Showing papers in "Communications in Mathematical Physics in 2007"

An Index for 4 dimensional super conformal theories

Abstract: We present a trace formula for an index over the spectrum of four dimensional superconformal field theories on S 3 × time. Our index receives contributions from states invariant under at least one supercharge and captures all information – that may be obtained purely from group theory – about protected short representations in 4 dimensional superconformal field theories. In the case of the $${\mathcal N}=4$$ theory our index is a function of four continuous variables. We compute it at weak coupling using gauge theory and at strong coupling by summing over the spectrum of free massless particles in AdS 5 × S 5 and find perfect agreement at large N and small charges. Our index does not reproduce the entropy of supersymmetric black holes in AdS 5, but this is not a contradiction, as it differs qualitatively from the partition function over supersymmetric states of the $${\mathcal N}=4$$ theory. We note that entropy for some small supersymmetric AdS 5 black holes may be reproduced via a D-brane counting involving giant gravitons. For big black holes we find a qualitative (but not exact) agreement with the naive counting of BPS states in the free Yang Mills theory. In this paper we also evaluate and study the partition function over the chiral ring in the $${\mathcal N}=4$$ Yang Mills theory.

Efficient Quantum Algorithms for Simulating Sparse Hamiltonians

Abstract: We present an efficient quantum algorithm for simulating the evolution of a quantum state for a sparse Hamiltonian H over a given time t in terms of a procedure for computing the matrix entries of H. In particular, when H acts on n qubits, has at most a constant number of nonzero entries in each row/column, and ||H|| is bounded by a constant, we may select any positive integer k such that the simulation requires O((log*n)t1+1/2k) accesses to matrix entries of H. We also show that the temporal scaling cannot be significantly improved beyond this, because sublinear time scaling is not possible.

Least Energy Solitary Waves for a System of Nonlinear Schrödinger Equations in \({\mathbb{R}^n}\)

Abstract: In this paper we consider systems of coupled Schrodinger equations which appear in nonlinear optics. The problem has been considered mostly in the one-dimensional case. Here we make a rigorous study of the existence of least energy standing waves (solitons) in higher dimensions. We give: conditions on the parameters of the system under which it possesses a solution with least energy among all multi-component solutions; conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie.

A Higher dimensional stationary rotating black hole must be axisymmetric

Abstract: A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is “rotating”—i.e., is such that the stationary Killing field is not everywhere normal to the horizon—must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, P. This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.

Full-Wave Invisibility of Active Devices at All Frequencies

Abstract: There has recently been considerable interest in the possibility, both theoretical and practical, of invisibility (or “cloaking”) from observation by electromagnetic (EM) waves. Here, we prove invisibility with respect to solutions of the Helmholtz and Maxwell’s equations, for several constructions of cloaking devices. The basic idea, as in the papers [GLU2, GLU3, Le, PSS1], is to use a singular transformation that pushes isotropic electromagnetic parameters forward into singular, anisotropic ones. We define the notion of finite energy solutions of the Helmholtz and Maxwell’s equations for such singular electromagnetic parameters, and study the behavior of the solutions on the entire domain, including the cloaked region and its boundary. We show that, neglecting dispersion, the construction of [GLU3, PSS1] cloaks passive objects, i.e., those without internal currents, at all frequencies k. Due to the singularity of the metric, one needs to work with weak solutions. Analyzing the behavior of such solutions inside the cloaked region, we show that, depending on the chosen construction, there appear new “hidden” boundary conditions at the surface separating the cloaked and uncloaked regions. We also consider the effect on invisibility of active devices inside the cloaked region, interpreted as collections of sources and sinks or internal currents. When these conditions are overdetermined, as happens for Maxwell’s equations, generic internal currents prevent the existence of finite energy solutions and invisibility is compromised.

Estimates of Heat Kernel of Fractional Laplacian Perturbed by Gradient Operators

Abstract: We construct a continuous transition density of the semigroup generated by $${\Delta^{\alpha/2} + b(x)\cdot abla}$$ for $${1 < \alpha < 2, d\ge 1}$$ and b in the Kato class $${\mathcal{K}_d^{\alpha-1}}$$ on $${\mathbb{R}^d}$$ . For small time the transition density is comparable with that of the fractional Laplacian.

The Largest Eigenvalue of Rank One Deformation of Large Wigner Matrices

Abstract: The purpose of this paper is to establish universality of the fluctuations of the largest eigenvalue for some non-necessarily Gaussian complex Deformed Wigner Ensembles. The real model is also considered. Our approach is close to the one used by A. Soshnikov (cf. [11]) in the investigations of classical real or complex Wigner Ensembles. It is based on the computation of moments of traces of high powers of the random matrices under consideration.

Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm Equation

Abstract: It is shown that a strong solution of the Camassa-Holm equation, initially decaying exponentially together with its spacial derivative, must be identically equal to zero if it also decays exponentially at a later time. In particular, a strong solution of the Cauchy problem with compact initial profile can not be compactly supported at any later time unless it is the zero solution.

