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Showing papers on "Space (mathematics) published in 2017"


Journal ArticleDOI
TL;DR: In this article, the authors present a formula which extracts the spectrum and three-point functions of local operators as an analytic function of spin and converges to the high-energy Regge limit.
Abstract: Conformal theory correlators are characterized by the spectrum and three-point functions of local operators. We present a formula which extracts this data as an analytic function of spin. In analogy with a classic formula due to Froissart and Gribov, it is sensitive only to an “imaginary part” which appears after analytic continuation to Lorentzian signature, and it converges thanks to recent bounds on the high-energy Regge limit. At large spin, substituting in cross-channel data, the formula yields 1/J expansions with controlled errors. In large-N theories, the imaginary part is saturated by single-trace operators. For a sparse spectrum, it manifests the suppression of bulk higher-derivative interactions that constitutes the signature of a local gravity dual in Anti-de-Sitter space.

453 citations


Journal ArticleDOI
TL;DR: In this article, the authors present a formula which extracts the spectrum and three-point functions of local operators as an analytic function of spin and converges to the high-energy Regge limit.
Abstract: Conformal theory correlators are characterized by the spectrum and three- point functions of local operators. We present a formula which extracts this data as an analytic function of spin. In analogy with a classic formula due to Froissart and Gribov, it is sensitive only to an "imaginary part" which appears after analytic continuation to Lorentzian signature, and it converges thanks to recent bounds on the high-energy Regge limit. At large spin, substituting in cross-channel data, the formula yields 1/J expansions with controlled errors. In large-N theories, the imaginary part is saturated by single-trace operators. For a sparse spectrum, it manifests the suppression of bulk higher-derivative interactions that constitutes the signature of a local gravity dual in Anti-de-Sitter space.

343 citations


Journal ArticleDOI
TL;DR: In this article, the exact solution for the scattering problem in the flat space Jackiw-Teitelboim (JT) gravity coupled to an arbitrary quantum field theory was presented.
Abstract: We present the exact solution for the scattering problem in the flat space Jackiw-Teitelboim (JT) gravity coupled to an arbitrary quantum field theory. JT gravity results in a gravitational dressing of field theoretical scattering amplitudes. The exact expression for the dressed $S$-matrix was previously known as a solvable example of a novel UV asymptotic behavior, dubbed asymptotic fragility. This dressing is equivalent to the $T\bar{T}$ deformation of the initial quantum field theory. JT gravity coupled to a single massless boson provides a promising action formulation for an integrable approximation to the worldsheet theory of confining strings in 3D gluodynamics. We also derive the dressed $S$-matrix as a flat space limit of the near $AdS_2$ holography. We show that in order to preserve the flat space unitarity the conventional Schwarzian dressing of boundary correlators needs to be slightly extended. Finally, we propose a new simple expression for flat space amplitudes of massive particles in terms of correlators of holographic CFT's.

232 citations


Journal ArticleDOI
TL;DR: In this paper, the exact solution for the scattering problem in the flat space Jackiw-Teitelboim (JT) gravity coupled to an arbitrary quantum field theory was presented.
Abstract: We present the exact solution for the scattering problem in the flat space Jackiw-Teitelboim (JT) gravity coupled to an arbitrary quantum field theory. JT gravity results in a gravitational dressing of field theoretical scattering amplitudes. The exact expression for the dressed S-matrix was previously known as a solvable example of a novel UV asymptotic behavior, dubbed asymptotic fragility. This dressing is equivalent to the $$ T\overline{T} $$ deformation of the initial quantum field theory. JT gravity coupled to a single mass-less boson provides a promising action formulation for an integrable approximation to the worldsheet theory of confining strings in 3D gluodynamics. We also derive the dressed S-matrix as a flat space limit of the near AdS2 holography. We show that in order to preserve the flat space unitarity the conventional Schwarzian dressing of boundary correlators needs to be slightly extended. Finally, we propose a new simple expression for flat space amplitudes of massive particles in terms of correlators of holographic CFT’s.

