Showing papers in "Communications in Mathematical Physics in 2012"

Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops

Abstract: We prove conjecture due to Erickson-Semenoff-Zarembo and Drukker-Gross which relates supersymmetric circular Wilson loop operators in the \({\mathcal N=4}\) supersymmetric Yang-Mills theory with a Gaussian matrix model. We also compute the partition function and give a new matrix model formula for the expectation value of a supersymmetric circular Wilson loop operator for the pure \({\mathcal N=2}\) and the \({\mathcal N=2^*}\) supersymmetric Yang-Mills theory on a four-sphere. A four-dimensional \({\mathcal N=2}\) superconformal gauge theory is treated similarly.

Models for Gapped Boundaries and Domain Walls

Abstract: We define a class of lattice models for two-dimensional topological phases with boundary such that both the bulk and the boundary excitations are gapped. The bulk part is constructed using a unitary tensor category C as in the Levin-Wen model, whereas the boundary is associated with amodule category over C.We also consider domainwalls (or defect lines) between different bulk phases.Adomainwall is transparent to bulk excitations if the corresponding unitary tensor categories are Morita equivalent. Defects of higher codimension will also be studied. In summary, we give a dictionary between physical ingredients of lattice models and tensor-categorical notions.

Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues

Abstract: We consider the ensemble of adjacency matrices of Erdős-Renyi random graphs, i.e. graphs on N vertices where every edge is chosen independently and with probability p ≡ p(N). We rescale the matrix so that its bulk eigenvalues are of order one. Under the assumption $${p N \gg N^{2/3}}$$ , we prove the universality of eigenvalue distributions both in the bulk and at the edge of the spectrum. More precisely, we prove (1) that the eigenvalue spacing of the Erdős-Renyi graph in the bulk of the spectrum has the same distribution as that of the Gaussian orthogonal ensemble; and (2) that the second largest eigenvalue of the Erdős-Renyi graph has the same distribution as the largest eigenvalue of the Gaussian orthogonal ensemble. As an application of our method, we prove the bulk universality of generalized Wigner matrices under the assumption that the matrix entries have at least 4 + e moments.

Automorphic Equivalence within Gapped Phases of Quantum Lattice Systems

Abstract: Gapped ground states of quantum spin systems have been referred to in the physics literature as being ‘in the same phase’ if there exists a family of Hamiltonians H(s), with finite range interactions depending continuously on s Є [0,1], such that for each s, H(s) has a non-vanishing gap above its ground state and with the two initial states being the ground states of H(0) and H(1), respectively. In this work, we give precise conditions under which any two gapped ground states of a given quantum spin system that ’belong to the same phase’ are automorphically equivalent and show that this equivalence can be implemented as a flow generated by an s-dependent interaction which decays faster than any power law (in fact, almost exponentially). The flow is constructed using Hastings’ ‘quasi-adiabatic evolution’ technique, of which we give a proof extended to infinite-dimensional Hilbert spaces. In addition, we derive a general result about the locality properties of the effect of perturbations of the dynamics for quantum systems with a quasi-local structure and prove that the flow, which we call the spectral flow, connecting the gapped ground states in the same phase, satisfies a Lieb-Robinson bound. As a result, we obtain that, in the thermodynamic limit, the spectral flow converges to a co-cycle of automorphisms of the algebra of quasi-local observables of the infinite spin system. This proves that the ground state phase structure is preserved along the curve of models H(s), 0 ≤ s ≤ 1.

From the Ginzburg-Landau Model to Vortex Lattice Problems

Abstract: We introduce a “Coulombian renormalized energy” W which is a logarithmic type of interaction between points in the plane, computed by a “renormalization.” We prove various of its properties, such as the existence of minimizers, and show in particular, using results from number theory, that among lattice configurations the triangular lattice is the unique minimizer. Its minimization in general remains open.

Index Theory of One Dimensional Quantum Walks and Cellular Automata

Abstract: If a one-dimensional quantum lattice system is subject to one step of a reversible discrete-time dynamics, it is intuitive that as much “quantum information” as moves into any given block of cells from the left, has to exit that block to the right. For two types of such systems — namely quantum walks and cellular automata — we make this intuition precise by defining an index, a quantity that measures the “net flow of quantum information” through the system. The index supplies a complete characterization of two properties of the discrete dynamics. First, two systems S 1, S 2 can be “pieced together”, in the sense that there is a system S which acts like S 1 in one region and like S 2 in some other region, if and only if S 1 and S 2 have the same index. Second, the index labels connected components of such systems: equality of the index is necessary and sufficient for the existence of a continuous deformation of S 1 into S 2. In the case of quantum walks, the index is integer-valued, whereas for cellular automata, it takes values in the group of positive rationals. In both cases, the map $${S \mapsto {\rm ind} S}$$ is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems. Systems with trivial index are precisely those which can be realized by partitioned unitaries, and the prototypes of systems with non-trivial index are shifts.

