Higher-Dimensional BF Theories¶in the Batalin–Vilkovisky Formalism:¶The BV Action and Generalized Wilson Loops

This paper analyzes in detail the Batalin–Vilkovisky quantization procedure for BF theories on an n-dimensional manifold and describes a suitable superformalism to deal with the master equation and the search of observables. In particular, generalized Wilson loops for BF theories with additional polynomial B-interactions are discussed in any dimensions. The paper also contains the explicit proofs to the theorems stated in [16]. Received: 25 October 2000 / Accepted: 30 March 2001     

Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) space-times. We develop a purely local version of the Stückelberg–Bogoliubov–Epstein–Glaser method of renormalization by using techniques from microlocal analysis. Relying on recent results of Radzikowski, K?hler and the authors about a formulation of a local spectrum condition in terms of wave front sets of correlation functions of quantum fields on curved space-times, we construct time-ordered operator-valued products of Wick polynomials of free fields. They serve as building blocks for a local (perturbative) definition of interacting fields. Renormalization in this framework amounts to extensions of expectation values of time-ordered products to all points of space-time. The extensions are classified according to a microlocal generalization of Steinmann scaling degree corresponding to the degree of divergence in other renormalization schemes. As a result, we prove that the usual perturbative classification of interacting quantum field theories holds also on curved space-times. Finite renormalizations are deferred to a subsequent paper. As byproducts, we describe a perturbative construction of local algebras of observables, present a new definition of Wick polynomials as operator-valued distributions on a natural domain, and we find a general method for the extension of distributions which were defined on the complement of some surface. Received: 31 March 1999 / Accepted: 10 June 1999     

Donaldson Invariants of Product Ruled Surfaces¶and Two-Dimensional Gauge Theories

Using the u-plane integral of Moore and Witten, we derive a simple expression for the Donaldson invariants of product ruled surfaces Σ _g ×S ², where Σ _g is a Riemann surface of genus g. This expression generalizes a theorem of Morgan and Szabó for g=1 to any genus g. We give two applications of our results: (1) We derive Thaddeus' formulae for the intersection pairings on the moduli space of rank two stable bundles over a Riemann surface. (2) We derive the eigenvalue spectrum of the Fukaya–Floer cohomology of Σ _g ×S ¹. Received: 22 July 1999 / Accepted: 12 June 2000     

The K-Orbits, Identities, and Classification of Four-Dimensional Homogeneous Spaces with the Group of Poincaré and de Sitter Transforms

The method of orbits traditionally applied to geometric quantization problems is used to study homogeneous spaces. Based on the proposed classification of the orbits of co-adjoint representation (K-orbits), a classification of homogeneous spaces is constructed. This classification allows one, in particular, to point out the explicit form of identities – functional relations between the transform-group generators – which are of great importance in applied problems (e.g., in the theory of separation of variables). All four-dimensional homogeneous spaces with the group of Poincaré and de Sitter transforms are classified and all independent identities on these spaces are given in explicit form.     

In Appreciation¶A Lifetime of Connections: Otto Herbert Schmitt, 1913 - 1998

Otto H. Schmitt was born in St. Louis, Missouri, in 1913. As a youth, he displayed an affinity for electrical engineering but also pursued a wide range of other interests. He applied his multi-disciplinary talents as an undergraduate and graduate student at Washington University, where he worked in three departments: physics, zoology, and mathematics. For his doctoral research, Schmitt designed and built an electronic device to mimic the propagation of action potentials along nerve fibers. His most famous invention, now called the Schmitt trigger, arose from this early research. Schmitt spent most of his career at the University of Minnesota, where he did pioneering work in biophysics and bioengineering. He also worked at national and international levels to place biophysics and bioengineering on sound institutional footings. His years at Minnesota were interrupted by World War II. During that conflict - and the initial months of the Cold War to follow - Schmitt carried out defense-related research at the Airborne Instruments Laboratory in New York. Toward the end of his career at Minnesota, Schmitt coined the term biomimetics. He died in 1998. RID="*" ID="*"Jon M. Harkness received his Ph.D. degree in the history of science from the University of Wisconsin in 1996. During the spring of 2002, he is an adjunct assistant professor of the history of medicine at the University of Minnesota.     

Sets of Fibre Homotopy Classes¶and Twisted Order Parameter Spaces

We propose a machinery to calculate the set of global defect indices of a regularly defected order taking values in arbitrary parameter bundles. Received: 26 January 1999/ Accepted: 7 December 1999     

Master Spaces and the Coupling Principle:¶From Geometric Invariant Theory to Gauge Theory

We introduce a general mathematical principle, with roots in Geometric Invariant Theory, which provides a unified way for understanding several celebrated results and conjectures like e. g. the Verlinde formula, the Vafa-Intriligator formula, and Witten's conjecture about the relation between Donaldson theory and Seiberg–Witten theory. This principle also suggests new results about Gromov invariants of moduli spaces of stable bundles over curves, and shows that gauge theoretical invariants associated with moduli spaces of PU(2)-monopoles are determined by Seiberg–Witten and Donaldson invariants. Received: 17 November 1998 / Accepted: 7 March 1999     

Strong Connections and Chern–Connes Pairing¶in the Hopf–Galois Theory

We reformulate the concept of connection on a Hopf–Galois extension B⊆P in order to apply it in computing the Chern–Connes pairing between the cyclic cohomology HC ^{2 n} (B) and K ₀ (B). This reformulation allows us to show that a Hopf–Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a Cuntz–Quillen-type bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the Chern–Connes pairing for the super line bundles associated to a super Hopf fibration. Received: 8 March 2000 / Accepted: 5 January 2001     

Gravitational Anomalies, Gerbes, and¶Hamiltonian Quantization

In ref. [1], Schwinger terms in hamiltonian quantization of chiral fermions coupled to vector potentials were computed, using some ideas from the theory of gerbes, with the help of the family index theorem for a manifold with boundary. Here, we generalize this method to include gravitational Schwinger terms. Received: 5 May 1999 / Accepted: 30 January 2000     

Partition Functions, Loop Measure, and Versions of SLE

We discuss the partition function view of the Schramm-Loewner evolution. After reviewing a number of known results in the framework of Brownian loop measures and scaling rules for partition functions, we give some speculation about multiply connected domains.     

