Leibniz-Yang-Mills Gauge Theories and the 2-Higgs Mechanism

Abstract

A quadratic Leibniz algebra $(\mathbb{V},[\cdot ,\cdot ],\kappa )$ gives rise to a canonical Yang-Mills type functional $S$ over every space-time manifold. The gauge fields consist of 1-forms $A$ taking values in $\mathbb{V}$ and 2-forms $B$ with values in the subspace $\mathbb{W}\subset \mathbb{V}$ generated by the symmetric part of the bracket. If the Leibniz bracket is anti-symmetric, the quadratic Leibniz algebra reduces to a quadratic Lie algebra, $B\equiv 0$, and $S$ becomes identical to the usual Yang-Mills action functional. We describe this gauge theory for a general quadratic Leibniz algebra. We then prove its (classical and quantum) equivalence to a Yang-Mills theory for the Lie algebra $\mathfrak{g}=\mathbb{V}/\mathbb{W}$ to which one couples massive 2-form fields living in a $\mathfrak{g}$-representation. Since in the original formulation the B-fields have their own gauge symmetry, this equivalence can be used as an elegant mass-generating mechanism for 2-form gauge fields, thus providing a 'higher Higgs mechanism' for those fields.

6 pages; a subsection with remarks on finite gauge transformations added. Version to be published in Phys. Rev. D

References in corpus (6)

• The maximal D=4 supergravities
• Higher Yang-Mills Theory
• General Yang-Mills type gauge theories for p-form gauge fields: From physics-based ideas to a mathematical framework OR From Bianchi identities to twisted Courant algebroids
• Hidden Q-structure and Lie 3-algebra for non-abelian superconformal models in six dimensions
• The end of the p-form hierarchy
• Non-abelian Gerbes and Enhanced Leibniz Algebras

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• On Maximal, Universal and Complete Extensions of Yang-Mills-Type Theories