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 $${\mathbb{C}\mathbb{P}^2}$$ 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 $${\mathbb{Z}_3}$$ symmetry, alongside another action related to the Kähler 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.
Highlights
Introduction and SummaryA perturbative expansion lies at the very root of closed string theory; the familiar topological expansion in Riemann surfaces of given genus
Some important features of the transseries solution, such as the number of its parameters, the starting powers of the string coupling gs in the asymptotic series associated to the many sectors, or, as already mentioned, the instanton actions themselves; all these have to be obtained from a resurgence study of the perturbative free energy
We have constructed in detail a very explicit example, beginning to work out the structure of the resurgent transseries describing the nonperturbative free-energy of closed topological-strings in the mirror of the local CP2 Calabi–Yau background
Summary
A perturbative expansion lies at the very root of closed string theory; the familiar topological expansion in Riemann surfaces of given genus. Some important features of the transseries solution, such as the number of its parameters (equivalent to the number of different instanton actions explicitly appearing in the transseries), the starting powers of the string coupling gs in the asymptotic series associated to the many (nonperturbative) sectors, or, as already mentioned, the instanton actions themselves; all these have to be obtained from a resurgence study of the perturbative free energy This (apparent) lack of computational power lies in the fact that the holomorphic anomaly equations are equations in the B-model complex structure moduli, instead of “string equations” in the string coupling; and this sums up to the fact that, in this class of problems, the resurgent analysis itself must provide information which will later be checked against analytically computed expressions! We include a schematic description of the nonperturbative free energies we computed for local CP2
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