Exact instanton-anti-instanton solutions and semiclassical expansion in the ? model with fermions

Abstract

To the example of the two-dimensional Euclidean sigma model with fermions, it is shown that the effective action obtained after exact integration over the fermion fields has extrema that can be interpreted as instanton-anti-instanton configurations. The effective action does not depend on the parameters of the solutions, and the second variation has zero-frequency modes corresponding to the positions of the instantons and anti-instantons. The obtained solutions are unstable, but the authors estimate the single-loop contribution of these configurations to the functional integral using Feynman's variational principle.