A Reduction Theorem for Highest Weight Modules over Toroidal Lie Algebras

Let be a toroidal Lie algebra corresponding to a semisimple Lie algebra We describe all Borel subalgebras of which contain the Cartan subalgebra where is a fixed Cartan subalgebra of We show that each such Borel subalgebra determines a parabolic decomposition where is a proper toroidal subalgebra of and Our first main result is that, for any weight which does not vanish on , an arbitrary subquotient of the Verma module is induced from its submodule of invariant vectors. This reduces the study of subquotients of to the study of subquotients of Verma modules over . We then introduce categories and and their respective blocks and corresponding to a central charge which is nonzero on . Our second main result is that the functors of induction and invariants are mutually inverse equivalences of the category and the full subcategory of whose objects are generated by their invariants.     

Topological Sectors and a Dichotomy in Conformal Field Theory

Let be a local conformal net of factors on S¹ with the split property. We provide a topological construction of soliton representations of the n-fold tensor product that restrict to true representations of the cyclic orbifold We prove a quantum index theorem for our sectors relating the Jones index to a topological degree. Then is not completely rational iff the symmetrized tensor product has an irreducible representation with infinite index. This implies the following dichotomy: if all irreducible sectors of have a conjugate sector then either is completely rational or has uncountably many different irreducible sectors. Thus is rational iff is completely rational. In particular, if the -index of is finite then turns out to be strongly additive. By [31], if is rational then the tensor category of representations of is automatically modular, namely the braiding symmetry is non-degenerate. In interesting cases, we compute the fusion rules of the topological solitons and show that they determine all twisted sectors of the cyclic orbifold.Supported in part by GNAMPA-INDAM and MIURSupported in part by NSF     

Quasicrystals and Almost Periodicity

We give in this paper topological and dynamical characterizations of mathematical quasicrystals. Let denote the space of uniformly discrete subsets of the Euclidean space. Let denote the elements of that admit an autocorrelation measure. A Patterson set is an element of such that the Fourier transform of its autocorrelation measure is discrete. Patterson sets are mathematical idealizations of quasicrystals. We prove that S is a Patterson set if and only if S is almost periodic in (,), where denotes the Besicovitch topology. Let be an ergodic random element of . We prove that is almost surely a Patterson set if and only if the dynamical system has a discrete spectrum. As an illustration, we study deformed model sets.     

Poisson Geometrical Symmetries Associated to Non-Commutative Formal Diffeomorphisms

Let be the group of all formal power series starting with x with coefficients in a field of zero characteristic (with the composition product), and let F [ ] be its function algebra. In [BF] a non-commutative, non-cocommutative graded Hopf algebra was introduced via a direct process of disabelianisation of F [ ], taking the like presentation of the latter as an algebra but dropping the commutativity constraint. In this paper we apply a general method to provide four one-parameter deformations of , which are quantum groups whose semiclassical limits are Poisson geometrical symmetries such as Poisson groups or Lie bialgebras, namely two quantum function algebras and two quantum universal enveloping algebras. In particular the two Poisson groups are extensions of , isomorphic as proalgebraic Poisson varieties but not as proalgebraic groups.Acknowledgements. The author thanks Alessandra Frabetti and Loic Foissy for many helpful discussions.     

A Semiclassical Egorov Theorem and Quantum Ergodicity for Matrix Valued Operators

We study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the Hilbert space L²(^d)ⁿ into a finite number of almost invariant subspaces. For a certain class of observables, that is preserved by the time evolution, we prove an Egorov theorem. We then associate with each almost invariant subspace of L²(^d)ⁿ a classical system on a product phase space T^*d×, where is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented. For the projections of eigenvectors of the quantum Hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold, if the associated classical systems are ergodic.     

