Showing papers in "Communications in Mathematical Physics in 2013"

Gauge Theories and Macdonald Polynomials

Abstract: We study the $${\mathcal{N}=2}$$ four-dimensional superconformal index in various interesting limits, such that only states annihilated by more than one supercharge contribute. Extrapolating from the SU(2) generalized quivers, which have a Lagrangian description, we conjecture explicit formulae for all A-type quivers of class $${\mathcal S}$$ , which in general do not have one. We test our proposals against several expected dualities. The index can always be interpreted as a correlator in a two-dimensional topological theory, which we identify in each limit as a certain deformation of two-dimensional Yang-Mills theory. The structure constants of the topological algebra are diagonal in the basis of Macdonald polynomials of the holonomies.

Wave-Breaking and Peakons for a Modified Camassa-Holm Equation

Abstract: In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.

Bulk-Edge Correspondence for Two-Dimensional Topological Insulators

Abstract: Topological insulators can be characterized alternatively in terms of bulk or edge properties. We prove the equivalence between the two descriptions for two-dimensional solids in the single-particle picture. We give a new formulation of the \({\mathbb{Z}_{2}}\)-invariant, which allows for a bulk index not relying on a (two-dimensional) Brillouin zone. When available though, that index is shown to agree with known formulations. The method also applies to integer quantum Hall systems. We discuss a further variant of the correspondence, based on scattering theory.

Stability of Black Holes and Black Branes

Abstract: We establish a new criterion for the dynamical stability of black holes in D ≥ 4 spacetime dimensions in general relativity with respect to axisymmetric perturbations: Dynamical stability is equivalent to the positivity of the canonical energy, $${\mathcal{E}}$$ , on a subspace, $${\mathcal{T}}$$ , of linearized solutions that have vanishing linearized ADM mass, momentum, and angular momentum at infinity and satisfy certain gauge conditions at the horizon. This is shown by proving that—apart from pure gauge perturbations and perturbations towards other stationary black holes— $${\mathcal{E}}$$ is nondegenerate on $${\mathcal{T}}$$ and that, for axisymmetric perturbations, $${\mathcal{E}}$$ has positive flux properties at both infinity and the horizon. We further show that $${\mathcal{E}}$$ is related to the second order variations of mass, angular momentum, and horizon area by $${\mathcal{E} = \delta^2 M -\sum_A \Omega_A \delta^2 J_A - \frac{\kappa}{8\pi}\delta^2 A}$$ , thereby establishing a close connection between dynamical stability and thermodynamic stability. Thermodynamic instability of a family of black holes need not imply dynamical instability because the perturbations towards other members of the family will not, in general, have vanishing linearized ADM mass and/or angular momentum. However, we prove that for any black brane corresponding to a thermodynamically unstable black hole, sufficiently long wavelength perturbations can be found with $${\mathcal{E} < 0}$$ and vanishing linearized ADM quantities. Thus, all black branes corresponding to thermodynmically unstable black holes are dynamically unstable, as conjectured by Gubser and Mitra. We also prove that positivity of $${\mathcal{E}}$$ on $${\mathcal{T}}$$ is equivalent to the satisfaction of a “ local Penrose inequality,” thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability. Although we restrict our considerations in this paper to vacuum general relativity, most of the results of this paper are derived using general Lagrangian and Hamiltonian methods and therefore can be straightforwardly generalized to allow for the presence of matter fields and/or to the case of an arbitrary diffeomorphism covariant gravitational action.

A Renormalizable 4-Dimensional Tensor Field Theory

Abstract: We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on U(1)4 is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the $${\phi^6}$$ rather than of the $${\phi^4}$$ type, since two different $${\phi^6}$$ -type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous log-divergent $${(\int \phi^2)^2}$$ term, which can be interpreted as the generation of a scalar matter field out of pure gravity.

