A poisson—lie structure on the diffeomorphism group of a circle

Starting from the Gelfand-Fuks-Virasoro cocycle on the Lie algebraX(S ¹) of the vector fields on the circleS ¹ and applying the standard procedure described by Drinfel'd in a finite dimension, we obtain a classicalr-matrix (i.e. an elementr X(S ¹) X(S ¹) satisfying the classical Yang-Baxter equation), a Lie bialgebra structure onX(S ¹), and a sort of Poisson-Lie structure on the group of diffeomorphisms. Quantizations of such Lie bialgebra structures may lead to quantum diffeomorphism groups.Research supported by the Erwin Schrödinger International Institute for Mathematical Physics.     

Classical string dynamics with non-trivial topology

The possibility of branching processes for classical strings is investigated on the basis of the Nambu-Goto action. We parametrize the world sheet by a Riemann surface M and introduce a degenerate, semi-Riemannian metric η on M. Well-known results about the conformal group Diff(S¹) × Diff(S¹) are generalized to the case of (M, η). We provide an infinite dimensional Hamiltonian setting for branching processes of strings. Finally, the classical background for the theory of quantum strings as developed by Krichever and Novikov is discussed within this classical framework.     

Local Index Formula on the Equatorial Podleś Sphere

We discuss spectral properties of the equatorial Podleś sphere S _q ². As a preparation we also study the ‘degenerate’ (i.e. q=0) case (related to the quantum disk). Over S _q ² we consider two different spectral triples:one related to the Fock representation of the Toeplitz algebra and the isopectral one given in [7]. After the identification of the smooth pre-C ^*-algebra we compute the dimension spectrum and residues. We check the nontriviality of the (noncommutative) Chern character of the associated Fredholm modules by computing the pairing with the fundamental projector of the C ^*-algebra (the nontrivial generator of the K ₀-group) as well as the pairing with the q-analogue of the Bott projector. Finally, we show that the local index formula is trivially satisfied.     

Versal Deformations of a Dirac Type Differential Operator

Abstract

If we are given a smooth differential operator in the variable x ∈ R/2πZ, its normal form, as is well known, is the simplest form obtainable by means of the Diff(S ¹)-group action on the space of all such operators. A versal deformation of this operator is a normal form for some parametric infinitesimal family including the operator. Our study is devoted to analysis of versal deformations of a Dirac type differential operator using the theory of induced Diff(S ¹)-actions endowed with centrally extended Lie-Poisson brackets. After constructing a general expression for tranversal deformations of a Dirac type differential operator, we interpret it via the Lie-algebraic theory of induced Diff(S ¹)-actions on a special Poisson manifold and determine its generic moment mapping. Using a Marsden-Weinstein reduction with respect to certain Casimir generated distributions, we describe a wide class of versally deformed Dirac type differential operators depending on complex parameters.     

Torus Equivariant Spectral Triples for Odd-Dimensional Quantum Spheres Coming from <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Extensions

The torus group (S ¹)^ℓ+1 has a canonical action on the odd-dimensional sphere S _q ^2ℓ+1. We take the natural Hilbert space representation where this action is implemented and characterize all odd spectral triples acting on that space and equivariant with respect to that action. This characterization gives a construction of an optimum family of equivariant spectral triples having nontrivial K-homology class thus generalizing our earlier results for SU _q (2). We also relate the triple we construct with the C ^*-extension      

Representations of Nets of C*-Algebras over S 1

In recent times a new kind of representations has been used to describe superselection sectors of the observable net over a curved spacetime, taking into account the effects of the fundamental group of the spacetime. Using this notion of representation, we prove that any net of C^*-algebras over S ¹ admits faithful representations, and when the net is covariant under Diff(S ¹), it admits representations covariant under any amenable subgroup of Diff(S ¹).     

Classical solutions of the quantum Yang-Baxter equation

The classical analogue is developed here for part of the construction in which knot and link invariants are produced from representations of quantum groups. Whereas previous work begins with a quantum group obtained by deforming the multiplication of functions on a Poisson Lie group, we work directly with a Poisson Lie groupG and its associated symplectic groupoid. The classical analog of the quantumR-matrix is a lagrangian submanifold in the cartesian square of the symplectic groupoid. For any symplectic leafS inG, induces a symplectic automorphism ofS×S which satisfies the set-theoretic Yang-Baxter equation. When combined with the flip map exchanging components and suitably implanted in each cartesian powerS ⁿ , generates a symplectic action of the braid groupB _n onS ⁿ . Application of a symplectic trace formula to the fixed point set of the action of braids should lead to link invariants, but work on this last step is still in progress.Research partially supported by NSF Grant DMS-90-01089Research partially supported by NSF Grant DMS 90-01956 and Research Foundation of University of Pennsylvania     

Positive Energy Representations of the Loop Groups of Non-Simply Connected Lie Groups

