Holomorphic curves from matrices

Membranes holomorphically embedded in flat non-compact space are constructed in terms of the degrees of freedom of an infinite collection of 0-branes. To each holomorphic curve we associate infinite-dimensional matrices which are static solutions to the matrix theory equations of motion, and which can be interpreted as the matrix theory representation of the holomorphically embedded membrane. The problem of finding such matrix representations can be phrased as a problem in geometric quantization, where ϵ ∝ l_P³/R plays the role of the Planck constant and parametrizes families of solutions. The concept of Bergman projection is used as a basic tool, and a local expansion for the action of the projection in inverse powers of curvature is derived. This expansion is then used to compute the required matrices perturbatively in ϵ. The first two terms in the expansion correspond to the standard geometric quantization result and to the result obtained using the metaplectic correction to geometric quantization.     

Response of rotating rings to harmonic and periodic loading and comparison with the inverted problem

The harmonic and periodic forced vibrations of rotating rings are derived and investigated. The modal expansion technique yields the forced solution, which is characterized by four generalized co-ordinates associated with each n (circumferential wave number). The inextensional assumption is presumed, when flexural vibration is the only important component, to reduce the order of the system. The closed form solutions to the harmonic load cases, once concentrated, once distributed, are demonstrated and interpreted. The approach is then extended to periodic loads, where Fourier sine and cosine series is applied. Examples depict the numerical responses to all the cases being derived. The solutions of a stationary ring subjected to traveling loads are also solved for comparison. Their difference is investigated and interpreted from various viewpoints.     

On the possibility of an analytic solution of the three-body problem

Three-body Faddeev equations in the Noyes-Fiedeldey form are rewritten as a matrix analog of a one-dimensional nonrelativistic Schrödinger equation. Unlike the method of K-harmonics, where a similar equation was obtained by expansion of a three-body Schrödinger equation wavefunction into the orthogonal set of functions of two variables (K-harmonics), the use of the Noyes-Fiedeldey form of Faddeev equations allows us to limit ourselves to the expansion in functions of one variable only. The solutions of the above mentioned matrix equation are obtained. These solutions converge uniformly within every interval of continuity of the matrix, which corresponds to the potential of that equation. Their asymptotic behavior for large interparticle distances is discussed. The solutions for the harmonic oscillator, inverse-square, and Coulomb-Kepler potentials are found. It is shown that energy levels in the last case may be calculated from a simple formula which is very similar to the corresponding formula for the two-body Coulomb-Kepler problem. This formula can be easily generalized to the case of n particles interacting with inverse distance potentials.     

Effective Lagrangian for quantum black holes

We discuss the most general effective Lagrangian obtained from the assumption that the degrees of freedom to be quantized, in a black hole, are on the horizon. The effective Lagrangian depends only on the induced metric and the extrinsic curvature of the (fluctuating) horizon, and the possible operators can be arranged in an expansion in powers of M_P1/M, where M_P1 is the Planck mass and M the black hole mass. We perform a semiclassical expansion of the action with a formalism which preserves general covariance explicitly. Quantum fluctuations over the classical solutions are described by a single scalar field living in the (2 + 1)-dimensional world-volume swept by the horizon, with a given coupling to the background geometry. We discuss the resulting field theory and we compute the black hole entropy with our formalism.     

Finite width in out-of-equilibrium propagators and kinetic theory

We derive solutions to the Schwinger–Dyson equations on the Closed-Time-Path for a scalar field in the limit where backreaction is neglected. In Wigner space, the two-point Wightman functions have the curious property that the equilibrium component has a finite width, while the out-of equilibrium component has zero width. This feature is confirmed in a numerical simulation for scalar field theory with quartic interactions. When substituting these solutions into the collision term, we observe that an expansion including terms of all orders in gradients leads to an effective finite-width. Besides, we observe no breakdown of perturbation theory, that is sometimes associated with pinch singularities. The effective width is identical with the width of the equilibrium component. Therefore, reconciliation between the zero-width behaviour and the usual notion in kinetic theory, that the out-of-equilibrium contributions have a finite width as well, is achieved. This result may also be viewed as a generalisation of the fluctuation–dissipation relation to out-of-equilibrium systems with negligible backreaction.     

Effective method for calculating the analytic QCD coupling constant

The analytic running coupling constant α _an for strong interactions is considered for approximations of standard perturbation theory up to the three-loop level. Nonperturbative contributions are singled out explicitly in α _an. They are represented in the form of an expansion in a series in inverse powers of the Euclidean momentum squared. It is shown that two-and three-loop corrections lead to a partial compensation of the nonperturbative one-loop contribution of order 1/q ², which is leading in the ultraviolet region. An efficient method for calculating the analytic running coupling constant for all q>Λ is developed on the basis of the above expansion. A comparative analysis of perturbative and nonperturbative contributions is performed in the infrared region, where the latter play the most important role. A simultaneous consideration of the momentum dependence of α _an and its perturbative component for one-to three-loop cases leads to the conclusion that the analytic running coupling constant is stable with respect to higher corrections and that it depends only slightly on conditions imposed in matching solutions that involve different numbers n _f of active-quark flavors.     

