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 $q$--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

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

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 $p$-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

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

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

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

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

Understanding Potential Challenges in Demolition Robot Teleoperation to Inform Interface Design: Insights from Industry Professionals

Teleoperation is receiving intense attention due to its potential to address health and safety concerns for construction workers caused by on-site hazards. It allows operators to control robots from a distance outside their field of view using wireless communication technologies. Currently one of the most frequently deployed robotic technologies in construction is the remotely operated demolition robot within the field of view. While distance teleoperation offers great potential for enhancing worker safety and addressing labor shortages, the complex and dynamic demolition sites present unique challenges. This paper introduces the preliminary results of a study that explored potential challenges in demolition robot teleoperation, informed by the challenges identified in traditional demolition machine operations and the key concerns raised by industry professionals. A user-centered approach was employed through narrative interviews and focus groups with demolition professionals. The findings indicate that the challenges faced in traditional demolition machine operations may persist in teleoperation and could be further exacerbated by the sensory degradation inherent in teleoperation. Besides, the interactions among challenges increase the complexity of their overall impact. Moreover, enhancing operators’ situational awareness without inducing cognitive overload or distraction is critical for effectively addressing these challenges. Additionally, the results suggest the need for context-aware, multimodal teleoperation interfaces to assist operators in managing operational challenges. This study contributes to understanding challenges that operators may face during teleoperation, offering valuable insights for developing teleoperation interfaces adaptable to diverse demolition contexts