798 research outputs found
Gromov-Witten classes, quantum cohomology, and enumerative geometry
The paper is devoted to the mathematical aspects of topological quantum field
theory and its applications to enumerative problems of algebraic geometry. In
particular, it contains an axiomatic treatment of Gromov-Witten classes, and a
discussion of their properties for Fano varieties. Cohomological Field Theories
are defined, and it is proved that tree level theories are determined by their
correlation functions. Applications to counting rational curves on del Pezzo
surfaces and projective spaces are given.Comment: 44 p, amste
Theta Vectors and Quantum Theta Functions
In this paper, we clarify the relation between Manin's quantum theta function
and Schwarz's theta vector in comparison with the kq representation, which is
equivalent to the classical theta function, and the corresponding coordinate
space wavefunction. We first explain the equivalence relation between the
classical theta function and the kq representation in which the translation
operators of the phase space are commuting. When the translation operators of
the phase space are not commuting, then the kq representation is no more
meaningful. We explain why Manin's quantum theta function obtained via algebra
(quantum tori) valued inner product of the theta vector is a natural choice for
quantum version of the classical theta function (kq representation). We then
show that this approach holds for a more general theta vector with constant
obtained from a holomorphic connection of constant curvature than the simple
Gaussian one used in the Manin's construction. We further discuss the
properties of the theta vector and of the quantum theta function, both of which
have similar symmetry properties under translation.Comment: LaTeX 21 pages, give more explicit explanations for notions given in
the tex
Quantum Mechanics on the h-deformed Quantum Plane
We find the covariant deformed Heisenberg algebra and the Laplace-Beltrami
operator on the extended -deformed quantum plane and solve the Schr\"odinger
equations explicitly for some physical systems on the quantum plane. In the
commutative limit the behaviour of a quantum particle on the quantum plane
becomes that of the quantum particle on the Poincar\'e half-plane, a surface of
constant negative Gaussian curvature. We show the bound state energy spectra
for particles under specific potentials depend explicitly on the deformation
parameter . Moreover, it is shown that bound states can survive on the
quantum plane in a limiting case where bound states on the Poincar\'e
half-plane disappear.Comment: 16pages, Latex2e, Abstract and section 4 have been revise
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
Supersymmetric Harry Dym Type Equations
A supersymmetric version is proposed for the well known Harry Dym system. A
general class super Lax operator which leads to consistent equations is
considered.Comment: 4 pages, latex, no figure
Del Pezzo surfaces of degree 1 and jacobians
We construct absolutely simple jacobians of non-hyperelliptic genus 4 curves,
using Del Pezzo surfaces of degree 1. This paper is a natural continuation of
author's paper math.AG/0405156.Comment: 24 page
Covariant differential complexes on quantum linear groups
We consider the possible covariant external algebra structures for Cartan's
1-forms on GL_q(N) and SL_q(N). We base upon the following natural postulates:
1. the invariant 1-forms realize an adjoint representation of quantum group;
2. all monomials of these forms possess the unique ordering.
For the obtained external algebras we define the exterior derivative
possessing the usual nilpotence condition, and the generally deformed version
of Leibniz rules. The status of the known examples of GL_q(N)-differential
calculi in the proposed classification scheme, and the problems of
SL_q(N)-reduction are discussed.Comment: 23 page
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
Relations Among Universal Equations For Gromov-Witten Invariants
In this paper, we study relations among known universal equations for
Gromov-Witten invariants at genus 1 and 2.Comment: LaTex file, 13 page
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