We classify and construct all irreducible positive energy representations of the loop group of a compact, connectedand simple Lie group and show that they admit an intertwining action of Diff(S ¹). Received: 7 April 1998 / Accepted: 26 April 1999     

Diff (S 1) and the Teichmüller spaces

Precisely two of the homogeneous spaces that appear as coadjoint orbits of the group of string reparametrizations, , carry in a natural way the structure of infinite dimensional, holomorphically homogeneous complex analytic Kähler manifolds. These areN=Diff(S ¹)/Rot(S ¹) andM=Diff(S ¹)/Möb(S ¹). Note thatN is a holomorphic disc fiber space overM. Now,M can be naturally considered as embedded in the classical universal Teichmüller spaceT(1), simply by noting that a diffeomorphism ofS ¹ is a quasisymmetric homeomorphism.T(1) is itself a homomorphically homogeneous complex Banach manifold. We prove in the first part of the paper that the inclusion ofM inT(1) iscomplex analytic.In the latter portion of this paper it is shown that theunique homogeneous Kähler metric carried byM = Diff (S ¹/SL(2, ) induces preciselythe Weil-Petersson metric on the Teichmüller space. This is via our identification ofM as a holomorphic submanifold of universal Teichmüller space. Now recall that every Teichmüller spaceT(G) of finite or infinite dimension is contained canonically and holomorphically withinT(1). Our computations allow us also to prove that everyT(G), G any infinite Fuchsian group, projects out ofM transversely. This last assertion is related to the fractal nature ofG-invariant quasicircles, and to Mostow rigidity on the line.Our results thus connect the loop space approach to bosonic string theory with the sum-over-moduli (Polyakov path integral) approach.     

Closed geodesics on Riemannian manifolds via the heat flow

The main topic discussed in this paper is the following question: Given a Riemannian manifold M and a closed C¹ curve f: S¹ → M does there exist a (unique) solution of the heat equation ?_tf_t = τ(f_t) defined for all t ≧ 0 which is continuous at t = 0 along with its first S¹-derivative and which coincides with f at t = 0.     

C*-Algebras associated with one dimensional almost periodic tilings

For each irrational number, 0<α<1, we consider the space of one dimensional almost periodic tilings obtained by the projection method using a line of slope α. On this space we put the relation generated by translation and the identification of the “singular pairs”. We represent this as a topological spaceX _α with an equivalence relationR _α. OnR _α there is a natural locally Hausdorff topology from which we obtain a topological groupoid with a Haar system. We then construct the C^*-algebra of this groupoid and show that it is the irrational rotation C^*-algebra,A _α. Research supported by the Natural Sciences and Engineering Research Council of Canada and the Fields Institute for Research in Mathematical Sciences.     

Curvature of super diff(S1)/S1

Motivated by the work of Bowick and Rajeev, we calculate the curvature of the infinite-dimensional flag manifolds Diff(S¹)/S¹ and Super Diff(S¹)/S¹ using standard finite-dimensional coset space techniques. We regularize the infinity by ζ-function regularization and recover the conformal and superconformal anomalies respectively for a specific choice of the torsion.     

Predissociation by rotation and long-range interactions

A formula of Bernstein for the limiting curve of dissociation is generalized to the case where the interaction potential in the dispersion region is represented by the [1, 0] Padé approximant to the multipole expansion of the energy. The determination of the asymptotic properties of the potential from pre-dissociation data is discussed. Recent data of Herzberg for the state B' ¹σ _u ⁺ of H₂ indicate that the Padé approximant gives a consistent representation of the potential. The classical example of pre-dissociation by rotation (HgH) is studied and for the state X ²σ⁺ a value of C ₆ similar to Bernstein is obtained. In addition C ₈ is estimated to be 89 a.u.     

Conditional entropy in algebraic statistical physics

The notion of conditional entropy as entropy of conditional state on C^*-algebra with respect to its C^*-subalgebra ₁ is introduced. It is proved that for a compatible state σ on (which admits the conditional expectation of Umegaki-Takesaki) the mean conditional entropy in an a priori state σ₁ on ₁ is equal to the difference of the entropy of the state σ on and the entropy of the state σ₁ on ₁. The conditional entropy enables us to define the input-output information of a quantum communication channel in analogy to the classical Shannon formula.     

Can one ADM quantize relativistic bosonicstrings and membranes?