Quantum Tomography under Prior Information

Abstract: We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information? We show that if the prior information restricts the possible states to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that the minimal number of measurement outcomes (POVM elements) necessary to identify all pure states in a d-dimensional Hilbert space is 4d−3−c(d) α(d) for some \({c(d)\in[1,2]}\) and α(d) being the number of ones appearing in the binary expansion of (d−1).

Stability of Frustration-Free Hamiltonians

Abstract: We prove stability of the spectral gap for gapped, frustration-free Hamiltonians under general, quasi-local perturbations We present a necessary and sufficient condition for stability, which we call Local Topological Quantum Order and show that this condition implies an area law for the entanglement entropy of the groundstate subspace This result extends previous work by Bravyi et al on the stability of topological quantum order for Hamiltonians composed of commuting projections with a common zero-energy subspace We conclude with a list of open problems relevant to spectral gaps and topological quantum order

Minimality via Second Variation for a Nonlocal Isoperimetric Problem

Abstract: We discuss the local minimality of certain configurations for a nonlocal isoperimetric problem used to model microphase separation in diblock copolymer melts. We show that critical configurations with positive second variation are local minimizers of the nonlocal area functional and, in fact, satisfy a quantitative isoperimetric inequality with respect to sets that are L1-close. The link with local minimizers for the diffuse-interface Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative estimate, we get new results concerning periodic local minimizers of the area functional and a proof, via second variation, of the sharp quantitative isoperimetric inequality in the standard Euclidean case. As a further application, we address the global and local minimality of certain lamellar configurations.

Asymptotic Expansion of β Matrix Models in the One-cut Regime

Abstract: We prove the existence of a 1/N expansion to all orders in β matrix models with a confining, offcritical potential corresponding to an equilibrium measure with a connected support. Thus, the coefficients of the expansion can be obtained recursively by the “topological recursion” derived in Chekhov and Eynard (JHEP 0612:026, 2006). Our method relies on the combination of a priori bounds on the correlators and the study of Schwinger-Dyson equations, thanks to the uses of classical complex analysis techniques. These a priori bounds can be derived following (Boutet de Monvel et al. in J Stat Phys 79(3–4):585–611, 1995; Johansson in Duke Math J 91(1):151–204, 1998; Kriecherbauer and Shcherbina in Fluctuations of eigenvalues of matrix models and their applications, 2010) or for strictly convex potentials by using concentration of measure (Anderson et al. in An introduction to random matrices, Sect. 2.3, Cambridge University Press, Cambridge, 2010). Doing so, we extend the strategy of Guionnet and Maurel-Segala (Ann Probab 35:2160–2212, 2007), from the hermitian models (β = 2) and perturbative potentials, to general β models. The existence of the first correction in 1/N was considered in Johansson (1998) and more recently in Kriecherbauer and Shcherbina (2010). Here, by taking similar hypotheses, we extend the result to all orders in 1/N.

Non-Isothermal Boundary in the Boltzmann Theory and Fourier Law

Abstract: In the study of the heat transfer in the Boltzmann theory, the basic problem is to construct solutions to the following steady problem: $$v \cdot abla _{x}F =\frac{1}{{\rm K}_{\rm n}}Q(F,F),\qquad (x,v)\in \Omega \times \mathbf{R}^{3}, \quad \quad (0.1) $$ $$F(x,v)|_{n(x)\cdot v 0}F(x,v^{\prime})(n(x)\cdot v^{\prime})dv^{\prime},\quad x \in\partial \Omega,\quad \quad (0.2) $$ where Ω is a bounded domain in $${\mathbf{R}^{d}, 1 \leq d \leq 3}$$ , Kn is the Knudsen number and $${\mu _{\theta}=\frac{1}{2\pi \theta ^{2}(x)} {\rm exp} [-\frac{|v|^{2}}{2\theta (x)}]}$$ is a Maxwellian with non-constant(non-isothermal) wall temperature θ(x). Based on new constructive coercivity estimates for both steady and dynamic cases, for $${|\theta -\theta_{0}|\leq \delta \ll 1}$$ and any fixed value of Kn, we construct a unique non-negative solution F s to (0.1) and (0.2), continuous away from the grazing set and exponentially asymptotically stable. This solution is a genuine non-equilibrium stationary solution differing from a local equilibrium Maxwellian. As an application of our results we establish the expansion $${F_s=\mu_{\theta_0}+\delta F_{1}+O(\delta ^{2})}$$ and we prove that, if the Fourier law holds, the temperature contribution associated to F 1 must be linear, in the slab geometry.

