$L_{\infty }$ algebras for extended geometry from Borcherds superalgebras

Abstract

We examine the structure of gauge transformations in extended geometry, the framework unifying double geometry, exceptional geometry, etc. This is done by giving the variations of the ghosts in a Batalin-Vilkovisky framework, or equivalently, an $L_{\infty }$ algebra. The $L_{\infty }$ brackets are given as derived brackets constructed using an underlying Borcherds superalgebra ${\scr B}({\mathfrak g}_{r+1})$, which is a double extension of the structure algebra ${\mathfrak{g}}_r$. The construction includes a set of "ancillary" ghosts. All brackets involving the infinite sequence of ghosts are given explicitly. All even brackets above the 2-brackets vanish, and the coefficients appearing in the brackets are given by Bernoulli numbers. The results are valid in the absence of ancillary transformations at ghost number 1. We present evidence that in order to go further, the underlying algebra should be the corresponding tensor hierarchy algebra.

arXiv commentv2: 59 pp. Version published in Commun. Math. Phys. References added. Section 5 extended with a more detailed explanation of the equivalence between the BV ghost variation and the L_\infty brackets

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Cited by in corpus (13)

• The Geometry, Branes and Applications of Exceptional Field Theory
• Tensor hierarchy algebras and extended geometry II: Gauge structure and dynamics
• Tensor hierarchy algebras and extended geometry I: Construction of the algebra
• Tensor hierarchy algebra extensions of over-extended Kac--Moody algebras
• Infinity-enhancing of Leibniz algebras
• Teleparallelism in the algebraic approach to extended geometry
• Extended geometry of magical supergravities
• Towards an extended/higher correspondence -- Generalised geometry, bundle gerbes and global Double Field Theory
• The teleparallel complex
• Unfolding $E_{11}$
• Higher curvature Bianchi identities, generalised geometry and $L_{\infty }$ algebras
• On the $L_{\infty }$ formulation of Chern-Simons theories
• Nonlinear realisations of Lie superalgebras