Resumming the string perturbation series

TL;DR

The paper investigates whether perturbative string and gauge theory expansions can be resummed to reproduce exact non-perturbative results. By applying Borel–Padé resummation to ABJM's 1/N expansion and to the genus expansion of topological strings, and by using a quartic-oscillator toy model, the authors uncover a consistent pattern: even when a series is Borel summable, complex instantons can yield non-perturbative corrections that the resummation misses. They show that membrane instantons or related non-perturbative sectors must be included via a trans-series to recover the exact answers. The work highlights limitations of naive Borel summation in string theory contexts and motivates a trans-series framework, potentially guided by holomorphic anomaly methods, to achieve a complete non-perturbative definition.

Abstract

We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local $\mathbb P^1 \times \mathbb P^1$, which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.

Figures (9)

• Figure 1: The paths ${\cal C}_{\pm}$ avoiding the singularities of the Borel transform from above (respectively, below).
• Figure 2: Two different situations for Borel summability: in the case depicted on the left, the singularity occurs in the negative real axis of the Borel plane, but the corresponding instanton can not contribute to the final answer, since it would lead to an exponentially enhanced correction for $z>0$. However, in the case depicted on the right, we have two complex conjugate instantons whose actions have a real positive part. Although they do not obstruct Borel summability, they might lead to explicit non-perturbative corrections.
• Figure 3: The poles of the Padé approximant (\ref{['paden']}) in the $\zeta$ plane, with $n=320$, for the series (\ref{['borelb']}).
• Figure 4: The red dots signal the location of the poles of the Padé approximant (\ref{['paden']}), for the series (\ref{['btfg']}) with $\lambda\approx 2.61$. In the figure on the left we have included the full $F_g(\lambda)$, while in the figure on the right we have subtracted the contribution of constant maps. The blue circle corresponds to the numerical value of the complex instanton action $A_s(\lambda)$. The degree of the Padé approximant is $n=54$ (left) and $n=60$ (right).
• Figure 5: The red dots signal the location of the poles of the Padé approximant (\ref{['paden']}), for the series $F_g(\lambda)-c_g$ and $\lambda\approx 0.128$. The blue circle corresponds to the numerical value of the purely imaginary instanton action $A_w(\lambda)$ in (\ref{['w-inst']}). The degree of the Padé approximant is $n=54$.