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Motivated by the recent interest growing in contact structures, this paper is devoted to even-dimensional counterpart of contact geometry, namely the locally conformal symplectic one. In physics, these structures come natural from endowing the phase-space with a 2-form that is non-degenerate but locally closed up to a multiplicative non-vanishing function. In view of this, we initially adopt the trivial line bundle situation and show that a given locally conformal simplectic over an even-dimensional manifold leads to a transitive Jacobi pair and conversely. Then, we introduce the geometric perspective on locally conformal symplectic structures in terms of a line bundle over an even-dimensional manifold, L → M endowed with a flat-connection, ∇ and an L-valued non-degenerate and closed 2-form, ω ∈ Ω (M; L). In this framework we also exhibit the connection between locally conformal symplectic structures and transitive Jacobi ones. [ABSTRACT FROM AUTHOR]

Starting with an interacting SU(2) x U(1)-gauge model comprising four 1-forms and six real scalar fields, we closely investigate the standard mass-generation mechanism. Assuming that the scalar fields transform under an irreducible and orthogonal representation of the gauge group, we show that the outputs strongly rely on the choice of the vacuum expectation values for the scalar fields, as follows. Initially, by choosing a vacuum expectation configuration with a single non-vanishing component and applying the standard mass-generation mechanism we get a physical field spectrum consisting in three massive 1-forms, a massless 1-form and three real scalar fields, among which only one is massive. Repeating the same procedure, but for a vacuum expectation configuration with two different non-vanishing components, we exhibit a physical field spectrum consisting in four massive 1-forms with distinct masses and two real scalar fields, among which only one is massive. Finally, by considering a vacuum expectation configuration with two equal non-vanishing components, the same mechanism displays three massive 1-forms, among which two with the same mass, a massless 1-form and three physical scalar fields, among which only one is massive. [ABSTRACT FROM AUTHOR]

Motivated by the recent physicists’ interest in contact geometry, this review paper is devoted to some modern geometric insights upon the contact structures. In view of this, we start from the initial perspective on contact manifolds, namely that of an odd-dimensional orientable manifold whose volume form is generated by a 1-form θ and its differential dθ. This naturally arises from a coorientable maximally non-integrable hyperplane distribution. In this picture, we establish a one-to-one correspondence between the transitive Jacobi pairs over odd-dimensional manifolds and the coorientable contact structures over the same manifolds. Then, we introduce the geometric perspective on contact manifolds by omitting the coorientability of the maximally non-integrable hyperplane distribution, and we define the contact structure via an L-valued 1-form [with (L, π, M) a line bundle over an odd-dimensional manifold] with non-degenerate curvature. In this realm, it is shown that there exists a one-to-one correspondence between the transitive Jacobi line bundles over odd-dimensional manifolds and the contact structures over the same manifolds. This faithful ‘representation’ of contact structures brings them nearer to symplectic-like ones through the canonical [bracket] structures inherited from the corresponding Jacobi structures. [ABSTRACT FROM AUTHOR]

Starting with an interacting theory comprising an Abelian 1-form and four real scalar fields, we closely investigate the standard mass-generation mechanism. We show that the outputs strongly rely on the rigid symmetry group of the interacting potential for the scalar fields, as follows. Initially, by considering of an interacting potential that is S O (4)-invariant and applying the standard mass-generation mechanism, we get a physical massive one-form and three physical scalar fields, among which only one is massive. Repeating the same procedure, but for a potential that is invariant only under S O (2) × S O (2), we obtain the same physical field spectrum, namely, a massive one-form and three massive scalar fields, but among which at least two display the same mass. Finally, by taking into account a potential that is manifestly S O (2)-invariant, we exhibit a Zeeman-like effect in the mass-spectrum of the physical scalar fields, i.e., all the three masses corresponding to the scalar degrees of freedom are distinct. [ABSTRACT FROM AUTHOR]

In this paper we implement a new procedure by which one may generate mass for a vector field in the context of its interactions to a system of five real scalar fields. This purpose will be achieved by means of the general multi-step program from [1] adapted to the present situation: (1) we begin with a free theory in four space-time dimensions whose Lagrangian action is given by the sum between the standard Maxwell action and that for a collection consisting in five massless real scalar fields; (2) we construct a general class of gauge theories whose free limit is that from step (1) by means of the deformation of the solution to the master equation [2, 3] with the help of local BRST cohomology [4-6]; (3) we perform some suitable redefinitions of the free parameters that label interacting theories from (2) such that the mass terms become manifest in the new free limit. The outputs of our procedure can be synthesized in: (A) the vector field acquires mass; (B) the scalar fields gain gauge transformations; (C) the gauge algebras of the interacting theories are Abelian; (D) the propagator of the massive vector field emerging from the gauge-fixed actions behaves, in the limit of large Euclidean momenta, like that from the massless case. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to apply a procedure, different from the Higgs mechanism, by which one may generate mass for a vector field in the context of its interactions to a set of N = 4 real scalar fields. This goal will be attained via implementing the general multi-step program from [1] adapted to the present context: (1) we start from a free theory in D = 4 whose Lagrangian action is expressed like the sum between the Maxwell action for a single vector field and that for a collection of N = 4 massless real scalar fields; (2) we construct a general class of gauge theories whose free limit is that from step (1) by means of the deformation of the solution to the master equation [2] and [3] with the help of local BRST cohomology [4-6]. On the one hand, the setup described so far does not account in any way for the Higgs mechanism. On the other hand, it will be proved to produce the next results: (A) the vector field acquires mass; (B) the scalar fields gain gauge transformations, by contrast to the free limit, but the associated gauge algebra remains Abelian; (C) the propagator of the massive vector field emerging from the gauge-fixed action behaves, in the limit of large Euclidean momenta, like that from the massless case. [ABSTRACT FROM AUTHOR]

The scope of this paper consists in the construction of several versions of first-order formalisms corresponding to a generic second-order Lagrangian that describes the dynamics of an Abelian tensor gauge field of degree (k+1). The starting second-order dynamics depends on two arbitrary real constants. In view of this, one uses the generating sets of gauge transformations derived in [1] and some specific auxiliary matter/gauge fields. [ABSTRACT FROM AUTHOR]

Starting from a generic second-order Lagrangian (that describes the dynamics of an Abelian tensor gauge field of degree two and depends on two arbitrary real constants) we perform the Dirac analysis. This procedure determines the number of physical degrees of freedom and also a generating set of gauge transformations for the initial theory. [ABSTRACT FROM AUTHOR]