Abstract
A quadratic Leibniz algebra $(\mathbb{V},[\ifmmode\cdot\else\textperiodcentered\fi{},\ifmmode\cdot\else\textperiodcentered\fi{}],\ensuremath{\kappa})$ 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 $\mathbb{W}\ensuremath{\subset}\mathbb{V}$ generated by the symmetric part of the bracket. If the Leibniz bracket is antisymmetric, the quadratic Leibniz algebra reduces to a quadratic Lie algebra, $B\ensuremath{\equiv}0$, 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.
Highlights
Leibniz algebras are a simple generalization of Lie algebras, even though they have been much less studied than their antisymmetric specialization
What we find is that in this reformulation, one obtains an ordinary Yang-Mills gauge theory for a Lie algebra g, to which, 2-form fields are coupled in a g-representation
That in addition to the condition (2), which is needed for gauge invariance, we require the restriction κjW of κ to W to be nondegenerate, which is needed for the nontrivial propagation of the B-gauge fields in (9)
Summary
Leibniz algebras are a simple generalization of Lie algebras, even though they have been much less studied than their antisymmetric specialization. What we find is that in this reformulation, one obtains an ordinary Yang-Mills gauge theory for a Lie algebra g, to which, 2-form fields are coupled in a g-representation. We make use precisely of the restrictions posed on a Leibniz algebra by κ and turn to the main finding of this paper, namely the above-mentioned equivalence of the Leibniz-Yang-Mills functional with a YangMills theory coupled to massive 2-form fields that one obtains by an appropriate field redefinition. If the symmetric part of the bracket vanishes, and only the Leibniz algebra becomes a Lie algebra, W becomes the 0-vector inside V, and these gauge fields reduce to the Lie algebra valued connection 1-forms we are used to from ordinary Yang-Mills gauge theories. ΛG, multiplying the two contributions to S appropriately, tuning their relative weight and possibly introducing the appropriate physical dimensions
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