We build a conformal field theory on multiply connected domains out of correlation functions of the Gaussian free field. A key ingredient is the derivation of the generalized Eguchi-Ooguri equations, which allow us to derive an explicit form of Ward's equations. Our version of the Ward equation relates the insertion of a stress tensor in the computation of a correlation function to the application of a Lie derivative to the correlation function, along with a differential operator that accounts for the change in the Teichmuller modular parameters. By applying an appropriate operator product expansion to the Ward equation, we also produce an infinite family of martingale observables for an associated SLE process on the multiply connected domain.