Showing papers in "Letters in Mathematical Physics in 2003"

Deformation Quantization of Poisson Manifolds

Abstract: I prove that every finite-dimensional Poisson manifold X admits a canonical deformation quantization. Informally, it means that the set of equivalence classes of associative algebras close to the algebra of functions on X is in one-to-one correspondence with the set of equivalence classes of Poisson structures on X modulo diffeomorphisms. In fact, a more general statement is proven (the ‘Formality conjecture’), relating the Lie superalgebra of polyvector fields on X and the Hochschild complex of the algebra of functions on X. Coefficients in explicit formulas for the deformed product can be interpreted as correlators in a topological open string theory, although I do not explicitly use the language of functional integrals.

Formality of Chain Operad of Little Discs

Abstract: We prove that the chain operad of little disks is formal in characteristic zero, and discuss briefly the relation with Kontsevich formality in deformation quantization.

Analysis of the Global Relation for the Nonlinear Schrödinger Equation on the Half-line

Abstract: It has been shown recently that the unique, global solution of the Dirichlet problem of the nonlinear Schrodinger equation on the half-line can be expressed through the solution of a 2×2 matrix Riemann–Hilbert problem This problem is specified by the spectral functions {a(k),b(k)} which are defined in terms of the initial condition q(x,0)=q 0(x), and by the spectral functions {A(k),B(k)} which are defined in terms of the specified boundary condition q(0,t)=g 0(t) and the unknown boundary value q x (0,t)=g 1(t) Furthermore, it has been shown that given q 0 and g 0, the function g 1 can be characterized through the solution of a certain 'global relation' coupling q 0, g 0, g 1, and Φ(t,k), where Φ satisfies the t-part ofthe associated Lax pair evaluated at x=0 We show here that, by using a Gelfand–Levitan–Marchenko triangular representation of Φ, the global relation can be explicitly solved for g 1

Natural Star Products on Symplectic Manifolds and Quantum Moment Maps

Abstract: We define a natural class of star products: those which are given by a series of bidifferential operators which at order k in the deformation parameter have at most k derivatives in each argument. This class includes all the standard constructions of star products. We show that any such star product on a symplectic manifold defines a unique symplectic connection. We parametrise such star products, study their invariance properties and give necessary and sufficient conditions for them to have a quantum moment map. We show that Kravchenko's sufficient condition for a moment map for a Fedosov star product is also necessary.

Polarization-Free Quantum Fields and Interaction

Abstract: A new approach to the inverse scattering problem proposed by Schroer, is applied to two-dimensional integrable quantum field theories. For any two-particle S-matrix S_2 which is analytic in the physical sheet, quantum fields are constructed which are localizable in wedge-shaped regions of Minkowski space and whose two-particle scattering is described by the given S_2. These fields are polarization-free in the sense that they create one-particle states from the vacuum without polarization clouds. Thus they provide examples of temperate polarization-free generators in the presence of non-trivial interaction.

Moduli Space and Structure of Noncommutative 3-Spheres

Abstract: We analyze the moduli space and the structure of noncommutative 3-spheres. We develop the notion of central quadratic form for quadratic algebras, and prove a general algebraic result which considerably refines the classical homomorphism from a quadratic algebra to a cross-product algebra associated to the characteristic variety and lands in a richer cross-product. It allows to control the C*-norm on involutive quadratic algebras and to construct the differential calculus in the desired generality. The moduli space of noncommutative 3-spheres is identified with equivalence classes of pairs of points in a symmetric space of unitary unimodular symmetric matrices. The scaling foliation of the moduli space is identified to the gradient flow of the character of a virtual representation of SO(6). Its generic orbits are connected components of real parts of elliptic curves which form a net of biquadratic curves with eight points in common. We show that generically these curves are the same as the characteristic variety of the associated quadratic algebra. We then apply the general theory of central quadratic forms to show that the noncommutative 3-spheres admit a natural ramified covering π by a noncommutative three-dimensional nilmanifold. This yields the differential calculus. We then compute the Jacobian of the ramified covering π by pairing the direct image of the fundamental class of the noncommutative three-dimensional nilmanifold with the Chern character of the defining unitary and obtain the answer as the product of a period (of an elliptic integral) by a rational function. Finally, we show that the hyperfinite factor of type II1 appears as cross-product of the field K q of meromorphic functions on an elliptic curve by a subgroup of its Galois group $${\text{Aut}}_\mathbb{C} \left( {K_q } \right)$$ .

Analytic Functions and Integrable Hierarchies–Characterization of Tau Functions

Abstract: We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.

Degenerate Integrability of the Spin Calogero–Moser Systems and the Duality with the Spin Ruijsenaars Systems

Abstract: It is shown that spin Calogero–Moser systems are completely integrable in a sense of degenerate integrability. Their Liouville tori have dimension less than half of the dimension of the phase space. It is also shown that rational spin Ruijsenaars systems are degenerately integrable and dual to spin Calogero–Moser systems in a sense that action-angle variables of one are angle-action variables of the other.

