scholarly journals Why boundary conditions do not generally determine the universality class for boundary critical behavior

2020 ◽  
Vol 93 (10) ◽  
Author(s):  
Hans Werner Diehl
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Universality Class ◽  
Field Theories ◽  
Boundary Field ◽  
Universality Classes ◽  
The Continuum

Abstract Interacting field theories for systems with a free surface frequently exhibit distinct universality classes of boundary critical behaviors depending on gross surface properties. The boundary condition satisfied by the continuum field theory on some scale may or may not be decisive for the universality class that applies. In many recent papers on boundary field theories, it is taken for granted that Dirichlet or Neumann boundary conditions decide whether the ordinary or special boundary universality class is observed. While true in a certain sense for the Dirichlet boundary condition, this is not the case for the Neumann boundary condition. Building on results that have been worked out in the 1980s, but have not always been appropriately appreciated in the literature, the subtle role of boundary conditions and their scale dependence is elucidated and the question of whether or not they determine the observed boundary universality class is discussed. Graphical abstract

scholarly journals Abelian mirror symmetry of $$ \mathcal{N} $$ = (2, 2) boundary conditions

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Tadashi Okazaki
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Mirror Symmetry ◽  
Dirichlet Boundary ◽  
Triangular Matrix ◽  
Neumann Boundary ◽  
Dirichlet Boundary Conditions ◽  
Verma Modules ◽  
Abelian Gauge ◽  
Mirror Theory

Abstract We evaluate half-indices of $$ \mathcal{N} $$ N = (2, 2) half-BPS boundary conditions in 3d $$ \mathcal{N} $$ N = 4 supersymmetric Abelian gauge theories. We confirm that the Neumann boundary condition is dual to the generic Dirichlet boundary condition for its mirror theory as the half-indices perfectly match with each other. We find that a naive mirror symmetry between the exceptional Dirichlet boundary conditions defining the Verma modules of the quantum Coulomb and Higgs branch algebras does not always hold. The triangular matrix obtained from the elliptic stable envelope describes the precise mirror transformation of a collection of half-indices for the exceptional Dirichlet boundary conditions.

scholarly journals Charge algebra in Al(A)dSn spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Adrien Fiorucci ◽  
Romain Ruzziconi
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Three Dimensional ◽  
Dirichlet Boundary Conditions ◽  
Boundary Structure ◽  
Asymptotic Symmetry ◽  
Renormalization Procedure ◽  
Field Dependent ◽  
Odd Dimensions

Abstract The gravitational charge algebra of generic asymptotically locally (A)dS spacetimes is derived in n dimensions. The analysis is performed in the Starobinsky/Fefferman-Graham gauge, without assuming any further boundary condition than the minimal falloffs for conformal compactification. In particular, the boundary structure is allowed to fluctuate and plays the role of source yielding some symplectic flux at the boundary. Using the holographic renormalization procedure, the divergences are removed from the symplectic structure, which leads to finite expressions. The charges associated with boundary diffeomorphisms are generically non-vanishing, non-integrable and not conserved, while those associated with boundary Weyl rescalings are non-vanishing only in odd dimensions due to the presence of Weyl anomalies in the dual theory. The charge algebra exhibits a field-dependent 2-cocycle in odd dimensions. When the general framework is restricted to three-dimensional asymptotically AdS spacetimes with Dirichlet boundary conditions, the 2-cocycle reduces to the Brown-Henneaux central extension. The analysis is also specified to leaky boundary conditions in asymptotically locally (A)dS spacetimes that lead to the Λ-BMS asymptotic symmetry group. In the flat limit, the latter contracts into the BMS group in n dimensions.

The Heat Equation for the -Neumann Problem, II

1988 ◽  
Vol 40 (2) ◽  
pp. 502-512 ◽  
Author(s):  
Richard Beals ◽  
Nancy K. Stanton
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Neumann Problem ◽  
Elliptic Boundary ◽  
Neumann Boundary ◽  
Hermitian Manifold ◽  
Neumann Boundary Conditions ◽  
Compact Manifolds ◽  
Elliptic Boundary Value ◽  
Image Position

