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     

Renormalization and Periodic Orbits¶for Hamiltonian Flows

We consider a renormalization group transformation for analytic Hamiltonians in two or more dimensions, and use this transformation to construct invariant tori, as well as sequences of periodic orbits with rotation vectors approaching that of the invariant torus. The construction of periodic and quasiperiodic orbits is limited to near-integrable Hamiltonians. But as a first step toward a non-perturbative analysis, we extend the domain of to include any Hamiltonian for which a certain non-resonance condition holds. Received: 5 October 1999 / Accepted: 2 February 2000     

Spin and Abelian Electromagnetic Duality¶on Four-Manifolds

We investigate the electromagnetic duality properties of an Abelian gauge theory on a compact oriented four-manifold by analysing the behaviour of a generalised partition function under modular transformations of the dimensionless coupling constants. The true partition function is invariant under the full modular group but the generalised partition function exhibits more complicated behaviour depending on topological properties of the four-manifold concerned. It is already known that there may be “modular weights” which are linear combinations of the Euler number and Hirzebruch signature of the four-manifold. But sometimes the partition function transforms only under a subgroup of the modular group (the Hecke subgroup). In this case it is impossible to define real spinor wave-functions on the four-manifold. But complex spinors are possible provided the background magnetic fluxes are appropriately fractional rather than integral. This gives rise to a second partition function which enables the full modular group to be realised by permuting the two partition functions, together with a third. Thus the full modular group is realised in all cases. The demonstration makes use of various constructions concerning integral lattices and theta functions that seem to be of intrinsic interest. Received: 5 June 2000 / Accepted: 9 October 2000     

Superconformal Invariance and the Geography¶of Four-Manifolds

The correlation functions of supersymmetric gauge theories on a four-manifold X can sometimes be expressed in terms of topological invariants of X. We show how the existence of superconformal fixed points in the gauge theory can provide nontrivial information about four-manifold topology. In particular, in the example of gauge group SU(2) with one doublet hypermultiplet, we derive a theorem relating classical topological invariants such as the Euler character and signature to sum rules for Seiberg–Witten invariants. A short account of this paper can be found in [1]. Received: 19 December 1998 / Accepted: 7 March 1999     

First KdV Integrals¶and Absolutely Continuous Spectrum¶for 1-D Schrödinger Operator

We consider 1-D Schr?dinger operators on L ²(R ₊) with slowly decaying potentials. Under some conditions on the potential, related to the first integrals of the KdV equation, we prove that the a.c. spectrum of the operator coincides with the positive semiaxis and the singular spectrum is unstable. Examples show that for special classes of sparse potentials these results can not be improved. Received: 16 June 2000 / Accepted: 11 August 2000     

WKB and Spectral Analysis¶of One-Dimensional Schrödinger Operators¶with Slowly Varying Potentials

Consider a Schr?dinger operator on L ² of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a finite sum of terms, each of which has a derivative of some order in L ¹+L ^p for some exponent p<2, then an essential support of the the absolutely continuous spectrum equals ℝ⁺. Almost every generalized eigenfunction is bounded, and satisfies certain WKB-type asymptotics at infinity. If moreover these derivatives belong to L ^p with respect to a weight |x|^γ with γ >0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one. Received: 27 July 2000 / Accepted: 23 October 2000     

The Oscillator Representation and¶Groups of Heisenberg Type

We obtain the explicit reduction of the Oscillator representation of the symplectic group, on the subgroups of automorphisms of certain vector-valued skew forms | of "Clifford type"-equivalently, of automorphisms of Lie algebras of Heisenberg type. These subgroups are of the form G · \Spin(k), with G a real reductive matrix group, in general not compact, commuting with Spin(k) with finite intersection. The reduction turns out to be free of multiplicity in all the cases studied here, which include some where the factors do not form a Howe pair. If G is maximal compact in G, the restriction to K · \Spin(k) is essentially the action on the symmetric algebra on a space of spinors. The cases when this is multiplicity-free are listed in [R]; our examples show that replacing K by G does make a difference. Our question is motivated to a large extent by the geometric object that comes with such a |: a Fock-space bundle over a sphere, with G acting fiberwise via the oscillator representation. It carries a Dirac operator invariant under G and determines special derivations of the corresponding gauge algebra.     

Renormalization Group¶and the Melnikov Problem for PDE's

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions. Received: 29 January 2001 / Accepted: 8 March 2001     

Stochastic Stability¶for Contracting Lorenz Maps and Flows

In a previous work [M], we proved the existence of absolutely continuous invariant measures for contracting Lorenz-like maps, and constructed Sinai–Ruelle–Bowen measures f or the flows that generate them. Here, we prove stochastic stability for such one-dimensional maps and use this result to prove that the corresponding flows generating these maps are stochastically stable under small diffusion-type perturbations, even though, as shown by Rovella [Ro], they are persistent only in a measure theoretical sense in a parameter space. For the one-dimensional maps we also prove strong stochastic stability in the sense of Baladi and Viana[BV]. Received: 24 February 1999 / Accepted: 7 January 2000     

Elliptic Gromov–Witten Invariants¶and Virasoro Conjecture

We study some necessary and sufficient conditions for the genus-1 Virasoro conjecture proposed by Eguchi–Hori–Xiong and S. Katz. Received: 22 August 1999 / Accepted: 7 October 2000     

