Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence

Understand

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.
• Motivated in part by the semi-classical expansion of QCD with adjoint matter on ${\mathbb R}^3\times S^1$, we study quantum-mechanical systems with bosonic and fermionic (Grassmann) degrees of freedom with harmonic degenerate minima, as well as (related) purely bosonic systems with harmonic non-degenerate minima.
• We find exact finite action non-BPS bounce and bion solutions to the holomorphic Newton equations.