Loop Groups, Anyons and the Calogero–Sutherland Model

The positive energy representations of the loop group of U(1) are used to construct a boson-anyon correspondence. We compute all the correlation functions of our anyon fields and study an anyonic W-algebra of unbounded operators with a common dense domain. This algebra contains an operator with peculiar exchange relations with the anyon fields. This operator can be interpreted as a second quantized Calogero–Sutherland (CS) Hamiltonian and may be used to solve the CS model. In particular, we inductively construct all eigenfunctions of the CS model from anyon correlation functions, for all particle numbers and positive couplings. Received: 12 May 1998 / Accepted: 4 August 1998     

Atom charge states, Z and comparing the ADK and cADK—Theories

During the process of adjusting the ADK-theory for the superstrong laser fields we took some part in few past years [1, 7, 8], mainly in analyzing the consequences of the influence of atom charge Z, being changed during the ionization of atoms, on the transition rate of ejected electrons. In this activity we introduced a slightly changed variant of ADK-theory [3], which we began to call corrected ADK-theory, cADK, for short [8]. Now, cADK-theory is not experimentally challenged yet, but it’s results are in accordance with many predictions [see, for instance 1, 7–9]. In present work, we used calculations of modified ionization potential of atom E _i , in order to improve formula for transition rate W _cADK. As we already discussed the transition rate dependence on the atom charges state Z [1], now we explained better the differences of the two variants of the theory, ADK and cADK. Of course, our predictions need experimental check.     

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ ⁿ . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres S ⁴ _θ. We construct the noncommutative algebras ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2). We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g _μν on S ⁴ whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,

Application of the τ-Function Theory¶of Painlevé Equations to Random Matrices:¶PIV, PII and the GUE

Tracy and Widom have evaluated the cumulative distribution of the largest eigenvalue for the finite and scaled infinite GUE in terms of a PIV and PII transcendent respectively. We generalise these results to the evaluation of , where for and otherwise, and the average is with respect to the joint eigenvalue distribution of the GUE, as well as to the evaluation of . Of particular interest are and F _N (λ;2), and their scaled limits, which give the distribution of the largest eigenvalue and the density respectively. Our results are obtained by applying the Okamoto τ-function theory of PIV and PII, for which we give a self contained presentation based on the recent work of Noumi and Yamada. We point out that the same approach can be used to study the quantities and F _N (λ;a) for the other classical matrix ensembles. Received: 27 June 2000 / Accepted: 8 December 2000     

On Fractional Lévy Processes: Tempering,Sample Path Properties and Stochastic Integration

Journal of Statistical Physics - We define two new classes of stochastic processes, called tempered fractional Lévy process of the first and second kinds (TFLP and TFLP II, respectively). TFLP...     

Dynamical Yang–Baxter Equations,Quasi-Poisson Homogeneous Spaces,and Quantization

The purpose of this Letter is twofold. First, we generalize the correspondence between dynamical r-matrices and Poisson homogeneous spaces provided in Karolinsky, E. and Stolin, A [Lett. Math. Phys. 60 (2002), 257–274]. Secondly, we propose a quantization of this quasi-classical result. In particular, we explain the relationship between irreducible highest weight modules and equivariant quantization of coadjoint orbits.*Supported in part by the Royal Swedish Academy of Sciences**Supported in part by RFFI grant 02-01-00085a and CRDF grant RM1-2334-MO-02Mathematical Subject Classification (2000). 17B10, 17B20, 17B35, 17B62, 53C30.     

k-Cosymplectic Classical Field Theories: Tulczyjew and Skinner–Rusk Formulations

The k-cosymplectic Lagrangian and Hamiltonian formalisms of first-order classical field theories are reviewed and completed. In particular, they are stated for singular and almost-regular systems. Subsequently, several alternative formulations for k-cosymplectic first-order field theories are developed: First, generalizing the construction of Tulczyjew for mechanics, we give a new interpretation of the classical field equations. Second, the Lagrangian and Hamiltonian formalisms are unified by giving an extension of the Skinner–Rusk formulation on classical mechanics.     

On α-Induction, Chiral Generators and¶Modular Invariants for Subfactors

We consider a type III subfactor N⊂N of finite index with a finite system of braided N-N morphisms which includes the irreducible constituents of the dual canonical endomorphism. We apply α-induction and, developing further some ideas of Ocneanu, we define chiral generators for the double triangle algebra. Using a new concept of intertwining braiding fusion relations, we show that the chiral generators can be naturally identified with the α-induced sectors. A matrix Z is defined and shown to commute with the S- and T-matrices arising from the braiding. If the braiding is non-degenerate, then Z is a “modular invariant mass matrix” in the usual sense of conformal field theory. We show that in that case the fusion rule algebra of the dual system of M-M morphisms is generated by the images of both kinds of α-induction, and that the structural information about its irreducible representations is encoded in the mass matrix Z. Our analysis sheds further light on the connection between (the classifications of) modular invariants and subfactors, and we will construct and analyze modular invariants from SU(n) _k loop group subfactors in a forthcoming publication, including the treatment of all SU(2) _k modular invariants. Received: 13 April 1999 / Accepted: 13 July 1999