A New Bernstein’s Inequality and the 2D Dissipative Quasi-Geostrophic Equation

Abstract: We show a new Bernstein’s inequality which generalizes the results of Cannone-Planchon, Danchin and Lemarie-Rieusset As an application of this inequality, we prove the global well-posedness of the 2D quasi-geostrophic equation with the critical and super-critical dissipation for the small initial data in the critical Besov space, and local well-posedness for the large initial data

One-and-a-half quantum de Finetti theorems

Abstract: When n − k systems of an n-partite permutation-invariant state are traced out, the resulting state can be approximated by a convex combination of tensor product states. This is the quantum de Finetti theorem. In this paper, we show that an upper bound on the trace distance of this approximation is given by \({2\frac{kd^2}{n}}\) , where d is the dimension of the individual system, thereby improving previously known bounds. Our result follows from a more general approximation theorem for representations of the unitary group. Consider a pure state that lies in the irreducible representation \({U_{\mu + u} \subset U_\mu \otimes U_ u}\) of the unitary group U(d), for highest weights μ, ν and μ + ν. Let ξμ be the state obtained by tracing out Uν. Then ξμ is close to a convex combination of the coherent states \({U_\mu(g)|{v_\mu\rangle}}\) , where \({g\in U(d)}\) and \({|v_\mu\rangle}\) is the highest weight vector in Uμ.

Remarks about the Inviscid Limit of the Navier–Stokes System

Abstract: In this paper we prove two results about the inviscid limit of the Navier-Stokes system. The first one concerns the convergence in H s of a sequence of solutions to the Navier-Stokes system when the viscosity goes to zero and the initial data is in H s . The second result deals with the best rate of convergence for vortex patch initial data in 2 and 3 dimensions. We present here a simple proof which also works in the 3D case. The 3D case is new.

Obstructions to the Existence of Sasaki–Einstein Metrics

Abstract: We describe two simple obstructions to the existence of Ricci-flat Kahler cone metrics on isolated Gorenstein singularities or, equivalently, to the existence of Sasaki-Einstein metrics on the links of these singularities. In particular, this also leads to new obstructions for Kahler–Einstein metrics on Fano orbifolds. We present several families of hypersurface singularities that are obstructed, including 3-fold and 4-fold singularities of ADE type that have been studied previously in the physics literature. We show that the AdS/CFT dual of one obstruction is that the R–charge of a gauge invariant chiral primary operator violates the unitarity bound.

Large n Limit of Gaussian Random Matrices with External Source, Part III: Double Scaling Limit

Abstract: We consider the double scaling limit in the random matrix ensemble with an external source $${1\over{Z_n}} e^{-n \hbox{Tr}({1\over 2}M^2 -AM)} dM$$ defined on n × n Hermitian matrices, where A is a diagonal matrix with two eigenvalues ±a of equal multiplicities. The value a = 1 is critical since the eigenvalues of M accumulate as n → ∞ on two intervals for a > 1 and on one interval for 0 < a < 1. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case a = 1 new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Brezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a 3 × 3-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.

Boson Stars as Solitary Waves

Abstract: We study the nonlinear equation i theta t psi = (root-Delta+m(2) - m) psi - (vertical bar x vertical bar(-1) * vertical bar psi vertical bar(2)) psi on R-3, which is known to describe the dynamics ...

Generalized Kähler manifolds and off-shell supersymmetry

Abstract: We solve the long standing problem of finding an off-shell supersymmetric formulation for a general N = (2, 2) nonlinear two dimensional sigma model. Geometrically the problem is equivalent to proving the existence of special coordinates; these correspond to particular superfields that allow for a superspace description. We construct and explain the geometric significance of the generalized Kahler potential for any generalized Kahler manifold; this potential is the superspace Lagrangian.

Fast soliton scattering by delta impurities

Abstract: We study the Gross-Pitaevskii equation with a repulsive delta function potential. We show that a high velocity incoming soliton is split into a transmitted component and a reflected component. The transmitted mass (L 2 norm squared) is shown to be in good agreement with the quantum transmission rate of the delta function potential. We further show that the transmitted and reflected components resolve into solitons plus dispersive radiation, and quantify the mass and phase of these solitons.

Hidden Grassmann Structure in the XXZ Model

Abstract: For the critical XXZ model, we consider the space \(\mathcal{W}_{[\alpha]}\) of operators which are products of local operators with a disorder operator. We introduce two anti-commutative families of operators \({\bf {b}}(\zeta), {\bf {c}}(\zeta)\) which act on \(\mathcal{W}_{[\alpha]}\). 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 \(\mathcal{W}_{[\alpha]}\) can be expressed in terms of an exponential of a quadratic form of \({\bf {b}}(\zeta), {\bf {c}}(\zeta)\).

Contour Dynamics of Incompressible 3-D Fluids in a Porous Medium with Different Densities

Abstract: We consider the problem of the evolution of the interface given by two incompressible fluids through a porous medium, which is known as the Muskat problem and in two dimensions it is mathematically analogous to the two-phase Hele–Shaw cell. We focus on a fluid interface given by a jump of densities, being the equation of the evolution obtained using Darcy’s law. We prove local well-posedness when the smaller density is above (stable case) and in the unstable case we show ill-posedness.

