Critical points at infinity, non-Gaussian saddles, and bions

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 [IĪ] 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 [IĪ] 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.

Instantons in the Hofstadter butterfly: difference equation, resurgence and quantum mirror curves

“…it is independent of the constant B which parametrizes the instanton-anti-instanton interactions. This is in fact clarified by the thimble integration procedure in [27], summarized above. The ambiguity comes from the vicinity of the critical point at infinity, which, for a finite temporal extent, is the instantonanti-instanton pair at opposing ends of the temporal circle.…”

“…But there are several issues here. Firstly the semi-classics of path-integrals is to this day not a completely understood subject, but it has become clear recently that the correct interpretation of it is via the Picard-Lefschetz (PL) theory [19][20][21][22][23][24][25][26][27][28]. The PL theory analysis is by far not a straightforward matter, and requires the identification of saddles which contribute in the semi-classical expansion.…”

“…Particularly tricky is the instanton-anti-instanton contribution [IĪ], which is a priori ill-defined. This is because when instanton and anti-instanton are close to each other the configuration is indistinguishable from the perturbative vacuum, and it is not clear how such configurations should be taken into account (see [27] for an incomplete list of references on the topic). If we naively superpose the well-separated instanton and anti-instanton, where we label their separation by τ , the action will be an increasing function of τ .…”

“…Since uncorrelated instantons have already been taken into account by the instanton gas approximation it should be subtracted here to avoid double counting. this puzzle was proposed by the interpretation of the instanton-anti-instanton pair as a saddle point at infinity [27], which establishes a concrete method for a systematic calculation of the semi-classical expansion in path integrals. The procedure is roughly as follows (We refer the reader to [27] for details.…”

“…this puzzle was proposed by the interpretation of the instanton-anti-instanton pair as a saddle point at infinity [27], which establishes a concrete method for a systematic calculation of the semi-classical expansion in path integrals. The procedure is roughly as follows (We refer the reader to [27] for details. ): 1) Consider an instanton-anti-instanton configuration for the case of finite time β.…”