Reflection factors and exact g-functions for purely elastic scattering theories

We discuss reflection factors for purely elastic scattering theories and relate them to perturbations of specific conformal boundary conditions, using recent results on exact off-critical g-functions. For the non-unitary cases, we support our conjectures using a relationship with quantum group reductions of the sine-Gordon model. Our results imply the existence of a variety of new flows between conformal boundary conditions, some of them driven by boundary-changing operators.     

Boundary renormalisation group flows of the supersymmetric Lee–Yang model and its extensions

In this paper we examine the supersymmetric Lee–Yang model in the presence of boundaries. We determine the reflection factors for the Neveu–Schwarz type boundary conditions from the reduction of the supersymmetric sine-Gordon model and check them by using boundary truncated conformal space approach in the massless case. We explore the boundary renormalisation groups flows using boundary TBA and TCSA.     

Quantum brownian motion on a triangular lattice and c=2 boundary conformal field theory

We study a single particle diffusing on a triangular lattice and interacting with a heat bath, using boundary conformal field theory (CFT) and exact integrability techniques. We derive a correspondence between the phase diagram of this problem and that recently obtained for the 2-dimensional 3-state Potts model with a boundary. Exact results are obtained on phases with intermediate mobilities. These correspond to nontrivial boundary states in a conformal field theory with 2 free bosons which we explicitly construct for the first time. These conformally invariant boundary conditions are not simply products of Dirichlet and Neumann ones and unlike those trivial boundary conditions, are not invariant under a Heisenberg algebra.     

Exact one-point function of N=1 super-Liouville theory with boundary

In this paper, exact one-point functions of N=1 super-Liouville field theory in two-dimensional space–time with appropriate boundary conditions are presented. Exact results are derived both for the theory defined on a pseudosphere with discrete (NS) boundary conditions and for the theory with explicit boundary actions which preserves super conformal symmetries. We provide various consistency checks. We also show that these one-point functions can be related to a generalized Cardy conditions along with corresponding modular S-matrices. Using this result, we conjecture the dependence of the boundary two-point functions of the (NS) boundary operators on the boundary parameter.     

Boundary states in c=−2 logarithmic conformal field theory

Starting from first principles, a constructive method is presented to obtain boundary states in conformal field theory. It is demonstrated that this method is well suited to compute the boundary states of logarithmic conformal field theories. By studying the logarithmic conformal field theory with central charge c=−2 in detail, we show that our method leads to consistent results. In particular, it allows to define boundary states corresponding to both, indecomposable representations as well as their irreducible subrepresentations.     

Integrable and Conformal Boundary Conditions for \widehat{s\ell}(2) A–D–E Lattice Models and Unitary Minimal Conformal Field Theories

Integrable boundary conditions are studied for critical A–D–E and general graph-based lattice models of statistical mechanics. In particular, using techniques associated with the Temperley–Lieb algebra and fusion, a set of boundary Boltzmann weights which satisfies the boundary Yang–Baxter equation is obtained for each boundary condition. When appropriately specialized, these boundary weights, each of which depends on three spins, decompose into more natural two-spin edge weights. The specialized boundary conditions for the A–D–E cases are naturally in one-to-one correspondence with the conformal boundary conditions of $\widehat{s\ell }$ (2) unitary minimal conformal field theories. Supported by this and further evidence, we conclude that, in the continuum scaling limit, the integrable boundary conditions provide realizations of the complete set of conformal boundary conditions in the corresponding field theories.     

Conformal field theories and critical phenomena

In this article we present a brief review of the conformal symmetry and the two-dimensional conformal quantum field theories. As concrete applications of the conformal theories to the critical phenomena in statistical systems, we calculate the value of central charge and the anomalous scale dimensions of the Z ₂ symmetric quantum chain with boundary condition. The results are compatible with the prediction of the conformal field theories.     

On the renormalisation group for the boundary truncated conformal space approach

In this paper we continue the study of the truncated conformal space approach to perturbed boundary conformal field theories. This approach to perturbation theory suffers from a renormalisation of the coupling constant and a multiplicative renormalisation of the Hamiltonian. We show how these two effects can be predicted by both physical and mathematical arguments and prove that they are correct to leading order for all states in the TCSA system. We check these results using the TCSA applied to the tri-critical Ising model and the Yang–Lee model. We also study the TCSA of an irrelevant (non-renormalisable) perturbation and find that, while the convergence of the coupling constant and energy scales are problematic, the renormalised and rescaled spectrum remain a very good fit to the exact result, and we find a numerical relationship between the IR and UV couplings describing a particular flow. Finally we study the large coupling behaviour of TCSA and show that it accurately encompasses several different fixed points.     

Anti-de Sitter-space/conformal-field-theory Casimir energy for rotating black holes

We show that, if one chooses the Einstein static universe as the metric on the conformal boundary of Kerr-anti-de Sitter spacetime, then the Casimir energy of the boundary conformal field theory can easily be determined. The result is independent of the rotation parameters, and the total boundary energy then straightforwardly obeys the first law of thermodynamics. Other choices for the metric on the conformal boundary will give different, more complicated, results. As an application, we calculate the Casimir energy for free self-dual tensor multiplets in six dimensions and compare it with that of the seven-dimensional supergravity dual. They differ by a factor of 5/4.     