On Pointwise Decay of Linear Waves on a Schwarzschild Black Hole Background

Abstract: We prove sharp pointwise t −3 decay for scalar linear perturbations of a Schwarzschild black hole without symmetry assumptions on the data. We also consider electromagnetic and gravitational perturbations for which we obtain decay rates t −4, and t −6, respectively. We proceed by decomposition into angular momentum l and summation of the decay estimates on the Regge-Wheeler equation for fixed l. We encounter a dichotomy: the decay law in time is entirely determined by the asymptotic behavior of the Regge-Wheeler potential in the far field, whereas the growth of the constants in l is dictated by the behavior of the Regge-Wheeler potential in a small neighborhood around its maximum. In other words, the tails are controlled by small energies, whereas the number of angular derivatives needed on the data is determined by energies close to the top of the Regge-Wheeler potential. This dichotomy corresponds to the well-known principle that for initial times the decay reflects the presence of complex resonances generated by the potential maximum, whereas for later times the tails are determined by the far field. However, we do not invoke complex resonances at all, but rely instead on semiclassical Sigal-Soffer type propagation estimates based on a Mourre bound near the top energy.

A weak spectral condition for the controllability of the bilinear Schrödinger equation with application to the control of a rotating planar molecule

Abstract: In this paper we prove an approximate controllability result for the bilinear Schrodinger equation. This result requires less restrictive non-resonance hypotheses on the spectrum of the uncontrolled Schrodinger operator than those present in the literature. The control operator is not required to be bounded and we are able to extend the controllability result to the density matrices. The proof is based on fine controllability properties of the finite dimensional Galerkin approximations and allows to get estimates for the $L^{1}$ norm of the control. The general controllability result is applied to the problem of controlling the rotation of a bipolar rigid molecule confined on a plane by means of two orthogonal external fields.

A Certain Necessary Condition of Potential Blow up for Navier-Stokes Equations

Abstract: We show that a necessary condition for T to be a potential blow up time is \({\lim olimits_{t\uparrow T} \|v(\cdot,t)\|_{L_3}=\infty}\).

Batalin-Vilkovisky Formalism in the Functional Approach to Classical Field Theory

Abstract: We develop the Batalin-Vilkovisky formalism for classical field theory on generic globally hyperbolic spacetimes. A crucial aspect of our treatment is the incorporation of the principle of local covariance which amounts to formulate the theory without reference to a distinguished spacetime. In particular, this allows a homological construction of the Poisson algebra of observables in classical gravity. Our methods heavily rely on the differential geometry of configuration spaces of classical fields.

Lagrange Structure and Dynamics for Solutions to the Spherically Symmetric Compressible Navier-Stokes Equations

Abstract: The compressible Navier-Stokes system (CNS) with density-dependent viscosity coefficients is considered in multi-dimension, the prototype of the system is the viscous Saint-Venat model for the motion of shallow water. A spherically symmetric weak solution to the free boundary value problem for CNS with stress free boundary condition and arbitrarily large data is shown to exist globally in time with the free boundary separating fluids and vacuum and propagating at finite speed as particle path, which is continuous away from the symmetry center. Detailed regularity and Lagrangian structure of this solution have been obtained. In particular, it is shown that the particle path is uniquely defined starting from any non-vacuum region away from the symmetry center, along which vacuum states shall not form in any finite time and the initial regularities of the solution is preserved. Starting from any non-vacuum point at a later-on time, a particle path is also uniquely defined backward in time, which either reaches at some initial non-vacuum point, or stops at a small middle time and connects continuously with vacuum. In addition, the free boundary is shown to expand outward at an algebraic rate in time, and the fluid density decays and tends to zero almost everywhere away from the symmetry center as the time grows up. This finally leads to the formation of vacuum state almost everywhere as the time goes to infinity.