229 citations


Journal ArticleDOI
23 Mar 2017
TL;DR: In this article, the concept of extended b-metric space is introduced, inspired by the concepts of b-means and b-space, and some fixed point theorems for self-mappings defined on such spaces are established.
Abstract: In this paper, inspired by the concept of b-metric space, we introduce the concept of extended b-metric space. We also establish some fixed point theorems for self-mappings defined on such spaces. Our results extend/generalize many pre-existing results in literature.

170 citations


Proceedings Article
01 Jan 2017

165 citations


Proceedings Article
01 Jan 2017
TL;DR: This paper describes methods used to reduce storage usage in RocksDB and shows that RocksDB uses less than half the storage that InnoDB uses, yet performs well and in many cases even better than the B-tree-based Inno DB storage engine.
Abstract: RocksDB is an embedded, high-performance, persistent keyvalue storage engine developed at Facebook. Much of our current focus in developing and configuring RocksDB is to give priority to resource efficiency instead of giving priority to the more standard performance metrics, such as response time latency and throughput, as long as the latter remain acceptable. In particular, we optimize space efficiency while ensuring read and write latencies meet service-level requirements for the intended workloads. This choice is motivated by the fact that storage space is most often the primary bottleneck when using Flash SSDs under typical production workloads at Facebook. RocksDB uses log-structured merge trees to obtain significant space efficiency and better write throughput while achieving acceptable read performance. This paper describes methods we used to reduce storage usage in RocksDB. We discuss how we are able to trade off storage efficiency and CPU overhead, as well as read and write amplification. Based on experimental evaluations of MySQL with RocksDB as the embedded storage engine (using TPC-C and LinkBench benchmarks) and based on measurements taken from production databases, we show that RocksDB uses less than half the storage that InnoDB uses, yet performs well and in many cases even better than the B-tree-based InnoDB storage engine. To the best of our knowledge, this is the first time a Log-structured merge treebased storage engine has shown competitive performance when running OLTP workloads at large scale.

149 citations


Journal ArticleDOI
TL;DR: In this article, the authors apply analytic conformal bootstrap ideas in Mellin space to conformal field theories with O(N) symmetry and cubic anisotropy and derive new results up to O(ϵ 3).
Abstract: We apply analytic conformal bootstrap ideas in Mellin space to conformal field theories with O(N) symmetry and cubic anisotropy. We write down the conditions arising from the consistency between the operator product expansion and crossing symmetry in Mellin space. We solve the constraint equations to compute the anomalous dimension and the OPE coefficients of all operators quadratic in the fields in the epsilon expansion. We reproduce known results and derive new results up to O(ϵ 3). For the O(N) case, we also study the large N limit in general dimensions and reproduce known results at the leading order in 1/N.

115 citations


Proceedings ArticleDOI
01 Jul 2017
TL;DR: In this paper, a learning scheme to construct a Hilbert space (i.e., a vector space along its inner product) to address both unsupervised and semi-supervised domain adaptation problems is introduced by learning projections from each domain to a latent space along the Mahalanobis metric.
Abstract: This paper introduces a learning scheme to construct a Hilbert space (i.e., a vector space along its inner product) to address both unsupervised and semi-supervised domain adaptation problems. This is achieved by learning projections from each domain to a latent space along the Mahalanobis metric of the latent space to simultaneously minimizing a notion of domain variance while maximizing a measure of discriminatory power. In particular, we make use of the Riemannian optimization techniques to match statistical properties (e.g., first and second order statistics) between samples projected into the latent space from different domains. Upon availability of class labels, we further deem samples sharing the same label to form more compact clusters while pulling away samples coming from different classes. We extensively evaluate and contrast our proposal against state-of-the-art methods for the task of visual domain adaptation using both handcrafted and deep-net features. Our experiments show that even with a simple nearest neighbor classifier, the proposed method can outperform several state-of-the-art methods benefitting from more involved classification schemes.

