Higher curvature Bianchi identities, generalised geometry and $L_{\infty }$ algebras

Abstract

The Bianchi identities for bosonic fluxes in supergravity can receive higher derivative quantum and string corrections, the most well known being that of Heterotic theory $dH=\frac{1}{4}{\alpha }^{\prime }(\text{tr }{F}^2-\text{tr }{R}^2)$. Less studied are the modifications at order $R^4$ that may arise, for example, in the Bianchi identity for the seven-form flux of M theory compactifications. We argue that such corrections appear to be incompatible with the exceptional generalised geometry description of the lower order supergravity, and seem to imply a gauge algebra for the bosonic potentials that cannot be written in terms of an (exceptional) Courant bracket. However, we show that this algebra retains the form of an $L_{\infty }$ gauge field theory, which terminates at a level ten multibracket for the case involving just the seven-form flux.

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