Infinite-Dimensional Schur–Weyl Duality and the Coxeter–Laplace Operator

We extend the classical Schur–Weyl duality between representations of the groups ${SL(n, \mathbb{C})}$ and ${\mathfrak{S}_N}$ to the case of ${SL(n, \mathbb{C})}$ and the infinite symmetric group ${\mathfrak{S}_\mathbb{N}}$ . Our construction is based on a “dynamic,” or inductive, scheme of Schur–Weyl dualities. It leads to a new class of representations of the infinite symmetric group, which has not appeared earlier. We describe these representations and, in particular, find their spectral types with respect to the Gelfand–Tsetlin algebra. The main example of such a representation acts in an incomplete infinite tensor product. As an important application, we consider the weak limit of the so-called Coxeter–Laplace operator, which is essentially the Hamiltonian of the XXX Heisenberg model, in these representations.     

Gauge quantization for spin-32 fields

The free massless Rarita-Schwinger equation and a recently constructed interacting field theory known as supergravity are invariant under fermionic gauge transformations. Gauge field quantization techniques are applied in both cases. For the free field the Faddeev-Popov ansatz for the generating functional is justified by showing that it is equivalent to canonical quantization in a particular gauge. Propagators are obtained in several gauges and are shown to be ghost-free and causal. For supergravity the Faddeev-Popov ansatz is presented and the gauge fixing and determinant terms are discussed in detail in a Lorentz covariant gauge. The Slavnov-Taylor identity is obtained. It is argued that supergravity theory is free from the difficulty of acausal wave propagation of the type found by Velo and Zwanziger and that pole residues in tree approximation S-matrix elements are positive as required by unitarity.     

ADM-like Hamiltonian formulation of gravity in the teleparallel geometry

We present a new Hamiltonian formulation of the teleparallel equivalent of general relativity (TEGR) meant to serve as the departure point for canonical quantization of the theory. TEGR is considered here as a theory of a cotetrad field on a spacetime. The Hamiltonian formulation is derived by means of an ADM-like $3+1$ decomposition of the field and without any gauge fixing. A complete set of constraints on the phase space and their algebra are presented. The formulation is described in terms of differential forms.     

How does anticommutator\left\{ {Q_\alpha \bar Q_\beta } \right\} realize energy-momentumP μ in quantum supergravity?

It is investigated to what extent the well-known algebra \(\left\{ {Q^S ,\bar Q^S } \right\} = \gamma ^\mu P_\mu \) in the rigid supersymmetry theory holds in quantum supergravity: The anti-commutator \(\left\{ {Q_\alpha ^S ,\bar Q_\beta ^S } \right\} = \gamma ^m \tilde P_m \) defines an “internal” translation generator \(\tilde P_m \) , quite another from the “external” translation generatorP _μ. It is, however, shown that those two operators give the same matrix elements between any two physical states aside from a proportional factor. Such a “miracle” is caused by some particular properties of global gauge transformation charge universal in gauge theories. These properties are fully clarified in a general manner.     

Superintegrability in Two Dimensions and the Racah–Wilson Algebra

The analysis of the most general second-order superintegrable system in two dimensions: the generic 3-parameter model on the 2-sphere is cast in the framework of the Racah problem for the \({\mathfrak{su}(1,1)}\) algebra. The Hamiltonian of the 3-parameter system and the generators of its quadratic symmetry algebra are seen to correspond to the total and intermediate Casimir operators of the combination of three \({\mathfrak{su}(1,1)}\) algebras, respectively. The construction makes explicit the isomorphism between the Racah–Wilson algebra, which is the fundamental algebraic structure behind the Racah problem for \({\mathfrak{su}(1, 1)}\) , and the invariance algebra of the generic 3-parameter system. It also provides an explanation for the occurrence of the Racah polynomials as overlap coefficients in this context. The irreducible representations of the Racah–Wilson algebra are reviewed as well as their connection with the Askey scheme of classical orthogonal polynomials.     

