Generalizations of 3-Sasakian manifolds and skew torsion

In the first part, we define and investigate new classes of almost 3-contact metric manifolds, with two guiding ideas in mind: first, what geometric objects are best suited for capturing the key properties of almost 3-contact metric manifolds, and second, the new classes should admit ‘good’ metric connections with skew torsion. In particular, we introduce the Reeb commutator function and the Reeb Killing function, we define the new classes of canonical almost 3-contact metric manifolds and of 3-(α, δ)-Sasaki manifolds (including as special cases 3-Sasaki manifolds, quaternionic Heisenberg groups, and many others) and prove that the latter are hypernormal, thus generalizing a seminal result of Kashiwada. We study their behaviour under a new class of deformations, called 𝓗-homothetic deformations, and prove that they admit an underlying quaternionic contact structure, from which we deduce the Ricci curvature. For example, a 3-(α, δ)-Sasaki manifold is Einstein either if α = δ (the 3-α-Sasaki case) or if δ = (2n + 3)α, where dim M = 4n + 3.In the second part we find these adapted connections. We start with a very general notion of φ-compatible connections, where φ denotes any element of the associated sphere of almost contact structures, and make them unique by a certain extra condition, thus yielding the notion of canonical connection (they exist exactly on canonical manifolds, hence the name). For 3-(α, δ)-Sasaki manifolds, we compute the torsion of this connection explicitly and we prove that it is parallel, we describe the holonomy, the ∇-Ricci curvature, and we show that the metric cone is a HKT-manifold. In dimension 7, we construct a cocalibrated G2-structure inducing the canonical connection and we prove the existence of four generalized Killing spinors.

BRAIDED STATISTICS FROM ABELIAN TWIST IN κ-MINKOWSKI SPACETIME

κ-deformed commutation relation between quantum operators is constructed via Abelian twist deformation in κ-Minkowski spacetime. The commutation relation is written in terms of universal R-matrix satisfying braided statistics. The equal-time commutator function vanishes in this framework.

COVARIANT TWO-POINT FUNCTION FOR MINIMALLY COUPLED SCALAR FIELD IN de SITTER SPACE–TIME

In a recent paper,1 it has been shown that negative norm states are indispensable for a fully covariant quantization of the minimally coupled scalar field in de Sitter space. Their presence, while leaving unchanged the physical content of the theory, offers an automatic and covariant renormalization of the vacuum energy divergence. This paper is a completion of our previous work. An explicit construction of the covariant two-point function of the "massless" minimally coupled scalar field in de Sitter space is given, which is free of any infrared divergence. The associated Schwinger commutator function and retarded Green's function are calculated in a fully gauge-invariant way, which also means coordinate independent.

Multiple semidirect products of associative systems

Suppose that a group G is the semidirect product of a subgroup N and a normal subgroup M. Then the elements of G have unique expressions mn (m ∈ M, n ∈ N) and the commutator functionmaps N x M into M. In fact there is an action (by automorphisms) of N on M given byConversely, if one is given an action of a group N on a group M then one can construct a semidirect product.

Various Aspects of Divergent Electromagnetic Masses: Analyticity, Scaling, and Light-Cone

Various features associated with the divergent electromagnetic self-mass of a particle are elucidated. Bjorken's original analysis is implemented with a simple convergence theorem which serves as a basis on which various models are discussed regarding their implications on the divergence problem as well as on the scaling property. A simple model of the current commutator function is proposed, on the basis of which it is argued that the quark-model light-cone commutator gives a wrong sign to the divergent self-mass due to its lack of the current conservation. It is also shown that it is the next-to-leading singularities on the light-cone of the commutator that bring forth a (logarithmically) divergent mass shift.