Integrable sigma models with theta=pi

A fundamental result relevant to spin chains and two-dimensional disordered systems is that the sphere sigma model with instanton coupling theta=pi has a non-trivial low-energy fixed point and a gapless spectrum. This result is extended to two series of sigma models with theta=pi: the SU(N)/SO(N) sigma models flow to the SU(N)_1 WZW theory, while the O(2N)/O(N)\times O(N) models flow to O(2N)_1 (2N free Majorana fermions). These models are integrable, and the exact quasiparticle spectra and S matrices are found. One interesting feature is that charges fractionalize when theta=pi. I compute the energy in a background field, and verify that the perturbative expansions for \theta=0 and pi are the same as they must be. I discuss the flows between the two sequences of models, and also argue that the analogous sigma models with Sp(2N) symmetry, the Sp(2N)/U(N) models, flow to Sp(2N)_1.