Ferromagnetic Ordering of Energy Levels for {U_q(\mathfrak{sl}_2)} Symmetric Spin Chains

We consider the class of quantum spin chains with arbitrary ${U_{q}(\mathfrak{sl}_{2})}$ -invariant nearest-neighbor interactions, sometimes called SU _q (2) for the quantum deformation of SU(2), for q >?0. We derive sufficient conditions for the Hamiltonian to satisfy the property we call ferromagnetic ordering of energy levels. This is the property that the ground state energy restricted to a fixed total spin subspace is a decreasing function of the total spin. Using the Perron?CFrobenius theorem, we show sufficient conditions are positivity of all interactions in the dual canonical basis of Lusztig. We characterize the cone of positive interactions, showing that it is a simplicial cone consisting of all non-positive linear combinations of ??cascade operators,?? a special new basis of ${U_{q}(\mathfrak{sl}_2)}$ intertwiners we define.     

Higher Spin Polynomial Solutions of Quantum Knizhnik–Zamolodchikov Equation

We provide explicit formulae for highest-weight to highest-weight correlation functions of perfect vertex operators of ${U_q(\widehat{\mathfrak{sl}(2)})}$ in arbitrary integer level ?. They are given in terms of certain Macdonald polynomials. We apply this construction to the computation of the ground state of higher spin vertex models, spin chains (spin ?/2 XXZ) or loop models in the root of unity case ${q=-e^{-i\pi/(\ell+2)}}$ .     

Vortex Counting and Lagrangian 3-Manifolds

To every 3-manifold M one can associate a two-dimensional ${\mathcal{N}=(2, 2)}$ supersymmetric field theory by compactifying five-dimensional ${\mathcal{N}=2}$ super-Yang?CMills theory on M. This system naturally appears in the study of half-BPS surface operators in four-dimensional ${\mathcal{N}=2}$ gauge theories on one hand, and in the geometric approach to knot homologies, on the other. We study the relation between vortex counting in such two-dimensional ${\mathcal{N}=(2, 2)}$ supersymmetric field theories and the refined BPS invariants of the dual geometries. In certain cases, this counting can also be mapped to the computation of degenerate conformal blocks in two-dimensional CFT??s. Degenerate limits of vertex operators in CFT receive a simple interpretation via geometric transitions in BPS counting.     

Mass mixing,CP violation and left-right symmetry for heavy neutral mesons

We investigate \(M^0 - \bar M^0 \) mixing and CP violation in the minimal left-right symmetric gauge model with spontaneous P and CP violation. The dominant contributions to the mixing amplitude including QCD corrections are calculated explicitly for \(B^0 - \bar B^0 \) . While the amount of mixing is not much changed with respect to the standard model leftright symmetry can give rise to significantly larger CP violation in the \(B_s^0 - \bar B_s^0 \) system (up to two orders of magnitude for the dilepton charge asymmetry). Sizable CP violating effects require that the left-right contribution to theK _L K _S mass difference has the same sign as the standard model contribution. We also comment on \(D^0 - \bar D^0 \) mixing including a careful discussion of the standard model prediction for the short distance part.     

Mixing properties of the one-atom maser

We study the relaxation properties of the quantized electromagnetic field in a cavity under repeated interactions with single two-level atoms, so-called one-atom maser. We improve the ergodic results obtained in Bruneau and Pillet (J Stat Phys 134(5–6):1071–1095, 2009) and prove that, whenever the atoms are initially distributed according to the canonical ensemble at temperature $T>0$ , all the invariant states are mixing. Under some non-resonance condition this invariant state is known to be thermal equilibirum at some renormalized temperature $T^*$ and we prove that the mixing is then arbitrarily slow, in other words that there is no lower bound on the relaxation speed.     