Differential Forms and the Noncommutative Residue for Manifolds with Boundary in the Non-product Case

In this Letter, for an even-dimensional compact manifold with boundary which has the non-product metric near the boundary, we use the noncommutative residue to define a conformal invariant pair. For a four-dimensional manifold, we compute this conformal invariant pair under some conditions and point out the way of computations in the general.     

Dynamics of Finger Formation in Laplacian Growth Without Surface Tension

We study the dynamics of “finger” formation in Laplacian growth without surface tension in a channel geometry (the Saffman–Taylor problem). We present a pedagogical derivation of the dynamics of the conformal map from a strip in the complex plane to the physical channel. In doing so we pay attention to the boundary conditions (no flux rather than periodic) and derive a field equation of motion for the conformal map. We first consider an explicit analytic class of conformal maps that form a basis for solutions in infinitely long channels, characterized by meromorphic derivatives. The great bulk of these solutions can lose conformality due to finite time singularities. By considerations of the nature of the analyticity of these solutions, we show that those solutions which are free of such singularities inevitably result in a single asymptotic “finger” whose width is determined by initial conditions. This is in contradiction with the experimental results that indicate selection of a finger of width 1/2. In the last part of this paper we show that such a solution might be determined by the boundary conditions of a finite body of fluid, e.g. finiteness can lead to pattern selection.     

Casimir Effect for Moving Branes in Static dS4+1 Bulk

In this paper we study the Casimir effect for conformally coupled massless scalar fields on background of Static dS₄₊₁ spacetime. We will consider the general plane–symmetric solutions of the gravitational field equations and boundary conditions of the Dirichlet type on the branes. Then we calculate the vacuum energy-momentum tensor in a configuration in which the boundary branes are moving by uniform proper acceleration in static de Sitter background. Static de Sitter space is conformally related to the Rindler space, as a result we can obtain vacuum expectation values of energy-momentum tensor for conformally invariant field in static de Sitter space from the corresponding Rindler counterpart by the conformal transformation.     

Conformal invariance, Noether symmetry, Lie symmetry and conserved quantities of Hamilton systems

In this paper, the relation of the conformal invariance, the Noether symmetry, and the Lie symmetry for the Hamilton system is discussed in detail. The definition of the conformal invariance for Hamilton systems is given. The relation between the conformal invariance and the Noether symmetry is discussed, the conformal factors of the determining expressions are found by using the Noether symmetry, and the Noether conserved quantity resulted from the conformal invariance is obtained. The relation between the conformal invariance and the Lie symmetry is discussed, the conformal factors are found by using the Lie symmetry, and the Hojman conserved quantity resulted from the conformal invariance of the system is obtained. Two examples are given to illustrate the application of the results.     

谱域OCT的印刷电路板三防漆厚度测量

三防漆是一种广泛应用在印刷电路板（PCB）的保护性涂层,可以有效保护PCB使其免受恶劣环境的损害。三防漆的厚度是评价三防漆涂层质量的关键指标,因此需要对三防漆涂层进行厚度检测。提出了将谱域光学相干断层扫描成像技术（spectral domain optical coherence tomography,SD-OCT）与图像分割算法相结合,实现对三防漆涂层厚度快速无损测量。为了提高测量精度,选用了宽带SLD光源（带宽：180 nm）来设计SD-OCT系统,系统轴向分辨率达到1.72 μm。同时设计了基于边缘跟踪的涂层分割算法来实现三防漆涂层的快速准确分割。为了评估所设计方法的准确性,利用传统的金相切片方法进行了三防漆的厚度测量,将测量结果与该方法测量结果进行比较,分析了两种方法检测到的涂层上下边界吻合程度以及厚度差异。此外,还将所提出的涂层分割算法与我们组之前研究的基于图像梯度的边缘检测算法进行对比,分析了两种方法在测量结果的准确性和运行效率方面的差异,以此来评估该方法的优劣势。结果表明,所设计的三防漆厚度测量方法与传统的金相切片方法的测量结果具有很好的一致性,可以准确地实现三防漆的厚度测量;基于SD-OCT系统的三维成像能力可以直观地看到三防漆厚度地形图,克服了传统的金相切片方法无法进行区域性的三防漆厚度测量的缺陷;相比之前提出的基于图像梯度的边缘检测算法,此方法测量结果更加准确,效率显著提升,具备实时测量的潜力。     

Conformal boundary state for the rectangular geometry

We discuss conformal field theories (CFTs) in rectangular geometries, and develop a formalism that involves a conformal boundary state for the 1+1d$1+1\mathrm{d}$ open system. We focus on the case of homogeneous boundary conditions (no insertion of a boundary condition changing operator), for which we derive an explicit expression of the associated boundary state, valid for any arbitrary CFT. We check the validity of our solution, comparing it with known results for partition functions, numerical simulations of lattice discretizations, and coherent state expressions for free theories.     