Classification of Subsystems for Graded-Local Nets with Trivial Superselection Structure

We classify Haag-dual Poincaré covariant subsystems of a graded-local net on 4D Minkowski spacetime which satisfies standard assumptions and has trivial superselection structure. The result applies to the canonical field net of a net of local observables satisfying natural assumptions. As a consequence, provided that it has no nontrivial internal symmetries, such an observable net is generated by (the abstract versions of) the local energy-momentum tensor density and the observable local gauge currents which appear in the algebraic formulation of the quantum Noether theorem. Moreover, for a net of local observables as above, we also classify the Poincaré covariant local extensions which preserve the dynamics.Partially supported by the Italian MIUR and GNAMPA-INDAM.Acknowledgement We thank H.-J. Borchers, D. R. Davidson, S. Doplicher, R. Longo, G. Piacitelli, and J. E. Roberts for several comments and discussions at different stages of this research. A part of this work was done while the first named author (S. C.) was at the Department of Mathematics of the Università di Roma 3 thanks to a post-doctoral grant of this university. The final part was carried out while the second named author (R. C.) was visiting the Mittag-Leffler Institute in Stockholm during the year devoted to Noncommutative Geometry. He would like to thank the Organizers for the kind invitation and the Staff for providing a friendly atmosphere and perfect working conditions.     

Existence and Uniqueness of the Solution to the Dissipative 2D Quasi-Geostrophic Equations in the Sobolev Space

We study the two dimensional dissipative quasi-geostrophic equations in the Sobolev space Existence and uniqueness of the solution local in time is proved in H^s when s>2(1–). Existence and uniqueness of the solution global in time is also proved in H^s when s2(1–) and the initial data is small. For the case, s>2(1–), we also obtain the unique large global solution in H^s provided that is small enough.Acknowledgement The author thanks Professor Jiahong Wu for useful conversations, Professor Antonio Cordoba for kindly providing their preprints and Professor Peter Constantin for kind suggestions and encouragement. This work is partially supported by the Oklahoma State University, School of Art and Science new faculty start-up fund and by the Deans Incentive Grant.     

Higher-Level Appell Functions,Modular Transformations,and Characters

We study modular transformation properties of a class of indefinite theta series involved in characters of infinite-dimensional Lie superalgebras. The level- Appell functions satisfy open quasiperiodicity relations with additive theta-function terms emerging in translating by the period. Generalizing the well-known interpretation of theta functions as sections of line bundles, the function enters the construction of a section of a rank-(+1) bundle . We evaluate modular transformations of the functions and construct the action of an SL(2,) subgroup that leaves the section of constructed from invariant.Modular transformation properties of are applied to the affine Lie superalgebra at a rational level k>–1 and to the N=2 super-Virasoro algebra, to derive modular transformations of admissible characters, which are not periodic under the spectral flow and cannot therefore be rationally expressed through theta functions. This gives an example where constructing a modular group action involves extensions among representations in a nonrational conformal model.Acknowledgement We are grateful to B.L. Feigin for interesting discussions, to J. Fuchs for a useful suggestion, and to V.I. Ritus for his help with the small-t asymptotic expansion. AMS acknowledges support from the Royal Society through a grant RCM/ExAgr and the kind hospitality in Durham. AT acknowledges support from a Small Collaborative Grant of the London Mathematical Society that made a trip to Moscow possible, and the warm welcome extended to her during her visit. AMS & IYuT were supported in part by the grant LSS-1578.2003.2, by the Foundation for Support of Russian Science, and by the RFBR Grant 04-01-00303. IYuT was also supported in part by the RFBR Grant 03-01-06135 and the INTAS Grant 00-01-254.     

Arithmetic Properties of Eigenvalues of Generalized Harper Operators on Graphs

Let denote the field of algebraic numbers in A discrete group G is said to have the σ-multiplier algebraic eigenvalue property, if for every matrix A ∈ M_d((G, σ)), regarded as an operator on l²(G)^d, the eigenvalues of A are algebraic numbers, where σ ∈ Z²(G, ) is an algebraic multiplier, and denotes the unitary elements of . Such operators include the Harper operator and the discrete magnetic Laplacian that occur in solid state physics. We prove that any finitely generated amenable, free or surface group has this property for any algebraic multiplier σ. In the special case when σ is rational (σⁿ=1 for some positive integer n) this property holds for a larger class of groups containing free groups and amenable groups, and closed under taking directed unions and extensions with amenable quotients. Included in the paper are proofs of other spectral properties of such operators. The second and third authors acknowledge support from the Australian Research Council.     

