Showing papers in "Journal of Mathematical Physics in 2001"

Superstrings and topological strings at large N

Abstract: We embed the large N Chern–Simons/topological string duality in ordinary superstrings. This corresponds to a large N duality between generalized gauge systems with N=1 supersymmetry in four dimensions and superstrings propagating on noncompact Calabi–Yau manifolds with certain fluxes turned on. We also show that in a particular limit of the N=1 gauge theory system, certain superpotential terms in the N=1 system (including deformations if spacetime is noncommutative) are captured to all orders in 1/N by the amplitudes of noncritical bosonic strings propagating on a circle with self-dual radius. We also consider D-brane/anti-D-brane system wrapped over vanishing cycles of compact Calabi–Yau manifolds and argue that at large N they induce a shift in the background to a topologically distinct Calabi–Yau, which we identify as the ground state system of the brane/anti-brane system.

Strings in AdS3 and the SL(2,R) WZW model. I: The spectrum

Abstract: In this paper we study the spectrum of bosonic string theory on AdS3 We study classical solutions of the SL(2,R) WZW model, including solutions for long strings with nonzero winding number We show that the model has a symmetry relating string configurations with different winding numbers We then study the Hilbert space of the WZW model, including all states related by the above symmetry This leads to a precise description of long strings We prove a no-ghost theorem for all the representations that are involved and discuss the scattering of the long string

An exact prediction of N=4 supersymmetric Yang–Mills theory for string theory

Abstract: We propose that the expectation value of a circular BPS-Wilson loop in [script N] = 4 supersymmetric Yang–Mills can be calculated exactly, to all orders in a 1/N expansion and to all orders in g2N. Using the AdS/CFT duality, this result yields a prediction of the value of the string amplitude with a circular boundary to all orders in alpha[prime] and to all orders in gs. We then compare this result with string theory. We find that the gauge theory calculation, for large g2N and to all orders in the 1/N2 expansion, does agree with the leading string theory calculation, to all orders in gs and to lowest order in alpha[prime]. We also find a relation between the expectation value of any closed smooth Wilson loop and the loop related to it by an inversion that takes a point along the loop to infinity, and compare this result, again successfully, with string theory.

D-branes, categories and N=1 supersymmetry

Abstract: We show that boundary conditions in topological open string theory on Calabi–Yau (CY) manifolds are objects in the derived category of coherent sheaves, as foreseen in the homological mirror symmetry proposal of Kontsevich. Together with conformal field theory considerations, this leads to a precise criterion determining the supersymmetry preserving branes at any point in CY moduli space, completing the proposal of II-stability.

D=4 chiral string compactifications from intersecting branes

Abstract: Intersecting Dp-branes often give rise to chiral fermions living on their intersections. We study the construction of four-dimensional chiral gauge theories by considering configurations of type II D(3+n)-branes wrapped on nontrivial n-cycles on T2n×(R2(3−n)/ZN), for n=1, 2, 3. The gauge theories on the four noncompact dimensions of the brane world-volume are generically chiral and nonsupersymmetric. We analyze consistency conditions (RR tadpole cancellation) for these models, and their relation to four-dimensional anomaly cancellation. Cancellation of U(1) gauge anomalies involves a Green–Schwarz mechanism mediated by RR partners of untwisted and/or twisted moduli. This class of models is of potential phenomenological interest, and we construct explicit examples of SU(3)×SU(2)×U(1) three-generation models. The models are nonsupersymmetric, but the string scale may be lowered close to the weak scale so that the standard hierarchy problem is avoided. We also comment on the presence of scalar tachyons and p...

The geometry of photon surfaces

Abstract: The photon sphere concept in Schwarzschild space–time is generalized to a definition of a photon surface in an arbitrary space–time. A photon sphere is then defined as an SO(3)×R-invariant photon surface in a static spherically symmetric space–time. It is proved, subject to an energy condition, that a black hole in any such space–time must be surrounded by a photon sphere. Conversely, subject to an energy condition, any photon sphere must surround a black hole, a naked singularity or more than a certain amount of matter. A second order evolution equation is obtained for the area of an SO(3)-invariant photon surface in a general nonstatic spherically symmetric space–time. Many examples are provided.

Strings in AdS3 and the SL(2,R) WZW model. II: Euclidean black hole

Abstract: We consider the one-loop partition function for Euclidean BTZ black hole back-grounds or equivalently thermal AdS3 backgrounds which are quotients of H3 (Euclidean AdS3). The one-loop partition function is modular invariant and we can read off the spectrum which is consistent to that found in hep-th/0001053. We see long strings and discrete states in agreement with the expectations.

