Quantum probability and Levy Laplacians

The formula Δ _L = lim_?→0 ∫_{∥s–t∥<?} b _s b _t dsdt for the Levy Laplacian is obtained, where b _t stands for an annihilation process. The formula is extended to some generalizations of the Levy Laplacian.     

Fiber Bundles in Non-relativistic Quantum Mechanics

The problem of describing a quantum mechanical system with symmetry by a fiber bundle is considered. The quantization of a fiber bundle is introduced. Fiber bundles for the Kepler problem and the rotator are constructed. The fiber bundle concept provides a new model for a physical system: it provides us with a model for an elementary particle with extension having integral values of spin.     

Dynamical Yang-Baxter Equation and Quantum Vector Bundles

We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for quantum dynamical R-matrices, dynamical twists, etc. In this context, we define dynamical associative algebras and show that such algebras give quantizations of vector bundles on coadjoint orbits. We build a dynamical twist for any pair of a reductive Lie algebra and its Levi subalgebra. Using this twist, we obtain an equivariant star product quantization of vector bundles on semisimple coadjoint orbits of reductive Lie groups.The research is supported in part by the Israel Academy of Sciences grant no. 8007/99-03, the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center Group Theoretic Methods in the study of Algebraic Varieties of the Israel Science foundation, and by Russian Foundation for Basic Research grant no. 03-01-00593.Deceased January 2004Acknowledgement We are grateful to J. Bernstein, V. Ostapenko, and S. Shnider for stimulating discussions within the Quantum groups seminar at the Department of Mathematics, Bar Ilan University. We appreciate useful remarks by M. Gorelik, V. Hinich, and A. Joseph during a talk at the Weizmann Institute. Our special thanks to P. Etingof for his comments on various aspects of the subject.     

Quantum Chains of Hopf Algebras with Quantum Double Cosymmetry

Given a finite dimensional C ^*-Hopf algebra H and its dual Ĥ we construct the infinite crossed product and study its superselection sectors in the framework of algebraic quantum field theory. is the observable algebra of a generalized quantum spin chain with H-order and Ĥ-disorder symmetries, where by a duality transformation the role of order and disorder may also appear interchanged. If is a group algebra then becomes an ordinary G-spin model. We classify all DHR-sectors of – relative to some Haag dual vacuum representation – and prove that their symmetry is described by the Drinfeld double . To achieve this we construct localized coactions and use a certain compressibility property to prove that they are universal amplimorphisms on . In this way the double can be recovered from the observable algebra as a universal cosymmetry. Received: 4 September 1995\,/\,Accepted: 3 December 1996     

Gauged Q-stars

We derive the theory about gauged non-topological soliton stars and their black holes,and find that the gauged Q-stars with maximum particle number Q_max in a definite range of mass are cold, stable and in coherent states of very large mass.Their characteristics are similar to those of general soliton stars.When Q>Q_max,the gauged Q-stars are not stable.     

Higgs Bundles,Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the N-particle sector. This implies the full equivalence between the above gauge theory and the N-particle sub-sector of the quantum theory of the Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra.     

Gauged Kac-Moody algebras

We construct a non-Abelian field theory by gauging a Kac-Moody algebra. One obtains an infinite tower of interacting vector fields and associated ghosts obeying slightly modified Feynman rules.     

Differential and Twistor Geometry of the Quantum Hopf Fibration

We study a quantum version of the SU(2) Hopf fibration and its associated twistor geometry. Our quantum sphere arises as the unit sphere inside a q-deformed quaternion space . The resulting four-sphere is a quantum analogue of the quaternionic projective space . The quantum fibration is endowed with compatible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space and use it to study a system of anti-self-duality equations on , for which we find an ‘instanton’ solution coming from the natural projection defining the tautological bundle over .     

Gauged galileons from branes

We show how the coupling of SO(N)$\mathit{SO}(N)$ gauge fields to galileons arises from a probe brane construction. The galileons arise from the brane bending modes of a brane probing a co-dimension N bulk, and the gauge fields arise by turning on certain off-diagonal components in the zero mode of the bulk metric. By construction, the equations of motion for both the galileons and gauge fields remain second order. Covariant gauged galileons are derived as well.     