On the total mass of closed universes

The total mass, the Witten type gauge conditions and the spectral properties of the Sen–Witten and the 3-surface twistor operators in closed universes are investigated. It has been proven that a recently suggested expression $\mathtt{M}$ M for the total mass density of closed universes is vanishing if and only if the spacetime is flat with toroidal spatial topology; it coincides with the first eigenvalue of the Sen–Witten operator; and it is vanishing if and only if Witten’s gauge condition admits a non-trivial solution. Here we generalize slightly the result above on the zero-mass configurations: $\mathtt{M}=0$ M = 0 if and only if the spacetime is holonomically trivial with toroidal spatial topology. Also, we show that the multiplicity of the eigenvalues of the (square of the) Sen–Witten operator is even, and a potentially viable gauge condition is suggested. The monotonicity properties of $\mathtt{M}$ M through the examples of closed Bianchi I and IX cosmological spacetimes are also discussed. A potential spectral characterization of these cosmological spacetimes, in terms of the spectrum of the Riemannian Dirac operator and the Sen–Witten and the 3-surface twistor operators, is also indicated.     

On Baxter {\mathcal Q} -Operators and their Arithmetic Implications

We consider Baxter ${\mathcal Q}$ -operators for various versions of quantum affine Toda chain. The interpretation of eigenvalues of the finite Toda chain Baxter operators as local Archimedean L-functions proposed recently is generalized to the case of affine Lie algebras. We also introduce a simple generalization of Baxter operators and local L-functions compatible with this identification. This gives a connection of the Toda chain Baxter ${\mathcal Q}$ -operators with an Archimedean version of the Polya–Hilbert operator proposed by Berry-Keating. We also elucidate the Dorey–Tateo spectral interpretation of eigenvalues of ${\mathcal Q}$ -operators. Using explicit expressions for eigenfunctions of affine/relativistic Toda chain we obtain an Archimedean analog of Casselman–Shalika–Shintani formula for Whittaker function in terms of characters.     

Light Hypernuclei Based on Chiral Interactions at Next-to-Leading Order

We present predictions for the binding energies of the light hypernuclei \({^3_\Lambda{\rm H},\, ^4_\Lambda{\rm He}}\) and \({^4_\Lambda{\rm H}}\) based on Faddeev- and Yakubovsky equations in momentum space. We discuss how such results can help to test the existing hyperon–nucleon (Y N) potential models and effective field theory based Y N interactions. Especially, we show results for the chiral interactions at next-to-leading order.     

Relating the Quantum Mechanics of Discrete Systems to Standard Canonical Quantum Mechanics

Standard canonical quantum mechanics makes much use of operators whose spectra cover the set of real numbers, such as the coordinates of space, or the values of the momenta. Discrete quantum mechanics uses only strictly discrete operators. We show how one can transform systems with pairs of integer-valued, commuting operators $P_i$ and $Q_i$ , to systems with real-valued canonical coordinates $q_i$ and their associated momentum operators $p_i$ . The discrete system could be entirely deterministic while the corresponding (p, q) system could still be typically quantum mechanical.     

Classes of Invariant Subspaces for Some Operator Algebras

New results showing connections between structural properties of von Neumann algebras and order theoretic properties of structures of invariant subspaces given by them are proved. We show that for any properly infinite von Neumann algebra M there is an affiliated subspace \({\mathcal{L}} \) such that all important subspace classes living on \({\mathcal{L}} \) are different. Moreover, we show that \({\mathcal{L}} \) can be chosen such that the set of σ-additive measures on subspace classes of \({\mathcal{L}} \) are empty. We generalize measure theoretic criterion on completeness of inner product spaces to affiliated subspaces corresponding to Type I factor with finite dimensional commutant. We summarize hitherto known results in this area, discuss their importance for mathematical foundations of quantum theory, and outline perspectives of further research.     

Evidence for hybrid mesonsQ\bar QG

We point out that the recent CESR data on γ(5 S) and γ(6 S) show evidence for \(b\bar b\) and \(b\bar bg\) mixing. We show that ψ(4159) and ψ(4030) are not2D and 3S, \(c\bar c\) states but mixtures of 3S, \(c\bar c\) and \(c\bar cg\) states. We show various ways to find further evidence for hybrid states experimentally.     