The Resurgence of Instantons: Multi-Cut Stokes Phases and the Painlevé II Equation

Resurgent transseries have recently been shown to be a very powerful construction for completely describing nonperturbative phenomena in both matrix models and topological or minimal strings. These solutions encode the full nonperturbative content of a given gauge or string theory, where resurgence relates every (generalized) multi-instanton sector to each other via large-order analysis. The Stokes phase is the adequate gauge theory phase where a ’t Hooft large N expansion exists and where resurgent transseries are most simply constructed. This paper addresses the nonperturbative study of Stokes phases associated to multi-cut solutions of generic matrix models, constructing nonperturbative solutions for their free energies and exploring the asymptotic large-order behavior around distinct multi-instanton sectors. Explicit formulae are presented for the \({\mathbb{Z}_2}\) symmetric two-cut set-up, addressing the cases of the quartic matrix model in its two-cut Stokes phase; the “triple” Penner potential which yields four-point correlation functions in the AGT framework; and the Painlevé II equation describing minimal superstrings.     

On the ground state and infrared divergences of Goldstone bosons in two dimensions

We study the O(N) invariant Goldstone field theory in two dimensions where rigorous theorems forbid the occurrence of spontaneous symmetry breaking. We argue that for computation of the ground state energy at weak coupling it is still the standard Goldstone perturbation expansion that is applicable. This happens due to cancellation of infrared divergences and we demonstrate this fact explicitly at the two-loop level.     

Generalized asymptotic expansions for coupled wavenumbers in fluid-filled cylindrical shells

Analytical expressions are found for the coupled wavenumbers in an infinite fluid-filled cylindrical shell using the asymptotic methods. These expressions are valid for any general circumferential order (n). The shallow shell theory (which is more accurate at higher frequencies) is used to model the cylinder. Initially, the in vacuo shell is dealt with and asymptotic expressions are derived for the shell wavenumbers in the high- and the low-frequency regimes. Next, the fluid-filled shell is considered. Defining a relevant fluid-loading parameter μ, we find solutions for the limiting cases of small and large μ. Wherever relevant, a frequency scaling parameter along with some ingenuity is used to arrive at an elegant asymptotic expression. In all cases, Poisson's ratio ν is used as an expansion variable. The asymptotic results are compared with numerical solutions of the dispersion equation and the dispersion relation obtained by using the more general Donnell-Mushtari shell theory (in vacuo and fluid-filled). A good match is obtained. Hence, the contribution of this work lies in the extension of the existing literature to include arbitrary circumferential orders (n).     

Higher order solutions of Lieb-Liniger integral equation

We study higher order solutions of Lieb-Liniger integral equation for a one-dimensional δ-function Bose gas. By use of the power series expansion method, the integral equation is solved and the correction terms which improve the Bogoliubov theory are calculated analytically in the weak coupling regime. Physical quantities such as the ground state energy and the chemical potential are represented by a dimensionless parameter γ=c/ρ, where c is the interaction strength and ρ is the number density of particles while the quasi-momentum distribution function is expressed in terms of a dimensionless parameter λ=c/K, where K is the cut-off momentum.     

Effective Field Theory for Fermi Systems in a Large N Expansion

A system of fermions with short-range interactions at finite density is studied using the framework of effective field theory. The effective action formalism for fermions with auxiliary fields leads to a loop expansion in which particle-hole bubbles are resummed to all orders. For spin-independent interactions, the loop expansion is equivalent to a systematic expansion in 1/N, where N is the spin-isospin degeneracy g. Numerical results at next-to-leading order are presented and the connection to the Bose limit of this system is elucidated.     

On the number of solutions for the two-point boundary value problem on Riemannian manifolds

We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M,g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p,q∈M that are non-conjugate in a suitable sense, under the assumption that (M,g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.     

Extended Thomas-Fermi theory at finite temperature

We present a generalization of the extended Thomas-Fermi (ETF) theory to finite temperatures T. Starting from the Wigner-Kirkwood expansion of the Bloch density in powers of , we derive the gradient expansion of the free energy and entropy density functionals $\text{F}$[ρ] and σ[ρ] up to fourth order with their correct temperature-dependent coefficients. (Effective mass and spin-orbit contributions are taken into account up to second order.) For a harmonic-oscillator potential we show that both the h-expansion of the free energy and the entropy and the gradient expansion of the functionals [ρ] and σ[ρ] converge very fast and yield the exact quantum-mechanical results for kT ? 3 MeV, where the shell effects are washed out. Finally we discuss the Euler variational equation obtained with the new functionals and use its numerical solutions for semi-infinite symmetric nuclear matter to test the quality of parametrized trial densities. As an application, we present liquid-drop model parameters, calculated with a realistic Skyrme interaction, as functions of the temperature.     

