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.     

Non-unitary minimal models,Bailey's Lemma and <Emphasis Type="Italic">N</Emphasis>=1,2 Superconformal algebras

Using the Bailey flow construction, we derive character identities for the N=1 superconformal models SM(p′,2p+p′) and SM(p′,3p′−2p), and the N=2 superconformal model with central charge c=3 from the nonunitary minimal models M(p,p′). A new Ramond sector character formula for representations of N=2 superconformal algebras with central element c=3 is given. Supported in part by NSF grant DMS-0200774.     

Differentiating the Absolutely Continuous Invariant Measure of an Interval Map <Emphasis Type="Italic">f</Emphasis> with Respect to <Emphasis Type="Italic">f</Emphasis>

Let the map f:[−1,1]→[−1,1] have a.c.i.m. ρ (absolutely continuous f-invariant measure with respect to Lebesgue). Let δρ be the change of ρ corresponding to a perturbation X=δf○f⁻¹ of f. Formally we have, for differentiable A,but this expression does not converge in general. For f real-analytic and Markovian in the sense of covering (−1,1) m times, and assuming an analytic expanding condition, we show thatis meromorphic in C, and has no pole at λ=1. We can thus formally write δρ(A)=Ψ(1).     

Uniformly Local <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> Estimate for 2-D Vorticity Equation and Its Application to Euler Equations with Initial Vorticity in bmo

We shall prove the global existence theorem for the 2 dimensional Euler equations in with the initial vorticity in bmo containing functions which do not decay at infinity and have logarithmic singularities.     

Self-Dual Strings in Six Dimensions: Anomalies,the <Emphasis Type="Italic">ADE</Emphasis>-Classification,and the World-Sheet <Emphasis Type="Italic">WZW</Emphasis>-Model

We consider the (2,0) supersymmetric theory of tensor multiplets and self-dual strings in six space-time dimensions. Space-time diffeomorphisms that leave the string world-sheet invariant appear as gauge transformations on the normal bundle of the world-sheet. The naive invariance of the model under such transformations is however explicitly broken by anomalies: The electromagnetic coupling of the string to the two-form gauge field of the tensor multiplet suffers from a classical anomaly, and there is also a one-loop quantum anomaly from the chiral fermions on the string world-sheet. Both of these contributions are proportional to the Euler class of the normal bundle of the string world-sheet, and consistency of the model requires that they cancel. This imposes strong constraints on possible models, which are found to obey an ADE-classification. We then consider the decoupled world-sheet theory that describes low-energy fluctuations (compared to the scale set by the string tension) around a configuration with a static, straight string. The anomaly structure determines this to be a supersymmetric version of the level one Wess-Zumino-Witten model based on the group      

T-Duality: Topology Change from <Emphasis Type="Italic">H</Emphasis>-Flux

T-duality acts on circle bundles by exchanging the first Chern class with the fiberwise integral of the H-flux, as we motivate using E₈ and also using S-duality. We present known and new examples including NS5-branes, nilmanifolds, lens spaces, both circle bundles over Pⁿ, and the AdS⁵×S⁵ to AdS⁵×P²×S¹ with background H-flux of Duff, Lü and Pope. When T-duality leads to M-theory on a non-spin manifold the gravitino partition function continues to exist due to the background flux, however the known quantization condition for G₄ receives a correction. In a more general context, we use correspondence spaces to implement isomorphisms on the twisted K-theories and twisted cohomology theories and to study the corresponding Grothendieck-Riemann-Roch theorem. Interestingly, in the case of decomposable twists, both twisted theories admit fusion products and so are naturally rings.     

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.     

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.     

Non-Semisimple and Complex Gaugings of <Emphasis Type="Italic">N</Emphasis>=16 Supergravity

Maximal and non-maximal supergravities in three dimensions allow for a large variety of semisimple (Chern-Simons) gauge groups. In this paper, we analyze non-semisimple and complex gauge groups that satisfy the pertinent consistency relations for a maximal (N=16) gauged supergravity to exist. We give a general procedure how to generate non-semisimple gauge groups from known admissible semisimple gauge groups by a singular boost within E₈₍₈₎. Examples include the theories with gauge group SO(8)×T₂₈ that describe the reduction of IIA/IIB supergravity on the seven-sphere. In addition, we exhibit two strange embeddings of the complex gauge group into (real) E₈₍₈₎ and prove that both can be consistently gauged. We discuss the structure of the associated scalar potentials as well as their relation to those of D4 gauged supergravities.This work is partly supported by EU contract HPRN-CT-2000-00122 and HPRN-CT-2000-00131.     

