567 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
The symplectic origin of conformal and Minkowski superspaces
Supermanifolds provide a very natural ground to understand and handle
supersymmetry from a geometric point of view; supersymmetry in and
dimensions is also deeply related to the normed division algebras.
In this paper we want to show the link between the conformal group and
certain types of symplectic transformations over division algebras. Inspired by
this observation we then propose a new\,realization of the real form of the 4
dimensional conformal and Minkowski superspaces we obtain, respectively, as a
Lagrangian supermanifold over the twistor superspace and a
big cell inside it.
The beauty of this approach is that it naturally generalizes to the 6
dimensional case (and possibly also to the 10 dimensional one) thus providing
an elegant and uniform characterization of the conformal superspaces.Comment: 15 pages, references added, minor change
Why the general Zakharov-Shabat equations form a hierarchy?
The totality of all Zakharov-Shabat equations (ZS), i.e., zero-curvature
equations with rational dependence on a spectral parameter, if properly
defined, can be considered as a hierarchy. The latter means a collection of
commuting vector fields in the same phase space. Further properties of the
hierarchy are discussed, such as additional symmetries, an analogue to the
string equation, a Grassmannian related to the ZS hierarchy, and a Grassmannian
definition of soliton solutions.Comment: 13p
h-deformation of Gr(2)
The -deformation of functions on the Grassmann matrix group is
presented via a contraction of . As an interesting point, we have seen
that, in the case of the -deformation, both R-matrices of and
are the same
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
Harmonic Superspaces in Low Dimensions
Harmonic superspaces for spacetimes of dimension are constructed.
Some applications are given.Comment: 16, kcl-th-94-15. Two further references have been added (12 and 13)
and a few typographical errors have been correcte
On Macroscopic Energy Gap for -Quantum Mechanical Systems
The q-deformed harmonic oscillator within the framework of the recently
introduced Schwenk-Wess -Heisenberg algebra is considered. It is shown, that
for "physical" values , the gap between the energy levels decreases
with growing energy. Comparing with the other (real) -deformations of the
harmonic oscillator, where the gap instead increases, indicates that the
formation of the macroscopic energy gap in the Schwenk-Wess -Quantum
Mechanics may be avoided.Comment: 6 pages, TeX, PRA-HEP-92/1
Elliptic Schlesinger system and Painlev{\'e} VI
We construct an elliptic generalization of the Schlesinger system (ESS) with
positions of marked points on an elliptic curve and its modular parameter as
independent variables (the parameters in the moduli space of the complex
structure). ESS is a non-autonomous Hamiltonian system with pair-wise commuting
Hamiltonians. The system is bihamiltonian with respect to the linear and the
quadratic Poisson brackets. The latter are the multi-color generalization of
the Sklyanin-Feigin-Odeskii classical algebras. We give the Lax form of the
ESS. The Lax matrix defines a connection of a flat bundle of degree one over
the elliptic curve with first order poles at the marked points.
The ESS is the monodromy independence condition on the complex structure for
the linear systems related to the flat bundle.
The case of four points for a special initial data is reduced to the
Painlev{\'e} VI equation in the form of the Zhukovsky-Volterra gyrostat,
proposed in our previous paper.Comment: 16 pages; Dedicated to the centenary of the publication of the
Painleve VI equation in the Comptes Rendus de l'Academie des Sciences de
Paris by Richard Fuchs in 190
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