Approximately analytical solutions of the Manning-Rosen potential with the spin-orbit coupling term and spin symmetry

In this Letter the approximately analytical bound state solutions of the Dirac equation with the Manning-Rosen potential for arbitrary spin-orbit coupling quantum number k are carried out by taking a properly approximate expansion for the spin-orbit coupling term. In the case of exact spin symmetry, the associated two-component spinor wave functions of the Dirac equation for arbitrary spin-orbit quantum number k are presented and the corresponding bound state energy equation is derived. We study briefly two special cases; the general s-wave problem and the equal scalar and vector Manning-Rosen potential.     

Black holes in N=8 supergravity

Solutions of the bosonic field equations of the ungauged, N=8 supergravity which describe black holes with no scalar hairs are obtained. It is found that, in contrast to the Einstein-Maxwell theory where a uniqueness theorem exists, there are two distinct families of black holes in N=8 supergravity. There are also two distinct generalizations of Majumdar-Papapertrou solutions which describe the static equilibrium of many black holes.     

A generalization of Wigner's R-matrix method

A generalization of the Wigner's non-relativistic R-matrix theory of scattering by a central potential field is proposed. The idea is to use an R-matrix expansion basis generated by a Sturm-Liouville problem with an eigenparameter included both in a differential equation and in a boundary condition (in the standard theory an R-matrix basis is obtained by solving an eigenvalue problem with fixed boundary conditions). A partial fraction expansion of an R^(η)-matrix introduced is derived and shown to converge faster than a partial fraction expansion of Wigner's R-matrix used in the standard theory.     

First order green-function theory of isotropic Heisenberg ferromagnet

The Heisenberg ferromagnet with general spin S is considered within Green-function theory and spectral density method. New difference equations of the first order determining one- and two-particle correlation functions are derived and solved. The spectral density method is used to close Oguchi's variational theory without additional decoupling assumptions. The temperature renormalized spectrum is found to be a series expansion in that the first term coincide with RPA result and the first two terms correspond essentially to Callen's result. Low temperature expansions for the renormalization factor and the magnetization are given and shown to coincide with Callen's result.     

Renormalization group and Fermi liquid theory

We give a Hamiltonian-based interpretation of microscopic Fermi liquid theory within a renormalization group framework. The Fermi liquid fixed-point Hamiltonian with its leading-order corrections is identified and we show that the mean field calculations for this model correspond to the Landau phenomenological approach. This is illustrated first of all for the Kondo and Anderson models of magnetic impurities which display Fermi liquid behaviour at low temperatures. We then show how these results can be deduced by a reorganization of perturbation theory, in close parallel to that for the renormalized φ⁴ field theory. The Fermi liquid results follow from the two lowest order diagrams of the renormalized perturbation expansion. The calculations for the impurity models are simpler than for the general case because the self-energy depends on frequency only. We show, however, that a similar renormalized expansion can be derived also for the case of a translationally invariant system. The parameters specifying the Fermi liquid fixed-point Hamiltonian are related to the renormalized vertices appearing in the perturbation theory. The collective zero sound modes appear in the quasiparticle-quasihole ladder sum of the renormalized perturbation expansion. The renormalized perturbation expansion can in principle be used beyond the Fermi liquid regime to higher temperatures. This approach should be particularly useful for heavy fermions and other strongly correlated electron systems, where the renormalization of the single-particle excitations are particularly large.

We briefly look at the breakdown of Fermi liquid theory from a renormalized perturbation theory point of view. We show how a modified version of the approach can be used in some situations, such as the spinless Luttinger model, where Fermi liquid theory is not applicable. Other examples of systems where the Fermi liquid theory breaks down are also briefly discussed.     

Spontaneous compactification in six-dimensional Einstein-Maxwell theory

A discrete set of solutions to the classical Einstein-Maxwell equations in six-dimensional space-time is considered. These solutions have the form of a product of four-dimensional constant curvature space-time with a 2-sphere. The Maxwell field has support on the 2-sphere where it represents a monopole of magnetic charge, n = ±1, ±2, …. The spectrum of massless and massive states is obtained for the special case of flat 4-space, and the solution is shown to be classically stable. The limiting case where the radius of the 2-sphere becomes small is considered and a dimensionally reduced effective lagrangian for the long range modes is derived. This turns out to be an SU(2) × U(1) gauge theory with chiral couplings.