Defining string theory in backgrounds with non-perturbative formulation has been an ongoing challenge. Inspired by the Aharony-Bergmann-Jafferis-Maldecana (ABJM) theory which has a rich non-perturbative structure as pointed out by the Marino et al. over the last few years (in terms of a matrix integral or as a fermi gas), especially by its relation to topological string theory (the ’t Hooft expansion of the former being that of the genus expansion of the latter) on the local Calabi-Yau manifold as a cone over \(\mathbb{P}^1 \times \mathbb{P}^1\), the current paper nicely proposes a general correspondence which associates a non-perturbative quantum mechanical operator to a toric Calabi-Yau manifold.
The authors conjecture an explicit formula for its spectral Fredholm determinant (to be distinguished from the usual quantum-mechanical determinant) in terms of an M-theoretic version of the topological string free energy. It is given as an explicit theta-function-like sum (cf. Equations 3.44 and 3.45). The perturbative part of this quantization condition is given by the Nekrasov-Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. The conjecture is tested to great numerical accuracy for genus 1 mirror Calabi-Yu threefolds, corresponding to toric diagrams which are reflexive polygons.
The results provide a non-perturbative formulation of topological strings on toric Calabi-Yau manifolds, which is background independent. Moreover, the conjectures relate, in a mathematically surprising way, the spectral theory of functional difference operators to enumerative geometry.