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 .     

$${\mathsf{N}}$$ -Complexes as Functors,Amplitude Cohomology and Fusion Rules

We consider -complexes as functors over an appropriate linear category in order to show first that the Krull-Schmidt Theorem holds, then to prove that amplitude cohomology (called generalized cohomology by M. Dubois-Violette) only vanishes on injective functors providing a well defined functor on the stable category. For left truncated -complexes, we show that amplitude cohomology discriminates the isomorphism class up to a projective functor summand. Moreover amplitude cohomology of positive -complexes is proved to be isomorphic to an Ext functor of an indecomposable -complex inside the abelian functor category. Finally we show that for the monoidal structure of -complexes a Clebsch-Gordan formula holds, in other words the fusion rules for -complexes can be determined. This work has been supported by the projects PICT 08280 (ANPCyT), UBACYTX169, PIP-CONICET 5099 and the German Academic Exchange Service (DAAD). The second author is a research member of CONICET (Argentina) and a Regular Associate of ICTP Associate Scheme.     

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 Uniform Quantum Version of the Cherry Theorem

Consider in the operator family . P ₀ is the quantum harmonic oscillator with diophantine frequency vector ω, F ₀ a bounded pseudodifferential operator with symbol decreasing to zero at infinity in phase space, and . Then there exist independent of and an open set such that if and , the quantum normal form near P ₀ converges uniformly with respect to . This yields an exact quantization formula for the eigenvalues, and for the classical Cherry theorem on convergence of Birkhoff’s normal form for complex frequencies is recovered. Partially supported by PAPIIT-UNAM IN106106-2.     

Spectral Analysis and Zeta Determinant on the Deformed Spheres

We consider a class of singular Riemannian manifolds, the deformed spheres , defined as the classical spheres with a one parameter family g[k] of singular Riemannian structures, that reduces for k = 1 to the classical metric. After giving explicit formulas for the eigenvalues and eigenfunctions of the metric Laplacian , we study the associated zeta functions . We introduce a general method to deal with some classes of simple and double abstract zeta functions, generalizing the ones appearing in . An application of this method allows to obtain the main zeta invariants for these zeta functions in all dimensions, and in particular and . We give explicit formulas for the zeta regularized determinant in the low dimensional cases, N = 2,3, thus generalizing a result of Dowker [25], and we compute the first coefficients in the expansion of these determinants in powers of the deformation parameter k. Partially supported by FAPESP: 2005/04363-4     

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.     

Linking and Causality in Globally Hyperbolic Space-times

The classical linking number lk is defined when link components are zero homologous. In [15] we constructed the affine linking invariant alk generalizing lk to the case of linked submanifolds with arbitrary homology classes. Here we apply alk to the study of causality in Lorentzian manifolds. Let M ^m be a spacelike Cauchy surface in a globally hyperbolic space-time (X ^m+1, g). The spherical cotangent bundle ST ^* M is identified with the space of all null geodesics in (X,g). Hence the set of null geodesics passing through a point gives an embedded (m−1)-sphere in called the sky of x. Low observed that if the link is nontrivial, then are causally related. This observation yielded a problem (communicated by R. Penrose) on the V. I. Arnold problem list [3,4] which is basically to study the relation between causality and linking. Our paper is motivated by this question. The spheres are isotopic to the fibers of They are nonzero homologous and the classical linking number lk is undefined when M is closed, while alk is well defined. Moreover, alk if M is not an odd-dimensional rational homology sphere. We give a formula for the increment of alk under passages through Arnold dangerous tangencies. If (X,g) is such that alk takes values in and g is conformal to that has all the timelike sectional curvatures nonnegative, then are causally related if and only if alk . We prove that if alk takes values in and y is in the causal future of x, then alk is the intersection number of any future directed past inextendible timelike curve to y and of the future null cone of x. We show that x,y in a nonrefocussing (X, g) are causally unrelated if and only if can be deformed to a pair of S ^m-1-fibers of by an isotopy through skies. Low showed that if (X, g) is refocussing, then M is compact. We show that the universal cover of M is also compact.     

Counting Regions with Bounded Surface Area

Define a cubical complex to be a collection of integer-aligned unit cubes in d dimensions. Lebowitz and Mazel (1998) proved that there are between and complexes containing a fixed cube with connected boundary of (d − 1)-volume n. In this paper we narrow these bounds to between and . We also show that there are connected complexes containing a fixed cube with (not necessarily connected) boundary of volume n.     

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.     

A Strong Szegő Theorem for Jacobi Matrices

We use a classical result of Golinskii and Ibragimov to prove an analog of the strong Szegő theorem for Jacobi matrices on . In particular, we consider the class of Jacobi matrices with conditionally summable parameter sequences and find necessary and sufficient conditions on the spectral measure such that and lie in , the linearly-weighted l ² space. An erratum to this article can be found at     

Zero Modes’ Fusion Ring and Braid Group Representations for the Extended Chiral <Emphasis Type="Italic">su</Emphasis>(2) WZNW Model

A zero modes’ Fock space is constructed for the extended chiral WZNW model. It gives room to a realization of the fusion ring of representations of the restricted quantum universal enveloping algebra at an even root of unity, and of its infinite dimensional extension by the Lusztig operators We provide a streamlined derivation of the characteristic equation for the Casimir invariant from the defining relations of A central result is the characterization of the Grothendieck ring of both and in Theorem 3.1. The properties of the fusion ring in are related to the braiding properties of correlation functions of primary fields of the conformal current algebra model.      