Nonrelativistic closed string theory

Abstract: We construct a Galilean invariant nongravitational closed string theory whose excitations satisfy a nonrelativistic dispersion relation. This theory can be obtained by taking a consistent low energy limit of any of the conventional string theories, including the heterotic string. We give a finite first order worldsheet Hamiltonian for this theory and show that this string theory has a sensible perturbative expansion, interesting high energy behavior of scattering amplitudes and a Hagedorn transition of the thermal ensemble. The strong coupling duals of the Galilean superstring theories are considered and are shown to be described by an eleven-dimensional Galilean invariant theory of light membrane fluctuations. A new class of Galilean invariant nongravitational theories of light-brane excitations are obtained. We exhibit dual formulations of the strong coupling limits of these Galilean invariant theories and show that they exhibit many of the conventional dualities of M theory in a nonrelativistic setting.

An M theory flop as a large N duality

Abstract: We show how a recently proposed large N duality in the context of type IIA strings with N=1 supersymmetry in 4 dimensions can be derived from purely geometric considerations by embedding type IIA strings in M-theory. The phase structure of M-theory on G2 holonomy manifolds and an S3 flop are the key ingredients in this derivation.

Temporally stable coherent states for infinite well and Poschl-Teller potentials

Abstract: This article is a direct illustration of a construction of coherent states which has been recently proposed by two of us (JPG and JK). We have chosen the example of a particle trapped in an infinite square-well and also in Poschl–Teller potentials of the trigonometric type. In the construction of the corresponding coherent states, we take advantage of the simplicity of the solutions, which ultimately stems from the fact they share a common SU(1,1) symmetry a la Barut-Girardello. Many properties of these states are then studied, both from mathematical and from physical points of view.

Quadratic Poisson algebras of two-dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems

Abstract: The integrals of motion of the classical two-dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic Poisson algebra, whose the general form is investigated Each classical superintegrable problem has a quantum counterpart, a quantum superintegrable system The quadratic Poisson algebra is deformed into a quantum associative algebra, the finite-dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique It is shown that the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system The calculation of energy eigenvalues is reduced to the solution of algebraic equations, which are universal, that is for all two-dimensional superintegrable systems with quadratic integrals of motion

Fractional Fokker-Planck Equation for Nonlinear Stochastic Differential Equations Driven by Non-Gaussian Levy Stable Noises

Abstract: The Fokker–Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. In this paper, we therefore derive a fractional Fokker–Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevin-type equation, which is driven by a Levy stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the fractional Fokker–Planck equation.

Exact solvability of superintegrable systems

Abstract: It is shown that all four superintegrable quantum systems on the Euclidean plane possess the same underlying hidden algebra sl(3). The gauge-rotated Hamiltonians, as well as their integrals of motion, once rewritten in appropriate coordinates, preserve a flag of polynomials. This flag corresponds to highest-weight finite-dimensional representations of the sl(3)-algebra, realized by first-order differential operators.

On the Classification of N-extended Supersymmetric Quantum Mechanical Systems

Abstract: In this paper some properties of the irreducible multiplets of representation for the N=(p,q)-extended supersymmetry in one dimension are discussed. Essentially two results are presented. At first a peculiar property of the one dimension is exhibited, namely that any multiplet containing 2d (d bosonic and d fermionic) particles in M different spin states is equivalent to a {d,d} multiplet of just two spin states (all bosons and all fermions being grouped in the same spin). Later, it is shown that the classification of all multiplets of this kind carrying an irreducible representation of the N-extended supersymmetry is in one-to-one correspondence with the classification of real-valued Clifford Γ-matrices of Weyl type. In particular, p+q is mapped into D, the space–time dimensionality, while 2d is determined to be the dimensionality of the corresponding Γ-matrices. The implications of these results to the theory of spinning particles are analyzed.

Sigma model approach to string theory effective actions with tachyons

Abstract: Motivated by recent discussions of actions for tachyon and vector fields related to tachyon condensation in open string theory we review and clarify some aspects of their derivation within the sigma model approach In particular, we demonstrate that the renormalized partition function Z(T,A) of the boundary sigma model gives the effective action for massless vectors which is consistent with the string S-matrix and beta function, resolving an old problem with this suggestion in the bosonic string case at the level of the leading F2(dF)2 derivative corrections to Born–Infeld action We give a manifestly gauge invariant definition of Z(T,A) in non-Abelian NSR open string theory and check that its derivative reproduces the tachyon beta function in a particular scheme We also discuss the derivation of similar actions for tachyon and massless modes in closed bosonic and NSR (type 0) string theories In the bosonic case the tachyon potential has the structure −T2e−T, but it vanishes in the NSR string case

