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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]

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]

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]