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 $\mathrm{diff}(S^2)⋉\mathrm{vect}(S^2{)}_{\mathrm{ab}}$ 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 Plebański simplicity constraints.

References in corpus (15)

• The Weyl BMS group and Einstein's equations
• Edge modes of gravity -- II: Corner metric and Lorentz charges
• Embeddings and Integrable Charges for Extended Corner Symmetry
• Isolated Surfaces and Symmetries of Gravity
• Gravitational Edge Modes, Coadjoint Orbits, and Hydrodynamics
• Coadjoint representation of the BMS group on celestial Riemann surfaces
• Universal Corner Symmetry and the Orbit Method for Gravity
• Topological Higher Gauge Theory - from BF to BFCG theory
• Bi-metric theory of gravity from the non-chiral Plebanski action
• Asymptotic gravitational charges
• Hamiltonian derivation of dual gravitational charges
• Most general theory of 3d gravity: Covariant phase space, dual diffeomorphisms, and more
• Kinematical Gravitational Charge Algebra
• Dual diffeomorphisms and finite distance asymptotic symmetries in 3d gravity
• (2-)Drinfel'd Double and (2-)BF Theory

Cited by in corpus (2)

• Higher Gauge Theory and Integrability
• Probing $\Delta L=2$ lepton number violating SMEFT operators at the same-sign muon collider