On the Number of Negative Eigenvalues of a One-Dimensional Schrödinger Operator with Point Interactions

Abstract: An effective algorithm is provided for determining the number of negative eigenvalues of a one-dimensional Schrodinger operator with point interactions in terms of the intensities and the distances between the interactions

Generalized Transvectants-Rankin-Cohen Brackets

Abstract: We introduce oð p þ 1; q þ 1Þ-invariant bilinear differential operators on the space of tensor densities on R n generalizing the well-known bilinear sl2-invariant differential opera- tors in the one-dimensional case, called Transvectants or Rankin-Cohen brackets. We also consider already known linear oð p þ 1; q þ 1Þ-invariant differential operators given by powers of the Laplacian.

Selberg-Type Integrals Associated with SL3

Abstract: We present several formulae for Selberg-type integrals associated with the Lie algebra \(\mathfrak{s}\mathfrak{l}_3\).

A Formality Theorem for Poisson Manifolds

Abstract: Let M be a differential manifold. Using different methods, Kontsevich and Tamarkin have proved a formality theorem, which states the existence of a Lie homomorphism 'up to homotopy' between the Lie algebra of Hochschild cochains on C ∞(M) and its cohomology (Γ(M, ΛTM), [−, −] S ). Suppose M is a Poisson manifold equipped with a Poisson tensor π; then one can deduce from this theorem the existence of a star product $$\star$$ on C ∞(M). In this Letter we prove that the formality theorem can be extended to a Lie (and even Gerstenhaber) homomorphism 'up to homotopy' between the Lie (resp. Gerstenhaber 'up to homtoptopy') algebra of Hochschild cochains on the deformed algebra (C ∞(M), *) and the Poisson complex (Γ(M, ΛTM), [π, −] S ). We will first recall Tamarkin's proof and see how the formality maps can be deduced from Etingof and Kazhdan's theorem using only homotopies formulas. The formality theorem for Poisson manifolds will then follow.

Exact Solutions of the Bogoyavlensky–Konoplechenko Equation

Abstract: We give a Darboux transformation for the Bogoyavlensky–Konoplechenko equation, which is a two-dimensional generalisation of the Korteweg–deVries equation. This transformation is used to construct a family of solutions of this equation.

On a Generalized Frobenius–Stickelberger Addition Formula

Abstract: In this Letter we obtain a generalization of the Frobenius–Stickelberger addition formula for the (hyperelliptic) σ-function of a genus 2 curve in the case of three vector-valued variables. The result is given explicitly in the form of a polynomial in Kleinian ℘-functions.

Strict Quantizations of Symplectic Manifolds

Abstract: We introduce the notion of a strict quantization of a symplectic manifold and show its existence under a topological condition

Quantization of Poisson Families and of Twisted Poisson Structures

Abstract: We describe a deformation quantization of a modification of Poisson geometry by a closed 3-form. Under suitable conditions, it gives rise to a stack of algebras. The basic object used for this aim is a kind of families of Poisson structures given by a Maurer–Cartan equation; they are easily quantized using Kontsevich's formality theorem. We conclude with a section on quantization of complex manifolds.

R-matrices for leibniz algebras

Abstract: Leibniz agebras are a generalization of Lie algebras, where no symmetry properties of the bracket are required In this Letter we introduce a notion of R-matrices for this structure and the related Yang–Baxter equations, and discuss some of their basic properties

Improved Bounds on the Spectral Gap Above Frustration-Free Ground States of Quantum Spin Chains

Abstract: Nachtergaele obtained explicit lower bounds for the spectral gap above many frustration free quantum spin chains by using the ‘martingale method’. We present simple improvements to his main bounds which allow one to obtain a sharp lower bound for the spectral gap above the spin-1/2 ferromagnetic XXZ chain. As an illustration of the method, we also calculate a lower bound for the spectral gap of the AKLT model, which is about 1/3 the size of the expected gap.

Trigonometric Solutions of the WDVV Equations from Root Systems

Abstract: By introduction of an additional variable and addition of a Weyl invariant correction term to the perturbative prepotential in five-dimensional Seiberg-Witten theory we construct solutions of the Witten–Dijkgraaf–Verlinde–Verlinde equations of trigonometric type for all crystallographic root systems.

Equations in Dual Variables for Whittaker Functions

Abstract: It is known that the Whittaker functions w(qλ) associated with the group GL(N) are eigenfunctions of the Hamiltonians of the open Toda chain, hence satisfy a set of differential equations in the Toda variables q i . Using the expression of the q i for the closed Toda chain in terms of Sklyanin variables λ i , and the known relations between the open and the closed Toda chains, we show that Whittaker functions also satisfy a set of new difference equations in λ i .

