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Hidden topological angles and Lefschetz thimbles

Alireza Behtash, Tin Sulejmanpasic, Thomas Schaefer, Mithat Unsal

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.

Hidden topological angles and Lefschetz thimbles

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 to non-integer number of flavors, reducing in the integer limit to a valued phase difference between dominant saddles. In 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 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.

Paper Structure

This paper contains 14 equations, 2 figures.

Figures (2)

  • 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 over-all phase difference between the dominant saddles leading to a subtle cancellation for integer $k$.
  • Figure 2: A snap-shot of the euclidean vacuum of ${\cal N}=1$ SYM on small $\mathbb R^3 \times S^1_L$. 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.