arXiv: High Energy Physics - Theory | 2019
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.