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.     

Heteroclinic Connections Between Periodic Orbits in Planar Restricted Circular Three-Body Problem – A Computer Assisted Proof

 The restricted circular three-body problem is considered for the following parameter values C=3.03, μ=0.0009537 – the values for the Oterma comet in the Sun-Jupiter system. We present a computer assisted proof of an existence of a homo- and heteroclinic cycle between two Lyapunov orbits and an existence of symbolic dynamics on four symbols built on this cycle. Received: 15 April 2002 / Accepted: 16 May 2002 Published online: 20 January 2003 RID="*" ID="*" Research supported in part by Polish KBN grant 2 P03A 011 18, NSF grant DGE-98-04459 and PRODYN. Communicated by G. Gallavotti     

Dimension and Dynamical Entropy for Metrized <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">*</Emphasis></Superscript>-Algebras

 We introduce notions of dimension and dynamical entropy for unital C ^* -algebras ``metrized' by means of , which are complex-scalar versions of the Lip-norms constitutive of Rieffel's compact quantum metric spaces. Our examples involve the UHF algebras and noncommutative tori. In particular we show that the entropy of a noncommutative toral automorphism with respect to the canonical coincides with the topological entropy of its commutative analogue. Received: 13 February 2002 / Accepted: 20 August 2002 Published online: 22 November 2002     

Quantum aspects of GMS solutions of non-commutative field theory and large <Emphasis Type="Italic">N</Emphasis> limit of matrix models

We investigate quantum aspects of Gopakumar–Minwalla–Strominger (GMS) solutions of non-commutative field theory (NCFT) in the large non-commutativity limit, . Building upon a quantitative map between the operator formulation of 2- (respectively, (2+1)-) dimensional NCFTs and large-N matrix models of c=0 (respectively,c=1) non-critical strings, we show that GMS solutions are quantum mechanically sensible only if we make an appropriate joint scaling of and N. For 't Hooft's scaling, GMS solutions are replaced by large-N saddle-point solutions. GMS solutions are recovered from saddle-point solutions in the small 't Hooft coupling regime, but are destabilized in the large 't Hooft coupling regime by quantum effects. We make comparisons between these large-N effects and the recently studied infrared effects in NCFTs. We estimate the U(N) symmetry breaking effects of the gradient term and argue that they are suppressed only in the small 't Hooft coupling regime. Received: 2 January 2002 / Published online: 26 April 2002     

A Stochastic Model Associated with Enskog Equation and Its Kinetic Limit

 We examine a system of particles in which the particles travel deterministically in between stochastic collisions. The collisions are elastic and occur with probability ɛ ^d when two particles are at a distance σ. When the number of particles N goes to infinity and Nɛ ^d goes to a nonzero constant, we show that the particle density converges to a solution of the Enskog Equation. Received: 29 January 2002 / Accepted: 30 July 2002 Published online: 14 November 2002 RID="*" ID="*" Research supported in part by NSF Grant DMS-0072666     

On the Sutherland Spin Model of <Emphasis Type="Italic">B</Emphasis><Subscript><Emphasis Type="Italic">N</Emphasis></Subscript> Type and Its Associated Spin Chain

 The B _N hyperbolic Sutherland spin model is expressed in terms of a suitable set of commuting Dunkl operators. This fact is exploited to derive a complete family of commuting integrals of motion of the model, thus establishing its integrability. The Dunkl operators are shown to possess a common flag of invariant finite-dimensional linear spaces of smooth scalar functions. This implies that the Hamiltonian of the model preserves a corresponding flag of smooth spin functions. The discrete spectrum of the restriction of the Hamiltonian to this spin flag is explicitly computed by triangularization. The integrability of the hyperbolic Sutherland spin chain of B _N type associated with the dynamical model is proved using Polychronakos's ``freezing trick'. Received: 14 February 2002 / Accepted: 19 June 2002 Published online: 10 December 2002 RID="*" ID="*" Corresponding author. E-mail: artemio@fis.ucm.es RID="**" ID="**" On leave of absence from Institute of Mathematics, 3 Tereschenkivska St., 01601 Kyiv-4 Ukraine Communicated by L. Takhtajan     

Passive Advection and the Degenerate Elliptic Operators <Emphasis Type="Italic">M</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>

 We prove estimates for the stationary state n-point functions at zero molecular diffusivity in the Kraichnan model [13]. This is done by proving upper bounds for the heat kernels and Green's functions of the degenerate elliptic operators M _n that occur in the Hopf equations for the n-point functions. Received: 25 August 2001 / Accepted: 30 September 2002 Published online: 20 January 2003 Communicated by A. Kupiainen     

