Séminaire Physique mathématique ICJ

From Lie algebra crossed modules to tensor hierarchies, and beyond

by Dr Sylvain Lavau

Fokko du Cloux (Bâtiment Braconnier)

Fokko du Cloux (Bâtiment Braconnier)

Gauging procedures in supergravity theories depart from classical gauge theories as in the former, the gauge fields take values in the fundamental representation V of the Lie algebra g of global symmetries of the system. The consistency of the theory relies on a pairing V-->g called the embedding tensor, allowing to lift the Lie algebra structure of g to a Leibniz algebra structure on V. As is usually met in higher gauge theories, if the gauge algebra is not Lie (here it is Leibniz), it is replaced by some higher form of Lie algebras. Here, such a higher structure is materialized by a differential graded Lie algebra on a chain complex of g-modules, called the tensor hierarchy. 

In the present talk we explain how tensor hierarchies are genetically related to Lie algebra crossed modules. Indeed, two such algebras V and g, together with their embedding tensor, form a triple called a Lie-Leibniz triple, of which Lie algebra crossed modules are particular cases. The canonical assignment (functor) associating to any Lie algebra crossed module its corresponding unique 2-term differential graded Lie algebra can be extended to the category of Lie-Leibniz triples, giving their associated tensor hierarchies. This shows that Lie-Leibniz triples form natural generalizations of Lie algebra crossed modules and that their associated tensor hierarchies can be considered as some kind of 'lie-ization' of the former. Possible applications of such a functor will be outlined.