95 research outputs found
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
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
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
Quantum deformations of associative algebras and integrable systems
Quantum deformations of the structure constants for a class of associative
noncommutative algebras are studied. It is shown that these deformations are
governed by the quantum central systems which has a geometrical meaning of
vanishing Riemann curvature tensor for Christoffel symbols identified with the
structure constants. A subclass of isoassociative quantum deformations is
described by the oriented associativity equation and, in particular, by the
WDVV equation. It is demonstrated that a wider class of weakly (non)associative
quantum deformations is connected with the integrable soliton equations too. In
particular, such deformations for the three-dimensional and
infinite-dimensional algebras are described by the Boussinesq equation and KP
hierarchy, respectively.Comment: Numeration of the formulas is correcte
Modules of Abelian integrals and Picard-Fuchs systems
We give a simple proof of an isomorphism between the two
-modules: the module of relative cohomologies and the module of Abelian integrals corresponding to a regular at
infinity polynomial in two variables. Using this isomorphism, we prove
existence and deduce some properties of the corresponding Picard-Fuchs system.Comment: A separate section discusses Fuchsian properties of the Picard-Fuchs
system, Morse condition exterminated. Few errors were correcte
Seiberg-Witten Theory and Extended Toda Hierarchy
The quasiclassical solution to the extended Toda chain hierarchy,
corresponding to the deformation of the simplest Seiberg-Witten theory by all
descendants of the dual topological string model, is constructed explicitly in
terms of the complex curve and generating differential. The first derivatives
of prepotential or quasiclassical tau-function over the extra times, extending
the Toda chain, are expressed through the multiple integrals of the
Seiberg-Witten one-form. We derive the corresponding quasiclassical Virasoro
constraints, discuss the functional formulation of the problem and propose
generalization of the extended Toda hierarchy to the nonabelian theory.Comment: 32 pages, LaTe
Mirror symmetry and quantization of abelian varieties
The paper consists of two sections. The first section provides a new
definition of mirror symmetry of abelian varieties making sense also over
-adic fields. The second section introduces and studies quantized
theta-functions with two-sided multipliers, which are functions on
non-commutative tori. This is an extension of an earlier work by the author. In
the Introduction and in the Appendix the constructions of this paper are put
into a wider context.Comment: 24 pp., amstex file, no figure
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