113 citations


Journal ArticleDOI
TL;DR: In this paper, a planar tree-level four-point function is computed in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; additionally, for the half BPS operators a co-moving frame is chosen in flavour space.
Abstract: We consider a class of planar tree-level four-point functions in $$ \mathcal{N} $$ = 4 SYM in a special kinematic regime: one BMN operator with two scalar excitations and three half-BPS operators are put onto a line in configuration space; additionally, for the half-BPS operators a co-moving frame is chosen in flavour space. In configuration space, the four-punctured sphere is naturally triangulated by tree-level planar diagrams. We demonstrate on a number of examples that each tile can be associated with a modified hexagon form-factor in such a way as to efficiently reproduce the tree-level four-point function. Our tessellation is not of the OPE type, fostering the hope of finding an independent, integrability-based approach to the computation of planar four-point functions.

110 citations


Journal ArticleDOI
TL;DR: In this paper, the authors proposed a distributional scale setting, which allows one to minimize logarithmic contributions in the boundary terms of the solution, and to obtain the full distributional logrithmic structure from the solution's evolution kernel directly in distribution space.
Abstract: Differential spectra in observables that resolve additional soft or collinear QCD emissions exhibit Sudakov double logarithms in the form of logarithmic plus distributions. Important examples are the total transverse momentum q T in color-singlet production, N -jettiness (with thrust or beam thrust as special cases), but also jet mass and more complicated jet substructure observables. The all-order logarithmic structure of such distributions is often fully encoded in differential equations, so-called (renormalization group) evolution equations. We introduce a well-defined technique of distributional scale setting, which allows one to treat logarithmic plus distributions like ordinary logarithms when solving these differential equations. In particular, this allows one (through canonical scale choices) to minimize logarithmic contributions in the boundary terms of the solution, and to obtain the full distributional logarithmic structure from the solution’s evolution kernel directly in distribution space. We apply this technique to the q T distribution, where the two-dimensional nature of convolutions leads to additional difficulties (compared to one-dimensional cases like thrust), and for which the resummation in distribution (or momentum) space has been a long-standing open question. For the first time, we show how to perform the RG evolution fully in momentum space, thereby directly resumming the logarithms [ln n (q 2 /Q 2)/q 2 ]+ appearing in the physical q T distribution. The resummation accuracy is then solely determined by the perturbative expansion of the associated anomalous dimensions.


Journal ArticleDOI
TL;DR: In this paper, the most general asymptotically flat boundary conditions in three-dimensional Einstein gravity are considered, where the boundary charges and chemical potentials are restricted to the maximal number of independent free functions in the metric.
Abstract: We consider the most general asymptotically flat boundary conditions in three-dimensional Einstein gravity in the sense that we allow for the maximal number of independent free functions in the metric, leading to six towers of boundary charges and six associated chemical potentials. We find as associated asymptotic symmetry algebra an isl(2)_k current algebra. Restricting the charges and chemical potentials in various ways recovers previous cases, such as BMS_3, Heisenberg or Detournay-Riegler, all of which can be obtained as contractions of corresponding AdS_3 constructions. Finally, we show that a flat space contraction can induce an additional Carrollian contraction. As examples we provide two novel sets of boundary conditions for Carroll gravity.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the 3D homogeneous stationary Navier-Stokes equations in the whole space R 3 and proved the Liouville theorem in the marginal case of scaling invariance.


