Abstract
We formulate gauge theories based on Leibniz(-Loday) algebras and uncover their underlying mathematical structure. Various special cases have been developed in the context of gauged supergravity and exceptional field theory. These are based on ‘tensor hierarchies’, which describe towers of p-form gauge fields transforming under non-abelian gauge symmetries and which have been constructed up to low levels. Here we define ‘infinity-enhanced Leibniz algebras’ that guarantee the existence of consistent tensor hierarchies to arbitrary level. We contrast these algebras with strongly homotopy Lie algebras (L_{infty } algebras), which can be used to define topological field theories for which all curvatures vanish. Any infinity-enhanced Leibniz algebra carries an associated L_{infty } algebra, which we discuss.
Highlights
In this paper we construct the general gauge theory of Leibniz-Loday algebras [1,2,3,4,5,6], which are algebraic structures generalizing the notion of Lie algebras
This gauge theory construction has a certain degree of universality in that it is based on an algebraic structure encoding the most general bilinear ‘product’ defining transformations whose closure is governed by the same product, thereby generalizing the adjoint action of a Lie algebra
We use various observations made along the way, generalized further to arbitrary degrees, in order to motivate the general axioms of ‘infinity-enhanced Leibniz algebras’ that will be used in Sect. 4 to construct exact tensor hierarchies. These are not restricted to finite degrees, and we prove the consistency of the tensor hierarchy to all orders
Summary
In this paper we construct the general gauge theory of Leibniz-Loday algebras [1,2,3,4,5,6], which are algebraic structures generalizing the notion of Lie algebras. Since the associated antisymmetric bracket [·, ·] does not obey the Jacobi identity, one cannot define a gauge covariant field strength as in Yang-Mills theory This can be resolved by introducing two-form potentials taking values in the space of trivial parameters and coupling it to the naive field strength. Leibniz algebras’ [38,39], which extend a Leibniz algebra by an additional vector space, together with a new algebraic operation satisfying suitable compatibility conditions with the Leibniz product This structure is sufficient in order to define tensor hierarchies that end with two-forms. This gives an improved motivation for the axioms of infinity-enhanced Leibniz algebras as being obtained through a derived construction from a differential graded Lie algebra and a subsequent truncation to spaces of non-negative degree This is reflected in the new Sect.
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