Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators

We construct a continuous transition density of the semigroup generated by for and b in the Kato class on . For small time the transition density is comparable with that of the fractional Laplacian. Partially supported by KBN and MEN.     

Lie Superalgebras and Irreducibility of $$A_1^{(1)}$$ –Modules at the Critical Level

We introduce the infinite-dimensional Lie superalgebra and construct a family of mappings from a certain category of –modules to the category of –modules at the critical level. Using this approach, we prove the irreducibility of a large family of –modules at the critical level parameterized by . As a consequence, we present a new proof of irreducibility of certain Wakimoto modules. We also give natural realizations of irreducible quotients of relaxed Verma modules and calculate characters of these representations. Partially supported by the MZOS grant 0037125 of the Republic of Croatia     

(Co)cyclic (Co)homology of Bialgebroids: An Approach via (Co)monads

For a (co)monad T _l on a category , an object X in , and a functor , there is a (co)simplex in . The aim of this paper is to find criteria for para-(co)cyclicity of Z ^*. Our construction is built on a distributive law of T _l with a second (co)monad T _r on , a natural transformation , and a morphism in . The (symmetrical) relations i and w need to satisfy are categorical versions of Kaygun’s axioms of a transposition map. Motivation comes from the observation that a (co)ring T over an algebra R determines a distributive law of two (co)monads and on the category of R-bimodules. The functor Π can be chosen such that is the cyclic R-module tensor product. A natural transformation is given by the flip map and a morphism is constructed whenever T is a (co)module algebra or coring of an R-bialgebroid. The notion of a stable anti-Yetter-Drinfel’d module over certain bialgebroids, the so-called  ×  _R -Hopf algebras, is introduced. In the particular example when T is a module coring of a  ×  _R -Hopf algebra and X is a stable anti-Yetter-Drinfel’d -module, the para-cyclic object Z _* is shown to project to a cyclic structure on . For a -Galois extension , a stable anti-Yetter-Drinfel’d -module T _S is constructed, such that the cyclic objects and are isomorphic. This extends a theorem by Jara and Ştefan for Hopf Galois extensions. As an application, we compute Hochschild and cyclic homologies of a groupoid with coefficients in a stable anti-Yetter-Drinfel’d module, by tracing it back to the group case. In particular, we obtain explicit expressions for (coinciding relative and ordinary) Hochschild and cyclic homologies of a groupoid. The latter extends results of Burghelea on cyclic homology of groups.     

Uniqueness Theorem for BMS-Invariant States of Scalar QFT on the Null Boundary of Asymptotically Flat Spacetimes and Bulk-Boundary Observable Algebra Correspondence

This work concerns some features of scalar QFT defined on the causal boundary of an asymptotically flat at null infinity spacetime and based on the BMS-invariant Weyl algebra .(a) (i) It is noticed that the natural BMS invariant pure quasifree state λ on , recently introduced by Dappiaggi, Moretti and Pinamonti, enjoys positivity of the self-adjoint generator of u-translations with respect to every Bondi coordinate frame on , ( being the affine parameter of the complete null geodesics forming and complex coordinates on the transverse 2-sphere). This fact may be interpreted as a remnant of the spectral condition inherited from QFT in Minkowski spacetime (and it is the spectral condition for free fields when the bulk is the very Minkowski space). (ii) It is also proved that the cluster property under u-displacements is valid for every (not necessarily quasifree) pure state on which is invariant under u displacements. (iii) It is established that there is exactly one algebraic pure quasifree state which is invariant under u-displacements (of a fixed Bondi frame) and has positive self-adjoint generator of u-displacements. It coincides with the GNS-invariant state λ. (iv) Finally it is shown that in the folium of a pure u-displacement invariant state ω (like λ but not necessarily quasifree) on is the only state invariant under u-displacement.(b) It is proved that the theory can be formulated for spacetimes asymptotically flat at null infinity which also admit future time completion i ⁺ (and fulfill other requirements related with global hyperbolicity). In this case a -isomorphism ı exists - with a natural geometric meaning - which identifies the (Weyl) algebra of observables of a linear field propagating in the bulk spacetime with a sub algebra of . Using ı a preferred state on the field algebra in the bulk spacetime is induced by the BMS-invariant state λ on .     

