scholarly journals On the extraction of the boundary conditions and the boundary ports in second-order field theories

2018 ◽  
Vol 59 (10) ◽  
pp. 102902 ◽  
Author(s):  
Markus Schöberl ◽  
Kurt Schlacher
Keyword(s):  
Boundary Conditions ◽  
Second Order ◽  
Field Theories

scholarly journals Boundary conditions in topological AdS4/CFT3

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Pietro Benetti Genolini ◽  
Matan Grinberg ◽  
Paul Richmond
Keyword(s):  
Field Theory ◽  
Free Energy ◽  
Boundary Conditions ◽  
Smooth Solution ◽  
Three Dimensional ◽  
Field Theories ◽  
Legendre Transformation ◽  
Generating Functional ◽  
Holographic Duals

Abstract We revisit the construction in four-dimensional gauged Spin(4) supergravity of the holographic duals to topologically twisted three-dimensional $$ \mathcal{N} $$ N = 4 field theories. Our focus in this paper is to highlight some subtleties related to preserving supersymmetry in AdS/CFT, namely the inclusion of finite counterterms and the necessity of a Legendre transformation to find the dual to the field theory generating functional. Studying the geometry of these supergravity solutions, we conclude that the gravitational free energy is indeed independent from the metric of the boundary, and it vanishes for any smooth solution.

scholarly journals Sharp Lyapunov-type inequalities for second-order half-linear difference equations with different kinds of boundary conditions

2021 ◽  
Vol 115 (3) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Difference Equation ◽  
Boundary Conditions ◽  
Difference Equations ◽  
Periodic Boundary ◽  
Periodic Boundary Conditions ◽  
Second Order ◽  
Linear Difference Equations ◽  
Order Difference ◽  
Second Order Difference Equation ◽  
Order Difference Equation

AbstractIn this work, we establish optimal Lyapunov-type inequalities for the second-order difference equation with p-Laplacian $$\begin{aligned} \Delta (\left| \Delta u(k-1)\right| ^{p-2}\Delta u(k-1))+a(k)\left| u(k)\right| ^{p-2}u(k)=0 \end{aligned}$$ Δ ( Δ u ( k - 1 ) p - 2 Δ u ( k - 1 ) ) + a ( k ) u ( k ) p - 2 u ( k ) = 0 with Dirichlet, Neumann, mixed, periodic and anti-periodic boundary conditions.

ANOMALY FREE SUPERGRAVITY IN D=10: I) THE BIANCHI IDENTITIES AND THE BOSONIC LAGRANGIAN

1988 ◽  
Vol 03 (04) ◽  
pp. 953-1021 ◽  
Author(s):  
RICCARDO D’AURIA ◽  
PIETRO FRÉ ◽  
MARIO RACITI ◽  
FRANCO RIVA
Keyword(s):  
Differential Equation ◽  
Boundary Conditions ◽  
Variational Equation ◽  
Second Order ◽  
Effective Theory ◽  
Spin Connection ◽  
Bianchi Identities ◽  
First Order ◽  
Interaction Terms ◽  
Starting Point

Using a theorem by Bonora-Pasti and Tonin on the existence of a solution for D=10N=1 Bianchi identities in the presence of a Lorentz Chern Simons term, we find an explicit parametrization of the superspace curvatures. Our solution depends only on one free parameter which can be reabsorbed in a field redefinition of the dilaton and of the gravitello. We emphasize that the essential point which enables us to obtain a closed form for the curvature parametrizations and hence for the supersymmetry transformation rules is the use of first order formalism. The spin connection is known once the torsion is known. This latter, rather than being identified with Hµνρ as it is usually done in the literature, is related to it by a differential equation which reduces to the algebraic relation Hµνρ = - 3Tµνρ e4/3σ only at γ1=0 (γ1 being proportional to κ/g2). The solution of the Bianchi identities exhibited in this paper corresponds to a D=10 anomaly free supergravity (AFS). This theory is unique in first order formalism but corresponds to various theories in second order formalism. Indeed the torsion equation is a differential equation which, in order to be solved must be supplemented with boundary conditions. One wonders whether supplemented with a judicious choice of boundary conditions for the torsion equation, AFS yields all the interaction terms found in the effective theory of the heterotic string (ETHS). In this respect two remarks are in order. Firstly it appears that solving the torsion equation iteratively with Tµνρ = -1/3Hµνρ e-4/3σ as starting point all the terms of ETHS except those with a ζ(3) coefficient show up. (Whether the coefficient agree is still to be checked.) Secondly, as shown in this paper the rheonomic solution of the super Poincaré Bianchi identities is unique. Hence additional interaction terms can be added to the Lagrangian only by modifying the rheonomic parametrization of the [Formula: see text]-curvature. The only assumption made in our paper is that [Formula: see text] has at most ψ∧ψ∧V components (sector (1,2)). Correspondingly the only room left for a modification of the present theory is the addition of a (0, 3) part in the rheonomic parametrization of [Formula: see text]. When this work was already finished a conjecture was published by Lechner Pasti and Tonin that such a generalization of AFS might exist and be responsible for the ζ(3) missing term. Indeed if we were able to solve the [Formula: see text]-Bianchi with this new (0, 3)-part then the torsion equation would be modified via new terms which, in second order formalism, lead to additional gravitational interactions. The equation of motion of Anomaly Free Supergravity can be worked out from the Bianchi identities: we indicate through which steps. The corresponding Lagrangian could be constructed with the standard procedures of the rheonomy approach. In this paper we limit ourselves to the bosonic sector of such a Lagrangian and we show that it can indeed be constructed in such a way as to produce the relation between Hµνρ and Tµνρ as a variational equation.

A mechanical method for the solution of second-order linear differential equations

1931 ◽  
Vol 27 (4) ◽  
pp. 546-552 ◽  
Author(s):  
E. C. Bullard ◽  
P. B. Moon
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Order Differential Equation ◽  
Second Order ◽  
Linear Differential Equations ◽  
Mechanical Method ◽  
Order Linear ◽  
Second Order Differential Equation ◽  
Linear Differential

A mechanical method of integrating a second-order differential equation, with any boundary conditions, is described and its applications are discussed.

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