On the expansion of a quantum field theory around a topological sector

The idea of treating quantum general relativistic theories in a perturbative expansion around a topological theory has recently received attention, in the quantum gravity literature. We investigate the viability of this idea by applying it to conventional Yang–Mills theory on flat spacetime. This theory admits indeed a formulation as a modified topological theory, like general relativity. We find that the expansion around the topological theory coincides with the usual expansion around the free abelian theory, though the equivalence is non-trivial. In this context, the technique appears therefore to be viable, but not to bring particularly new insights. On the other hand, we point out that the relation of this expansion with the actual quantum BF theory is far from being transparent. Some implications for gravity are discussed.