Differential forms and the Wodzicki residue for manifolds with boundary

In [A. Connes, Quantized calculus and applications, XIth International Congress of Mathematical Physics (Paris,1994), 15–36, Internat Press, Cambridge, MA, 1995], Connes found a conformal invariant using Wodzicki’s 1-density and computed it in the case of 4-dimensional manifold without boundary. In [W. J. Ugalde, Differential forms and the Wodzicki residue, arXiv: Math, DG/0211361], Ugalde generalized the Connes’ result to n-dimensional manifold without boundary. In this paper, we generalize the results of [A. Connes, Quantized calculus and applications, XIth International Congress of Mathematical Physics (Paris,1994), 15–36, Internat Press, Cambridge, MA, 1995] and [W. J. Ugalde, Differential forms and the Wodzicki residue, arXiv: Math, DG/0211361] to the case of manifolds with boundary.     

Near-field performance analysis of locally-conformal perfectly matched absorbers via Monte Carlo simulations

In the numerical solution of some boundary value problems by the finite element method (FEM), the unbounded domain must be truncated by an artificial absorbing boundary or layer to have a bounded computational domain. The perfectly matched layer (PML) approach is based on the truncation of the computational domain by a reflectionless artificial layer which absorbs outgoing waves regardless of their frequency and angle of incidence. In this paper, we present the near-field numerical performance analysis of our new PML approach, which we call as locally-conformal PML, using Monte Carlo simulations. The locally-conformal PML method is an easily implementable conformal PML implementation, to the problem of mesh truncation in the FEM. The most distinguished feature of the method is its simplicity and flexibility to design conformal PMLs over challenging geometries, especially those with curvature discontinuities, in a straightforward way without using artificial absorbers. The method is based on a special complex coordinate transformation which is ‘locally-defined’ for each point inside the PML region. The method can be implemented in an existing FEM software by just replacing the nodal coordinates inside the PML region by their complex counterparts obtained via complex coordinate transformation. We first introduce the analytical derivation of the locally-conformal PML method for the FEM solution of the two-dimensional scalar Helmholtz equation arising in the mathematical modeling of various steady-state (or, time-harmonic) wave phenomena. Then, we carry out its numerical performance analysis by means of some Monte Carlo simulations which consider both the problem of constructing the two-dimensional Green’s function, and some specific cases of electromagnetic scattering.     

Intertwining operator realization of non-relativistic holography

We give a group-theoretic interpretation of non-relativistic holography as equivalence between representations of the Schrödinger algebra describing bulk fields and boundary fields. Our main result is the explicit construction of the boundary-to-bulk operators in the framework of representation theory (without specifying any action). Further we show that these operators and the bulk-to-boundary operators are intertwining operators. In analogy to the relativistic case, we show that each bulk field has two boundary fields with conjugated conformal weights. These fields are related by another intertwining operator given by a two-point function on the boundary. Analogously to the relativistic result of Klebanov–Witten we give the conditions when both boundary fields are physical. Finally, we recover in our formalism earlier non-relativistic results for scalar fields by Son and others.     

Complete affine connection in the causal boundary: static,spherically symmetric spacetimes

The boundary at \(\mathcal {I}^+\), future null infinity, for a standard static, spherically symmetric spactime is examined for possible linear connections. Two independent methods are employed, one for treating \(\mathcal {I}^+\) as the future causal boundary, and one for treating it as a conformal boundary (the latter is subsumed in the former, which is of greater generality). Both methods provide the same result: a constellation of various possible connections, depending on an arbitrary choice of a certain function, a sort of gauge freedom in obtaining a natural connection on \(\mathcal {I}^+\); choosing that function to be constant (for instance) results in a complete connection. Treating \(\mathcal {I}^+\) as part of the future causal boundary, the method is to impute affine connections on null hypersurfaces going out to \(\mathcal {I}^+\), in terms of a transverse vector field on each null hypersurface (there is much gauge freedom on choice of the transverse vector fields). Treating \(\mathcal {I}^+\) as part of a conformal boundary, the method is to make a choice of conformal factor that makes the boundary totally geodesic in the enveloping manifold (there is much gauge freedom in choice of that conformal factor). Similar examination is made of other boundaries, such as timelike infinity and timelike and spacelike singularities. These are much simpler, as they admit a unique connection from a similar limiting process (i.e., no gauge freedom); and that connection is complete.     

Models in Boundary Quantum Field Theory Associated with Lattices and Loop Group Models

In this article we give new examples of models in boundary quantum field theory, i.e. local time-translation covariant nets of von Neumann algebras, using a recent construction of Longo and Witten, which uses a local conformal net on the real line together with an element of a unitary semigroup associated with . Namely, we compute elements of this semigroup coming from H?lder continuous symmetric inner functions for a family of (completely rational) conformal nets which can be obtained by starting with nets of real subspaces, passing to its second quantization nets and taking local extensions of the former. This family is precisely the family of conformal nets associated with lattices, which as we show contains as a special case the level 1 loop group nets of simply connected, simply laced groups. Further examples come from the loop group net of at level 2 using the orbifold construction.