Complex Path Integrals and Saddles in Two-Dimensional Gauge Theory.

We study numerically the saddle point structure of two-dimensional lattice gauge theory, represented by the Gross-Witten-Wadia unitary matrix model. The saddle points are, in general, complex valued, even though the original integration variables and action are real. We confirm the trans-series and instanton gas structure in the weak-coupling phase, and we identify a new complex-saddle interpretation of nonperturbative effects in the strong-coupling phase. In both phases, eigenvalue tunneling refers to eigenvalues moving off the real interval, into the complex plane, and the weak-to-strong coupling phase transition is driven by saddle condensation.