## Abstract

It has been argued that many non-perturbative phenomena in quantum mechanics (QM) and quantum field theory (QFT) are determined by complex field configurations, and that these contributions should be understood in terms of Picard-Lefschetz theory. In this work we compute the contribution from non-BPS multi-instanton configurations, such as instanton-anti-instanton \( \left[\mathrm{\mathcal{I}}\overline{\mathrm{\mathcal{I}}}\right] \) pairs, and argue that these contributions should be interpreted as exact critical points at infinity. The Lefschetz thimbles associated with such critical points have a specific structure arising from the presence of non-Gaussian, quasi-zero mode (QZM), directions. When fermion degrees of freedom are present, as in supersymmetric theories, the effective bosonic potential can be written as the sum of a classical and a quantum potential. We show that in this case the semi-classical contribution of the critical point at infinity vanishes, but there is a non-trivial contribution that arises from its associated non-Gaussian QZM-thimble. This approach resolves several puzzles in the literature concerning the semi-classical contribution of correlated \( \left[\mathrm{\mathcal{I}}\overline{\mathrm{\mathcal{I}}}\right] \) pairs. It has the surprising consequence that the configurations dominating the expansion of observables, and the critical points defining the Lefschetz thimble decomposition need not be the same, a feature not present in the traditional Picard-Lefschetz approach.

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Behtash, A., Dunne, G.V., Schäfer, T. *et al.* Critical points at infinity, non-Gaussian saddles, and bions.
*J. High Energ. Phys.* **2018**, 68 (2018). https://doi.org/10.1007/JHEP06(2018)068

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DOI: https://doi.org/10.1007/JHEP06(2018)068