A Note on the Dispersionless Dym Hierarchy

Abstract: We discuss the Miura map as well as the Poisson algebras associated with the dispersionless Dym hierarchy. Particularly, we study explicitly the bi-Hamiltonian structure of a truncated Dym system with two variables, in which a new hierarchy flow generated by logarithmic Hamiltonians appears. We then show that this new hierarchy emerges naturally from the topological recursion relation in the Landau–Ginzburg formulation.

On the Dequantization of Fedosov's Deformation Quantization

Abstract: To each natural deformation quantization on a Poisson manifold M we associate a Poisson morphism from the formal neighborhood of the zero section of T*M to the formal neighborhood of the diagonal of the product M×\({\tilde M}\), where \({\tilde M}\) is a copy of M with the opposite Poisson structure. We call it dequantization of the natural deformation quantization. Then we 'dequantize' Fedosov's quantization.

Multisymplectic geometry, local conservation laws and a multisymplectic integrator for the Zakharov-Kuznetsov equation

Abstract: The multisymplectic geometry for the Zakharov–Kuznetsov equation is presented in this Letter. The multisymplectic form and the local energy and momentum conservation laws are derived directly from the variational principle. Based on the multisymplectic Hamiltonian formulation, we derive a 36-point multisymplectic integrator.

An Inverse Problem for an Harmonic Oscillator Perturbed by Potential: Uniqueness

Abstract: Consider the perturbed harmonic oscillator Ty = -y" + x2y + q(x)y on L2(R) where the real potential q satisfy some assumption on infinity (the case q ∈ L2(R), (∣t∣+1)-rdt), r < 1 is covered).

Absolute Continuity in Periodic Thin Tubes and Strongly Coupled Leaky Wires

Abstract: Using a perturbative argument, we show that in any finite region containing the lowest transverse eigenmode, the spectrum of a periodically curved smooth Dirichlet tube in two or three dimensions is absolutely continuous provided the tube is sufficiently thin. In a similar way we demonstrate absolute continuity at the bottom of the spectrum for generalized Schrodinger operators with a sufficiently strongly attractive δ interaction supported by a periodic curve in Rd = 2, 3.

The action functional for Moyal planes

Abstract: Modulo some natural generalizations to noncompact spaces, we show in this Letter that Moyal planes are nonunital spectral triples in the sense of Connes. The action functional of these triples is computed, and we obtain the expected result, i.e. the noncommutative Yang–Mills action associated with the Moyal product. In particular, we show that Moyal gauge theory naturally fits into the rigorous framework of noncommutative geometry.

Eigenvalue Asymptotics for the Schrödinger Operator with a δ-Interaction on a Punctured Surface

Abstract: Given n≥2, we put r=min $$\left\{ {i \in \mathbb{N};i > n/2} \right\}$$ . Let Σ be a compact, C r -smooth surface in ℝn which contains the origin. Let further $$\left\{ {S_\varepsilon } \right\}_{0 \leqslant \varepsilon < \eta } $$ be a family of measurable subsets of Σ such that $$\sup _{x \in S_\varepsilon } |x| = \mathcal{O}(\varepsilon )$$ as $$\varepsilon \to {\text{0}}$$ . We derive an asymptotic expansion for the discrete spectrum of the Schrodinger operator $$ - \Delta - \beta \delta \left( { \cdot - \sum \backslash S_\varepsilon } \right)$$ in L 2(ℝ n ), where β is a positive constant, as $$\varepsilon \to {\text{0}}$$ . An analogous result is given also for geometrically induced bound states due to a δ interaction supported by an infinite planar curve.

Quantization of Linear Poisson Structures and Degrees of Maps

Abstract: Kontsevich's formula for a deformation quantization of Poisson structures involves a Feynman series of graphs, with the weights given by some complicated integrals (using certain pullbacks of the standard angle form on a circle). We explain the geometric meaning of this series as degrees of maps of some grand configuration spaces; the associativity proof is also interpreted in purely homological terms. An interpretation in terms of degrees of maps shows that any other 1-form on the circle also leads to a star-product and allows one to compare these products.

Noncommutative Ricci Curvature and Dirac Operator on Cq[SL2] at Roots of Unity

Abstract: We find a unique torsion free Riemannian spin connection for the natural Killing metric on the quantum group C q [ SL2], using a recent frame bundle formulation. We find that its covariant Ricci curvature is essentially proportional to the metric (i.e. an Einstein space). We compute the Dirac operator and find for q an odd rth root of unity that its eigenvalues are given by q-integers [m] q for m = 0,1...,r − 1 offset by the constant background curvature. We fully solve the Dirac equation for r = 3.

Multiplicity Free Actions of Quantum Groups and Generalized Howe Duality

Abstract: Noncommutative associative algebras are constructed which have the structure of module algebras over tensor products of pairs of quantized universal enveloping algebras. These module algebras decompose into multiplicity free direct sums of irreducible modules, yielding quantum analogues of generalized Howe dualities.