
Physics 2012
Resurgence and Transseries in Quantum Field Theory: The CP(N1) ModelAbstract: This work is a step towards a nonperturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional nonlinear sigmamodels, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the transseries framework, and BorelEcalle resummation. The ideas from QFT use continuity on R^1 x S^1_L, i.e, the absence of any phase transition as N \to infinity, or rapidcrossovers for finiteN, and the smallL weak coupling limit to render the semiclassical sector welldefined and calculable. We classify semiclassical configurations with actions 1/N (kinkinstantons), 2/N (bions and bikinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IRrenormalon ambiguity that arises due to nonBorel summability of the largeorders perturbation series (of Gevrey1 type), for which a microscopic cancellation mechanism was unknown. This divergence must be present because the corresponding expansion is on a singular Stokes ray in the complexified coupling constant plane, and the sum exhibits the Stokes phenomenon crossing the ray. We show that there is also a nonperturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bionantibions) in the semiclassical expansion. We find a set of "confluence equations" that encode the exact cancellation of the two different type of ambiguities. We show that a new notion of "graded resurgence triangle" is necessary to capture the path integral approach to resurgence, and that graded resurgence underlies a potentially rigorous definition of general QFTs. The mass gap and the Theta angle dependence of vacuum energy are calculated from first principles, and are in accord with largeN and lattice results.
