Leibniz Gauge Theories and Infinity Structures

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.

50 pages, v2: refs added, new subsection 3.2, version to appear in Comm. Math. Phys

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Cited by in corpus (9)

• The controlling $L_{\infty }$-algebra, cohomology and homotopy of embedding tensors and Lie-Leibniz triples
• Non-associative algebraic structures: classification and structure
• Infinity-enhancing of Leibniz algebras
• From Lie algebra crossed modules to tensor hierarchies
• Lie and Leibniz Algebras of Lower-Degree Conservation Laws
• On the $L_{\infty }$ formulation of Chern-Simons theories
• The puzzle of global Double Field Theory: open problems and the case for a Higher Kaluza-Klein perspective
• Nonlinear realisations of Lie superalgebras
• Nonabelian embedding tensors on 3-Lie algebras and 3-Leibniz-Lie algebras