The Existence and Stability of Noncommutative Scalar Solitons

 We establish existence and stability results for solitons in noncommutative scalar field theories in even space dimension 2d. In particular, for any finite rank spectral projection P of the number operator 𝒩 of the d-dimensional harmonic oscillator and sufficiently large noncommutativity parameter θ we prove the existence of a rotationally invariant soliton which depends smoothly on θ and converges to a multiple of P as θ→∞. In the two-dimensional case we prove that these solitons are stable at large θ, if P=P _N , where P _N projects onto the space spanned by the N+1 lowest eigenstates of 𝒩, and otherwise they are unstable. We also discuss the generalisation of the stability results to higher dimensions. In particular, we prove stability of the soliton corresponding to P=P ₀ for all θ in its domain of existence. Finally, for arbitrary d and small values of θ, we prove without assuming rotational invariance that there do not exist any solitons depending smoothly on θ. Received: 13 July 2001 / Accepted: 9 July 2002 Published online: 10 January 2003     

The Thermodynamic Limit in Mean Field Spin Glass Models

 We present a simple strategy in order to show the existence and uniqueness of the infinite volume limit of thermodynamic quantities, for a large class of mean field disordered models, as for example the Sherrington-Kirkpatrick model, and the Derrida p-spin model. The main argument is based on a smooth interpolation between a large system, made of N spin sites, and two similar but independent subsystems, made of N ₁ and N ₂ sites, respectively, with N ₁ +N ₂ =N. The quenched average of the free energy turns out to be subadditive with respect to the size of the system. This gives immediately convergence of the free energy per site, in the infinite volume limit. Moreover, a simple argument, based on concentration of measure, gives the almost sure convergence, with respect to the external noise. Similar results hold also for the ground state energy per site. Received: 19 April 2002 / Accepted: 22 April 2002 Published online: 6 August 2002     

Rigorous Solution of the Gardner Problem

 We prove rigorously the well-known result of Gardner about the typical fractional volume of interactions between N spins which solve the problem of storing a given set of p random patterns. The Gardner formula for this volume in the limit N, p→∞, p/N→α is proven for all values of α. Besides, we prove a useful criterion for the factorisation of all correlation functions for a class of models of classical statistical mechanics. Received: 2 December 2001 / Accepted: 12 June 2002 Published online: 31 January 2003 Communicated by A. Kupiainen     

Conformally Invariant Powers of the Laplacian, <Emphasis Type="Italic">Q</Emphasis>-Curvature,and Tractor Calculus

 We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the conformal Cartan connection. Applications discussed include standard formulae for these operators in terms of the Levi-Civita connection and its curvature and a direct definition and formula for T. Branson's so-called Q-curvature (which integrates to a global conformal invariant) as well as generalisations of the operators and the Q-curvature. Among examples, the operators of order 4, 6 and 8 and the related Q-curvatures are treated explicitly. The algorithm exploits the ambient metric construction of Fefferman and Graham and includes a procedure for converting the ambient curvature and its covariant derivatives into tractor calculus expressions. This is partly based on [12], where the relationship of the normal standard tractor bundle to the ambient construction is described. Received: 24 January 2002 / Accepted: 1 November 2002 Published online: 18 February 2003 Communicated by P. Sarnak     

On the Structure of Stationary Solutions of the Navier-Stokes Equations

 This paper is a supplementary section to [1]. We show that without any additional hypothesis the main result in [1] (Theorem 1) can be considerably strengthened. Note. This paper cannot be read independently of [1]. The numbering of equations, theorems and propositions as well as cross-references used here have to be understood as if this paper were an additional section to [1]. Received: 7 May 2002 / Accepted: 15 October 2002 Published online: 10 February 2003 RID="*" ID="*" Supported in part by the Fonds National Suisse. Communicated by A. Kupiainen     

Equivariant <Emphasis Type="Italic">K</Emphasis>-Theory,Generalized Symmetric Products,and Twisted Heisenberg Algebra

 For a space X acted on by a finite group Γ, the product space X ⁿ affords a natural action of the wreath product Γ _n =Γ ⁿ ⋊S _n . The direct sum of equivariant K-groups were shown earlier by the author to carry several interesting algebraic structures. In this paper we study the K-groups of Γ _n -equivariant Clifford supermodules on X ⁿ . We show that is a Hopf algebra and it is isomorphic to the Fock space of a twisted Heisenberg algebra. Twisted vertex operators make a natural appearance. The algebraic structures on ℱ⁻ _Γ(X), when Γ is trivial and X is a point, specialize to those on a ring of symmetric functions with the Schur Q-functions as a linear basis. As a by-product, we present a novel construction of K-theory operations using the spin representations of the hyperoctahedral groups. Received: 3 February 2001 / Accepted: 17 August 2002 Published online: 10 January 2003     