The Influence of Fractional Diffusion in Fisher-KPP Equations

Abstract: We study the Fisher-KPP equation where the Laplacian is replaced by the generator of a Feller semigroup with power decaying kernel, an important example being the fractional Laplacian. In contrast with the case of the standard Laplacian where the stable state invades the unstable one at constant speed, we prove that with fractional diffusion, generated for instance by a stable Levy process, the front position is exponential in time. Our results provide a mathematically rigorous justification of numerous heuristics about this model.

BPS Quivers and Spectra of Complete N=2 Quantum Field Theories

Abstract: We study the BPS spectra of \({\mathcal{N}=2}\) complete quantum field theories in four dimensions. For examples that can be described by a pair of M5 branes on a punctured Riemann surface we explain how triangulations of the surface fix a BPS quiver and superpotential for the theory. The BPS spectrum can then be determined by solving the quantum mechanics problem encoded by the quiver. By analyzing the structure of this quantum mechanics we show that all asymptotically free examples, Argyres-Douglas models, and theories defined by punctured spheres and tori have a chamber with finitely many BPS states. In all such cases we determine the spectrum.

Bicategories for boundary conditions and for surface defects in 3-d TFT

Abstract: We analyze topological boundary conditions and topological surface defects in three-dimensional topological field theories of Reshetikhin-Turaev type based on arbitrary modular tensor categories. Boundary conditions are described by central functors that lift to trivializations in the Witt group of modular tensor categories. The bicategory of boundary conditions can be described through the bicategory of module categories over any such trivialization. A similar description is obtained for topological surface defects. Using string diagrams for bicategories we also establish a precise relation between special symmetric Frobenius algebras and Wilson lines involving special defects. We compare our results with previous work of Kapustin-Saulina and of Kitaev-Kong on boundary conditions and surface defects in abelian Chern-Simons theories and in Turaev-Viro type TFTs, respectively.

On the continuum limit for discrete NLS with long-range lattice interactions

Abstract: We consider a general class of discrete nonlinear Schrodinger equations (DNLS) on the lattice \({h\mathbb{Z}}\) with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrodinger equation (NLS) on \({\mathbb{R}}\) with the fractional Laplacian (−Δ)α as dispersive symbol. In particular, we obtain that fractional powers \({\frac{1}{2} < \alpha < 1}\) arise from long-range lattice interactions when passing to the continuum limit, whereas the NLS with the usual Laplacian −Δ describes the dispersion in the continuum limit for short-range or quick-decaying interactions (e. g., nearest-neighbor interactions).

Batalin-Vilkovisky Formalism in Perturbative Algebraic Quantum Field Theory

Abstract: On the basis of a thorough discussion of the Batalin-Vilkovisky formalism for classical field theory presented in our previous publication, we construct in this paper the Batalin-Vilkovisky complex in perturbatively renormalized quantum field theory. The crucial technical ingredient is an extended notion of the renormalized time-ordered product as a binary product equivalent to the pointwise product of classical field theory. Originally, in causal perturbation theory, the time-ordered product is understood merely as a sequence of multilinear maps on the space of local functionals. Our extended notion of the renormalized time-ordered product (denoted by \({\cdot_{{}^{\mathcal{T}_{\rm r}}}}\)) is consistent with the old one and we found a subspace of the quantum algebra which is closed with respect to \({\cdot_{{}^{\mathcal{T}_{\rm r}}}}\) . On this space the renormalized Batalin-Vilkovisky algebra is then the classical algebra but written in terms of the time-ordered product, together with an operator which replaces the ill defined graded Laplacian of the unrenormalized theory. We identify it with the anomaly term of the anomalous Master Ward Identity of Brennecke and Dutsch. Contrary to other approaches we do not refer to the path integral formalism and do not need to use regularizations in intermediate steps.