Journal ArticleDOI
TL;DR: In this article, the authors investigate how super-Planckian axions can arise when type IIB 3-form flux is used to restrict a two-axion field space to a one-dimensional winding trajectory.
Abstract: We investigate how super-Planckian axions can arise when type IIB 3-form flux is used to restrict a two-axion field space to a one-dimensional winding trajectory. If one does not attempt to address notoriously complicated issues like Kahler moduli stabilization, SUSY-breaking and inflation, this can be done very explicitly. We show that the presence of flux generates flat monodromies in the moduli space which we therefore call ‘Monodromic Moduli Space’. While we do indeed find long axionic trajectories, these are non-geodesic. Moreover, the length of geodesics remains highly constrained, in spite of the (finite) monodromy group introduced by the flux. We attempt to formulate this in terms of a ‘Moduli Space Size Conjecture’. Interesting mathematical structures arise in that the relevant spaces turn out to be fundamental domains of congruence subgroups of the modular group. In addition, new perspectives on inflation in string theory emerge.

Journal ArticleDOI
TL;DR: In this paper, the authors describe the localization of three dimensional N = 2 supersymmetric theories on compact manifolds, including the squashed sphere, S^3_b, the lens space, and S^2 x S^1.
Abstract: In this review article we describe the localization of three dimensional N=2 supersymmetric theories on compact manifolds, including the squashed sphere, S^3_b, the lens space, S^3_b/Z_p, and S^2 x S^1. We describe how to write supersymmetric actions on these spaces, and then compute the partition functions and other supersymmetric observables by employing the localization argument. We briefly survey some applications of these computations.

Journal ArticleDOI
TL;DR: In this paper, the authors considered the infinite depth water wave equation in two dimensions, without gravity but with surface tension, and proved that small data solutions have at least cubic lifespan while small localized data leads to global solutions.
Abstract: This article is concerned with the incompressible, irrotational infinite depth water wave equation in two space dimensions, without gravity but with surface tension. We consider this problem expressed in position–velocity potential holomorphic coordinates, and prove that small data solutions have at least cubic lifespan while small localized data leads to global solutions.

Journal ArticleDOI
TL;DR: In this paper, the higher-spin algebra behind the type-A cubic couplings of the free O(N ) model was studied in generic dimensions, and it was shown that they coincide with the known structure constants for the unique higher spin algebra.
Abstract: In this article we study the higher-spin algebra behind the type-A cubic couplings recently extracted from the free O(N ) model in generic dimensions, demonstrating that they coincide with the known structure constants for the unique higher-spin algebra in generic dimensions. This provides an explicit check of the holographic reconstruction and of the duality between higher-spin theories and the free O(N ) model in generic dimensions, generalising the result of Giombi and Yin in AdS4. For completeness, we also address the same problem in the flat space for the cubic couplings derived by Metsaev in 1991, which are recovered from the flat limit of the AdS type-A cubic couplings. We observe that both flat and AdS4 higher-spin Lorentz subalgebras coincide, hinting towards the existence of a full higher-spin symmetry behind the flat-space cubic couplings of Metsaev.

Journal ArticleDOI
TL;DR: In this article, the Caffarelli-Silvestre extension of the Riesz kernel has been used to model a system of points in the Euclidean space of dimension 2.
Abstract: We study systems of points in the Euclidean space of dimension interacting via a Riesz kernel and confined by an external potential, in the regime where . We also treat the case of logarithmic interactions in dimensions 1 and 2. Our study includes and retrieves all cases previously studied in Sandier and Serfaty [2D Coulomb gases and the renormalized energy, Ann. Probab. (to appear); 1D log gases and the renormalized energy: crystallization at vanishing temperature (2013)] and Rougerie and Serfaty [Higher dimensional Coulomb gases and renormalized energy functionals, Comm. Pure Appl. Math. (to appear)]. Our approach is based on the Caffarelli–Silvestre extension formula, which allows one to view the Riesz kernel as the kernel of an (inhomogeneous) local operator in the extended space . As , we exhibit a next to leading order term in in the asymptotic expansion of the total energy of the system, where the constant term in factor of depends on the microscopic arrangement of the points and is expressed in terms of a ‘renormalized energy’. This new object is expected to penalize the disorder of an infinite set of points in whole space, and to be minimized by Bravais lattice (or crystalline) configurations. We give applications to the statistical mechanics in the case where temperature is added to the system, and identify an expected ‘crystallization regime’. We also obtain a result of separation of the points for minimizers of the energy.