6j Symbols for the Modular Double,Quantum Hyperbolic Geometry,and Supersymmetric Gauge Theories

We revisit the definition of the 6j symbols from the modular double of ${\mathcal{U}_q(\mathfrak{sl}(2, \mathbb{R}))}$ , referred to as b-6j symbols. Our new results are (1) the identification of particularly natural normalization conditions, and (2) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6j symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. Our new integral representations finally indicate a possible interpretation of the b-6j symbols as partition functions of non-abelian three-dimensional ${\mathcal{N}=2}$ supersymmetric gauge theories.     

Quantum Entanglement Under Lorentz Transformation

We study the properties of quantum entanglement in moving frames, with a non-maximally entangled bipartite state: $|\phi\rangle=\sqrt{1-\varepsilon}|{\uparrow\uparrow}\rangle +\sqrt{\varepsilon}|{\downarrow\downarrow}\rangle$ , (0<ε<1). We calculate the concurrence of this state under Lorentz transformation and show that if the momenta part of the spin-1/2 pair is appropriately correlated, the concurrence is invariant ( $\mathcal {C}(\rho)=2\sqrt{\varepsilon-\varepsilon^{2}}$ ); despite the entanglement of this state is not maximal, there is no transfer of entanglement between spin and momentum.     

Gauge Theories on ALE Space and Super Liouville Correlation Functions

We present a relation between ${\mathcal{N}=2}$ quiver gauge theories on the ALE space ${\mathcal{O}_{\mathbb{P}^1}(-2)}$ and correlators of ${\mathcal{N}=1}$ super Liouville conformal field theory, providing checks in the case of punctured spheres and tori. We derive a blow-up formula for the full Nekrasov partition function and show that, up to a U(1) factor, the ${\mathcal{N}=2^*}$ instanton partition function is given by the product of the character of ${\widehat{SU}(2)_2}$ times the super Virasoro conformal block on the torus with one puncture. Moreover, we match the perturbative gauge theory contribution with super Liouville three-point functions.     

Noncommutative gauge fields coupled to noncommutative gravity

We present a noncommutative (NC) version of the action for vielbein gravity coupled to gauge fields. Noncommutativity is encoded in a twisted $\star $ -product between forms, with a set of commuting background vector fields defining the (abelian) twist. A first order action for the gauge fields avoids the use of the Hodge dual. The NC action is invariant under diffeomorphisms and $\star $ -gauge transformations. The Seiberg–Witten map, adapted to our geometric setting and generalized for an arbitrary abelian twist, allows to re-express the NC action in terms of classical fields: the result is a deformed action, invariant under diffeomorphisms and usual gauge transformations. This deformed action is a particular higher derivative extension of the Einstein-Hilbert action coupled to Yang-Mills fields, and to the background vector fields defining the twist. Here noncommutativity of the original NC action dictates the precise form of this extension. We explicitly compute the first order correction in the NC parameter of the deformed action, and find that it is proportional to cubic products of the gauge field strength and to the symmetric anomaly tensor $D_{IJK}$ .     

Elliptic Genera of Two-Dimensional {\mathcal{N} = 2} Gauge Theories with Rank-One Gauge Groups

We compute the elliptic genera of two-dimensional ${\mathcal{N} = (2, 2)}$ and ${\mathcal{N} = (0, 2)}$ -gauged linear sigma models via supersymmetric localization, for rank-one gauge groups. The elliptic genus is expressed as a sum over residues of a meromorphic function whose argument is the holonomy of the gauge field along both the spatial and the temporal directions of the torus. We illustrate our formulas by a few examples including the quintic Calabi–Yau, ${\mathcal{N} = (2, 2)}$ SU(2) and O(2) gauge theories coupled to N fundamental chiral multiplets, and a geometric ${\mathcal{N} = (0, 2)}$ model.     

Mixing of intermediate bosons with Q\bar Q bound states

We discuss the effects of a close degeneracy between theZ ⁰ and vectorQ \(\bar Q\) bound states; large mixing effects can appear modifying mainly theQ \(\bar Q\) states (widths and couplings) and leading to curious structures inside theZ peak.