$$\alpha $$-Modulation Spaces for Step Two Stratified Lie Groups

We define and investigate $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) associated to a step two stratified Lie group G with rational structure constants. This is an extension of the Euclidean $$\alpha $$ α -modulation spaces $$M_{p,q}^{s,\alpha }({\mathbb {R}}^n)$$ M p , q s , α ( R n ) that act as intermediate spaces between the modulation spaces ($$\alpha = 0$$ α = 0 ) in time-frequency analysis and the Besov spaces ($$\alpha = 1$$ α = 1 ) in harmonic analysis. We will illustrate that the group structure and dilation structure on G affect the boundary cases $$\alpha = 0,1$$ α = 0 , 1 where the spaces $$M_{p,q}^{s}(G)$$ M p , q s ( G ) and $${\mathcal {B}}_{p,q}^{s}(G)$$ B p , q s ( G ) have non-standard translation and dilation symmetries. Moreover, we show that the spaces $$M_{p,q}^{s,\alpha }(G)$$ M p , q s , α ( G ) are non-trivial and generally distinct from their Euclidean counterparts. Finally, we examine how the metric geometry of the coverings $${\mathcal {Q}}(G)$$ Q ( G ) underlying the $$\alpha = 0$$ α = 0 case $$M_{p,q}^{s}(G)$$ M p , q s ( G ) allows for the existence of geometric embeddings $$\begin{aligned} F:M_{p,q}^{s}({\mathbb {R}}^k) \longrightarrow {} M_{p,q}^{s}(G), \end{aligned}$$ F : M p , q s ( R k ) ⟶ M p , q s ( G ) , as long as k (that only depends on G) is small enough. Our approach naturally gives rise to several open problems that is further elaborated at the end of the paper.

Three-point functions in ABJM and Bethe Ansatz

We develop an integrability-based framework to compute structure constants of two sub-determinant operators and a single-trace non-BPS operator in ABJM theory in the planar limit. In this first paper, we study them at weak coupling using a relation to an integrable spin chain. We first develop a nested Bethe ansatz for an alternating SU(4) spin chain that describes single-trace operators made out of scalar fields. We then apply it to the computation of the structure constants and show that they are given by overlaps between a Bethe eigenstate and a matrix product state. We conjecture that the determinant operator corresponds to an integrable matrix product state and present a closed-form expression for the overlap, which resembles the so-called Gaudin determinant. We also provide evidence for the integrability of general sub-determinant operators. The techniques developed in this paper can be applied to other quantities in ABJM theory including three-point functions of single-trace operators.

Широкополосная ЭПР-спектроскопия кристалла SrY-=SUB=-2-=/SUB=-O-=SUB=-4-=/SUB=-:Ho-=SUP=-3+-=/SUP=-

EPR spectra of impurity Ho3+ ions in oriented SrY2O4 single-crystals are registered at the temperature 4.2 K in the frequency range from 70 to 180 GHz. The results of measurements evidence for the substitution of Ho3+ ions for the Y3+ ions at the structurally nonequivalent sites R1 and R2 with the local Cs point symmetry. The values of g-factors, hyperfine structure constants and the energy gaps berween the ground and the first excited non-degenherate crystal-field sublevels of the ground 5I8 multiplet are determined. The observed specific features of the ground states of Но3+ ions (non-Kramers doublets with the zero-field splittings of 4.30 and 1.67 cm-1) open a possibility to identify transitions in optical spectra of SrY2O4:Ho and inelastic neutron scattering spectra of SrHo2O4 crystals.

Analytic conformal bootstrap and Virasoro primary fields in the Ashkin-Teller model

We revisit the critical two-dimensional Ashkin–Teller model, i.e. the \mathbb{Z}_2ℤ2 orbifold of the compactified free boson CFT at c=1c=1. We solve the model on the plane by computing its three-point structure constants and proving crossing symmetry of four-point correlation functions. We do this not only for affine primary fields, but also for Virasoro primary fields, i.e. higher twist fields and degenerate fields. This leads us to clarify the analytic properties of Virasoro conformal blocks and fusion kernels at c=1c=1. We show that blocks with a degenerate channel field should be computed by taking limits in the central charge, rather than in the conformal dimension. In particular, Al. Zamolodchikov’s simple explicit expression for the blocks that appear in four-twist correlation functions is only valid in the non-degenerate case: degenerate blocks, starting with the identity block, are more complicated generalized theta functions.

The AdS3 × S1 chiral ring

We study AdS3× S1× Y supersymmetric string theory backgrounds with Neveu-Schwarz-Neveu-Schwarz flux that are dual to $$ \mathcal{N} $$ N = 2 superconformal theories on the boundary. We classify all worldsheet vertex operators that correspond to space-time chiral primaries. We compute space-time chiral ring structure constants for operators in the zero spectral flow sector using the operator product expansion in the worldsheet theory. We find that the structure constants take a universal form that depends only on the topological data of the $$ \mathcal{N} $$ N = 2 superconformal theory on Y.

THE FIRST DETAILED ANALYSIS ON THE ASTROPHYSICAL PROPERTIES OF THE ECCENTRIC BINARY V990 HER

We present accurate physical parameters of the eccentric binary system V990 Her which has an orbital period of P=8.193315±0.000003 days using its photometric and spectroscopic data. The physical parameters of the components were derived as Teff1=8000±200 K, Teff2=7570±200 K, M1=2.01±0.07 Mʘ, M2=1.83±0.03 Mʘ, R1=2.22±0.02 Rʘ, R2=2.12±0.01 Rʘ, log(L1/Lʘ)=1.25±0.04, log(L2/Lʘ)=1.12±0.05. Our findings revealed that both components are slightly evolved from the zero-age main sequence with an age of 6.3×108 years. We estimated an apsidal motion with a period of U=14683±2936 years in the system and the internal structure constants of the components were derived for the first time.

Energy-Band Theory

Various methods to calculate energy bands and the electronic structure of solids are described in detail. Although the emphasis lies on linear methods well known for their transparency and high numerical speed, traditional methods are described to supply historical background and to point the way to modern methods. After introducing Bloch electrons and the reciprocal space, plane waves, orthogonalized plane waves, and pseudopotentials are discussed, followed by the important augmented plane wave method (APW). Multiple scattering theory defines scattering phase shifts encoding atomic properties and the structure constants that describe the crystal lattice. Linear combination of atomic orbitals (LCAO) and linear combination of muffin-tin orbitals (LMTO) result in efficient and fast methods as does the related augmented spherical waves (ASW) method. The treatment of arbitrary spin configurations using the ASW method and the formulation of incommensurate spiral structures on the basis of the unitary SU(2) group are developed in detail.

Jacobi-Lie T-plurality

We propose a Leibniz algebra, to be called DD^++, which is a generalization of the Drinfel’d double. We find that there is a one-to-one correspondence between a DD^++ and a Jacobi–Lie bialgebra, extending the known correspondence between a Lie bialgebra and a Drinfel’d double. We then construct generalized frame fields E_A{}^M\in\text{O}(D,D)\times\mathbb{R}^+EAM∈O(D,D)×ℝ+ satisfying the algebra \hat{\pounds}_{E_A}E_B = - X_{AB}{}^C\,E_C£̂EAEB=−XABCEC, where X_{AB}{}^CXABC are the structure constants of the DD^++ and \hat{\pounds}£̂ is the generalized Lie derivative in double field theory. Using the generalized frame fields, we propose the Jacobi–Lie TT-plurality and show that it is a symmetry of double field theory. We present several examples of the Jacobi–Lie TT-plurality with or without Ramond–Ramond fields and the spectator fields.