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.     

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.     

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.     

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.     

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.     

Nonassociative Tori and Applications to T-Duality

In this paper, we initiate the study of C^*-algebras endowed with a twisted action of a locally compact abelian Lie group , and we construct a twisted crossed product , which is in general a nonassociative, noncommutative, algebra. The duality properties of this twisted crossed product algebra are studied in detail, and are applied to T-duality in Type II string theory to obtain the T-dual of a general principal torus bundle with general H-flux, which we will argue to be a bundle of noncommutative, nonassociative tori. Nonassociativity is interpreted in the context of monoidal categories of modules. We also show that this construction of the T-dual includes the other special cases already analysed in a series of papers.     

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.     

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     

Wavelet Analysis of Fractal Boundaries. Part 1: Local Exponents

Let be a domain of . In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of . Classifications based on geometric accessibility conditions are introduced and compared; they are related to analytic criteria based either on local L^p regularity of the characteristic function or on its wavelet coefficients. Part 2 deals with the global analysis of the boundary of . We develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains.The first author is supported by the Institut Universitaire de France.This work was performed while the second author was at the Laboratoire d’Analyse et de Mathématiques Appliquées (University Paris XII, France) and at the Istituto di Matematica Applicata e Tecnologie Informatiche (Pavia, Italy) and partially supported by the Société de Secours des amis des Sciences and the TMR Research Network “Breaking Complexity”.     

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.     

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.     

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.     

Universality of the Edge Distribution of Eigenvalues of Wigner Random Matrices with Polynomially Decaying Distributions of Entries

We consider an ensemble of Wigner symmetric random matrices A_n={a_ij}, i,j=1, . . . ,n with matrix elements a_ij, being i.i.d. symmetrically distributed random variables We assume that and that for p>18. We prove that the distribution of the k (k=1,2, . . . ) largest (smallest) eigenvalues has a universal limit as n→∞ (the Tracy-Widom distribution).     

Spin Gromov-Witten Invariants

We define and study r-spin Gromov-Witten invariants and r-spin quantum cohomology of a projective variety V, where r≥2 is an integer. The main element of the construction is the space of r-spin maps, the stable maps into a variety V from n-pointed algebraic curves of genus g with the additional data of an r-spin structure on the curve. We prove that is a Deligne-Mumford stack and use it to define the r-spin Gromov-Witten classes of V. We show that these classes yield a cohomological field theory (CohFT) which is isomorphic to the tensor product of the CohFT associated to the usual Gromov-Witten invariants of V and the r-spin CohFT. Restricting to genus zero, we obtain the notion of an r-spin quantum cohomology of V, whose Frobenius structure is isomorphic to the tensor product of the Frobenius manifolds corresponding to the quantum cohomology of V and the r^th Gelfand-Dickey hierarchy (or, equivalently, the A_r−1 singularity). We also prove a generalization of the descent property which, in particular, explains the appearance of the ψ classes in the definition of gravitational descendants.Research of the first author was partially supported by NSA grant number MDA904-99-1-0039Research of the second author was partially supported by NSF grant number DMS-9803427Research of the third author was partially supported by NSF grant DMS-0104397     

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.     

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.     

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.     

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.