Quantum Geometry of Algebra Factorisations¶and Coalgebra Bundles

We develop the noncommutative geometry (bundles, connections etc.) associated to algebras that factorise into two subalgebras. An example is the factorisation of matrices M ₂(ℂ)=ℂℤ₂·ℂℤ₂. We also further extend the coalgebra version of theory introduced previously, to include frame resolutions and corresponding covariant derivatives and torsions. As an example, we construct q-monopoles on all the Podleś quantum spheres S ² _q,s . Received: 25 September 1998 / Accepted: 23 February 2000     

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     

Extended Diffeomorphism Algebras¶and Trajectories in Jet Space

Let the DRO (Diffeomorphism, Reparametrization, Observer) algebra?DRO(N) be the extension of diff(N)⊕ diff(1) by its four inequivalent Virasoro-like cocycles. Here diff(N) is the diffeomorphism algebra in N-dimensional spacetime and diff(1) describes reparametrizations of trajectories in the space of tensor-valued p-jets. DRO(N) has a Fock module for each p and each representation of gl(N). Analogous representations for gauge algebras (higher-dimensional Kac–Moody algebras) are also given. The reparametrization symmetry can be eliminated by a gauge fixing procedure, resulting in previously discovered modules. In this process, two DRO(N) cocycles transmute into anisotropic cocycles for diff(N). Thus the Fock modules of toroidal Lie algebras and their derivation algebras are geometrically explained. Received: 29 October 1998 / Accepted: 2 May 2000     

Entropy Production and Information Gain¶in Axiom-A Systems

Anosov systems are mathematical models for chaotic systems in statistical mechanics and fluid dynamics. Most of these systems enjoy the property of positive entropy production. We introduce the concept of specific information gain (or specific relative entropy) h(μ₊,μ₋) in Anosov systems and prove that it is identical to the entropy production rate e _p (μ₊) defined by Ruelle and Gallavotti in Anosov systems. From this point of view, the entropy production rate e _p (μ₊ characterizes the degree of macroscopic irreversibility of the system. Received: 2 August 1999 / Accepted: 14 April 2000     

Large Deviations and¶Variational Principle for Harmonic Crystals

We consider massless Gaussian fields with covariance related to the Green function of a long range random walk on Ê^d. These are viewed as Gibbs measures for a linear-quadratic interaction. We establish thermodynamic identities and prove a version of Gibbs' variational principle, showing that translation invariant Gibbs measures are characterized as minimizers of the relative entropy density. We then study the large deviations of the empirical field of a Gibbs measure. We show that a weak large deviation principle holds at the volume order, with rate given by the relative entropy density.     

Random Homogenization and Singular Perturbations¶in Perforated Domains

The paper considers the singularly perturbed Dirichlet problem −ɛΔu ^ɛ+u ^ɛ=f in a randomly perforated domain Ω^ɛ, which is obtained from a bounded open set Ω in R ^N after removing many holes of size ɛ ^q . The perforated domain is described in terms of an ergodic dynamical system acting on a probability space. Imposing certain conditions on the domain, the behaviour of u ^ɛ when ɛ→ 0 in Lebesgue spaces L ⁿ (Ω) is studied. Test functions together with the Birkhoff ergodic theorem are the main tools of analysis. The Poisson distribution of holes of size ɛ ^p with the intensity λɛ^{− r} is then considered. The above results apply in some cases; other cases are treated by the Wiener sausage approach. Received: 15 December 1999 / Accepted: 14 April 2000     

Bootstrap Multiscale Analysis and Localization¶in Random Media

We introduce an enhanced multiscale analysis that yields subexponentially decaying probabilities for bad events. For quantum and classical waves in random media, we obtain exponential decay for the resolvent of the corresponding random operators in boxes of side L with probability higher than 1 − e ^{− L ζ}, for any 0<ζ<1. The starting hypothesis for the enhanced multiscale analysis only requires the verification of polynomial decay of the finite volume resolvent, at some sufficiently large scale, with probability bigger than 1 − (d is the dimension). Note that from the same starting hypothesis we get conclusions that are valid for any 0 < ζ < 1. This is achieved by the repeated use of a bootstrap argument. As an application, we use a generalized eigenfunction expansion to obtain strong dynamical localization of any order in the Hilbert–Schmidt norm, and better estimates on the behavior of the eigenfunctions. Received: 29 November 2000 / Accepted: 21 June 2001     

Operator Algebras and Poisson Manifolds¶Associated to Groupoids

It is well known that a measured groupoid G defines a von Neumann algebra W ^*(G), and that a Lie groupoid G canonically defines both a C ^*-algebra C ^*(G) and a Poisson manifold A ^*(G). We construct suitable categories of measured groupoids, Lie groupoids, von Neumann algebras, C ^*-algebras, and Poisson manifolds, with the feature that in each case Morita equivalence comes down to isomorphism of objects. Subsequently, we show that the maps G↦W ^*(G), G↦C ^*(G), and G↦A ^*(G) are functorial between the categories in question. It follows that these maps preserve Morita equivalence. Received: 6 December 2000 / Accepted: 19 April 2001     

Polynomial Invariants for Torus Knots¶and Topological Strings

We make a precision test of a recently proposed conjecture relating Chern–Simons gauge theory to topological string theory on the resolution of the conifold. First, we develop a systematic procedure to extract string amplitudes from vacuum expectation values (vevs) of Wilson loops in Chern–Simons gauge theory, and then we evaluate these vevs in arbitrary irreducible representations of SU(N) for torus knots. We find complete agreement with the predictions derived from the target space interpretation of the string amplitudes. We also show that the structure of the free energy of topological open string theory gives further constraints on the Chern–Simons vevs. Our work provides strong evidence towards an interpretation of knot polynomial invariants as generating functions associated to enumerative problems. Received: 1 May 2000 / Accepted: 6 November 2000     

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,