The standard methods for quantizing relativistic strings diverge significantly from the Dirac-Wheeler-DeWitt program for quantization of generally covariant systems and one wonders whether the latter could be successfully implemented as an alternative to the former. As a first step in this direction, we consider the possibility of quantizing strings (and also relativistic membranes) via a partially gauge-fixed ADM (Arnowitt, Deser and Misner) formulation of the reduced field equations for these systems. By exploiting some (Euclidean signature) Hamilton-Jacobi techniques that Mike Ryan and I had developed previously for the quantization of Bianchi IX cosmological models, I show how to construct Diff(S ¹)-invariant (or Diff(Σ)-invariant in the case of membranes) ground state wave functionals for the cases of co-dimension one strings and membranes embedded in Minkowski spacetime. I also show that the reduced Hamiltonian density operators for these systems weakly commute when applied to physical (i.e. Diff(S ¹) or Diff(Σ)-invariant) states. While many open questions remain, these preliminary results seem to encourage further research along the same lines.     

On the structure of superfields in a field theory on a supermanifold

In view of applications to the formulation of gauge field theories on supermanifolds, we study the relation between the sheaves of functions on a supermanifold M and its body manifold M ₀, respectively. The nonuniqueness of the local injections t: M ₀M is analysed in consideration of its role in supersymmetric field theories. A Banach space structure is given to the set of bounded, supersmooth, C ^k fields on M in order to get a rigorous formulation of variational principles for the class of theories under consideration.Research work partly supported by the National Group for Mathematical Physics (GNFM) of the Italian Research Council (CNR) and by the Italian Ministry of Public Education through the research project Geometria e Fisica.     

Principal Fibrations from Noncommutative Spheres

We construct noncommutative principal fibrations S_θ⁷→S_θ⁴ which are deformations of the classical SU(2) Hopf fibration over the four sphere. We realize the noncommutative vector bundles associated to the irreducible representations of SU(2) as modules of coequivariant maps and construct corresponding projections. The index of Dirac operators with coefficients in the associated bundles is computed with the Connes-Moscovici local index formula. “The algebra inclusion is an example of a not-trivial quantum principal bundle.”     

On the Curie points and high temperature susceptibilities of Heisenberg model ferromagnetics

The first six coefficients in the expansion of the susceptibility χ, and its inverse, χ ^?1, in ascending powers of the reciprocal temperature, have been determined for the Heisenberg model of a ferromagnetic, for any spin value, S, and any lattice. The first five coefficients appropriate to the magnetic specific heat, C, have also been found. For the body-centred and face-centred cubic lattices, the χ and C coefficients are tabulated for half-integral S from 1/2 to 3.

From these coefficients estimates have been made of the reduced Curie temperatures, θs _c= k T _c/J. It is found that for the simple, body-centred and face-centred cubic lattices the formula reproduces the estimated Curie temperatures fairly accurately. Here X=S(S+1) and z is the lattice coordination-number.

It is found that, suitably scaled, the theoretical curves for inverse susceptibility against temperature above the Curie point are rather insensitive to the spin value and to the precise lattice structure. The ratio of their initial to their final gradients is approximately 0·3. A comparison is made with the experimental values of χ ^?1 for both iron and nickel. If iron is represented by the Heisenberg model with S=1, then the observed Curie temperature corresponds to a J value of 1·19×10^?2 ev.

Brief consideration is given to the use of the tabulated coefficients for antiferromagnetic problems.     

A Duality Theorem for Ergodic Actions of Compact Quantum Groups on <Emphasis Type="Italic">C</Emphasis><Superscript>*</Superscript>-Algebras

The spectral functor of an ergodic action of a compact quantum group G on a unital C ^*-algebra is quasitensor, in the sense that the tensor product of two spectral subspaces is isometrically contained in the spectral subspace of the tensor product representation, and the inclusion maps satisfy natural properties. We show that any quasitensor ^*-functor from Rep(G) to the category of Hilbert spaces is the spectral functor of an ergodic action of G on a unital C ^*-algebra. As an application, we associate an ergodic G-action on a unital C ^*-algebra to an inclusion of Rep(G) into an abstract tensor C ^*-category . If the inclusion arises from a quantum subgroup K of G, the associated G-system is just the quotient space K\G. If G is a group and has permutation symmetry, the associated G-system is commutative, and therefore isomorphic to the classical quotient space by a subgroup of G. If a tensor C ^*-category has a Hecke symmetry making an object ρ of dimension d and μ-determinant 1, then there is an ergodic action of S _μ U(d) on a unital C ^*-algebra having the as its spectral subspaces. The special case of is discussed.     

The Spectrum of D = 11 Supergravity via Harmonic Expansions on S4 X S7

We derive the complete particle spectrum of the d = 11 supergravity compactified on AdS X S⁷, by analytically continuing AdS to S⁴, and using the Salam-Strathdee method of harmonic expansion on S⁴ X S⁷. The spectrum is arranged into supermultiplets labelled by an integer l = 0, 1, 2,. The massless supermultiplet corresponds to l = 0. For the l'th supermultiplet we find a relation, involving the eigenvalues of the second order Casimir operators of SO(3, 2) and SO(8), given by 2C₂[SO(3, 2)] + C₂[SO(8)] = 3/2(l + 2) (l + 4).