Twisting of Quantum Differentials and¶the Planck Scale Hopf Algebra

We show that the crossed modules and bicovariant differential calculi on two Hopf algebras related by a cocycle twist are in 1-1 correspondence. In particular, for quantum groups which are cocycle deformation-quantisations of classical groups the calculi are obtained as deformation-quantisations of the classical ones. As an application, we classify all bicovariant differential calculi on the Planck scale Hopf algebra . This is a quantum group which has an limit as the functions on a classical but non-Abelian group and a limit as flat space quantum mechanics. We further study the noncommutative differential geometry and Fourier theory for this Hopf algebra as a toy model for Planck scale physics. The Fourier theory implements a T-duality-like self-duality. The noncommutative geometry turns out to be singular when and is therefore not visible in flat space quantum mechanics alone. Received: 28 October 1998 / Accepted: 7 March 1999     

Twisted Traces of Quantum Intertwiners¶and Quantum Dynamical R-Matrices¶Corresponding to Generalized Belavin–Drinfeld Triples

We consider weighted traces of products of intertwining operators for quantum groups U _q (?), suitably twisted by a “generalized Belavin–Drinfeld triple”. We derive two commuting sets of difference equations – the (twisted) Macdonald–Ruijsenaars system and the (twisted) quantum Knizhnik–Zamolodchikov–Bernard (qKZB) system. These systems involve the nonstandard quantum R-matrices defined in a previous joint work with T. Schedler ([ESS]). When the generalized Belavin–Drinfeld triple comes from an automorphism of the Lie algebra ?, we also derive two additional sets of difference equations, the dual Macdonald–Ruijsenaars system and the \textit{dual} qKZB equations. Received: 20 March 2000 / Accepted: 11 December 2000     

The Action of Outer Automorphisms on Bundles¶of Chiral Blocks

On the bundles of WZW chiral blocks over the moduli space of a punctured rational curve we construct isomorphisms that implement the action of outer automorphisms of the underlying affine Lie algebra. These bundle-isomorphisms respect the Knizhnik–Zamolodchikov connection and have finite order. When all primary fields are fixed points, the isomorphisms are endomorphisms; in this case, the bundle of chiral blocks is typically a reducible vector bundle. A conjecture for the trace of such endomorphisms is presented; the proposed relation generalizes the Verlinde formula. Our results have applications to conformal field theories based on non-simply connected groups and to the classification of boundary conditions in such theories. Received: 11 May 1998 / Accepted: 17 April 1999     

Symplectic Structures of Moduli Space¶of Higgs Bundles over a Curve and Hilbert Scheme¶of Points on the Canonical Bundle

The moduli space of triples of the form (E,θ,s) are considered, where (E,θ) is a Higgs bundle on a fixed Riemann surface X, and s is a nonzero holomorphic section of E. Such a moduli space admits a natural map to the moduli space of Higgs bundles simply by forgetting s. If (Y,L) is the spectral data for the Higgs bundle (E,θ), then s defines a section of the line bundle L over Y. The divisor of this section gives a point of a Hilbert scheme, parametrizing 0-dimensional subschemes of the total space of the canonical bundle K _X , since Y is a curve on K _X . The main result says that the pullback of the symplectic form on the moduli space of Higgs bundles to the moduli space of triples coincides with the pullback of the natural symplectic form on the Hilbert scheme using the map that sends any triple (E,θ,s) to the divisor of the corresponding section of the line bundle on the spectral curve. Received: 15 January 2000 / Accepted: 25 March 2001     

Noncommutative Geometry¶and Gauge Theory on Fuzzy Sphere

The differential algebra on the fuzzy sphere is constructed by applying Connes' scheme. The U(1) gauge theory on the fuzzy sphere based on this differential algebra is defined. The local U(1) gauge transformation on the fuzzy sphere is identified with the left U(N+1) transformation of the field, where a field is a bimodule over the quantized algebra . The interaction with a complex scalar field is also given. Received: 21 January 1998 / Accepted: 4 February 2000     

Marriage of Clifford Algebra and Finsler Geometry: A Lineage for Unification?

The teleparallel Finsler connections on Riemannian metrics are more compatible with algebraic structure than the usual metric-compatible connections. Tangent and cotangent Clifford algebras come together with Finslerian teleparallelism to give rise to a Kaua—Klein structure endowed with a canonical connection. The implications of this convergence for unification are explored.     