The AM(1) automata related to crystals of symmetric tensors

Abstract: A soliton cellular automaton associated with crystals of symmetric tensor representations of the quantum affine algebra Uq′(AM(1)) is introduced It is a crystal theoretic formulation of the generalized box–ball system in which capacities of boxes and carriers are arbitrary and inhomogeneous Scattering matrices of two solitons coincide with the combinatorial R matrices of Uq′(AM−1(1)) A piecewise linear evolution equation of the automaton is identified with an ultradiscrete limit of the nonautonomous discrete Kadomtsev–Petviashivili equation A class of N soliton solutions is obtained through the ultradiscretization of soliton solutions of the latter

The uniqueness theorem for entanglement measures

Abstract: We explore and develop the mathematics of the theory of entanglement measures. After a careful review and analysis of definitions, of preliminary results, and of connections between conditions on entanglement measures, we prove a sharpened version of a uniqueness theorem which gives necessary and sufficient conditions for an entanglement measure to coincide with the reduced von Neumann entropy on pure states. We also prove several versions of a theorem on extreme entanglement measures in the case of mixed states. We analyze properties of the asymptotic regularization of entanglement measures proving, for example, convexity for the entanglement cost and for the regularized relative entropy of entangle ment.

Two binary Darboux transformations for the KdV hierarchy with self-consistent sources

Abstract: Two binary (integral type) Darboux transformations for the KdV hierarchy with self-consistent sources are proposed. In contrast with the Darboux transformation for the KdV hierarchy, one of the two binary Darboux transformations provides non-auto-Backlund transformation between two nth KdV equations with self-consistent sources with different degrees. The formula for the m-times repeated binary Darboux transformations are presented. This enables us to construct the N-soliton solution for the KdV hierarchy with self-consistent sources.

Sequential quantum measurements

Abstract: A quantum effect is an operator A on a complex Hilbert space H that satisfies 0⩽A⩽I. We denote the set of quantum effects by E(H). The set of self-adjoint projection operators on H corresponds to sharp effects and is denoted by P(H). We define the sequential product of A,B∈E(H) by A∘B=A1/2BA1/2. The main purpose of this article is to study some of the algebraic properties of the sequential product. Many of our results show that algebraic conditions on A∘B imply that A and B commute for the usual operator product. For example, if A∘B satisfies certain distributive or associative laws, then AB=BA. Moreover, if A∘B∈P(H), then AB=BA and A∘B=B or B∘A=B if and only if AB=BA=B. A natural definition of stochastic independence is introduced and briefly studied.

Fundamental strings in open string theory at the tachyonic vacuum

Abstract: We show that the world-volume theory on a D-p-brane at the tachyonic vacuum has solitonic string solutions whose dynamics is governed by the Nambu–Goto action of a string moving in (25+1) dimensional space–time. This provides strong evidence for the conjecture that at this vacuum the full (25+1) dimensional Poincare invariance is restored. We also use this result to argue that the open string field theory at the tachyonic vacuum must contain closed string excitations.

Conformally flat anisotropic spheres in general relativity

Abstract: The condition for the vanishing of the Weyl tensor is integrated in the spherically symmetric case. Then, the resulting expression is used to find new, conformally flat, interior solutions to Einstein equations for locally anisotropic fluids. The slow evolution of these models is contrasted with the evolution of models with similar energy density or radial pressure distribution but nonvanishing Weyl tensor, thereby bringing out the different role played by the Weyl tensor, the local anisotropy of pressure, and the inhomogeneity of the energy density in the collapse of relativistic spheres.

Noncommutative tachyons and K-theory

Abstract: We show that the relation between D-branes and noncommutative tachyons leads very naturally to the relation between D-branes and K-theory. We also discuss some relations between D-branes and K-homology, provide a noncommutative generalization of the ABS construction, and give a simple physical interpretation of Bott periodicity. In addition, a framework for constructing Neveu–Schwarz fivebranes as noncommutative solitons is proposed.

Towards the classification of static vacuum spacetimes with negative cosmological constant

Abstract: We present a systematic study of static solutions of the vacuum Einstein equations with negative cosmological constant which asymptotically approach the generalized Kottler (“Schwarzschild–anti-de Sitter”) solution, within (mainly) a conformal framework. We show connectedness of conformal infinity for appropriately regular such spacetimes. We give an explicit expression for the Hamiltonian mass of the (not necessarily static) metrics within the class considered; in the static case we show that they have a finite and well-defined Hawking mass. We prove inequalities relating the mass and the horizon area of the (static) metrics considered to those of appropriate reference generalized Kottler metrics. Those inequalities yield an inequality which is opposite to the conjectured generalized Penrose inequality. They can thus be used to prove a uniqueness theorem for the generalized Kottler black holes if the generalized Penrose inequality can be established.