Pair Excitations and the Mean Field Approximation of Interacting Bosons, I

Abstract: In our previous work (Grillakis et al. in Commun Math Phys 294:273–301, 2010; Adv Math 228:1788–1815, 2011) we introduced a correction to the mean field approximation of interacting Bosons. This correction describes the evolution of pairs of particles that leave the condensate and subsequently evolve on a background formed by the condensate. In Grillakis et al. (Adv Math 228:1788–1815, 2011) we carried out the analysis assuming that the interactions are independent of the number of particles N. Here we consider the case of stronger interactions. We offer a new transparent derivation for the evolution of pair excitations. Indeed, we obtain a pair of linear equations describing their evolution. Furthermore, we obtain a priori estimates independent of the number of particles and use these to compare the exact with the approximate dynamics.

Delocalization and Diffusion Profile for Random Band Matrices

Abstract: We consider Hermitian and symmetric random band matrices H = (hxy) in \({d\,\geqslant\,1}\) dimensions. The matrix entries hxy, indexed by \({x,y \in (\mathbb{Z}/L\mathbb{Z})^d}\), are independent, centred random variables with variances \({s_{xy} = \mathbb{E} |h_{xy}|^2}\). We assume that sxy is negligible if |x − y| exceeds the band width W. In one dimension we prove that the eigenvectors of H are delocalized if \({W\gg L^{4/5}}\). We also show that the magnitude of the matrix entries \({|{G_{xy}}|^2}\) of the resolvent \({G=G(z)=(H-z)^{-1}}\) is self-averaging and we compute \({\mathbb{E} |{G_{xy}}|^2}\). We show that, as \({L\to\infty}\) and \({W\gg L^{4/5}}\), the behaviour of \({\mathbb{E} |G_{xy}|^2}\) is governed by a diffusion operator whose diffusion constant we compute. Similar results are obtained in higher dimensions.

On Blowup of Classical Solutions to the Compressible Navier-Stokes Equations

Abstract: In this paper, we study the finite time blow up of smooth solutions to the Compressible Navier-Stokes system when the initial data contain vacuums We prove that any classical solutions of viscous compressible fluids without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group (see Definition 22) The results hold regardless of either the size of the initial data or the far fields being vacuum or not This improves the blowup results of Xin (Comm Pure Appl Math 51:229–240, 1998) by removing the crucial assumptions that the initial density has compact support and the smooth solution has finite total energy Furthermore, the analysis here also yields that any classical solutions of viscous compressible fluids without heat conduction in bounded domains or periodic domains will blow up in finite time, if the initial data have an isolated mass group satisfying some suitable conditions

Enhancement of Near Cloaking Using Generalized Polarization Tensors Vanishing Structures. Part I: The Conductivity Problem

Abstract: The aim of this paper is to provide an original method of constructing very effective near-cloaking structures for the conductivity problem These new structures are such that their first Generalized Polarization Tensors (GPT) vanish We show that this in particular significantly enhances the cloaking effect We then present some numerical examples of Generalized Polarization Tensors vanishing structures

Correlators, Feynman Diagrams, and Quantum No-hair in deSitter Spacetime

Abstract: We provide a parametric representation for position-space Feynman integrals of a massive, self-interacting scalar field in deSitter spacetime, for an arbitrary graph. The expression is given as a multiple contour integral over a kernel whose structure is determined by the set of all trees (or forests) within the graph, and it belongs to a class of generalized hypergeometric functions. We argue from this representation that connected deSitter n-point vacuum correlation functions have exponential decay for large proper time-separation, and also decay for large spatial separation, to arbitrary orders in perturbation theory. Our results may be viewed as an analog of the so-called cosmic-no-hair theorem in the context of a quantized test scalar field. This work has significant overlap with a paper by Marolf and Morrison, which is being released simultaneously.