On Genus Two Riemann Surfaces Formed from Sewn Tori

We describe the period matrix and other data on a higher genus Riemann surface in terms of data coming from lower genus surfaces via an explicit sewing procedure. We consider in detail the construction of a genus two Riemann surface by either sewing two punctured tori together or by sewing a twice-punctured torus to itself. In each case the genus two period matrix is explicitly described as a holomorphic map from a suitable domain (parameterized by genus one moduli and sewing parameters) to the Siegel upper half plane . Equivariance of these maps under certain subgroups of is shown. The invertibility of both maps in a particular domain of is also shown. Support provided by the National Science Foundation DMS-0245225, and the Committee on Research at the University of California, Santa Cruz. Supported by The Millenium Fund, National University of Ireland, Galway.     

A study of the nuclear-medium influence on neutral-strange-particle production in deep-inelastic neutrino scattering

The influence of nuclear effects on the production of neutral strange particles (V ⁰) is investigated using the data obtained with SKAT propane-freon bubble chamber irradiated in the neutrino beam (with E _v =3−30 GeV) at the Serpukhov accelerator. The mean multiplicity of V ⁰ particles in nuclear interactions, , is found to exceed significantly that in “quasideuteron” interactions, . The ratio of is larger than that for π⁻ mesons, . It is shown that the multiplicity gain of V ⁰ particles can be explained by intranuclear interactions of produced pions. The text was submitted by the authors in English.     

A Variational Principle in the Dual Pair of Reproducing Kernel Hilbert Spaces and an Application

Given a positive definite, bounded linear operator A on the Hilbert space ₀≔l ²(E), we consider a reproducing kernel Hilbert space ₊ with a reproducing kernel A(x,y). Here E is any countable set and A(x,y), x,y∊ E, is the representation of A w.r.t. the usual basis of ₀. Imposing further conditions on the operator A, we also consider another reproducing kernel Hilbert space ₋ with a kernel function B(x,y), which is the representation of the inverse of A in a sense, so that ₋⊃ ₀⊃ ₊ becomes a rigged Hilbert space. We investigate the ratios of determinants of some partial matrices of A and B. We also get a variational principle on the limit ratios of these values. We apply this relation to show the Gibbsianness of the determinantal point process (or fermion point process) defined by the operator A(I+A)⁻¹ on the set E. 2000 Mathematics Subject Classification: Primary: 46E22 Secondary: 60K35     

Gravity’s immunity from vacuum: the holographic structure of semiclassical action

Principle of equivalence, general covariance and the demand that the variation of the action functional should be well defined, lead to a generic Lagrangian for semiclassical gravity of the form with . The expansion of in terms of the derivatives of the metric tensor determines the structure of the theory uniquely. The zeroth order term gives the Einstein–Hilbert action and the first order correction is given by the Gauss–Bonnet action. Remarkably, any such Lagrangian can be decomposed into a surface and bulk terms which are related holographically. The equations of motion can be obtained purely from a surface term in the gravity sector and hence gravity does not respond to the changes in the bulk vacuum energy density. Third award in the 2006 Essay Competition of the Gravity Research Foundation.     

Twistor Actions for Self-Dual Supergravities

We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravity theories in four dimensions, dealing with supersymmetries, the cases where different parts of the R-symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group of holomorphic Poisson transformations of supertwistor space where the form of the Poisson structure determines the amount of R-symmetry gauged and the value of the cosmological constant. We give a formulation in terms of a finite deformation of an integrable -operator on a supertwistor space, i.e., on regions in . For , we also give a formulation that does not require the choice of a background.     

An Infinite Genus Mapping Class Group and Stable Cohomology

We exhibit a finitely generated group whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface of infinite genus, and contains all the pure mapping class groups of compact surfaces of genus g with n boundary components, for any g ≥ 0 and n > 0. We construct a representation of into the restricted symplectic group of the real Hilbert space generated by the homology classes of non-separating circles on , which generalizes the classical symplectic representation of the mapping class groups. Moreover, we show that the first universal Chern class in is the pull-back of the Pressley-Segal class on the restricted linear group via the inclusion . L. F. was partially supported by the ANR Repsurf:ANR-06-BLAN-0311.     

A Counterexample to Dispersive Estimates for Schrödinger Operators in Higher Dimensions

In dimension n > 3 we show the existence of a compactly supported potential in the differentiability class , for which the solutions to the linear Schrödinger equation in,

Stability of Two Soliton Collision for Nonintegrable gKdV Equations

We continue our study of the collision of two solitons for the subcritical generalized KdV equations

Full Field Algebras

We introduce a notion of full field algebra which is essentially an algebraic formulation of the notion of genus-zero full conformal field theory. For any vertex operator algebras V ^L and V ^R , is naturally a full field algebra and we introduce a notion of full field algebra over . We study the structure of full field algebras over using modules and intertwining operators for V ^L and V ^R . For a simple vertex operator algebra V satisfying certain natural finiteness and reductivity conditions needed for the Verlinde conjecture to hold, we construct a bilinear form on the space of intertwining operators for V and prove the nondegeneracy and other basic properties of this form. The proof of the nondegenracy of the bilinear form depends not only on the theory of intertwining operator algebras but also on the modular invariance for intertwining operator algebras through the use of the results obtained in the proof of the Verlinde conjecture by the first author. Using this nondegenerate bilinear form, we construct a full field algebra over and an invariant bilinear form on this algebra.     