Quantum Conjugacy Classes of Simple Matrix Groups

Let G be a simple complex classical group and its Lie algebra. Let be the Drinfeld-Jimbo quantization of the universal enveloping algebra . We construct an explicit -equivariant quantization of conjugacy classes of G with Levi subgroups as the stabilizers. Dedicated to the memory of Joseph Donin This research is partially supported by the Emmy Noether Research Institute for Mathematics, the Minerva Foundation of Germany, the Excellency Center “Group Theoretic Methods in the study of Algebraic Varieties” of the Israel Science foundation, by the EPSRC grant C511166, and by the RFBR grant no. 06-01-00451.     

The Dynamics of Relativistic Strings Moving in the Minkowski Space $$\mathbb{R}^{1+n}$$

In this paper we investigate the dynamics of relativistic (in particular, closed) strings moving in the Minkowski space . We first derive a system with n nonlinear wave equations of Born-Infeld type which governs the motion of the string. This system can also be used to describe the extremal surfaces in . We then show that this system enjoys some interesting geometric properties. Based on this, we give a sufficient and necessary condition for the global existence of extremal surfaces without space-like point in with given initial data. This result corresponds to the global propagation of nonlinear waves for the system describing the motion of the string in . We also present an explicit exact representation of the general solution for such a system. Moreover, a great deal of numerical analyses are investigated, and the numerical results show that, in phase space, various topological singularities develop in finite time in the motion of the string. Finally, some important discussions related to the theory of extremal surfaces of mixed type in are given.     

Convergence of the Wick Star Product

We construct a Fréchet space as a subspace of where the Wick star product converges and is continuous. The resulting Fréchet algebra _ħ is studied in detail including a ^*-representation of _ħ in the Bargmann-Fock space and a discussion of star exponentials and coherent states.     

Dynamical Noncommutativity and Noether Theorem in Twisted $${\phi^{\star 4}}$$ Theory

A -product is defined via a set of commuting vector fields , and used in a theory coupled to the fields. The -product is dynamical, and the vacuum solution , reproduces the usual Moyal product. The action is invariant under rigid translations and Lorentz rotations, and the conserved energy–momentum and angular momentum tensors are explicitly derived.      

Boson Stars as Solitary Waves

We study the nonlinear equation

Hidden Grassmann Structure in the XXZ Model

For the critical XXZ model, we consider the space of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators which act on . These operators are constructed as traces over representations of the q-oscillator algebra, in close analogy with Baxter’s Q-operators. We show that the vacuum expectation values of operators in can be expressed in terms of an exponential of a quadratic form of . On leave of absence from Skobeltsyn Institute of Nuclear Physics, MSU, 119992, Moscow, Russia Membre du CNRS     

Group Orbits and Regular Partitions of Poisson Manifolds

We study a large class of Poisson manifolds, derived from Manin triples, for which we construct explicit partitions into regular Poisson submanifolds by intersecting certain group orbits. Examples include all varieties of Lagrangian subalgebras of reductive quadratic Lie algebras with Poisson structures defined by Lagrangian splittings of . In the special case of , where is a complex semi-simple Lie algebra, we explicitly compute the ranks of the Poisson structures on defined by arbitrary Lagrangian splittings of . Such Lagrangian splittings have been classified by P. Delorme, and they contain the Belavin–Drinfeld splittings as special cases.     

The Paraboson Fock Space and Unitary Irreducible Representations of the Lie Superalgebra {\mathfrak{osp}(1|2n)}

It is known that the defining relations of the orthosymplectic Lie superalgebra are equivalent to the defining (triple) relations of n pairs of paraboson operators . In particular, with the usual star conditions, this implies that the “parabosons of order p” correspond to a unitary irreducible (infinite-dimensional) lowest weight representation V(p) of . Apart from the simple cases p = 1 or n = 1, these representations had never been constructed due to computational difficulties, despite their importance. In the present paper we give an explicit and elegant construction of these representations V(p), and we present explicit actions or matrix elements of the generators. The orthogonal basis vectors of V(p) are written in terms of Gelfand-Zetlin patterns, where the subalgebra of plays a crucial role. Our results also lead to character formulas for these infinite-dimensional representations. Furthermore, by considering the branching , we find explicit infinite-dimensional unitary irreducible lowest weight representations of and their characters. NIS was supported by a project from the Fund for Scientific Research – Flanders (Belgium) and by project P6/02 of the Interuniversity Attraction Poles Programme (Belgian State – Belgian Science Policy). An erratum to this article can be found at     

Baxter Operator and Archimedean Hecke Algebra

In this paper we introduce Baxter integral -operators for finite-dimensional Lie algebras and . Whittaker functions corresponding to these algebras are eigenfunctions of the -operators with the eigenvalues expressed in terms of Gamma-functions. The appearance of the Gamma-functions is one of the manifestations of an interesting connection between Mellin-Barnes and Givental integral representations of Whittaker functions, which are in a sense dual to each other. We define a dual Baxter operator and derive a family of mixed Mellin-Barnes-Givental integral representations. Givental and Mellin-Barnes integral representations are used to provide a short proof of the Friedberg-Bump and Bump conjectures for G = GL(ℓ + 1) proved earlier by Stade. We also identify eigenvalues of the Baxter -operator acting on Whittaker functions with local Archimedean L-factors. The Baxter -operator introduced in this paper is then described as a particular realization of the explicitly defined universal Baxter operator in the spherical Hecke algebra , K being a maximal compact subgroup of G. Finally we stress an analogy between -operators and certain elements of the non-Archimedean Hecke algebra .