  A general quantum mechanical or quantum field theoretical system in the path
integral formulation has both real and complex saddles (instantons and
ghost-instantons). Resurgent asymptotic analysis implies that both types of
saddles contribute to physical observables, even if the complex saddles are not
on the integration path i.e., the associated Stokes multipliers are zero. We
show explicitly that instanton-anti-instanton and ghost--anti-ghost saddles
both affect the expansion around the perturbative vacuum. We study a self-dual
model in which the analytic continuation of the partition function to negative
values of coupling constant gives a pathological exponential growth, but a
homotopically independent combination of integration cycles (Lefschetz
thimbles) results in a sensible theory. These two choices of the integration
cycles are tied with a quantum phase transition. The general set of ideas in
our construction may provide new insights into non-perturbative QFT, string
theory, quantum gravity, and the theory of quantum phase transitions.
