Carroll contractions of Lorentz-invariant theories

We consider Carroll-invariant limits of Lorentz-invariant field theories. We show that just as in the case of electromagnetism, there are two inequivalent limits, one “electric” and the other “magnetic”. Each can be obtained from the corresponding Lorentz-invariant theory written in Hamiltonian form through the same “contraction” procedure of taking the ultrarelativistic limit c → 0 where c is the speed of light, but with two different consistent rescalings of the canonical variables. This procedure can be applied to general Lorentz-invariant theories (p-form gauge fields, higher spin free theories etc) and has the advantage of providing explicitly an action principle from which the electrically-contracted or magnetically-contracted dynamics follow (and not just the equations of motion). Even though not manifestly so, this Hamiltonian action principle is shown to be Carroll invariant. In the case of p-forms, we construct explicitly an equivalent manifestly Carroll-invariant action principle for each Carroll contraction. While the manifestly covariant variational description of the electric contraction is rather direct, the one for the magnetic contraction is more subtle and involves an additional pure gauge field, whose elimination modifies the Carroll transformations of the fields. We also treat gravity, which constitutes one of the main motivations of our study, and for which we provide the two different contractions in Hamiltonian form.

Weighted Homology of Bi-Structures over Certain Discrete Valuation Rings

An RNA bi-structure is a pair of RNA secondary structures that are considered as arc-diagrams. We present a novel weighted homology theory for RNA bi-structures, which was obtained through the intersections of loops. The weighted homology of the intersection complex X features a new boundary operator and is formulated over a discrete valuation ring, R. We establish basic properties of the weighted complex and show how to deform it in order to eliminate any 3-simplices. We connect the simplicial homology, Hi(X), and weighted homology, Hi,R(X), in two ways: first, via chain maps, and second, via the relative homology. We compute H0,R(X) by means of a recursive contraction procedure on a weighted spanning tree and H1,R(X) via an inflation map, by which the simplicial homology of the 1-skeleton allows us to determine the weighted homology H1,R(X). The homology module H2,R(X) is naturally obtained from H2(X) via chain maps. Furthermore, we show that all weighted homology modules Hi,R(X) are trivial for i>2. The invariant factors of our structure theorems, as well as the weighted Whitehead moves facilitating the removal of filled tetrahedra, are given a combinatorial interpretation. The weighted homology of bi-structures augments the simplicial counterpart by introducing novel torsion submodules and preserving the free submodules that appear in the simplicial homology.

Quantum E(2) groups for complex deformation parameters

We construct a family of [Formula: see text] deformations of E(2) group for nonzero complex parameters [Formula: see text] as locally compact braided quantum groups over the circle group [Formula: see text] viewed as a quasitriangular quantum group with respect to the unitary [Formula: see text]-matrix [Formula: see text] for all [Formula: see text]. For real [Formula: see text], the deformation coincides with Woronowicz’s [Formula: see text] groups. As an application, we study the braided analogue of the contraction procedure between [Formula: see text] and [Formula: see text] groups in the spirit of Woronowicz’s quantum analogue of the classic Inönü–Wigner group contraction. Consequently, we obtain the bosonization of braided [Formula: see text] groups by contracting [Formula: see text] groups.

A polynomial class of u(2) algebras

A r-parameter u{κ1,κ2,…,κr}(2) algebra is introduced. Finite unitary representations are investigated. This polynomial algebra reduces via a contraction procedure to the generalized Weyl–Heisenberg algebra 𝒜{κ1,κ2,…,κr} [M. Daoud and M. Kibler, J. Phys. A: Math. Theor.45 (2012) 244036]. A pair of nonlinear (quadratic) bosons of type 𝒜κ ≡ 𝒜{κ1=κ,κ2=0,…,κr=0} is used to construct, à la Schwinger, a one parameter family of (cubic) uκ(2) algebra. The corresponding Hilbert space is constructed. The analytical Bargmann representation is also presented.

THE JORDAN-SCHWINGER REALIZATION OF TWO-PARAMETRIC QUANTUM GROUP slq,s(2)

The Jordan-Schwinger realization of two-parametric quantum group sl q,s(2), is presented by introducing two-parametric deformed harmonic oscillator. The Heisenberg commutation relations of the two-parametric deformed oscillator are derived by virtue of the Schwinger’s contraction procedure.