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 investigate topological properties of Calabi-Yau four-folds and consider a wide class of explicit constructions in weighted projective spaces and, more generally, toric varieties. Divisors which lead to a non-perturbative superpotential in the effective theory have a very simple description in the toric construction. Relevant properties of them follow just by counting lattice points and can also be used to construct examples with negative Euler number. We study nets of transitions between cases with generically smooth elliptic fibres and cases with ADE gauge symmetries in the <i>N</i> = 1 theory due to degenerations of the fibre over codimension one loci in the base. Finally we investigate the quantum cohomology ring of this four-folds using Frobenius algebra.

This paper presents a 2×2 dynamical system to study the cyclic appearance and disappearance of the gas phase in a two-component (CO2, H2O), two phase (gas, liquid) flow in a single pore. Depending on injection rate, linearization of the dynamical system around equilibrium shows that the gas phase can exhibit two behaviors, either cyclically vanishing and appearing, or approaching a steady-state volume value. Numerical simulations were also run on the fully non-linear dynamical system to verify the results of the linearized model.

Extremal but non-supersymmetric charged black holes with SU(2)_L spin in IIB string theory compactified to five dimensions on K^3 x S^1 are considered. These have a near-horizon or NHEK region with an enhanced SL(2,R)_L conformal symmetry. It is shown that the NHEK geometry has a second, inequivalent, asymptotically flat extension in which the radius of the S^1 becomes infinite but the radius of the angular circles of SU(2)_L orbits approach a constant. The asymptotic charges associated to the second solution identify it as a 5D D1-D5-Taub-NUT black string with certain nonzero worldvolume charge densities, temperatures and chemical potentials. The dual of the NHEK geometry is then identified as an IR limit of this wrapped brane configuration.

This is the second part of a three part series of papers. In this paper, we consider a general class of $N\times N$ random band matrices $H=(H_{ij})$ whose entries are centered random variables, independent up to a symmetry constraint. We assume that the variances $\mathbb E |H_{ij}|^2$ form a band matrix with typical band width $1\ll W\ll N$. We consider the generalized resolvent of $H$ defined as $G(Z):=(H - Z)^{-1}$, where $Z$ is a deterministic diagonal matrix such that $Z_{ij}=\left(z\mathds{1}_{1\le i \le W}+\wt z\mathds{1}_{ i > W} \right) \delta_{ij}$, with two distinct spectral parameters $z\in \mathbb C_+:=\{z\in \mathbb C:\im z>0\}$ and $\wt z\in \mathbb C_+\cup \mathbb R$. In this paper, we prove a sharp bound for the local law of the generalized resolvent $G$ for $W\gg N^{3/4}$. This bound is a key input for the proof of delocalization and bulk universality of random band matrices in [2]. Our proof depends on a fluctuations averaging bound on certain averages of polynomials in the resolvent entries, which will be proved in [10].