariants in
terms of the genus-0 data. We limit our considerations to the case of a
projective line, whose Gromov-Witten invariants are well-known and easy to
compute. We make some simple checks supporting his conjecture.Comment: 13 pages, no figures; v.2: new title, minor change
The structure of 2D semi-simple field theories
I classify all cohomological 2D field theories based on a semi-simple complex
Frobenius algebra A. They are controlled by a linear combination of
kappa-classes and by an extension datum to the Deligne-Mumford boundary. Their
effect on the Gromov-Witten potential is described by Givental's Fock space
formulae. This leads to the reconstruction of Gromov-Witten invariants from the
quantum cup-product at a single semi-simple point and from the first Chern
class, confirming Givental's higher-genus reconstruction conjecture. The proof
uses the Mumford conjecture proved by Madsen and Weiss.Comment: Small errors corrected in v3. Agrees with published versio
The Abelian/Nonabelian Correspondence and Frobenius Manifolds
We propose an approach via Frobenius manifolds to the study (began in
math.AG/0407254) of the relation between rational Gromov-Witten invariants of
nonabelian quotients X//G and those of the corresponding ``abelianized''
quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses
the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove
this conjecture when the nonabelian quotients are partial flag manifolds.Comment: 35 pages, no figure
Open Virtual Structure Constants and Mirror Computation of Open Gromov-Witten Invariants of Projective Hypersurfaces
In this paper, we generalize Walcher's computation of the open Gromov-Witten
invariants of the quintic hypersurface to Fano and Calabi-Yau projective
hypersurfaces. Our main tool is the open virtual structure constants. We also
propose the generalized mirror transformation for the open Gromov-Witten
invariants, some parts of which are proven explicitly. We also discuss possible
modification of the multiple covering formula for the case of higher
dimensional Calabi-Yau manifolds. The generalized disk invariants for some
Calabi-Yau and Fano manifolds are shown and they are certainly integers after
re-summation by the modified multiple covering formula. This paper also
contains the direct integration method of the period integrals for higher
dimensional Calabi-Yau hypersurfaces in the appendix.Comment: 24pages, 5figure
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