Hidden Topological Angles in Path Integrals

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 Z 2 valued phase difference between dominant saddles. In 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.Introduction. Providing a non-perturbative continuum definition of the path integral in quantum field theory is a challenging but important problem [1]. There is growing evidence that, if an ordinary integral or a path integral admits a Lefschetz-thimble decomposition [2,3] or resurgent transseries expansion [4][5][6][7][8][9][10][11][12] then either of these methods gives this long-sought non-perturbative definition. If this is indeed the case, then we expect that these new methods will provide new and deep insight into quantum field theory and quantum mechanics formulated in terms of path integrals. In this article we introduce a new phenomenon of this kind, the appearance of hidden topological angles (HTAs).The main prescription associated with the Lefschetzthimble decomposition or the resurgent expansion is the following: Even if an ordinary integral or a path integral is formulated over real fields, the natural space that the critical points (saddles) ρ σ live in is the complexification of the original space of fields. However, the dimension of the crit-

Monte Carlo algorithm for simulating fermions on Lefschetz thimbles

“…Forμ = 0 this critical point is at z 0 = 0 but for nonzero chemical potential it is a pure imaginary, that is z 0 = ix for some real x, even though the original path integral is along the real values of z t , namely the interval [0, 2⇡]. This is an example of the situation where complex valued configurations which lie outside of the original integration region contribute to the semiclassical expansion [23][24][25]37]. In fact in this case the complex configuration constitutes the leading contribution.…”

“…The original integral is then expressed as a combination of integrals over those thimbles. This idea has ben fruitful in di↵erent aspects of quantum mechanics [22][23][24][25] and quantum field theories [26][27][28], especially in studying semiclassical expansions. In this paper, however, we will focus on the implementation of the Lefschetz thimble approach on lattice field theory.…”

New Nonperturbative Methods in Quantum Field Theory: From Large-NOrbifold Equivalence to Bions and Resurgence

“…A hidden topological angle (HTA) is an invariant angle associated with saddle points of the complexified path integral and their descent manifolds (Lefschetz thimbles) [99,100]. The HTA is distinct from theta-parameter in the lagrangian.…”

Complex Path Integrals and Saddles in Two-Dimensional Gauge Theory

“…The imaginary part of S (z) is always a multiple of π: Im S (z) = π m/2 , where · is the floor function, so that the weight exp (−S (z)) is always real, but can have either sign. This sign comes exclusively from the Vandermonde determinant ∆ 2 (z), and is interpreted as a hidden topological angle [37]. Vacuum saddle: we identify the m = 0 saddle with the planar (N = ∞) contribution.…”