Novel ansatzes and scalar quantities in gravito-electromagnetism

In this work, we focus on the theory of gravito-electromagnetism (GEM)—the theory that describes the dynamics of the gravitational field in terms of quantities met in electromagnetism—and we propose two novel forms of metric perturbations. The first one is a generalisation of the traditional GEM ansatz, and succeeds in reproducing the whole set of Maxwell’s equations even for a dynamical vector potential \(\mathbf {A}\). The second form, the so-called alternative ansatz, goes beyond that leading to an expression for the Lorentz force that matches the one of electromagnetism and is free of additional terms even for a dynamical scalar potential \(\varPhi \). In the context of the linearised theory, we then search for scalar invariant quantities in analogy to electromagnetism. We define three novel, 3rd-rank gravitational tensors, and demonstrate that the last two can be employed to construct scalar quantities that succeed in giving results very similar to those found in electromagnetism. Finally, the gauge invariance of the linearised gravitational theory is studied, and shown to lead to the gauge invariance of the GEM fields \(\mathbf {E}\) and \(\mathbf {B}\) for a general configuration of the arbitrary vector involved in the coordinate transformations.     

A note on the area spectrum of d-dimensional pure de Sitter Space-time

Consistent supercurrent multiplets are naturally associated with linearized off-shell supergravity models. In S.M. Kuzenko, J. High Energy Phys. 1004, 022 (2010) we presented the hierarchy of such supercurrents which correspond to all the models for linearized 4D $\mathcal{N}=1$ supergravity classified a few years ago. Here we analyze the correspondence between the most general supercurrent given in S.M. Kuzenko, J. High Energy Phys. 1004, 022 (2010) and the one obtained eight years ago in M. Magro et al., Ann. Phys. 298, 123 (2002) using the superfield Noether procedure. We apply the Noether procedure to the general $\mathcal{N}=1$ supersymmetric nonlinear sigma-model and show that it naturally leads to the so-called $\mathcal{S}$ -multiplet, revitalized in Z. Komargodski, N. Seiberg, J. High Energy Phys. 1007, 017 (2010).     

Localization of the Grover Walks on Spidernets and Free Meixner Laws

In the first part, we have constructed several families of interacting wedge-local nets of von Neumann algebras. In particular, we discovered a family of models based on the endomorphisms of the U(1)-current algebra ${\mathcal{A} ^{(0)}}$ of Longo-Witten. In this second part, we further investigate endomorphisms and interacting models. The key ingredient is the free massless fermionic net, which contains the U(1)-current net as the fixed point subnet with respect to the U(1) gauge action. Through the restriction to the subnet, we construct a new family of Longo-Witten endomorphisms on ${\mathcal{A} ^{(0)}}$ and accordingly interacting wedge-local nets in two-dimensional spacetime. The U(1)-current net admits the structure of particle numbers and the S-matrices of the models constructed here do mix the spaces with different particle numbers of the bosonic Fock space.     

The Weil Algebra of a Hopf Algebra I: A Noncommutative Framework

We generalize the notion, introduced by Henri Cartan, of an operation of a Lie algebra ${\mathfrak{g}}$ in a graded differential algebra Ω. We define the notion of an operation of a Hopf algebra ${\mathcal{H}}$ in a graded differential algebra Ω which is referred to as a ${\mathcal{H}}$ -operation. We then generalize for such an operation the notion of algebraic connection. Finally we discuss the corresponding noncommutative version of the Weil algebra: The Weil algebra ${W(\mathcal{H})}$ of the Hopf algebra ${\mathcal{H}}$ is the universal initial object of the category of ${\mathcal{H}}$ -operations with connections.     

Canonical Basis for Quantum {\mathfrak{osp}(1|2)}

We introduce a modified quantum enveloping algebra as well as a (modified) covering quantum algebra for the ortho-symplectic Lie superalgebra ${\mathfrak{osp}(1|2)}$ . Then we formulate and compute the corresponding canonical bases, and relate them to the counterpart for ${\mathfrak{sl}(2)}$ . This provides a first example of canonical basis for quantum superalgebras.     