Regular Strongly Typical Blocks of $${\mathcal{O}^{\mathfrak {q}}}$$

We use the technique of Harish-Chandra bimodules to prove that regular strongly typical blocks of the category for the queer Lie superalgebra are equivalent to the corresponding blocks of the category for the Lie algebra .     

On \mathbb{Z}-Gradations of Twisted Loop Lie Algebras of Complex Simple Lie Algebras

We define the twisted loop Lie algebra of a finite dimensional Lie algebra as the Fréchet space of all twisted periodic smooth mappings from to . Here the Lie algebra operation is continuous. We call such Lie algebras Fréchet Lie algebras. We introduce the notion of an integrable -gradation of a Fréchet Lie algebra, and find all inequivalent integrable -gradations with finite dimensional grading subspaces of twisted loop Lie algebras of complex simple Lie algebras.On leave of absence from the Institute for Nuclear Research of the Russian Academy of Sciences, 117312 Moscow, Russia.     

An Algebra of Deformation Quantization for Star-Exponentials on Complex Symplectic Manifolds

The cotangent bundle T ^* X to a complex manifold X is classically endowed with the sheaf of k-algebras of deformation quantization, where k := is a subfield of . Here, we construct a new sheaf of k-algebras which contains as a subalgebra and an extra central parameter t. We give the symbol calculus for this algebra and prove that quantized symplectic transformations operate on it. If P is any section of order zero of , we show that is well defined in .     

Measures on Banach Manifolds and Supersymmetric Quantum Field Theory

We show how to construct measures on Banach manifolds associated to supersymmetric quantum field theories. These measures are mathematically well-defined objects inspired by the formal path integrals appearing in the physics literature on quantum field theory. We give three concrete examples of our construction. The first example is a family of measures on a space of functions on the two-torus, parametrized by a polynomial P (the Wess-Zumino-Landau-Ginzburg model). The second is a family of measures on a space of maps from to a Lie group (the Wess-Zumino-Novikov-Witten model). Finally we study a family of measures on the product of a space of connections on the trivial principal bundle with structure group G on a three-dimensional manifold M with a space of -valued three-forms on M. We show that these measures are positive, and that the measures are Borel probability measures. As an application we show that formulas arising from expectations in the measures reproduce formulas discovered by Frenkel and Zhu in the theory of vertex operator algebras. We conjecture that a similar computation for the measures , where M is a homology three-sphere, will yield the Casson invariant of M. Dedicated to the memory of Raoul Bott Supported in part by NSF grant DMS 04/05670.     

Cardy Condition for Open-Closed Field Algebras

Let V be a vertex operator algebra satisfying certain reductivity and finiteness conditions such that , the category of V-modules, is a modular tensor category. We study open-closed field algebras over V equipped with nondegenerate invariant bilinear forms for both open and closed sectors. We show that they give algebras over a certain -extension of the so-called Swiss-cheese partial dioperad, and we can obtain Ishibashi states easily in such algebras. The Cardy condition can be formulated as an additional condition on such open-closed field algebras in terms of the action of the modular transformation on the space of intertwining operators of V. We then derive a graphical representation of S in the modular tensor category . This result enables us to give a categorical formulation of the Cardy condition and the modular invariance condition for 1-point correlation functions on the torus. Then we incorporate these two conditions and the axioms of the open-closed field algebra over V equipped with nondegenerate invariant bilinear forms into a tensor-categorical notion called the Cardy -algebra. In the end, we give a categorical construction of the Cardy -algebra in the Cardy case.     

Torelli Theorem for the Deligne–Hitchin Moduli Space

Fix integers g ≥ 3 and r ≥ 2, with r ≥ 3 if g = 3. Given a compact connected Riemann surface X of genus g, let denote the corresponding Deligne–Hitchin moduli space. We prove that the complex analytic space determines (up to an isomorphism) the unordered pair , where is the Riemann surface defined by the opposite almost complex structure on X.