On the Spectra of Carbon Nano-Structures

Abstract: An explicit derivation of dispersion relations and spectra for periodic Schrodinger operators on carbon nano-structures (including graphene and all types of single-wall nano-tubes) is provided.

Opening Mirror Symmetry on the Quintic

Abstract: Aided by mirror symmetry, we determine the number of holomorphic disks ending on the real Lagrangian in the quintic threefold. We hypothesize that the tension of the domainwall between the two vacua on the brane, which is the generating function for the open Gromov-Witten invariants, satisfies a certain extension of the Picard-Fuchs differential equation governing periods of the mirror quintic. We verify consistency of the monodromies under analytic continuation of the superpotential over the entire moduli space. We further check the conjecture by reproducing the first few instanton numbers by a localization computation directly in the A-model, and verifying Ooguri-Vafa integrality. This is the first exact result on open string mirror symmetry for a compact Calabi-Yau manifold.

Finite-time blow-up of solutions of an aggregation equation in R n

Abstract: We consider the aggregation equation $$u_t + abla \cdot(u abla K\,*\,u) = 0$$ in R n , n ≥ 2, where K is a rotationally symmetric, nonnegative decaying kernel with a Lipschitz point at the origin, e.g. K(x) = e −|x|. We prove finite-time blow-up of solutions from specific smooth initial data, for which the problem is known to have short time existence of smooth solutions.

The Beale-Kato-Majda Criterion for the 3D Magneto-Hydrodynamics Equations

Abstract: We study the blow-up criterion of smooth solutions to the 3D MHD equations. By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda type blow-up criterion of smooth solutions via the vorticity of velocity only, namely $$\sup_{j\in\mathbb{Z}}\int_0^T\|\Delta_j( abla\times u)\|_\infty dt,$$ where Δ j is the frequency localization operator in the Littlewood-Paley decomposition.

Vanishing Shear Viscosity Limit in the Magnetohydrodynamic Equations

Abstract: We study an initial boundary value problem for the equations of plane magnetohydrodynamic compressible flows, and prove that as the shear viscosity goes to zero, global weak solutions converge to a solution of the original equations with zero shear viscosity As a by-product, this paper improves the related results obtained by Frid and Shelukhin for the case when the magnetic effect is neglected

Determining a Magnetic Schrödinger Operator from Partial Cauchy Data

Abstract: In this paper we show, in dimension n ≥ 3, that knowledge of the Cauchy data for the Schrodinger equation in the presence of a magnetic potential, measured on possibly very small subsets of the boundary, determines uniquely the magnetic field and the electric potential. We follow the general strategy of [7] using a richer set of solutions to the Dirichlet problem that has been used in previous works on this problem.

Some Geometric Calculations on Wasserstein Space

Abstract: We compute the Riemannian connection and curvature for the Wasserstein space of a smooth compact Riemannian manifold.

Interior Regularity Criteria for Suitable Weak Solutions of the Navier-Stokes Equations

Abstract: We present new interior regularity criteria for suitable weak solutions of the 3-D Navier-Stokes equations: a suitable weak solution is regular near an interior point z if either the scaled \({L^{p,q}_{x,t}}\) -norm of the velocity with 3/p + 2/q ≤ 2, 1 ≤ q ≤ ∞, or the \({L^{p,q}_{x,t}}\) -norm of the vorticity with 3/p + 2/q ≤ 3, 1 ≤ q < ∞, or the \({L^{p,q}_{x,t}}\) -norm of the gradient of the vorticity with 3/p + 2/q ≤ 4, 1 ≤ q, 1 ≤ p, is sufficiently small near z.

Cluster Expansion for Abstract Polymer Models. New Bounds from an Old Approach

Abstract: We revisit the classical approach to cluster expansions, based on tree graphs, and establish a new convergence condition that improves those by Kotecký-Preiss and Dobrushin, as we show in some examples. The two ingredients of our approach are: (i) a careful consideration of the Penrose identity for truncated functions, and (ii) the use of iterated transformations to bound tree-graph expansions.

The Uncertainty of Fluxes

Abstract: In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3-manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We also develop the Hamilton theory of self-dual fields, noting that they are quantized by Pontrjagin self-dual cohomology theories and that the quantum Hilbert space is \({\mathbb{Z}/2\mathbb{Z}}\) -graded, so typically contains both bosonic and fermionic states. Significantly, these ideas apply to the Ramond-Ramond field in string theory, showing that its K-theory class cannot be measured.

Dimers on Surface Graphs and Spin Structures. II

Abstract: Partition functions for dimers on closed oriented surfaces are known to be alternating sums of Pfaffians of Kasteleyn matrices. In this paper, we obtain the formula for the coefficients in terms of discrete spin structures.

Existence and Regularity of Weak Solutions to the Quasi-Geostrophic Equations in the Spaces L p or $$\dot{H}^{-1/2}$$

Abstract: In this paper we study the 2D quasi-geostrophic equation with and without dissipation. We give global existence results of weak solutions for an initial data in the space Lp or \(\dot{H}^{-1/2}\) . In the dissipative case, when the initial data is in Lp, p > 2, we give a regularity result of these solutions.