Journal ArticleDOI
TL;DR: The Morse boundary as discussed by the authors is a quasi-isometry invariant boundary for proper geodesic spaces, which is constructed with rays that identify the "hyperbolic directions" in that space.
Abstract: We introduce a new type of boundary for proper geodesic spaces, called the Morse boundary, that is constructed with rays that identify the "hyperbolic directions" in that space. This boundary is a quasi-isometry invariant and thus produces a well-defined boundary for any finitely generated group. In the case of a proper $\mathrm{CAT}(0)$ space this boundary is the contracting boundary of Charney and Sultan and in the case of a proper Gromov hyperbolic space this boundary is the Gromov boundary. We prove three results about the Morse boundary of Teichm\"uller space. First, we show that the Morse boundary of the mapping class group of a surface is homeomorphic to the Morse boundary of the Teichm\"uller space of that surface. Second, using a result of Leininger and Schleimer, we show that Morse boundaries of Teichm\"uller space can contain spheres of arbitrarily high dimension. Finally, we show that there is an injective continuous map of the Morse boundary of Teichm\"uller space into the Thurston compactification of Teichm\"uller space by projective measured foliations.

Journal ArticleDOI
Joachim Toft1
TL;DR: In this paper, a broad family of test function spaces and their dual (distribution) spaces is considered, including Gelfand-Shilov spaces and a family of Test Function Spaces introduced by Pilipovic.
Abstract: We consider a broad family of test function spaces and their dual (distribution) spaces. The family includes Gelfand–Shilov spaces, and a family of test function spaces introduced by Pilipovic. We deduce different characterizations of such spaces, especially under the Bargmann transform and the Short-time Fourier transform. The family also include a test function space, whose dual space is mapped by the Bargmann transform bijectively to the set of entire functions.

Journal Article
TL;DR: In this article, the focusing cubic nonlinear Schrodinger equation with inverse-square potential in three dimensions was considered and a sharp threshold between scattering and blowup was identified, establishing a result analogous to that of Duyckaerts, Holmer and Roudenko for the standard focusing cubic NLS.
Abstract: We consider the focusing cubic nonlinear Schrodinger equation with inverse-square potential in three space dimensions. We identify a sharp threshold between scattering and blowup, establishing a result analogous to that of Duyckaerts, Holmer, and Roudenko for the standard focusing cubic NLS [7, 11]. We also prove failure of uniform space-time bounds at the

Journal ArticleDOI
TL;DR: In this paper, it was shown that any solution of the energy-critical wave equation in space dimensions 3, 4 or 5, which is bounded in the energy space decouples asymptotically, for a sequence of times going to its maximal time of existence, as a finite sum of modulated solitons and a dispersive term.
Abstract: In this paper, we prove that any solution of the energy-critical wave equation in space dimensions 3, 4 or 5, which is bounded in the energy space decouples asymptotically, for a sequence of times going to its maximal time of existence, as a finite sum of modulated solitons and a dispersive term. This is an important step towards the full soliton resolution in the nonradial case and without any size restrictions. The proof uses a Morawetz estimate very similar to the one known for energy-critical wave maps, a virial type identity and a new channels of energy argument based on a lower bound of the exterior energy for well-prepared initial data.

Journal ArticleDOI
TL;DR: In this article, it was shown that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute.
Abstract: The quantum speed limit is a fundamental upper bound on the speed of quantum evolution. However, the actual mathematical expression of this fundamental limit depends on the choice of a measure of distinguishability of quantum states. We show that quantum speed limits are qualitatively governed by the Schatten-p-norm of the generator of quantum dynamics. Since computing Schatten-p-norms can be mathematically involved, we then develop an alternative approach in Wigner phase space. We find that the quantum speed limit in Wigner space is fully equivalent to expressions in density operator space, but that the new bound is significantly easier to compute. Our results are illustrated for the parametric harmonic oscillator and for quantum Brownian motion.