Twisted Differential String and Fivebrane Structures

Abstract: In the background effective field theory of heterotic string theory, the Green-Schwarz anomaly cancellation mechanism plays a key role. Here we reinterpret it and its magnetic dual version in terms of, differential twisted String- and differential twisted Fivebrane-structures that generalize the notion of Spin-structures and Spin-lifting gerbes and their differential refinement to smooth Spin-connections. We show that all these structures can be encoded in terms of nonabelian cohomology, twisted nonabelian cohomology, and differential twisted nonabelian cohomology, extending the differential generalized abelian cohomology as developed by Hopkins and Singer and shown by Freed to formalize the global description of anomaly cancellation problems in higher gauge theories arising in string theory. We demonstrate that the Green-Schwarz mechanism for the H3-field, as well as its magnetic dual version for the H7-field define cocycles in differential twisted nonabelian cohomology that may be called, respectively, differential twisted Spin(n)-, String(n)- and Fivebrane(n)- structures on target space, where the twist in each case is provided by the obstruction to lifting the classifying map of the gauge bundle through a higher connected cover of U(n) or O(n). We show that the twisted Bianchi identities in string theory can be captured by the (nonabelian) L∞-algebra valued differential form data provided by the differential refinements of these twisted cocycles.

Dynamical Localization in Disordered Quantum Spin Systems

Abstract: We say that a quantum spin system is dynamically localized if the time-evolution of local observables satisfies a zero-velocity Lieb-Robinson bound. In terms of this definition we have the following main results: First, for general systems with short range interactions, dynamical localization implies exponential decay of ground state correlations, up to an explicit correction. Second, the dynamical localization of random xy spin chains can be reduced to dynamical localization of an effective one-particle Hamiltonian. In particular, the isotropic xy chain in random exterior magnetic field is dynamically localized.

Multipermutation Solutions of the Yang–Baxter Equation

Abstract: Set-theoretic solutions of the Yang–Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set X and a function r : X × X → X × X which satisfies the braid relation.

Global Well–Posedness of the 3D Primitive Equations with Partial Vertical Turbulence Mixing Heat Diffusion

Abstract: The three–dimensional incompressible viscous Boussinesq equations, under the assumption of hydrostatic balance, govern the large scale dynamics of atmospheric and oceanic motion, and are commonly called the primitive equations. To overcome the turbulence mixing a partial vertical diffusion is usually added to the temperature advection (or density stratification) equation. In this paper we prove the global regularity of strong solutions to this model in a three-dimensional infinite horizontal channel, subject to periodic boundary conditions in the horizontal directions, and with no-penetration and stress-free boundary conditions on the solid, top and bottom boundaries. Specifically, we show that short time strong solutions to the above problem exist globally in time, and that they depend continuously on the initial data.

Analytic Quasi-Perodic Cocycles with Singularities and the Lyapunov Exponent of Extended Harper’s Model

Abstract: We show how to extend (and with what limitations) Avila’s global theory of analytic SL(2,C) cocycles to families of cocycles with singularities. This allows us to develop a strategy to determine the Lyapunov exponent for the extended Harper’s model, for all values of parameters and all irrational frequencies. In particular, this includes the self-dual regime for which even heuristic results did not previously exist in physics literature. The extension of Avila’s global theory is also shown to imply continuous behavior of the LE on the space of analytic $${M_2(\mathbb{C})}$$ -cocycles. This includes rational approximation of the frequency, which so far has not been available.

Invariant Higher-Order Variational Problems

Abstract: We investigate higher-order geometric k-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in longitudinal studies of Computational Anatomy. Our approach formulates Euler-Poincare theory in higher-order tangent spaces on Lie groups. In particular, we develop the Euler-Poincare formalism for higher-order variational problems that are invariant under Lie group transformations. The theory is then applied to higher-order template matching and the corresponding curves on the Lie group of transformations are shown to satisfy higher-order Euler-Poincare equations. The example of SO(3) for template matching on the sphere is presented explicitly. Various cotangent bundle momentum maps emerge naturally that help organize the formulas. We also present Hamiltonian and Hamilton-Ostrogradsky Lie-Poisson formulations of the higher-order Euler-Poincare theory for applications on the Hamiltonian side.

