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).     

The Dirac Operator on <Emphasis Type="Italic">SU</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(2)

We construct a 3⁺-summable spectral triple over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.Partially supported by Polish State Committee for Scientific Research (KBN) under grant 2 P03B 022 25.Regular Associate of the Abdus Salam ICTP, Trieste.     

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.     

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.     

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.     

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      

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.     

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.     

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     

<Emphasis Type="Italic">G</Emphasis>-Structures and Wrapped NS5-Branes

We analyse the geometrical structure of supersymmetric solutions of type II supergravity of the form ^1,9–n×M_n with non-trivial NS flux and dilaton. Solutions of this type arise naturally as the near-horizon limits of wrapped NS fivebrane geometries. We concentrate on the case d=7, preserving two or four supersymmetries, corresponding to branes wrapped on associative or SLAG three-cycles. Given the existence of Killing spinors, we show that M₇ admits a G₂-structure or an SU(3)-structure, respectively, of specific type. We also prove the converse result. We use the existence of these geometric structures as a new technique to derive some known and new explicit solutions, as well as a simple theorem implying that we have vanishing NS three-form and constant dilaton whenever M₇ is compact with no boundary. The analysis extends simply to other type II examples and also to type I supergravity.     

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.     

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.     

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.     

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.     

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.     

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.     

Hamiltonian BRST and Batalin-Vilkovisky Formalisms for Second Quantization of Gauge Theories

Gauge theories that have been first quantized using the Hamiltonian BRST operator formalism are described as classical Hamiltonian BRST systems with a BRST charge of the form and with natural ghost and parity degrees for all fields. The associated proper solution of the classical Batalin-Vilkovisky master equation is constructed from first principles. Both of these formulations can be used as starting points for second quantization. In the case of time reparametrization invariant systems, the relation to the standard master action is established.Research Associate of the National Fund for Scientific Research (Belgium)Postdoctoral Visitor of the National Fund for Scientific Research (Belgium)     

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)     

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