Hidden Topological Angles in Path Integrals

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 angles 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 a noninteger number of flavors, reducing in the integer $n_f$ limit to a ${\mathbb{Z}}_2$ valued phase difference between dominant saddles. In $\mathcal{N}=1$ 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 semiclassical expansion. In quantum mechanics, a HTA leads to a difference in semiclassical expansion of integer and half-integer spin particles.

Figure 1

The blue areas show “good regions” in which the integrand falls sufficiently rapidly at infinity to guarantee convergence. The red dots give the locations of the saddle points, and the blue contours are the Lefschetz thimbles. If the boundary of integration is ($-\infty ,-\infty +2\pi i$), then the Lefschetz decomposition is ${\mathcal{J}}_1+{\mathcal{J}}_2+{\mathcal{J}}_3$. Here, ${\rho }_1$ and ${\rho }_3$ are equally dominant saddles over ${\rho }_2$, but there is an overall phase difference between the dominant saddles leading to a subtle cancellation for integer $k$.

Figure 2

A snapshot of the Euclidean vacuum of $\mathcal{N}=1$ SYM theory on small ${\mathbb{R}}^3\times S_L^1$. Both neutral and magnetic bions carry action $2S_0$, but their contribution to gluon condensate cancels exactly because of the presence of a HTA, a $\pi$-phase difference between the two saddles.