Dyson's instability in lattice gauge theory

We discuss Dyson's argument that the vacuum is unstable under a change g2 g2, in the context of lattice gauge theory. For compact gauge groups, the partition function is well defined at negative g2, but the average plaquette P has a discontinuity when g2 changes sign. This reflects a change of vacuumrather than a loss of vacuum. In addition, P has poles in the complex g2 plane, located at the complex zeros of the partition function (Fisher's zeros). We discuss the relevance of these singularities for lattice perturbation theory. We present new methods to locate Fisher's zeros using numerical values for the density of state in SU(2) and U(1) pure gauge theory. We briefly discuss similar issues for O(N) nonlinear sigma models where the local integrals are also over compact spaces.