Recurrence for Discrete Time Unitary Evolutions

Abstract: We consider quantum dynamical systems specified by a unitary operator U and an initial state vector \({\phi}\). In each step the unitary is followed by a projective measurement checking whether the system has returned to the initial state. We call the system recurrent if this eventually happens with probability one. We show that recurrence is equivalent to the absence of an absolutely continuous part from the spectral measure of U with respect to \({\phi}\). We also show that in the recurrent case the expected first return time is an integer or infinite, for which we give a topological interpretation. A key role in our theory is played by the first arrival amplitudes, which turn out to be the (complex conjugated) Taylor coefficients of the Schur function of the spectral measure. On the one hand, this provides a direct dynamical interpretation of these coefficients; on the other hand it links our definition of first return times to a large body of mathematical literature.

Serrin-Type Blowup Criterion for Viscous, Compressible, and Heat Conducting Navier-Stokes and Magnetohydrodynamic Flows

Abstract: This paper establishes a blowup criterion for the three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic (MHD) flows. It is essentially shown that for the Cauchy problem and the initial-boundary-value one of the three-dimensional compressible MHD flows with initial density allowed to vanish, the strong or smooth solution exists globally if the density is bounded from above and the velocity satisfies Serrin’s condition. Therefore, if the Serrin norm of the velocity remains bounded, it is not possible for other kinds of singularities (such as vacuum states vanishing or vacuum appearing in the non-vacuum region or even milder singularities) to form before the density becomes unbounded. This criterion is analogous to the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier-Stokes equations, in particular, it is independent of the temperature and magnetic field and is just the same as that of the barotropic compressible Navier-Stokes equations. As a direct application, it is shown that the same result also holds for the strong or smooth solutions to the three-dimensional full compressible Navier-Stokes system describing the motion of a viscous, compressible, and heat conducting fluid.

Enhancement of Near-Cloaking. Part II: The Helmholtz Equation

Abstract: The aim of this paper is to extend the method of Ammari et al (Commun Math Phys, 2012) to scattering problems We construct very effective near-cloaking structures for the scattering problem at a fixed frequency These new structures are, before using the transformation optics, layered structures and are designed so that their first scattering coefficients vanish Inside the cloaking region, any target has near-zero scattering cross section for a band of frequencies We analytically show that our new construction significantly enhances the cloaking effect for the Helmholtz equation

The Excitation Spectrum for Weakly Interacting Bosons in a Trap

Abstract: We investigate the low-energy excitation spectrum of a Bose gas confined in a trap, with weak long-range repulsive interactions. In particular, we prove that the spectrum can be described in terms of the eigenvalues of an effective one-particle operator, as predicted by the Bogoliubov approximation.

Instanton moduli spaces and bases in coset conformal field theory

Abstract: The recently proposed relation between conformal field theories in two dimensions and supersymmetric gauge theories in four dimensions predicts the existence of the distinguished basis in the space of local fields in CFT. This basis has a number of remarkable properties: one of them is the complete factorization of the coefficients of the operator product expansion. We consider a particular case of the U(r) gauge theory on $${\mathbb{C}^{2}/\mathbb{Z}_{p}}$$ which corresponds to a certain coset conformal field theory and describe the properties of this basis. We argue that in the case p = 2, r = 2 there exist different bases. We give an explicit construction of one of them. For another basis we propose the formula for matrix elements.

