Resurgence and dynamics of O(N) and Grassmannian sigma models

Abstract

We study the non-perturbative dynamics of the two dimensional O(N ) and Grassmannian sigma models by using compactification with twisted boundary conditions on R×S 1 , semi-classical techniques and resurgence. While the O(N ) model has no instantons for N > 3, it has (non-instanton) saddles on R 2 , which we call 2d-saddles. On R × S 1 , the resurgent relation between perturbation theory and non-perturbative physics is encoded in new saddles, which are associated with the affine root system of the o(N ) algebra. These events may be viewed as fractionalizations of the 2d-saddles. The first beta function coefficient, given by the dual Coxeter number, can then be intepreted as the sum of the multiplicities (dual Kac labels) of these fractionalized objects. Surprisingly, the new saddles in O(N ) models in compactified space are in one-to-one correspondence with monopole-instanton saddles in SO(N ) gauge theory on R 3 × S 1 . The Grassmannian sigma models Gr(N, M ) have 2d instantons, which fractionalize into N kink-instantons. The small circle dynamics of both sigma models can be described as a dilute gas of the one-events and two-events, bions. One-events are the leading source of a variety of non-perturbative effects, and produce the strong scale of the 2d theory in the compactified theory. We show that in both types of sigma models the neutral bion emulates the role of IR-renormalons. We also study the topological theta angle dependence in both the O(3) model and Gr(N, M ), and describe the multi-branched structure of the observables in terms of the theta-angle dependence of the saddle amplitudes.

Resurgence of the dressing phase for AdS5 × S5

“…This stresses once more that these new non-perturbative corrections we find in the dressing phase should have a different semi-classical origin compared to the cusp anomaly ones. It is interesting to note that the difference between these two types of leading non-perturbative effects is analogous to the relation between the mass gap of the O(N ) sigma model and the action of its minimal uniton-like saddle which is N − 2 times bigger than the mass gap [32], this precisely reproduces a factor of 4 difference in the O(6) case, i.e. the model relevant to our discussion.…”

“…At low energies we can describe the world-sheet theory in terms of an effective sigma model, containing an O(6) factor [31], with a non-trivial strongly coupled IR dynamics. In a suitable regime [32], this 2-d quantum field theory contains non-perturbative objects, i.e. finite action solutions to the classical equations of motion, that, in the semiclassical approximation, give rise to…”

From 4d Yang-Mills to 2d ℂℙN − 1 model: IR problem and confinement at weak coupling

“…It was considered in physics independently and later combined with the mathematical idea of resurgence: see e.g., [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. (See also literature on large N twisted Eguchi-Kawai model [35][36][37]).…”

Resurgence of the cusp anomalous dimension

“…See [38] for a recent discussion of the resurgence properties of the O(N) models and the connections with the IR renormalons problem. 3 One can always set Γ (n) 0 = 1 by factoring out an appropriate factor.…”