The Lie–Rinehart Universal Poisson Algebra of Classical and Quantum Mechanics

The Lie–Rinehart algebra of a (connected) manifold ${\mathcal {M}}$ , defined by the Lie structure of the vector fields, their action and their module structure over ${C^\infty({\mathcal {M}})}$ , is a common, diffeomorphism invariant, algebra for both classical and quantum mechanics. Its (noncommutative) Poisson universal enveloping algebra ${\Lambda_{R}({\mathcal {M}})}$ , with the Lie–Rinehart product identified with the symmetric product, contains a central variable (a central sequence for non-compact ${{\mathcal {M}}}$ ) ${Z}$ which relates the commutators to the Lie products. Classical and quantum mechanics are its only factorial realizations, corresponding to Z  =  i z, z  =  0 and ${z = \hbar}$ , respectively; canonical quantization uniquely follows from such a general geometrical structure. For ${z =\hbar \neq 0}$ , the regular factorial Hilbert space representations of ${\Lambda_{R}({\mathcal{M}})}$ describe quantum mechanics on ${{\mathcal {M}}}$ . For z  =  0, if Diff( ${{\mathcal {M}}}$ ) is unitarily implemented, they are unitarily equivalent, up to multiplicity, to the representation defined by classical mechanics on ${{\mathcal {M}}}$ .     

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,

Essential Self-Adjointness of¶Translation-Invariant Quantum Field Models¶for Arbitrary Coupling Constants

The Hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants. The Hamiltonian has a translation invariant part. By means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self-adjoint extension of the translation invariant part is shown. With a non-perturbative approach it is proved that the Hamiltonian is essentially self-adjoint on a domain. A typical example is the Pauli–Fierz model with spin 1/2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants. Received: 26 May 1999 / Accepted: 9 November 1999     

The Split Property and the Symmetry Breaking¶of the Quantum Spin Chain

We consider the relationship between the symmetry breaking and the split property of pure states of quantum spin chains. We obtain a representation theoretic condition implying that the half-sided uniform mixing condition leads to symmetry breaking of translationally invariant pure states. This is a mathematical generalization of Dichotomy previously found by I. Affleck and E. Lieb and M. Aizenman and B. Nachtergaele for ground states of a special class of Hamiltonians. Received: 1 February 1999 / Accepted: 5 December 2000     

The Geometry of¶(Super) Conformal Quantum Mechanics

N-particle quantum mechanics described by a sigma model with an N-dimensional target space with torsion is considered. It is shown that an SL(2,ℝ) conformal symmetry exists if and only if the geometry admits a homothetic Killing vector D ^a δ _a whose associated one-form D _a dX ^a is closed. Further, the SL(2,ℝ) can always be extended to Osp(1|2) superconformal symmetry, with a suitable choice of torsion, by the addition of N real fermions. Extension to SU(1,1|1) requires a complex structure I and a holomorphic U(1) isometry D ^a I _a ^b δ _b . Conditions for extension to the superconformal group D(2,1;α), which involve a triplet of complex structures and SU(2)×SU(2) isometries, are derived. Examples are given. Received: 3 September 1999 / Accepted: 30 January 2000     

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     

Rational Surfaces Associated with Affine Root Systems¶and Geometry of the Painlevé Equations

We present a geometric approach to the theory of Painlevé equations based on rational surfaces. Our starting point is a compact smooth rational surface X which has a unique anti-canonical divisor D of canonical type. We classify all such surfaces X. To each X, there corresponds a root subsystem of E ⁽¹⁾ ₈ inside the Picard lattice of X. We realize the action of the corresponding affine Weyl group as the Cremona action on a family of these surfaces. We show that the translation part of the affine Weyl group gives rise to discrete Painlevé equations, and that the above action constitutes their group of symmetries by B?cklund transformations. The six Painlevé differential equations appear as degenerate cases of this construction. In the latter context, X is Okamoto's space of initial conditions and D is the pole divisor of the symplectic form defining the Hamiltonian structure. Received: 18 September 1999 / Accepted: 29 January 2001     

Norton's Trace Formulae for the Griess Algebra¶of a Vertex Operator Algebra with Larger Symmetry

Formulae expressing the trace of the composition of several (up to five) adjoint actions of elements of the Griess algebra of a vertex operator algebra are derived under certain assumptions on the action of the automorphism group. They coincide, when applied to the moonshine module V , with the trace formulae obtained in a different way by S. Norton, and the spectrum of some idempotents related to 2A, 2B, 3A and 4A elements of the Monster is determined by the representation theory of the Virasoro algebra at c= 1/2, the W ₃ algebra at c= 4/5 or the W ₄ algebra at c= 1. The generalization to the trace function on the whole space is also given for the composition of two adjoint actions, which can be used to compute the McKay-Thompson series for a 2A involution of the Monster. Received: 24 July 2000 / Accepted: 15 June 2001     

Homogeneous Decoherence Functionals¶in Standard and History Quantum Mechanics

General history quantum theories are quantum theories without a globally defined notion of time. Decoherence functionals represent the states in the history approach and are defined as certain bivariate complex-valued functionals on the space of all histories. However, in practical situations – for instance in the history formulation of standard quantum mechanics – there often is a global time direction and the homogeneous decoherence functionals are specified by their values on the subspace of homogeneous histories. In this work we study the analytic properties of (i) the standard decoherence functional in the history version of standard quantum mechanics and (ii) homogeneous decoherence functionals in general history theories. We restrict ourselves to the situation where the space of histories is given by the lattice of projections on some Hilbert space ℋ. Among other things we prove the non-existence of a finitely valued extension for the standard decoherence functional to the space of all histories, derive a representation for the standard decoherence functional as an unbounded quadratic form with a natural representation on a Hilbert space and prove the existence of an Isham–Linden–Schreckenberg (ILS) type representation for the standard decoherence functional. Received: 26 November 1998 / Accepted: 2 December 1998     