Dynamical Collapse of Boson Stars

Abstract: We study the time evolution in a system of N bosons with a relativistic dispersion law interacting through a Newtonian gravitational potential with coupling constant G. We consider the mean field scaling where N tends to infinity, G tends to zero and λ = G N remains fixed. We investigate the relation between the many body quantum dynamics governed by the Schrodinger equation and the effective evolution described by a (semi-relativistic) Hartree equation. In particular, we are interested in the super-critical regime of large λ [the sub-critical case has been studied in Elgart and Schlein (Comm Pure Appl Math 60(4):500–545, 2007) and Knowles and Pickl (Commun Math Phys 298(1):101–138, 2010)], where the nonlinear Hartree equation is known to have solutions which blow up in finite time. To inspect this regime, we need to regularize the interaction in the many body Hamiltonian with an N dependent cutoff that vanishes in the limit N → ∞. We show, first, that if the solution of the nonlinear equation does not blow up in the time interval [−T, T], then the many body Schrodinger dynamics (on the level of the reduced density matrices) can be approximated by the nonlinear Hartree dynamics, just as in the sub-critical regime. Moreover, we prove that if the solution of the nonlinear Hartree equation blows up at time T (in the sense that the H 1/2 norm of the solution diverges as time approaches T), then also the solution of the linear Schrodinger equation collapses (in the sense that the kinetic energy per particle diverges) if t → T and, simultaneously, N → ∞ sufficiently fast. This gives the first dynamical description of the phenomenon of gravitational collapse as observed directly on the many body level.

Entanglement of the Antisymmetric State

Abstract: We analyse the entanglement of the antisymmetric state in dimension d×d and present two main results. First, we show that the amount of secrecy that can be extracted from the state is low, more precisely, the distillable key is bounded by O(1/d). Second, we show that the state is highly entangled in the sense that a large number of ebits are needed in order to create the state: entanglement cost is larger than a constant, independent of d. The second result is shown to imply that the regularised relative entropy with respect to separable states is also lower bounded by a constant. Finally, we note that the regularised relative entropy of entanglement is asymptotically continuous in the state. Elementary and advanced facts from the representation theory of the unitary group, including the concept of plethysm, play a central role in the proofs of the main results.

Baxter’s Q-operators and Operatorial Bäcklund Flow for Quantum (Super)-Spin Chains

Abstract: We propose the operatorial Baxter’s TQ-relations in a general form of the operatorial Backlund flow describing the nesting process for the inhomogeneous rational gl(K|M) quantum (super)spin chains with twisted periodic boundary conditions. The full set of Q-operators and T-operators on all levels of nesting is explicitly defined. The results are based on a generalization of the identities among the group characters and their group co-derivatives with respect to the twist matrix, found by one of the authors Kazakov and Vieira (JHEP 0810:050, 2008). Our formalism, based on this new “master” identity, allows a systematic and rather straightforward derivation of the whole set of nested Bethe ansatz equations for the spectrum of quantum integrable spin chains, starting from the R-matrix.

Quantum Subgroups of the Haagerup Fusion Categories

Abstract: We answer three related questions concerning the Haagerup subfactor and its even parts, the Haagerup fusion categories. Namely we find all simple module categories over each of the Haagerup fusion categories (in other words, we find the “quantum subgroups” in the sense of Ocneanu), we find all irreducible subfactors whose principal even part is one of the Haagerup fusion categories, and we compute the Brauer-Picard groupoid of Morita equivalences of the Haagerup fusion categories. In addition to the two even parts of the Haagerup subfactor, there is exactly one more fusion category which is Morita equivalent to each of them. This third fusion category has six simple objects and the same fusion rules as one of the even parts of the Haagerup subfactor, but has not previously appeared in the literature. We also find the full lattice of intermediate subfactors for every irreducible subfactor whose even part is one of these three fusion categories, and we discuss how our results generalize to Izumi subfactors.

Deformations of Quantum Field Theories and Integrable Models

Abstract: Deformations of quantum field theories which preserve Poincare covariance and localization in wedges are a novel tool in the analysis and construction of model theories. Here a general scenario for such deformations is discussed, and an infinite class of explicit examples is constructed on the Borchers-Uhlmann algebra underlying Wightman quantum field theory. These deformations exist independently of the space-time dimension, and contain the recently studied warped convolution deformation as a special case. In the special case of two-dimensional Minkowski space, they can be used to deform free field theories to integrable models with non-trivial S-matrix.

Adiabatic Theorems for Generators of Contracting Evolutions

Abstract: We develop an adiabatic theory for generators of contracting evolution on Banach spaces. This provides a uniform framework for a host of adiabatic theorems ranging from unitary quantum evolutions through quantum evolutions of open systems generated by Lindbladians all the way to classically driven stochastic systems. In all these cases the adiabatic evolution approximates, to lowest order, the natural notion of parallel transport in the manifold of instantaneous stationary states. The dynamics in the manifold of instantaneous stationary states and transversal to it have distinct characteristics: The former is irreversible and the latter is transient in a sense that we explain. Both the gapped and gapless cases are considered. Some applications are discussed.

