On the resurgent structure of quantum periods

Abstract

Quantum periods appear in many contexts, from quantum mechanics to local mirror symmetry. They can be described in terms of topological string free energies and Wilson loops, in the so-called Nekrasov-Shatashvili limit. We consider the trans-series extension of the holomorphic anomaly equations satisfied by these quantities, and we obtain exact multi-instanton solutions for these trans-series. Building on this result, we propose a unified perspective on the resurgent structure of quantum periods. We show for example that the Delabaere-Pham formula, which was originally obtained in quantum mechanical examples, is a generic feature of quantum periods. We illustrate our general results with explicit calculations for the double-well in quantum mechanics, and for the quantum mirror curve of local \mathbb{P}^2ℙ2.

Exact instanton transseries for quantum mechanics

“…The transseries extension to this series and its resurgent structure were studied in [24], and here we briefly recall some key aspects that will be necessary for what is to come. 10 The Borel transform of the quantum prepotential F (E; ħ h) will have rays along which evenly spaced singularities lie.…”

“…Here, the perturbative quantum prepotential F is encoded in F (0) = F − F 0 , where following standard conventions the leading term is subtracted to obtain nicer-looking holomorphic anomaly equations. The first three nonperturbative sectors [24] have the form…”

“…The operator D and the quantity G are quite subtle objects that depend on which Stokes line we are studying. For a complete and thorough explanation we refer to [23][24][25], but for our purposes it suffices to know that G has the generic form 10 The key result that we are after is the action of the alien derivatives on the perturbative quantum prepotential F (E; ħ h), shown in (46). For the purpose of understanding the main results of the present paper, the finer details of the quantum prepotential transeries F presented here can be skipped upon first reading.…”

“…It has been argued in [18] that quantum periods, which are the essential building blocks of the aforementioned quantisation conditions, are governed by the holomorphic anomaly equations [19] of the refined topological string in the Nekrasov-Shatashvili background [20]. Subsequently, building on ideas from [21,22], the nonperturbative solution to these holomorphic anomaly equations was studied in [23] and finally solved for in exact form in [24] -also in the so-called selfdual background [25]. This nonperturbative solution, that appears in the form of a transseries for the quantum prepotential or Nekrasov-Shatashvili free energy, relates directly to the ordinary energy E through a relation known as the PNP relation.…”

“…This nonperturbative solution, that appears in the form of a transseries for the quantum prepotential or Nekrasov-Shatashvili free energy, relates directly to the ordinary energy E through a relation known as the PNP relation. We will exploit this relation, allowing us to compare to, and utilise the results of [24] for our goals.…”