Chiral BRST Cohomology of N= 2 Strings at Arbitrary Ghost and Picture Number

We compute the BRST cohomology of the holomorphic part of the N= 1 string at arbitrary ghost and picture number. We confirm the expectation that the relative cohomology at non-zero momentum consists of a single massless state in each picture. The absolute cohomology is obtained by an independent method based on homological algebra. For vanishing momentum, the relative and absolute cohomologies both display a picture dependence – a phenomenon discovered recently also in the relative Ramond sector of N= 1 strings by Berkovits and Zwiebach [1]. Received: 5 January 1998 / Accepted: 16 November 1998     

The Classification of Non-Local Chiral CFT with <Emphasis Type="Italic">c</Emphasis> < 1

All non-local but relatively local irreducible extensions of Virasoro chiral CFTs with c < 1 are classified. The classification, which is a prerequisite for the classification of local c < 1 boundary CFTs on a two-dimensional half-space, turns out to be 1 to 1 with certain pairs of A-D-E graphs with distinguished vertices. Dedicated to Hans-Jürgen Borchers on the occasion of his 80th birthday Supported in part by JSPS. Supported in part by EU network “Quantum Spaces - Noncommutative Geometry” HPRN-CT-2002-00280. Supported in part by GNAMPA and MIUR.     

The Weyl Quantization and the Quantum Group Quantization of the Moduli Space of Flat <Emphasis Type="Italic">SU</Emphasis>(2)-Connections on the Torus are the Same

 We prove that, for the moduli space of flat SU(2)-connections on the 2-dimensional torus, the Weyl quantization and the quantization performed using the quantum group of SL(2,C) are the same. This is done by comparing the matrices of the operators associated through the two quantizations to cosine functions. We also discuss the *-product of the Weyl quantization and show that it satisfies the product-to-sum formula for noncommutative cosines on the noncommutative torus. Received: 27 January 2002 / Accepted: 9 September 2002 Published online: 19 December 2002 RID="*" ID="*" Research supported in part by the NSF, award No. DMS 0070690 Communicated by A. Connes     

Vertex Algebras,Mirror Symmetry,and D-Branes: The Case of Complex Tori

 A vertex algebra is an algebraic counterpart of a two-dimensional conformal field theory. We give a new definition of a vertex algebra which includes chiral algebras as a special case, but allows for fields which are neither meromorphic nor anti-meromorphic. To any complex torus equipped with a flat K?hler metric and a closed 2-form we associate an N=2 superconformal vertex algebra (N=2 SCVA) in the sense of our definition. We find a criterion for two different tori to produce isomorphic N=2 SCVA's. We show that for algebraic tori the isomorphism of N=2 SCVA's implies the equivalence of the derived categories of coherent sheaves corresponding to the tori or their noncommutative generalizations (Azumaya algebras over tori). We also find a criterion for two different tori to produce N=2 SCVA's related by a mirror morphism. If the 2-form is of type (1,1), this condition is identical to the one proposed by Golyshev, Lunts, and Orlov, who used an entirely different approach inspired by the Homological Mirror Symmetry Conjecture of Kontsevich. Our results suggest that Kontsevich's conjecture must be modified: coherent sheaves must be replaced with modules over Azumaya algebras, and the Fukaya category must be ``twisted' by a closed 2-form. We also describe the implications of our results for BPS D-branes on Calabi-Yau manifolds. Received: 3 May 2001 / Accepted: 17 August 2002 Published online: 8 January 2003     

The Z = 82 shell closure in neutron-deficient Pb isotopes

Recent mass measurements show a substantial weakening of the binding-energy difference δ_2p(Z, N) = E(Z - 2, N) - 2E(Z, N) + E(Z + 2, N) in the neutron-deficient Pb isotopes. As δ_2p is often attributed to the size of the proton magic gap, it might be speculated that reduction in δ_2p is related to a weakening of the spherical Z = 82 shell. We demonstrate that the observed trend is described quantitatively by self-consistent mean-field models in terms of deformed ground states of Hg and Po isotopes. Received: 25 October 2001 / Accepted: 28 February 2002     

On 2D <Emphasis Type="Italic">N</Emphasis> = (4,4) superspace supergravity

We review some recent results obtained in studying superspa.ee formulations of 2D N = (4,4) matter-coupled supergravity. For a superspace geometry described by the minimal supergravity multiplet, we first describe how to reduce to components the chiral integral by using “ectoplasm” superform techniques as in arXiv:0907.5264 and then we review the bi-projective superspace formalism introduced in arXiv:0911.2546. After that, we elaborate on the curved bi-projective formalism providing a new result: the solution of the covariant type-I twisted multiplet constraints in terms of a weight-(−1, −1) bi-projective superfield.