Surface Observables, $2$-Knot Invariants, and Nonabelian Electric Fluxes

This work introduces a surface observable for nonabelian four-dimensional $BF$ theory with a cosmological term. The surface observable yields new $2$-knot invariants that may extend beyond known examples such as the Alexander invariant. By BV pushforward, the surface observable induces an electric observable in nonabelian Yang-Mills theory, offering a concrete realization of 't~Hooft operators. An application to self-dual Yang-Mills theory is also discussed.