Scaling Limit for the Space-Time Covariance of the Stationary Totally Asymmetric Simple Exclusion Process

The totally asymmetric simple exclusion process (TASEP) on the one-dimensional lattice with the Bernoulli ρ measure as initial conditions, 0<ρ<1, is stationary in space and time. Let N_t(j) be the number of particles which have crossed the bond from j to j+1 during the time span [0,t]. For we prove that the fluctuations of N_t(j) for large t are of order t^1/3 and we determine the limiting distribution function , which is a generalization of the GUE Tracy-Widom distribution. The family of distribution functions have been obtained before by Baik and Rains in the context of the PNG model with boundary sources, which requires the asymptotics of a Riemann-Hilbert problem. In our work we arrive at through the asymptotics of a Fredholm determinant. is simply related to the scaling function for the space-time covariance of the stationary TASEP, equivalently to the asymptotic transition probability of a single second class particle. An erratum to this article can be found at     

On a Penrose Inequality with Charge

We construct a time-symmetric asymptotically flat initial data set to the Einstein-Maxwell Equations which satisfieswhere m is the total mass, is the area radius of the outermost horizon and Q is the total charge. This yields a counter-example to a natural extension of the Penrose Inequality for charged black holes.The research of the first author was supported in part by NSF Grant DMS-0205545.The research of the second author was supported in part by NSF Grant DMS-0222387.     

Space-Time Foam from Non-Commutative Instantons

We show that a U(1) instanton on non-commutative corresponds to a non-singular U(1) gauge field on a commutative Kähler manifold X which is a blowup of at a finite number of points. This gauge field on X obeys Maxwells equations in addition to the susy constraint F^0,2=0. For instanton charge k the manifold X can be viewed as a space-time foam with b₂k. A direct connection with integrable systems of Calogero-Moser type is established. We also make some comments on the non-abelian case.     

On a Class of Representations of the Yangian and Moduli Space of Monopoles

A new class of infinite dimensional representations of the Yangians Y and Y corresponding to a complex semisimple algebra and its Borel subalgebra is constructed. It is based on the generalization of the Drinfeld realization of in terms of quantum minors to the case of an arbitrary semisimple Lie algebra . The Poisson geometry associated with the constructed representations is described. In particular it is shown that the underlying symplectic leaves are isomorphic to the moduli spaces of G-monopoles defined as the components of the space of based maps of ℙ¹ into the generalized flag manifold . Thus the constructed representations of the Yangian may be considered as a quantization of the moduli space of the monopoles.     

Inverse Problem for Harmonic Oscillator Perturbed by Potential,Characterization

Consider the perturbed harmonic oscillator Ty=-y+x²y+q(x)y in L²(), where the real potential q belongs to the Hilbert space H={q, xq L²()}. The spectrum of T is an increasing sequence of simple eigenvalues _n(q)=1+2n+_n, n 0, such that _n 0 as n. Let _n(x,q) be the corresponding eigenfunctions. Define the norming constants _n(q)=lim_xlog |_n (x,q)/_n (-x,q)|. We show that for some real Hilbert space and some subspace Furthermore, the mapping :q(q)=({_n(q)}₀, {_n(q)}₀) is a real analytic isomorphism between H and is the set of all strictly increasing sequences s={s_n}₀ such that The proof is based on nonlinear functional analysis combined with sharp asymptotics of spectral data in the high energy limit for complex potentials. We use ideas from the analysis of the inverse problem for the operator -ypy, p L²(0,1), with Dirichlet boundary conditions on the unit interval. There is no literature about the spaces We obtain their basic properties, using their representation as spaces of analytic functions in the disk.     