Log-Gamma Polymer Free Energy Fluctuations via a Fredholm Determinant Identity

Abstract: We prove that under n1/3 scaling, the limiting distribution as n → ∞ of the free energy of Seppalainen’s log-Gamma discrete directed polymer is GUE Tracy-Widom. The main technical innovation we provide is a general identity between a class of n-fold contour integrals and a class of Fredholm determinants. Applying this identity to the integral formula proved in Corwin et al. (Tropical combinatorics and Whittaker functions. http://arxiv.org/abs/1110.3489v3 [math.PR], 2012) for the Laplace transform of the log-Gamma polymer partition function, we arrive at a Fredholm determinant which lends itself to asymptotic analysis (and thus yields the free energy limit theorem). The Fredholm determinant was anticipated in Borodin and Corwin (Macdonald processes. http://arxiv.org/abs/1111.4408v3 [math.PR], 2012) via the formalism of Macdonald processes yet its rigorous proof was so far lacking because of the nontriviality of certain decay estimates required by that approach.

On the global well-posedness for the Boussinesq system with horizontal dissipation

Abstract: In this paper, we investigate the Cauchy problem for the tridimensional Boussinesq equations with horizontal dissipation. Under the assumption that the initial data is axisymmetric without swirl, we prove the global well-posedness for this system. In the absence of vertical dissipation, there is no smoothing effect on the vertical derivatives. To make up this shortcoming, we first establish a magic relationship between \({\frac{u^{r}}{r}}\) and \({\frac{\omega_\theta}{r}}\) by taking full advantage of the structure of the axisymmetric fluid without swirl and some tricks in harmonic analysis. This together with the structure of the coupling of (1.2) entails the desired regularity.

Phase Transition and Level-Set Percolation for the Gaussian Free Field

Abstract: We consider level-set percolation for the Gaussian free field on \({\mathbb{Z}^{d}}\), d ≥ 3, and prove that, as h varies, there is a non-trivial percolation phase transition of the excursion set above level h for all dimensions d ≥ 3. So far, it was known that the corresponding critical level h*(d) satisfies h*(d) ≥ 0 for all d ≥ 3 and that h*(3) is finite, see Bricmont et al. (J Stat Phys 48(5/6):1249–1268, 1987). We prove here that h*(d) is finite for all d ≥ 3. In fact, we introduce a second critical parameter h** ≥ h*, show that h**(d) is finite for all d ≥ 3, and that the connectivity function of the excursion set above level h has stretched exponential decay for all h > h**. Finally, we prove that h* is strictly positive in high dimension. It remains open whether h* and h** actually coincide and whether h* > 0 for all d ≥ 3.

Thermodyamic Bounds on Drude Weights in Terms of Almost-conserved Quantities

Abstract: We consider one-dimensional translationally invariant quantum spin (or fermionic) lattices and prove a Mazur-type inequality bounding the time-averaged thermodynamic limit of a finite-temperature expectation of a spatio-temporal autocorrelation function of a local observable in terms of quasi-local conservation laws with open boundary conditions. Namely, the commutator between the Hamiltonian and the conservation law of a finite chain may result in boundary terms only. No reference to techniques used in Suzuki’s proof of Mazur bound is made (which strictly applies only to finite-size systems with exact conservation laws), but Lieb-Robinson bounds and exponential clustering theorems of quasi-local C* quantum spin algebras are invoked instead. Our result has an important application in the transport theory of quantum spin chains, in particular it provides rigorous non-trivial examples of positive finite-temperature spin Drude weight in the anisotropic Heisenberg XXZ spin 1/2 chain (Prosen, in Phys Rev Lett 106:217206, 2011).

Stability of Peakons for an Integrable Modified Camassa-Holm Equation with Cubic Nonlinearity

Abstract: Considered herein is the dynamical stability of the single peaked soliton and periodic peaked soliton for an integrable modified Camassa-Holm equation with cubic nonlinearity. The equation is known to admit a single peaked soliton and multi-peakon solutions, and is shown here to possess a periodic peaked soliton. By constructing certain Lyapunov functionals, it is demonstrated that the shapes of these waves are stable under small perturbations in the energy space.