Geometric Transitions and Open String Instantons

TL;DR

The paper analyzes a new class of geometric transitions where open string instanton corrections modify the Chern-Simons description of open strings on a lagrangian sphere. By combining local mirror symmetry, toric geometry, and Chern-Simons techniques, it computes closed-string amplitudes on the original Calabi–Yau X and open-string amplitudes on the transitioned space Y, including instanton effects. A precise open-closed duality is demonstrated by matching the free energies under a duality map between closed and open moduli, with a quantum correction interpreted as arising from open-string instantons. A toric localization analysis shows that all fixed open-string maps contributing to the invariants are multicovers of a single rigid disc, providing a solid geometric foundation for the duality and its computational framework.

Abstract

We investigate the physical and mathematical structure of a new class of geometric transitions proposed by Aganagic and Vafa. The distinctive aspect of these transitions is the presence of open string instanton corrections to Chern-Simons theory. We find a precise match between open and closed string topological amplitudes applying a beautiful idea proposed by Witten some time ago. The closed string amplitudes are reproduced from an open string perspective as a result of a fascinating interplay of enumerative techniques and Chern-Simons computations.

Figures (4)

• Figure 1: Triangulation of the toric variety (\ref{['eq:momentA']}) with $\hbox{Re}(s)>0$, $\hbox{Re}(t)>0$
• Figure 2: Mirror Riemann surface
• Figure 3: A Toric Compactification of $X$.
• Figure 4: A degenerate open string instanton of type $(g,h)=(0,3)$