Hidden topological angles and Lefschetz thimbles

TL;DR

This paper uncovers hidden topological angles in quantum theories, explaining their effects on phenomena like gluon condensates and spin differences, through complex saddle point analysis and Lefschetz thimbles.

Contribution

It introduces the concept of hidden topological angles in quantum field theories and quantum mechanics, linking them to Lefschetz thimbles and explaining their physical implications.

Findings

HTAs cause a Z2 phase difference between saddles.

Gluon condensate can be positive or negative due to HTAs.

HTAs explain small condensates in super Yang-Mills and QCD-like theories.

Abstract

We demonstrate the existence of hidden topological angles (HTAs) in a large class of quantum field theories and quantum mechanical systems. HTAs are distinct from theta-parameters in the lagrangian. They arise as invariant angle associated with saddle points of the complexified path integral and their descent manifolds (Lefschetz thimbles). Physical effects of HTAs become most transparent upon analytic continuation in $n_f$ to non-integer number of flavors, reducing in the integer $n_f$ limit to a $\mathbb Z_2$ valued phase difference between dominant saddles. In ${\cal N}=1$ super Yang-Mills theory we demonstrate the microscopic mechanism for the vanishing of the gluon condensate. The same effect leads to an anomalously small condensate in a QCD-like $SU(N)$ gauge theory with fermions in the two-index representation. The basic phenomenon is that, contrary to folklore, the gluon condensate can receive both positive and negative contributions in a semi-classical expansion. In quantum mechanics, a HTA leads to a difference in semi-classical expansion of integer and half-integer spin particles.