54 research outputs found
Coherent States for Quantum Compact Groups
Coherent states are introduced and their properties are discussed for all
simple quantum compact groups. The multiplicative form of the canonical element
for the quantum double is used to introduce the holomorphic coordinates on a
general quantum dressing orbit and interpret the coherent state as a
holomorphic function on this orbit with values in the carrier Hilbert space of
an irreducible representation of the corresponding quantized enveloping
algebra. Using Gauss decomposition, the commutation relations for the
holomorphic coordinates on the dressing orbit are derived explicitly and given
in a compact R--matrix formulation (generalizing this way the --deformed
Grassmann and flag manifolds). The antiholomorphic realization of the
irreducible representations of a compact quantum group (the analogue of the
Borel--Weil construction) are described using the concept of coherent state.
The relation between representation theory and non--commutative differential
geometry is suggested.}Comment: 25 page
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
Quantum Wall Crossing in N=2 Gauge Theories
We study refined and motivic wall-crossing formulas in N=2 supersymmetric
gauge theories with SU(2) gauge group and N_f < 4 matter hypermultiplets in the
fundamental representation. Such gauge theories provide an excellent testing
ground for the conjecture that "refined = motivic."Comment: 24 pages, 4 figure
Poisson structures on affine spaces and flag varieties. I. Matrix affine Poisson space
The standard Poisson structure on the rectangular matrix variety Mm,n(C) is
investigated, via the orbits of symplectic leaves under the action of the maximal torus T ⊂
GLm+n(C). These orbits, finite in number, are shown to be smooth irreducible locally closed
subvarieties of Mm,n(C), isomorphic to intersections of dual Schubert cells in the full flag
variety of GLm+n(C). Three different presentations of the T-orbits of symplectic leaves in
Mm,n(C) are obtained – (a) as pullbacks of Bruhat cells in GLm+n(C) under a particular
map; (b) in terms of rank conditions on rectangular submatrices; and (c) as matrix products
of sets similar to double Bruhat cells in GLm(C) and GLn(C). In presentation (a), the orbits
of leaves are parametrized by a subset of the Weyl group Sm+n, such that inclusions of
Zariski closures correspond to the Bruhat order. Presentation (b) allows explicit calculations
of orbits. From presentation (c) it follows that, up to Zariski closure, each orbit of leaves is
a matrix product of one orbit with a fixed column-echelon form and one with a fixed rowechelon
form. Finally, decompositions of generalized double Bruhat cells in Mm,n(C) (with
respect to pairs of partial permutation matrices) into unions of T-orbits of symplectic leaves
are obtained
Coherent states for Hopf algebras
Families of Perelomov coherent states are defined axiomatically in the
context of unitary representations of Hopf algebras possessing a Haar integral.
A global geometric picture involving locally trivial noncommutative fibre
bundles is involved in the construction. A noncommutative resolution of
identity formula is proved in that setup. Examples come from quantum groups.Comment: 19 pages, uses kluwer.cls; the exposition much improved; an example
of deriving the resolution of identity via coherent states for SUq(2) added;
the result differs from the proposals in literatur
Mirror duality and noncommutative tori
In this paper, we study a mirror duality on a generalized complex torus and a
noncommutative complex torus. First, we derive a symplectic version of Riemann
condition using mirror duality on ordinary complex tori. Based on this we will
find a mirror correspondence on generalized complex tori and generalize the
mirror duality on complex tori to the case of noncommutative complex tori.Comment: 22pages, no figure
Tetrahedron and 3D reflection equations from quantized algebra of functions
Soibelman's theory of quantized function algebra A_q(SL_n) provides a
representation theoretical scheme to construct a solution of the Zamolodchikov
tetrahedron equation. We extend this idea originally due to Kapranov and
Voevodsky to A_q(Sp_{2n}) and obtain the intertwiner K corresponding to the
quartic Coxeter relation. Together with the previously known 3-dimensional (3D)
R matrix, the K yields the first ever solution to the 3D analogue of the
reflection equation proposed by Isaev and Kulish. It is shown that matrix
elements of R and K are polynomials in q and that there are combinatorial and
birational counterparts for R and K. The combinatorial ones arise either at q=0
or by tropicalization of the birational ones. A conjectural description for the
type B and F_4 cases is also given.Comment: 26 pages. Minor correction
Quantum transfer matrices for discrete and continuous quasi-exactly solvable problems
We clarify the algebraic structure of continuous and discrete quasi-exactly
solvable spectral problems by embedding them into the framework of the quantum
inverse scattering method. The quasi-exactly solvable hamiltonians in one
dimension are identified with traces of quantum monodromy matrices for specific
integrable systems with non-periodic boundary conditions. Applications to the
Azbel-Hofstadter problem are outlined.Comment: 15 pages, standard LaTe
D-instantons and twistors: some exact results
We present some results on instanton corrections to the hypermultiplet moduli
space in Calabi-Yau compactifications of Type II string theories. Previously,
using twistor methods, only a class of D-instantons (D2-instantons wrapping
A-cycles) was incorporated exactly and the rest was treated only linearly. We
go beyond the linear approximation and give a set of holomorphic functions
which, through a known procedure, capture the effect of D-instantons at all
orders. Moreover, we show that for a sector where all instanton charges have
vanishing symplectic invariant scalar product, the hypermultiplet metric can be
computed explicitly.Comment: 32 pages, 3 figures, uses JHEP3.cls; some changes in section 3.3.3;
corrected formula for the contact potentia
Challenges of beta-deformation
A brief review of problems, arising in the study of the beta-deformation,
also known as "refinement", which appears as a central difficult element in a
number of related modern subjects: beta \neq 1 is responsible for deviation
from free fermions in 2d conformal theories, from symmetric omega-backgrounds
with epsilon_2 = - epsilon_1 in instanton sums in 4d SYM theories, from
eigenvalue matrix models to beta-ensembles, from HOMFLY to super-polynomials in
Chern-Simons theory, from quantum groups to elliptic and hyperbolic algebras
etc. The main attention is paid to the context of AGT relation and its possible
generalizations.Comment: 20 page
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