$L_\infty$-algebraic extensions of non-Lorentzian kinematical Lie algebras, gravities, and brane couplings

Abstract

The Newtonian limit of Newton-Cartan gravity relies crucially on the Lie-algebraic central extension to the Galilean algebra, namely the Bargmann algebra. Lie-algebraic central extensions naturally generalise to $L_\infty$-algebraic central extensions, which in turn classify branes in superstring theory via the brane bouquet. This paper classifies all $L_\infty$-algebraic central extensions of all kinematical Lie algebras that do not depend on the spatial rotation generators as well as all iterated central extensions thereof (for codimensions $\le3$). The Bargmann central extension of the Galilean algebra then appears as merely one term in a sequence of $L_\infty$-algebraic central extensions in each degree; a similar situation obtains for the Newton-Hooke algebra and the static algebra, but not for the Carrollian algebra nor those kinematical Lie algebras that are not Wigner-\.In\"on\"u deformations of a simple algebra. The sequence of $L_\infty$-algebraic central extensions in each degree then corresponds to a tower of $p$-form fields. After imposing conventional constraints, the zero-form field provides absolute time, and the higher-form fields are certain wedge products of the field strengths of the one-form (Bargmann) gravitational field. These then provide natural $(p-1)$-brane couplings to the corresponding non-Lorentzian gravities, which are found to produce velocity-dependent gravitational effects in the presence of torsion. The $L_\infty$-algebraic cocycles also provide Wess-Zumino-Witten terms for the $(p-1)$-brane action, which require the introduction of doubled spatial coordinates that are reminiscent of double field theory, but which (in some cases at least, and given appropriate kinetic terms) do not result in doubled physics.