Non-perturbative quantum geometry III

Daniel Krefl

doi:10.1007/jhep08(2016)020

Abstract

The Nekrasov-Shatashvili limit of the refined topological string on toric Calabi-Yau manifolds and the resulting quantum geometry is studied from a non-perturbative perspective. The quantum differential and thus the quantum periods exhibit Stokes phenomena over the combined string coupling and quantized Kaehler moduli space. We outline that the underlying formalism of exact quantization is generally applicable to points in moduli space featuring massless hypermultiplets, leading to non-perturbative band splitting. Our prime example is local P1xP1 near a conifold point in moduli space. In particular, we will present numerical evidence that in a Stokes chamber of interest the string based quantum geometry reproduces the non-perturbative corrections for the Nekrasov-Shatashvili limit of 4d supersymmetric SU(2) gauge theory at strong coupling found in the previous part of this series. A preliminary discussion of local P2 near the conifold point in moduli space is also provided.

Highlights

JHEP08(2016)020 contributions to the quantum B-periods and the analytic continuation thereof
The Nekrasov-Shatashvili limit of the refined topological string on toric CalabiYau manifolds and the resulting quantum geometry is studied from a non-perturbative perspective
We outline that the underlying formalism of exact quantization is generally applicable to points in moduli space featuring massless hypermultiplets, leading to non-perturbative band splitting

Summary

Quantum geometry

The NS condition ensures that we sit in a supersymmetric vacuum of the corresponding effective 2d theory It was further proposed in [1] to use this condition in general, including for toric Calabi-Yaus, to infer non-perturbative information, because (2.6) can only be satisfied if the quantum modulus E receives non-perturbative corrections. Note that it is clear from (2.5) that φ corresponds to the phase of the WKB wave-function under monodromy along the B-cycle. We like to stress here that the physical nature of the non-perturbative effects, ensuring that (3.8) holds, change over moduli space, see [2] and in particular [7]

Deformed conifold

A first look at a toric Calabi-Yau

Local P2

Numerical Mathieu spectrum

Oscillator basis expansion