Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model

TL;DR

This work advances a non-perturbative continuum definition of quantum field theory by applying resurgence and trans-series to the two-dimensional ${\mathbb {CP}}^{N-1}$ model, using spatially twisted compactification to render semi-classical sectors (kink-instantons and bions) calculable. It identifies confluence equations that cancel perturbative IR renormalon ambiguities with neutral topological molecules, organizes sectors in a graded resurgence triangle, and demonstrates explicit cancellations that render physical observables unambiguous. The analysis yields a microscopic origin for the mass gap and ${\Theta}$-dependence, with results consistent with large-$N$ and lattice studies, and connects these phenomena to a robust resurgence framework that could underpin a rigorous continuum definition of general QFTs. The approach also clarifies center symmetry, volume independence, and the role of twisted boundary conditions in controlling infrared behavior, potentially generalizing to gauge theories such as QCD(adj) and deformed Yang–Mills. Overall, the paper provides a concrete, calculational bridge between perturbation theory, semi-classical physics, and non-perturbative continuum definitions in QFTs.

Abstract

This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT. The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle 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 rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable. We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one. Perturbation theory possesses the IR-renormalon ambiguity that arises due to non-Borel summability of the large-orders perturbation series (of Gevrey-1 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 non-perturbative ambiguity inherent to certain neutral topological molecules (neutral bions and bion-anti-bions) 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 large-N and lattice results.

Figures (11)

• Figure 1: Three types of gauge holonomy in non-abelian gauge theories and analogously ${\sigma}$-connection holonomy (defined in (\ref{['hol']})) in non-linear sigma models, and their classification according to their eigenvalue distribution. a) is the weak coupling trivial holonomy. b) and c) are, respectively, the weak and strong coupling non-trivial holonomy, and they are continuously connected in the sense of gauge invariant order parameters. In b), the eigenvalues of holonomy are located at the roots of unity and their fluctuations are small. In c), the positions of the eigenvalues are uniform and randomized. In gauge theory on $\mathbb{R}^3 \times S^1$, b) is the counterpart of the weak coupling adjoint Higgs regime. There is no weak coupling or even a potential description for c). The difference between (b) and (c) in gauge theory is discussed in Ref. Unsal:2008ch.
• Figure 2: This figure depicts one of the main ideas of confluence equations and this work: i) For real positive $g^2$, perturbation theory is non-Borel summable, i.e., ill-defined. Continue to negative $g^2$ where the perturbation theory becomes Borel summable. Then, continue back to $|g^2| \pm i \epsilon$, where one obtains left(right) Borel sums, $\mathbb{B}_{0, {\theta}=0^\pm}$. The absence of a smooth $\theta =0$ limit means non-Borel summability, i.e., perturbation theory does not define the theory. ii) For real positive $g^2$, the neutral bion amplitude is also ill-defined. Continue to negative $g^2$ where it is well-defined. Then, continue it back, via $\tilde{C}_{\pm}$ to $|g^2| \pm i \epsilon$. Upon continuation, one obtains a left(right) neutral bion amplitude $[{\mathcal{B}}_{ii}]_{{\theta}=0^\pm}$, with an imaginary discontinuity between the two. This means an ambiguity at $\theta=0$. We demonstrate, analytically, the exact cancellation of these two ambiguities at order $e^{-2S_0}\equiv e^{-2S_I/N}$. This is the first of many such cancellation encoded into confluence equations. This means quantum field theory is non-perturbatively well-defined in continuum up to ambiguities at order $e^{-4S_0}$, which can further be improved systematically. iii)The mathematical reason behind this phenomenon is that $\mathbb{R}^{+}$ in the complex $g^2$-plane is a Stokes ray. The jump in the resummed perturbation theory is the Stokes jump, which is (remarkably) mirrored by a jump in the neutral bion amplitude, to render observables meaningful even along the Stokes ray.
• Figure 3: The perturbative spectrum of the $\mathbb{C}\mathbb{P}^{\rm N-1}$ theory as a function of the background ${\sigma}$-connection holonomy. (a) Weak-coupling trivial holonomy (as well as classical theory) gives the usual $2\pi/L$ level spacing. (b) Weak-coupling non-trivial holonomy ($\mathbb{Z}_N$ symmetric background) at finite-$N$ produces a finer level spacing. (c) $\mathbb{Z}_N$ symmetric background at $N=\infty$ leads to a continuous spectrum. Classically, the background for the ${\sigma}$-connection holonomy is equivalent to twisted boundary conditions on ${\mathbb C}{\mathbb P}^{N-1}$ fields. For $N_f \geq 1$ theories, quantum mechanically, (b) and (c) are stable upon spatial compactification and (a) is stable upon thermal compactification. To achieve (b) and (c) in the $N_f=0$ case, we deform the ${\mathbb C}{\mathbb P}^{N-1}$ Lagrangian appropriately. (b) admits a semi-classical analysis of the confined regime at finite $N$, and (c) satisfies large-$N$ volume independence at $N=\infty$. (a) is not suitable for the semi-classical study of the confined regime/phase.
• Figure 4: The topological charge density of the single untwisted ${\mathbb C}{\mathbb P}^{1}$ instanton (\ref{['untwisted1']}) on ${\mathbb R}^1$$\times$${\mathbb S}^1$ at fixed $\lambda_2=1$ and $L=1$. The plots are for $\lambda_1=10$, $\lambda_1=1/10$, and $\lambda_1=1/100$, respectively. The fractionalization of the instanton does not occur for an untwisted background.
• Figure 5: Small and large $Q=1$ instantons in ${\mathbb C}{\mathbb P}^{1}$ in a weak coupling center-symmetric background. Large instantons split into two $Q=\frac{1}{2}$ instantons.