Cosmological perturbations in SFT inspired non-local scalar field models

We clearly and consistently supersymmetrize the celebrated horizontality condition to derive the off-shell nilpotent and absolutely anticommuting Becchi?CRouet?CStora?CTyutin (BRST) and anti-BRST symmetry transformations for the supersymmetric system of a free spinning relativistic particle within the framework of superfield approach to BRST formalism. For the precise determination of the proper (anti-)BRST symmetry transformations for all the bosonic and fermionic dynamical variables of our system, we consider the present theory on a (1,2)-dimensional supermanifold parameterized by an even (bosonic) variable (??) and a pair of odd (fermionic) variables ?? and $\bar{\theta}$ (with $\theta^{2} = \bar{\theta}^{2} = 0$ , $\theta\bar{\theta}+ \bar{\theta}\theta= 0$ ) of the Grassmann algebra. One of the most important and novel features of our present investigation is the derivation of (anti-)BRST invariant Curci?CFerrari type restriction which turns out to be responsible for the absolute anticommutativity of the (anti-)BRST transformations and existence of the coupled (but equivalent) Lagrangians for the present theory of a supersymmetric system. These observations are completely new results for this model.     

Supersymmetric deformations of 3D SCFTs from tri-Sasakian truncation

We holographically study supersymmetric deformations of \(N=3\) and \(N=1\) superconformal field theories in three dimensions using four-dimensional \(N=4\) gauged supergravity coupled to three-vector multiplets with non-semisimple \(SO(3)\ltimes (\mathbf {T}^3,\hat{\mathbf {T}}^3)\) gauge group. This gauged supergravity can be obtained from a truncation of 11-dimensional supergravity on a tri-Sasakian manifold and admits both \(N=1,3\) supersymmetric and stable non-supersymmetric \(AdS_4\) critical points. We analyze the BPS equations for SO(3) singlet scalars in detail and study possible supersymmetric solutions. A number of RG flows to non-conformal field theories and half-supersymmetric domain walls are found, and many of them can be given analytically. Apart from these “flat” domain walls, we also consider \(AdS_3\)-sliced domain wall solutions describing two-dimensional conformal defects with \(N=(1,0)\) supersymmetry within the dual \(N=1\) field theory while this type of solutions does not exist in the \(N=3\) case.     

Free Field Construction for the ABF Models in Regime II

The Wakimoto construction for the quantum affine algebra U $_q$ ( $(\widehat{\mathfrak{s}\mathfrak{l}_2 })$ ) admits a reduction to the q-deformed parafermion algebras. We interpret the latter theory as a free field realization of the Andrews–Baxter–Forrester models in regime II. We give multi-particle form factors of some local operators on the lattice and compute their scaling limit, where the models are described by a massive field theory with $\mathbb{Z}$ $_k$ symmetric minimal scattering matrices.     

Weyl type N solutions with null electromagnetic fields in the Einstein–Maxwell <Emphasis Type="Italic">p</Emphasis>-form theory

We consider d-dimensional solutions to the electrovacuum Einstein–Maxwell equations with the Weyl tensor of type N and a null Maxwell \((p+1)\)-form field. We prove that such spacetimes are necessarily aligned, i.e. the Weyl tensor of the corresponding spacetime and the electromagnetic field share the same aligned null direction (AND). Moreover, this AND is geodetic, shear-free, non-expanding and non-twisting and hence Einstein–Maxwell equations imply that Weyl type N spacetimes with a null Maxwell \((p+1)\)-form field belong to the Kundt class. Moreover, these Kundt spacetimes are necessarily \({ CSI}\) and the \((p+1)\)-form is \({ VSI}\). Finally, a general coordinate form of solutions and a reduction of the field equations are discussed.     

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.     

Angles, Scales and Parametric Renormalization

We discuss the structure of renormalized Feynman rules. Regarding them as maps from the Hopf algebra of Feynman graphs to ${\mathbb{C}}$ originating from the evaluation of graphs by Feynman rules, they are elements of a group ${G=\mathrm{Spec}_{\mathrm{Feyn}}(H)}$ . We study the kinematics of scale and angle-dependence to decompose G into subgroups ${G_{\mathrm{\makebox{1-s}}}}$ and ${G_{\mathrm{fin}}}$ . Using parametric representations of Feynman integrals, renormalizability and the renormalization group underlying the scale dependence of Feynman amplitudes are derived and proven in the context of algebraic geometry.     

Conformally flat sources for the Linet–Tian spacetime

We investigate the matching, across cylindrical surfaces, of static cylindrically symmetric conformally flat spacetimes with a cosmological constant $\Lambda $ , satisfying regularity conditions at the axis, to an exterior Linet–Tian spacetime. We prove that for $\Lambda \le 0$ such matching is impossible. On the other hand, we show through simple examples that the matching is possible for $\Lambda >0$ . We suggest a physical argument that might explain these results.