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

TL;DR

This paper develops a Sugawara-type construction for boundary charges in 4d BF theory, revealing a quadratic algebra of boundary symmetries that generalizes known 3d results and suggests links to gravitational symmetries.

Contribution

It extends the Sugawara construction of boundary charges from 3d to 4d BF theory, identifying a two-dimensional space of quadratic generators and their associated symmetry algebra.

Findings

Quadratic boundary charges form a well-defined algebra of vector fields.

The symmetry algebra in 4d BF theory is isomorphic to iff(S^2) imesiff(S^2) or iff(S^2) timesct(S^2)_ab.

The construction hints at a reduction to gravitational symmetry algebra via simplicity constraints.

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.