Toward Picard-Lefschetz Theory of Path Integrals and the Physics of  Complex Saddles

We show that the semi-classical analysis of generic Euclidean path integrals necessarily requires complexification of the action and measure, and consideration of complex saddle solutions. We demonstrate that complex saddle points have a natural interpretation in terms of the Picard-Lefschetz theory. |In the supersymmetric theories, the inclusion of complex saddles is strictly necessary to prevent clash with the supersymmetry algebra. In calculable Yang-Mills theories, the non-perturbative stabilization of center-symmetry is due to complex saddles, called neutral bions