Ground state of a spin-1/2 charged particle in a two-dimensional magnetic field

Abstract: It is investigated that the structure of the kernel of the Dirac–Weyl operator D of a charged particle in the magnetic-field B=B0+B1, given by the sum of a strongly singular magnetic field B0(⋅)=Σνγνδ(⋅−aν) with some singular points aν and a magnetic-field B1 with a bounded support Here the magnetic field B1 may have some singular points with the order of the singularity less than 2 At a glance, it seems that, following “Aharonov–Casher Theorem” [Phys Rev A 19, 2461 (1979)], the dimension of the kernel of D, dim ker D, is a function of one variable of the total magnetic flux (=Σνγν+∫R2B1dxdy) of B However, since the influence of the strongly singular points works, dim ker D indeed is a function of several variables of the total magnetic flux and each of γν’s

Unimodular relativity and cosmological constant

Abstract: Unimodular relativity is a theory of gravity and space–time with a fixed absolute space–time volume element, the modulus, which we suppose is proportional to the number of microscopic modules in that volume element. In general relativity an arbitrary fixed measure can be imposed as a gauge condition, while in unimodular relativity it is determined by the events in the volume. Since this seems to break general covariance, some have suggested that it permits a nonzero covariant divergence of the material stress-energy tensor and a variable cosmological “constant.” In Lagrangian unimodular relativity, however, even with higher derivatives of the gravitational field in the dynamics, the usual covariant continuity holds and the cosmological constant is still a constant of integration of the gravitational field equations.

Intertwined isospectral potentials in an arbitrary dimension

Abstract: The method of intertwining with n-dimensional (nD) linear intertwining operator L is used to construct nD isospectral, stationary potentials. It has been proven that the differential part of L is a series in Euclidean algebra generators. Integrability conditions of the consistency equations are investigated and the general form of a class of potentials respecting all these conditions have been specified for each n=2, 3, 4, 5. The most general forms of 2D and 3D isospectral potentials are considered in detail and construction of their hierarchies is exhibited. The followed approach provides coordinate systems which make it possible to perform separation of variables and to apply the known methods of supersymmetric quantum mechanics for 1D systems. It has been shown that in choice of coordinates and L there are a number of alternatives increasing with n that enlarge the set of available potentials. Some salient features of higher dimensional extension as well as some applications of the results are presented.

A family of completely integrable multi-Hamiltonian systems explicitly related to some celebrated equations

Abstract: By introducing a spectral problem with an arbitrary parameter, we derive a Kaup–Newell-type hierarchy of nonlinear evolution equations, which is explicitly related to many important equations such as the Kundu equation, the Kaup–Newell (KN) equation, the Chen–Lee–Liu (CLL) equation, the Gerdjikov–Ivanov (GI) equation, the Burgers equation, the modified Korteweg-deVries (MKdV) equation and the Sharma–Tasso–Olver equation. It is shown that the hierarchy is integrable in Liouville’s sense and possesses multi-Hamiltonian structure. Under the Bargann constraint between the potentials and the eigenfunctions, the spectral problem is nonlinearized as a finite-dimensional completely integrable Hamiltonian system. The involutive representation of the solutions for the Kaup–Newell-type hierarchy is also presented. In addition, an N-fold Darboux transformation of the Kundu equation is constructed with the help of its Lax pairs and a reduction technique. According to the Darboux transformation, the solutions of the Ku...

Fractional differential forms

Abstract: A generalization of exterior calculus is considered by allowing the partial derivatives in the exterior derivative to assume fractional orders. That is, a fractional exterior derivative is defined. This is found to generate new vector spaces of finite and infinite dimension, fractional differential form spaces. The definitions of closed and exact forms are extended to the new fractional form spaces with closure and integrability conditions worked out for a special case. Coordinate transformation rules are also computed. The transformation rules are different from those of the standard exterior calculus due to the properties of the fractional derivative. The metric for the fractional form spaces is given, based on the coordinate transformation rules. All results are found to reduce to those of standard exterior calculus when the order of the coordinate differentials is set to one.

On upper bounds for infinite Prandtl number convection with or without rotation

Abstract: Bounds for the bulk heat transport in Rayleigh–Benard convection for an infinite Prandtl number fluid are derived from the primitive equations. The enhancement of heat transport beyond the minimal conduction value (the Nusselt number Nu) is bounded in terms of the nondimensional temperature difference across the layer (the Rayleigh number Ra) according to Nu⩽cRa2/5, where c<1 is an absolute constant. This rigorous upper limit is uniform in the rotation rate when a Coriolis force, corresponding to the rotating convection problem, is included.

Hofstadter butterfly as quantum phase diagram

Abstract: The Hofstadter butterfly is viewed as a quantum phase diagram with infinitely many phases, labeled by their (integer) Hall conductance, and a fractal structure. We describe various properties of this phase diagram: We establish Gibbs phase rules; count the number of components of each phase, and characterize the set of multiple phase coexistence.