Gauged Kaluza-Klein AdS pseudo-supergravity

We obtain the pseudo-supergravity extension of the D-dimensional Kaluza-Klein theory, which is the circle reduction of pure gravity in D+1 dimensions. The fermionic partners are pseudo-gravitino and pseudo-dilatino. The full Lagrangian is invariant under the pseudo-supersymmetric transformation, up to quadratic order in fermion fields. We find that the theory possesses a U(1) global symmetry that can be gauged so that all the fermions are charged under the Kaluza-Klein vector. The gauging process generates a scalar potential that has a maximum, leading to the AdS vacuum. Whist the highest dimension for gauged AdS supergravity is seven, our gauged AdS pseudo-supergravities can exist in arbitrary dimensions.     

Determinants of Laplacians

The determinant of the Laplacian on spinor fields on a Riemann surface is evaluated in terms of the value of the Selberg zeta function at the middle of the critical strip. A key role in deriving this relation is played by the Barnes double gamma function.This work was supported in part by the NSF Grant No. DMS-85-04329     

Lessons from Quantum Field Theory: Hopf Algebras and Spacetime Geometries

We discuss the prominence of Hopf algebras in recent progress in Quantum Field Theory. In particular, we will consider the Hopf algebra of renormalization, whose antipode turned out to be the key to a conceptual understanding of the subtraction procedure. We shall then describe several occurrences of this, or closely related Hopf algebras, in other mathematical domains, such as foliations, Runge-Kutta methods, iterated integrals and multiple zeta values. We emphasize the unifying role which the Butcher group, discovered in the study of numerical integration of ordinary differential equations, plays in QFT.     

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

We construct a quantum version of the SU(2) Hopf bundle S⁷→S⁴. The quantum sphere S⁷_q arises from the symplectic group Sp_q(2) and a quantum 4-sphere S⁴_q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S⁴_q) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S⁴. We compute the fundamental K-homology class of S⁴_q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU_q(2) on S⁷_q such that the algebra A(S⁷_q) is a non-trivial quantum principal bundle over A(S⁴_q) with structure quantum group A(SU_q(2)).     

Distribution of Zeros of Random and Quantum Chaotic Sections of Positive Line Bundles

We study the limit distribution of zeros of certain sequences of holomorphic sections of high powers M^N of a positive holomorphic Hermitian line bundle L over a compact complex manifold M. Our first result concerns "random" sequences of sections. Using the natural probability measure on the space of sequences of orthonormal bases {S^N_j} of H⁰(M, L^N), we show that for almost every sequence {S^N_j}, the associated sequence of zero currents &1/NZ_S^N_j; tends to the curvature form y of L. Thus, the zeros of a sequence of sections s_N ] H⁰(M, L^N) chosen independently and at random become uniformly distributed. Our second result concerns the zeros of quantum ergodic eigenfunctions, where the relevant orthonormal bases {S^N_j} of H⁰(M, L^N) consist of eigensections of a quantum ergodic map. We show that also in this case the zeros become uniformly distributed.     

Gauged supergravities in various spacetime dimensions

In this review article we study the gaugings of extended supergravity theories in various space‐time dimensions. These theories describe the low‐energy limit of non‐trivial string compactifications. For each theory under consideration we review all possible gaugings that are compatible with supersymmetry. They are parameterized by the so‐called embedding tensor which is a group theoretical object that has to satisfy certain representation constraints. This embedding tensor determines all couplings in the gauged theory that are necessary to preserve gauge invariance and supersymmetry. The concept of the embedding tensor and the general structure of the gauged supergravities are explained in detail. The methods are then applied to the half‐maximal (N = 4) supergravities in d = 4 and d = 5 and to the maximal supergravities in d = 2 and d = 7. Examples of particular gaugings are given. Whenever possible, the higher‐dimensional origin of these theories is identified and it is shown how the compactification parameters like fluxes and torsion are contained in the embedding tensor.     

Examples of gauged Laplacians on noncommutative spaces

We present two (classes of) examples of gauged Laplacian operators. The first one is a model of spin-Hall effect on a noncommutative four-sphere S _ϑ ⁴ with isospin degrees of freedom, coming from a noncommutative instanton, and invariant under the quantum group SO _ϑ (5). The second one, a Hall effect on a quantum 2-dimensional sphere S _q ², describes ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole with symmetry coming from the quantum group SU _q (2). For both models, ample symmetries provide a complete diagonalization.     

On determinants of Laplacians on Riemann surfaces

Determinants of Laplacians on tensors and spinors of arbitrary weights on compact hyperbolic Riemann surfaces are computed in terms of values of Selberg zeta functions at half integer points.Research supported in part by the U.S. Department of EnergyResearch supported in part by the National Science Foundation under Grant DMS-84-02710