The Structure of the Ladder Insertion-Elimination Lie Algebra

We continue our investigation into the insertion-elimination Lie algebra of Feynman graphs in the ladder case, emphasizing the structure of this Lie algebra relevant for future applications in the study of Dyson–Schwinger equations. We work out the relation to the classical infinite dimensional Lie algebra and we determine the cohomology of .D.K. supported by CNRS; both authors supported in parts by NSF grant DMS-0401262, Ctr. Math. Phys. at Boston Univ.; BUCMP/04-06.     

Phase Turbulence in the Complex Ginzburg-Landau Equation via Kuramoto–Sivashinsky Phase Dynamics

We study the Complex Ginzburg-Landau initial value problem for a complex field uC, with ,R. We consider the Benjamin–Feir linear instability region We show that for all and for all initial data u₀ sufficiently close to 1 (up to a global phase factor eⁱ₀,₀R) in the appropriate space, there exists a unique (spatially) periodic solution of space period L₀. These solutions are small even perturbations of the traveling wave solution, and s, have bounded norms in various L^p and Sobolev spaces. We prove that apart from corrections whenever the initial data satisfy this condition, and that in the linear instability range the dynamics is essentially determined by the motion of the phase alone, and so exhibits phase turbulence. Indeed, we prove that the phase satisfies the Kuramoto–Sivashinsky equation for times while the amplitude 1+² s is essentially constant.Supported in part by the Fonds National Suisse.     

Noncommutative Riemannian and Spin Geometry of the Standard <Emphasis Type="Italic">q</Emphasis>-Sphere

We study the quantum sphere as a quantum Riemannian manifold in the quantum frame bundle approach. We exhibit its 2-dimensional cotangent bundle as a direct sum ^0,11,0 in a double complex. We find the natural metric, volume form, Hodge * operator, Laplace and Maxwell operators and projective module structure. We show that the q-monopole as spin connection induces a natural Levi-Civita type connection and find its Ricci curvature and q-Dirac operator . We find the possibility of an antisymmetric volume form quantum correction to the Ricci curvature and Lichnerowicz-type formulae for We also remark on the geometric q-Borel-Weil-Bott construction.     

Smoothness of Time Functions and the Metric Splitting of Globally Hyperbolic Spacetimes

The folk questions in Lorentzian Geometry which concerns the smoothness of time functions and slicings by Cauchy hypersurfaces, are solved by giving simple proofs of: (a) any globally hyperbolic spacetime (M, g) admits a smooth time function whose levels are spacelike Cauchy hyperfurfaces and, thus, also a smooth global splitting if a spacetime M admits a (continuous) time function t then it admits a smooth (time) function with timelike gradient on all M.The second-named author has been partially supported by a MCyT-FEDER Grant, MTM2004-04934-C04-01.To Professor P.E. Ehrlich, wishing him a continued recovery and good health     

Perturbation of Singular Equilibria of Hyperbolic Two-Component Systems: A Universal Hydrodynamic Limit

We consider one-dimensional, locally finite interacting particle systems with two conservation laws which under the Eulerian hydrodynamic limit lead to two-by-two systems of conservation laws:with where is a convex compact polygon in ². The system is typically strictly hyperbolic in the interior of with possible non-hyperbolic degeneracies on the boundary . We consider the case of an isolated singular (i.e. non-hyperbolic) point on the interior of one of the edges of , call it (₀,u₀). We investigate the propagation of small nonequilibrium perturbations of the steady state of the microscopic interacting particle system, corresponding to the densities (₀,u₀) of the conserved quantities. We prove that for a very rich class of systems, under a proper hydrodynamic limit the propagation of these small perturbations are universally driven by the two-by-two systemwhere the parameter is the only trace of the microscopic structure.The proof relies on the relative entropy method and thus, it is valid only in the regime of smooth solutions of the pde. But there are essential new elements: in order to control the fluctuations of the terms with Poissonian (rather than Gaussian) decay coming from the low density approximations we have to apply refined pde estimates. In particular Lax entropies of these pde systems play a not merely technical key role in the main part of the proof.