22 research outputs found
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
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
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
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
Meromorphic tensor equivalence for Yangians and quantum loop algebras
Let be a complex semisimple Lie algebra, and , the corresponding Yangian and quantum loop algebra,
with deformation parameters related by . When is not a
rational number, we constructed in arXiv:1310.7318 a faithful functor
from the category of finite-dimensional representations of to those of . The functor is governed by the
additive difference equations defined by the commuting fields of the Yangian,
and restricts to an equivalence on a subcategory of
defined by choosing a branch of the logarithm. In this paper, we construct a
tensor structure on and show that, if , it yields an
equivalence of meromorphic braided tensor categories, when
and are endowed with the deformed Drinfeld coproducts and
the commutative part of the universal -matrix. This proves in particular the
Kohno-Drinfeld theorem for the abelian KZ equations defined by
. The tensor structure arises from the abelian KZ
equations defined by a appropriate regularisation of the commutative -matrix
of .Comment: Title changed, details added. 67 pages, 1 figure. Final version, to
appear in Publ. Math IHE
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
Quantum group A
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