Disk Instantons, Mirror Symmetry and the Duality Web

TL;DR

This paper applies mirror symmetry techniques to compute disk instantons for D-branes on non-compact Calabi-Yau 3-folds, clarifies boundary coordinates, and explores dualities linking string, Chern-Simons, and M-theory frameworks.

Contribution

It introduces a method for calculating disk instantons in complex geometries and clarifies the notion of flat coordinates for boundary theories, revealing an IR ambiguity related to framing in Chern-Simons theory.

Findings

Disk instanton computations for D-branes on non-compact Calabi-Yaus.

Identification of an IR ambiguity related to framing in Chern-Simons theory.

Mapping of disk instantons to string instantons in type IIB and superpotential calculations in M-theory.

Abstract

We apply the methods recently developed for computation of type IIA disk instantons using mirror symmetry to a large class of D-branes wrapped over Lagrangian cycles of non-compact Calabi-Yau 3-folds. Along the way we clarify the notion of ``flat coordinates'' for the boundary theory. We also discover an integer IR ambiguity needed to define the quantum theory of D-branes wrapped over non-compact Lagrangian submanifolds. In the large $N$ dual Chern-Simons theory, this ambiguity is mapped to the UV choice of the framing of the knot. In a type IIB dual description involving $(p,q)$ 5-branes, disk instantons of type IIA get mapped to $(p,q)$ string instantons. The M-theory lift of these results lead to computation of superpotential terms generated by M2 brane instantons wrapped over 3-cycles of certain manifolds of $G_2$ holonomy.