Spin foam models of Yang–Mills theory coupled to gravity

We construct a spin foam model of Yang–Mills theory coupled to gravity by using a discretized path integral of the BF theory with polynomial interactions and the Barrett–Crane ansatz. In the Euclidean gravity case, we obtain a vertex amplitude which is determined by a vertex operator acting on a simple spin network function. The Euclidean gravity results can be straightforwardly extended to the Lorentzian case, so that we propose a Lorentzian spin foam model of Yang–Mills theory coupled to gravity.