Diffeomorphisms as quadratic charges in 4d BF theory and related TQFTs

Abstract

We present a Sugawara-type construction for boundary charges in 4d BF theory and in a general family of related TQFTs. Starting from the underlying current Lie algebra of boundary symmetries, this gives rise to well-defined quadratic charges forming an algebra of vector fields. In the case of 3d BF theory (i.e. 3d gravity), it was shown in [PRD 106 (2022), arXiv:2012.05263 [hep-th]] that this construction leads to a two-dimensional family of diffeomorphism charges which satisfy a certain modular duality. Here we show that adapting this construction to 4d BF theory first requires to split the underlying gauge algebra. Surprisingly, the space of well-defined quadratic generators can then be shown to be once again two-dimensional. In the case of tangential vector fields, this canonically endows 4d BF theory with a $\mathrm{diff}(S^2)\times\mathrm{diff}(S^2)$ or $\mathrm{diff}(S^2)\ltimes\mathrm{vect}(S^2)_\mathrm{ab}$ algebra of boundary symmetries depending on the gauge algebra. The prospect is to then understand how this can be reduced to a gravitational symmetry algebra by imposing Pleba\'nski simplicity constraints.