Resurgent Transseries and the Holomorphic Anomaly: Nonperturbative Closed Strings in Local CP2

TL;DR

This work delivers a concrete nonperturbative completion of the holomorphic anomaly equations for closed topological strings by using resurgent transseries, with an explicit, highly detailed study of the mirror of local ${\mathbb{C}P^2}$. It uncovers a rich resurgence structure: three conifold instanton actions tied by a $Z_3$ symmetry, a Kahler‑type action at large radius, and resonance phenomena that force multi‑branched Borel structures. The authors derive nonperturbative holomorphic anomaly equations, compute higher instanton sectors, and perform extensive high‑precision numerical checks that confirm the large‑order relations and transseries predictions in both holomorphic and nonholomorphic frames. Their results demonstrate that nonperturbative information can be consistently extracted from holomorphic anomaly data and hint at a broader, highly resonant transseries framework for topological strings, with potential links to matrix models and modularity.

Abstract

The holomorphic anomaly equations describe B-model closed topological strings in Calabi-Yau geometries. Having been used to construct perturbative expansions, it was recently shown that they can also be extended past perturbation theory by making use of resurgent transseries. These yield formal nonperturbative solutions, showing integrability of the holomorphic anomaly equations at the nonperturbative level. This paper takes such constructions one step further by working out in great detail the specific example of topological strings in the mirror of the local CP2 toric Calabi-Yau background, and by addressing the associated (resurgent) large-order analysis of both perturbative and multi-instanton sectors. In particular, analyzing the asymptotic growth of the perturbative free energies, one finds contributions from three different instanton actions related by Z_3 symmetry, alongside another action related to the Kahler parameter. Resurgent transseries methods then compute, from the extended holomorphic anomaly equations, higher instanton sectors and it is shown that these precisely control the asymptotic behavior of the perturbative free energies, as dictated by resurgence. The asymptotic large-order growth of the one-instanton sector unveils the presence of resonance, i.e., each instanton action is necessarily joined by its symmetric contribution. The structure of different resurgence relations is extensively checked at the numerical level, both in the holomorphic limit and in the general nonholomorphic case, always showing excellent agreement with transseries data computed out of the nonperturbative holomorphic anomaly equations. The resurgence relations further imply that the string free energy displays an intricate multi-branched Borel structure, and that resonance must be properly taken into account in order to describe the full transseries solution.

Figures (24)

• Figure 1: Complex-structure moduli space of local ${\mathbb C}{\mathbb P}^2$ in different coordinates.
• Figure 2: The two images on top show the holomorphicity of the instanton action at fixed $\psi=2\,{\rm e}^{{\rm i}\pi/6}$, and with varying $S^{zz} = S^{zz}_{\textrm{hol}} \cdot \left( 1 + x - {\rm i}\, y \right)$. We display both real and imaginary components of $A^2$ and in both cases all (numerical) points intersect the (theoretical) constant-height surface of $A^2$. In the two images below, we show the real and imaginary parts of \ref{['eq:Asquareratio']} for the particular value of $(x,y) = (4,-4)$, along with several Richardson transforms. We can compare the numerical results after fourteen Richardson transforms, and find that they agree with the predicted value of the instanton action achieving a precision as high as $\sim 10^{-10}$. If $S^{zz}$ is too large the convergence towards $A^2$ starts only at higher values of the genus, and the precision is consequently lower. The "theoretical" value of the instanton action is given by \ref{['eq:Atc']} and marked with a horizontal line.
• Figure 3: The dominant instanton action as obtained from large-order \ref{['eq:Asquareratio']}, with (left) and without (right) the constant map contribution \ref{['eq:removeCM']}. In the plots we have chosen $\psi$ real and positive (or $z$ real and negative), so that the conifold point $\psi = 1$ is clearly shown (note that the instanton action vanishes at this point). Also, in this case of $\arg(\psi)=0$, the instanton action is purely imaginary. The continuous red line that fits the data is given by \ref{['eq:Atc']}. Four Richardson transforms on the numerical data were enough to give an agreement of about one part in $10^9$. Precision is lower close to the transition point $\psi = 7.71$, on the left plot, because the two instanton actions become of the same order at this point.
• Figure 4: Oscillatory behavior of the perturbative sector, due to complex contributions of both $A_1$ and $A_2$, for values of the modulus $\psi = 1.25\, {\rm e}^{{\rm i}\pi/3}$ and propagator $S^{zz} = 10^{-5} \sim 0.15\,|S^{zz}_{1,{\textrm{hol}}}|$.
• Figure 5: Branch points and cuts of the instanton actions $A_1$ (left), $A_2$ (center) and $A_3$ (right), in the complex $\psi$ plane. Each wedge of angle $2\pi/3$ is in correspondence with one full complex $z$ plane. The rightmost wedge, in red, includes the first conifold point, $\psi = 1$; the upper wedge, in green, includes the second conifold point, $\psi = {\rm e}^{2\pi {\rm i}/3}$; and the lower wedge, in blue, includes the third conifold point, $\psi = {\rm e}^{-2\pi {\rm i}/3}$.