[Paper Review] Non-perturbative real topological strings

Marcos Mariño, Maximilian Schwick|arXiv (Cornell University)|Sep 21, 2023

Geometry and complex manifolds4 citations

TL;DR

This paper develops a non-perturbative framework for Walcher's real topological string on Calabi–Yau manifolds by extending the operator formalism of resurgence to include open and non-orientable sectors. It constructs all-orders trans-series solutions to the holomorphic anomaly equations, identifies disk-counting invariants as Stokes constants, and provides explicit formulae for multi-instanton amplitudes, with verification in the local ℙ² case.

ABSTRACT

We study the resurgent structure of Walcher's real topological string on general Calabi--Yau manifolds. We find all-orders trans-series solutions to the corresponding holomorphic anomaly equations, by extending the operator formalism of the closed topological string, and we obtain explicit formulae for multi-instanton amplitudes. We find that the integer invariants counting disks appear as Stokes constants in the resurgent structure, and we provide experimental evidence for our results in the case of the real topological string on local $\mathbb P^2$.

Motivation & Objective

• To extend the resurgent structure of closed topological strings to the real topological string, which includes open and non-orientable amplitudes.
• To construct all-orders trans-series solutions to the holomorphic anomaly equations (HAEs) of the real topological string using an extended operator formalism.
• To identify the connection between Stokes constants in the resurgent structure and integer invariants counting holomorphic disks.
• To provide explicit formulae for multi-instanton amplitudes in the real topological string framework.
• To test the one-instanton amplitude against perturbative results in the case of local ℙ².

Proposed method

• Generalizing the operator formalism from closed topological strings (gm-multi; gkkm) to the real topological string, incorporating contributions from D-branes and orientifolds.
• Deriving trans-series solutions to the HAEs by treating shifts in flat coordinates as integer multiples of the string coupling constant.
• Using boundary conditions at the conifold and large radius points to fix the trans-series parameters.
• Applying resurgence theory to relate the asymptotic behavior of the genus expansion to exponentially small non-perturbative corrections.
• Verifying the one-instanton amplitude via perturbative checks in the local ℙ² background using known mirror curve integrals and power series expansions.
• Employing integral representations of the domain wall tension via the Liouville one-form on the mirror curve to cross-check results.

Experimental results

Research questions

• RQ1 How can the resurgent structure of the closed topological string be generalized to include open and non-orientable amplitudes in the real topological string?
• RQ2 What is the form of the all-orders trans-series solutions to the holomorphic anomaly equations in the real topological string?
• RQ3 Do the Stokes constants in the resurgent expansion of the real topological string free energy correspond to integer invariants counting holomorphic disks?
• RQ4 Can the multi-instanton amplitudes in the real topological string be systematically computed using an extended operator formalism?
• RQ5 To what extent do the non-perturbative corrections in the real topological string match known perturbative results in specific geometries like local ℙ²?

Key findings

• The trans-series solutions to the holomorphic anomaly equations for the real topological string are constructed in closed form using an extended operator formalism, generalizing prior results from closed strings.
• Multi-instanton amplitudes are shown to arise from shifts of the closed string background by integer multiples of the string coupling constant, analogous to eigenvalue tunneling in matrix models.
• The integer invariants counting holomorphic disks in the real topological string appear explicitly as Stokes constants in the resurgent structure.
• The one-instanton amplitude is successfully verified against perturbative data in the local ℙ² geometry using integral representations of the domain wall tension.
• The asymptotic behavior of the free energy at large genus is reproduced from the trans-series, confirming consistency with perturbative results.
• The connection between Stokes constants and BPS invariants, previously observed in closed strings, is extended to the real topological string setting.