Boundary Maps for <Emphasis Type="Italic">C</Emphasis>*-Crossed Products with with an Application to the Quantum Hall Effect

The boundary map in K-theory arising from the Wiener-Hopf extension of a crossed product algebra with is the Connes-Thom isomorphism. In this article the Wiener Hopf extension is combined with the Heisenberg group algebra to provide an elementary construction of a corresponding map on higher traces (and cyclic cohomology). It then follows directly from a non-commutative Stokes theorem that this map is dual w.r.t. Connes pairing of cyclic cohomology with K-theory. As an application, we prove equality of quantized bulk and edge conductivities for the integer quantum Hall effect described by continuous magnetic Schrödinger operators.     

Higher-Dimensional Generalizations of Affine Kac-Moody and Virasoro Conformal Lie Algebras

We discuss the generalizations of the notion of Conformal Algebra and Local Distribution Lie algebras for multi-dimensional bases. We replace the algebra of Laurent polynomials on by an infinite-dimensional representation (with some additional structures) of a simple finite-dimensional Lie algebra in the space of regular functions on the corresponding Grassmann variety that can be described as a ``right' higher-dimensional generalization of from the point of view of a corresponding group action. For it gives us the usual Vertex Algebra notion. We construct the higher dimensional generalizations of the Virasoro and the Affine Kac-Moody Conformal Lie algebras explicitly and in terms of the Operator Product Expansion.     

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.     

Large <Emphasis Type="Italic">n</Emphasis> Limit of Gaussian Random Matrices with External Source,Part II

We continue the study of the Hermitian random matrix ensemble with external source where A has two distinct eigenvalues ±a of equal multiplicity. This model exhibits a phase transition for the value a=1, since the eigenvalues of M accumulate on two intervals for a>1, and on one interval for 0<a<1. The case a>1 was treated in Part I, where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices, that is, limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum, and in terms of the Airy kernel at the edge. In this paper we establish the same results for the case 0<a<1. As in Part I we apply the Deift/Zhou steepest descent analysis to a 3×3-matrix Riemann-Hilbert problem. Due to the different structure of an underlying Riemann surface, the analysis includes an additional step involving a global opening of lenses, which is a new phenomenon in the steepest descent analysis of Riemann-Hilbert problems.The first and third author are supported in part by INTAS Research Network NeCCA 03-51-6637 and by NATO Collaborative Linkage Grant PST.CLG.979738. The first author is supported in part by RFBR 05-01-00522 and the program “Modern problems of theoretical mathematics” RAS(DMS). The second author is supported in part by the National Science Foundation (NSF) Grant DMS-0354962. The third author is supported in part by FWO-Flanders projects G.0176.02 and G.0455.04 and by K.U.Leuven research grant OT/04/24 and by the European Science Foundation Program Methods of Integrable Systems, Geometry, Applied Mathematics (MISGAM) and the European Network in Geometry, Mathematical Physics and Applications (ENIGMA)     

Orthomodular Lattices Generated by Graphs of Functions

In a subset where ℝ is the real line and is an arbitrary topological space, an orthogonality relation is constructed from a family of graphs of continuous functions from connected subsets of ℝ to . It is shown that under two conditions on this family a complete lattice of double orthoclosed sets is orthomodular.     

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.     

Fermionic Characters and Arbitrary Highest-Weight Integrable -Modules

This paper contains the generalization of the Feigin-Stoyanovsky construction to all integrable -modules. We give formulas for the q-characters of any highest-weight integrable module of as a linear combination of the fermionic q-characters of the fusion products of a special set of integrable modules. The coefficients in the sum are the entries of the inverse matrix of generalized Kostka polynomials in q⁻¹. We prove the conjecture of Feigin and Loktev regarding the q-multiplicities of irreducible modules in the graded tensor product of rectangular highest weight-modules in the case of . We also give the fermionic formulas for the q-characters of the (non-level-restricted) fusion products of rectangular highest-weight integrable -modules.     

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     

Symmetry Classes of Disordered Fermions

Building upon Dysons fundamental 1962 article known in random-matrix theory as the threefold way, we classify disordered fermion systems with quadratic Hamiltonians by their unitary and antiunitary symmetries. Important physical examples are afforded by noninteracting quasiparticles in disordered metals and superconductors, and by relativistic fermions in random gauge field backgrounds.The primary data of the classification are a Nambu space of fermionic field operators which carry a representation of some symmetry group. Our approach is to eliminate all of the unitary symmetries from the picture by transferring to an irreducible block of equivariant homomorphisms. After reduction, the block data specifying a linear space of symmetry-compatible Hamiltonians consist of a basic vector space V, a space of endomorphisms in End(), a bilinear form on which is either symmetric or alternating, and one or two antiunitary symmetries that may mix V with V*. Every such set of block data is shown to determine an irreducible classical compact symmetric space. Conversely, every irreducible classical compact symmetric space occurs in this way.This proves the correspondence between symmetry classes and symmetric spaces conjectured some time ago.     

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