Let Ω be a compact complex n + 1-dimensional Hermitian manifold with smooth boundary M. In [2] we proved the following.THEOREM 1. Suppose satisfies condition Z(q) with 0 ≦ q ≦ n. Let □p,q denote the -Laplacian on (p, q) forms onwhich satisfy the -Neumann boundary conditions. Then as t → 0;,(0.1)(If q = n + 1, the -Neumann boundary condition is the Dirichlet boundary condition and the corresponding result is classical.)Theorem 1 is a version for the -Neumann problem of results initiated by Minakshisundaram and Pleijel [8] for the Laplacian on compact manifolds and extended by McKean and Singer [7] to the Laplacian with Dirichlet or Neumann boundary conditions and by Greiner [5] and Seeley [9] to elliptic boundary value problems on compact manifolds with boundary. McKean and Singer go on to show that the coefficients in the trace expansion are integrals of local geometric invariants.

Gradient Estimates for Spacelike Mean Curvature Flow with Boundary Conditions

2018 ◽  
Vol 62 (2) ◽  
pp. 459-469
Author(s):  
Ben Lambert
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Mean Curvature ◽  
Neumann Boundary Condition ◽  
Mean Curvature Flow ◽  
Neumann Boundary ◽  
Curvature Flow ◽  
Gradient Estimate ◽  
Gradient Estimates ◽  
The Mean

AbstractWe prove a gradient estimate for graphical spacelike mean curvature flow with a general Neumann boundary condition in dimension n = 2. This then implies that the mean curvature flow exists for all time and converges to a translating solution.

On the Number of Solutions to Semilinear Boundary Value Problems

2004 ◽  
Vol 4 (3) ◽  
Author(s):  
Markus Kunze ◽  
Rafael Ortega
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Upper Bound ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Dirichlet Boundary Conditions ◽  
Number Of Solutions ◽  
Semilinear Elliptic ◽  
Semilinear Elliptic Problems

AbstractWe consider semilinear elliptic problems of the form Δu + g(u) = f(x) with Neumann boundary conditions or Δu+λ1u+g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.

scholarly journals A note on boundary conditions in Euclidean gravity

2021 ◽  
pp. 2140004
Author(s):  
Edward Witten
Keyword(s):  
Boundary Condition ◽  
General Relativity ◽  
Perturbation Theory ◽  
Quantum Gravity ◽  
Boundary Conditions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Conformal Geometry ◽  
Extrinsic Curvature ◽  
Euclidean Signature

We review what is known about boundary conditions in General Relativity on a spacetime of Euclidean signature. The obvious Dirichlet boundary condition, in which one specifies the boundary geometry, is actually not elliptic and in general does not lead to a well-defined perturbation theory. It is better-behaved if the extrinsic curvature of the boundary is suitably constrained, for instance if it is positive- or negative-definite. A different boundary condition, in which one specifies the conformal geometry of the boundary and the trace of the extrinsic curvature, is elliptic and always leads formally to a satisfactory perturbation theory. These facts might have interesting implications for semiclassical approaches to quantum gravity. April, 2018

scholarly journals Non-universal Casimir Forces at Approach to Bose–Einstein Condensation of an Ideal Gas: Effect of Dirichlet Boundary Conditions

2020 ◽  
Vol 181 (3) ◽  
pp. 944-951
Author(s):  
M. Napiórkowski ◽  
J. Piasecki ◽  
J. W. Turner
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Neumann Boundary ◽  
Ideal Gas ◽  
Einstein Condensate ◽  
Dirichlet Boundary Conditions ◽  
Einstein Condensation ◽  
Casimir Forces ◽  
Leading Term ◽  
Bose Einstein

Abstract We analyze the Casimir forces for an ideal Bose gas enclosed between two infinite parallel walls separated by the distance D. The walls are characterized by the Dirichlet boundary conditions. We show that if the thermodynamic state with Bose–Einstein condensate present is correctly approached along the path pertinent to the Dirichlet b.c. then the leading term describing the large-distance decay of thermal Casimir force between the walls is $$\sim 1/D^{2}$$ ∼ 1 / D 2 with a non-universal amplitude. The next order correction is $$\sim \ln D/D^3$$ ∼ ln D / D 3 . These observations remain in contrast with the decay law for both the periodic and Neumann boundary conditions for which the leading term is $$\sim 1/D^3$$ ∼ 1 / D 3 with a universal amplitude. We associate this discrepancy with the D-dependent positive value of the one-particle ground state energy in the case of Dirichlet boundary conditions.

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