A Closed Formula for the Asymptotic Expansion of the Bergman Kernel

Abstract: We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formula due to M. Englis and A. Loi.

A Rigorous Justification of the Modulation Approximation to the 2D Full Water Wave Problem.

Abstract: We consider the 2D inviscid incompressible irrotational infinite depth water wave problem neglecting surface tension. Given wave packet initial data of the form \({\epsilon B(\epsilon \alpha)e^{ik \alpha}}\) for k > 0, we show that the modulation of the solution is a profile traveling at group velocity and governed by a focusing cubic nonlinear Schrodinger equation, with rigorous error estimates in Sobolev spaces. As a consequence, we establish existence of solutions of the water wave problem in Sobolev spaces for times of order \({O(\epsilon^{-2})}\) provided the initial data differs from the wave packet by at most \({O(\epsilon^{3/2})}\) in Sobolev spaces. These results are obtained by directly applying modulational analysis to the evolution equation with no quadratic nonlinearity constructed in Wu (Invent Math 117(1):45–135, 2009) and by the energy method.

Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications

Abstract: In this work we study certain invariant measures that can be associated to the time averaged observation of a broad class of dissipative semigroups via the notion of a generalized Banach limit. Consider an arbitrary complete separable metric space X which is acted on by any continuous semigroup {S(t)}t ≥ 0. Suppose that {S(t)}t ≥ 0 possesses a global attractor \({\mathcal{A}}\). We show that, for any generalized Banach limit LIMT → ∞ and any probability distribution of initial conditions \({\mathfrak{m}_0}\), that there exists an invariant probability measure \({\mathfrak{m}}\), whose support is contained in \({\mathcal{A}}\), such that $$\int_{X} \varphi(x) {\rm d}\mathfrak{m}(x) = \underset{t \rightarrow \infty}{\rm LIM}\frac{1}{T} \int_0^T \int_X \varphi(S(t) x) {\rm d}\mathfrak{m}_0(x) {\rm d}t,$$ for all observables φ living in a suitable function space of continuous mappings on X.

Global Well-Posedness of the Energy-Critical Defocusing NLS on \({\mathbb{R} \times \mathbb{T}^3}\)

Abstract: We prove global well-posedness in H1 for the energy-critical defocusing initial-value problem \({(i\partial_t+\Delta_x)u=u|u|^2,\quad u(0)=\phi,}\) in the semiperiodic setting \({x\in\mathbb{R} \times \mathbb{T}^3}\) .

On the Positive Mass Theorem for Manifolds with Corners

Abstract: We study the positive mass theorem for certain non-smooth metrics following P. Miao’s work. Our approach is to smooth the metric using the Ricci flow. As well as improving some previous results on the behaviour of the ADM mass under the Ricci flow, we extend the analysis of the zero mass case to higher dimensions.

The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

Abstract: The Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of a quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains ν n of the n th eigenfunction satisfies n ≥ ν n . Here, we provide a new interpretation for the Courant nodal deficiency d n = n − ν n in the case of quantum graphs. It equals the Morse index — at a critical point — of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning — it is the number of unstable directions in the vicinity of the critical point corresponding to the n th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.

Comparison of Quantum Statistical Models: Equivalent Conditions for Sufficiency

Abstract: A family of probability distributions (i.e. a statistical model) is said to be sufficient for another, if there exists a transition matrix transforming the probability distributions in the former to the probability distributions in the latter. The Blackwell-Sherman-Stein (BSS) Theorem provides necessary and sufficient conditions for one statistical model to be sufficient for another, by comparing their information values in statistical decision problems. In this paper we extend the BSS Theorem to quantum statistical decision theory, where statistical models are replaced by families of density matrices defined on finite-dimensional Hilbert spaces, and transition matrices are replaced by completely positive, trace-preserving maps (i.e. coarse-grainings). The framework we propose is suitable for unifying results that previously were independent, like the BSS theorem for classical statistical models and its analogue for pairs of bipartite quantum states, recently proved by Shmaya. An important role in this paper is played by statistical morphisms, namely, affine maps whose definition generalizes that of coarse-grainings given by Petz and induces a corresponding criterion for statistical sufficiency that is weaker, and hence easier to be characterized, than Petz’s.