Consider electromagnetic waves in two-dimensional honeycomb structured media. The properties of transverse electric (TE) polarized waves are determined by the spectral properties of the elliptic operator L^A=-\nabla_x·A(x) \nabla_x, where A(x) is Λ_h− periodic (Λ_h denotes the equilateral triangular lattice), and such that with respect to some origin of coordinates, A(x) is PC− invariant (A(x)=\overline{A(-x)}) and 120° rotationally invariant (A(R*x)=R*A(x)R, where R is a 120° rotation in the plane). We first obtain results on the existence, stability and instability of Dirac points, conical intersections between two adjacent Floquet-Bloch dispersion surfaces. We then show that the introduction through small and slow variations of a domain wall across a line-defect gives rise to the bifurcation from Dirac points of highly robust (topologically protected) edge states. These are time-harmonic solutions of Maxwell's equations which are propagating parallel to the line-defect and spatially localized transverse to it. The transverse localization and strong robustness to perturbation of these edge states is rooted in the protected zero mode of a one-dimensional effective Dirac operator with spatially varying mass term. These results imply the existence of uni-directional propagating edge states for two classes of time-reversal invariant media in which C symmetry is broken: magneto-optic media and bi-anisotropic media. Our analysis applies and extends the tools previously developed in the context of honeycomb Schrödinger operators.

The aim of this paper is to provide a fast and efficient procedure for (real-time) target identification in imaging based on matching on a dictionary of precomputed generalized polarization tensors (GPTs). The approach is based on some important properties of the GPTs and new invariants. A new shape representation is given and numerically tested in the presence of measurement noise. The stability and resolution of the proposed identification algorithm is numerically quantified. We compare the proposed GPT-based shape representation with a moment-based one.

The positive mass theorem states that for a nontrivial isolated physical system, the total energy, which includes contributions from both matter and gravitation is positive. This assertion was demonstrated in our previous paper in the important case when the space-time admits a maximal slice. Here this assumption is removed and the general theorem is demonstrated. Abstracts of the results of this paper appeared in [11] and [13].

The boundary of symmetry-protected topological states (SPTs) can harbor new quantum anomaly phenomena. In this work, we characterize the bosonic anomalies introduced by the 1+1D non-onsite-symmetric gapless edge modes of 2+1D bulk bosonic SPTs with a generic finite Abelian group symmetry (isomorphic to G=∏iZNi=ZN1×ZN2×ZN3×...). We demonstrate that some classes of SPTs (termed "Type II") trap fractional quantum numbers (such as fractional ZN charges) at the 0D kink of the symmetry-breaking domain walls; while some classes of SPTs (termed "Type III") have degenerate zero energy modes (carrying the projective representation protected by the unbroken part of the symmetry), either near the 0D kink of a symmetry-breaking domain wall, or on a symmetry-preserving 1D system dimensionally reduced from a thin 2D tube with a monodromy defect 1D line embedded. More generally, the energy spectrum and conformal dimensions of gapless edge modes under an external gauge flux insertion (or twisted by a branch cut, i.e., a monodromy defect line) through the 1D ring can distinguish many SPT classes. We provide a manifest correspondence from the physical phenomena, the induced fractional quantum number and the zero energy mode degeneracy, to the mathematical concept of cocycles that appears in the group cohomology classification of SPTs, thus achieving a concrete physical materialization of the cocycles. The aforementioned edge properties are formulated in terms of a long wavelength continuum field theory involving scalar chiral bosons, as well as in terms of Matrix Product Operators and discrete quantum lattice models. Our lattice approach yields a regularization with anomalous non-onsite symmetry for the field theory description. We also formulate some bosonic anomalies in terms of the Goldstone-Wilczek formula.

We study S duality of four dimensional N= 2 Argyres-Douglas (AD) theory engineered from 6d A_ {N-1}(2, 0) theory. We find a (p, q) sequence of SCFTs, here (p, q) is co-prime and class S theory defined on sphere corresponds to class (0, 1) theory. We represent these theories by a sphere with marked points, and S duality is interpreted as different pants decompositions of the same punctured sphere. The weakly coupled gauge theory description involves gauging AD matter which is represented by three punctured sphere.