Journal ArticleDOI
TL;DR: In this article, a two-dimensional super-BMS3 invariant theory dual to three-dimensional asymptotically flat supergravity is constructed, which is described by a constrained or gauged chiral Wess-Zumino-Witten action based on the super-Poincare algebra in the Hamiltonian.
Abstract: The two-dimensional super-BMS3 invariant theory dual to three-dimensional asymptotically flat $$ \mathcal{N}=1 $$ supergravity is constructed. It is described by a constrained or gauged chiral Wess-Zumino-Witten action based on the super-Poincare algebra in the Hamiltonian, respectively the Lagrangian formulation, whose reduced phase space description corresponds to a supersymmetric extension of flat Liouville theory.

Posted Content
TL;DR: In this paper, an optimal mass transport framework on the space of Gaussian mixture models is presented, which leads to a natural way to compare, interpolate and average Gaussian Mixture Models.
Abstract: We present an optimal mass transport framework on the space of Gaussian mixture models, which are widely used in statistical inference. Our method leads to a natural way to compare, interpolate and average Gaussian mixture models. Basically, we study such models on a certain submanifold of probability densities with certain structure. Different aspects of this framework are discussed and several examples are presented to illustrate the results. This method represents our first attempt to study optimal transport problems for more general probability densities with structures.

Journal ArticleDOI
Benjamin Assel1
TL;DR: In this paper, the authors used brane techniques to study the space of vacua of abelian 3d and showed that it has Coulomb-like branches, a Higgs branch and mixed branches, and an extra branch parametrized by monopoles with equal magnetic charges in all U(1) nodes and meson operators.
Abstract: We use brane techniques to study the space of vacua of abelian 3d $$ \mathcal{N}=3 $$ gauge theories. The coordinates on these spaces are the vevs of chiral monopole and meson operators, which are realized in the type IIB brane configuration of the theory by adding semi-infinite (1, k) strings or F1 strings. The study of various brane setups allows us to determine a basis of chiral operators and chiral ring relations relevant to each branch of vacua, leading to the algebraic description of these branches. The method is mostly graphical and does not require actual computations. We apply it and provide explicit results in various examples. For linear quivers we find that the space of vacua has in general a collection of Coulomb-like branches, a Higgs branch and mixed branches. For circular quivers we find an extra branch, the geometric branch, parametrized by monopoles with equal magnetic charges in all U(1) nodes and meson operators. We explain how to include FI and mass deformations. We also study $$ \mathcal{N}=3 $$ theories realized with (p, q) 5-branes.

Journal ArticleDOI
TL;DR: In this article, the authors combine space group representation theory together with the scanning of closed subdomains of the Brillouin zone with Wilson loops to algebraically determine the global band-structure topology.
Abstract: We combine space group representation theory together with the scanning of closed subdomains of the Brillouin zone with Wilson loops to algebraically determine the global band-structure topology. Considering space group No. $19$ as a case study, we show that the energy ordering of the irreducible representations at the high-symmetry points ${\mathrm{\ensuremath{\Gamma}},S,T,U}$ fully determines the global band topology, with all topological classes characterized through their simple and double Dirac points.

Journal ArticleDOI
TL;DR: In this paper, it was shown that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space, there exists an area minimizing disc which moreover has a quasi-conformal parametrization.
Abstract: We solve the classical problem of Plateau in the setting of proper metric spaces. Precisely, we prove that among all disc-type surfaces with prescribed Jordan boundary in a proper metric space there exists an area minimizing disc which moreover has a quasi-conformal parametrization. If the space supports a local quadratic isoperimetric inequality for curves we prove that such a solution is locally Holder continuous in the interior and continuous up to the boundary. Our results generalize corresponding results of Douglas Rado and Morrey from the setting of Euclidean space and Riemannian manifolds to that of proper metric spaces.