Local Wick Polynomials and Time Ordered Products¶of Quantum Fields in Curved Spacetime

In order to have well defined rules for the perturbative calculation of quantities of interest in an interacting quantum field theory in curved spacetime, it is necessary to construct Wick polynomials and their time ordered products for the noninteracting theory. A construction of these quantities has recently been given by Brunetti, Fredenhagen, and K?hler, and by Brunetti and Fredenhagen, but they did not impose any “locality” or “covariance” condition in their constructions. As a consequence, their construction of time ordered products contained ambiguities involving arbitrary functions of spacetime point rather than arbitrary parameters. In this paper, we construct an “extended Wick polynomial algebra”– large enough to contain the Wick polynomials and their time ordered products – by generalizing a construction of Dütsch and Fredenhagen to curved spacetime. We then define the notion of a local, covariant quantum field, and seek a definition of local Wick polynomials and their time ordered products as local, covariant quantum fields. We introduce a new notion of the scaling behavior of a local, covariant quantum field, and impose scaling requirements on our local Wick polynomials and their time ordered products as well as certain additional requirements – such as commutation relations with the free field and appropriate continuity properties under variations of the spacetime metric. For a given polynomial order in powers of the field, we prove that these conditions uniquely determine the local Wick polynomials and their time ordered products up to a finite number of parameters. (These parameters correspond to the usual renormalization ambiguities occurring in Minkowski spacetime together with additional parameters corresponding to the coupling of the field to curvature.) We also prove existence of local Wick polynomials. However, the issue of existence of local time ordered products is deferred to a future investigation. Received: 27 March 2001 / Accepted: 6 June 2001     

Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota–Baxter Operators

We will introduce an associative (or quantum) version of Poisson structure tensors. This object is defined as an operator satisfying a “generalized” Rota–Baxter identity of weight zero. Such operators are called generalized Rota–Baxter operators. We will show that generalized Rota–Baxter operators are characterized by a cocycle condition so that Poisson structures are so. By analogy with twisted Poisson structures, we propose a new operator “twisted Rota–Baxter operators,” which is a natural generalization of generalized Rota–Baxter operators. It is known that classical Rota–Baxter operators are closely related with dendriform algebras. We will show that twisted Rota–Baxter operators induce NS-algebra, which is a twisted version of dendriform algebra. The twisted Poisson condition is considered as a Maurer–Cartan equation up to homotopy. We will show the twisted Rota–Baxter condition also is so. And we will study a Poisson-geometric reason, how the twisted Rota–Baxter condition arises.     

The Complex Geometry of Weak Piecewise Smooth Solutions of Integrable Nonlinear PDE's¶of Shallow Water and Dym Type

An extension of the algebraic-geometric method for nonlinear integrable PDE's is shown to lead to new piecewise smooth weak solutions of a class of N-component systems of nonlinear evolution equations. This class includes, among others, equations from the Dym and shallow water equation hierarchies. The main goal of the paper is to give explicit theta-functional expressions for piecewise smooth weak solutions of these nonlinear PDE's, which are associated to nonlinear subvarieties of hyperelliptic Jacobians. The main results of the present paper are twofold. First, we exhibit some of the special features of integrable PDE's that admit piecewise smooth weak solutions, which make them different from equations whose solutions are globally meromorphic, such as the KdV equation. Second, we blend the techniques of algebraic geometry and weak solutions of PDE's to gain further insight into, and explicit formulas for, piecewise-smooth finite-gap solutions. The basic technique used to achieve these aims is rather different from earlier papers dealing with peaked solutions. First, profiles of the finite-gap piecewise smooth solutions are linked to certain finite dimensional billiard dynamical systems and ellipsoidal billiards. Second, after reducing the solution of certain finite dimensional Hamiltonian systems on Riemann surfaces to the solution of a nonstandard Jacobi inversion problem, this is resolved by introducing new parametrizations. Amongst other natural consequences of the algebraic-geometric approach, we find finite dimensional integrable Hamiltonian dynamical systems describing the motion of peaks in the finite-gap as well as the limiting (soliton) cases, and solve them exactly. The dynamics of the peaks is also obtained by using Jacobi inversion problems. Finally, we relate our method to the shock wave approach for weak solutions of wave equations by determining jump conditions at the peak location. Received: 16 February 1999 / Accepted: 10 April 2001     

Generalized Kahler Geometry in View of?Supersymmetric Quantum Mechanics

This paper contains a detailed study of generalized Kahler geometry from the viewpoint of quantum 0+1-dimensional supersymmetric σ-model. Peierls brackets rather than canonical quantization are used to quantize the superclassical system. Supercharges (or relevant differential operators) are expressed explicitly and covariantly. Index theorems in this context are also discussed briefly.