Geometric Transitions, del Pezzo Surfaces and Open String Instantons

TL;DR

This paper investigates geometric transitions involving del Pezzo surfaces, revealing a deep connection between open and closed string amplitudes through exact computations and conjectured relations with coupled Chern-Simons theories.

Contribution

It introduces an extremal transition for a local del Pezzo model and proposes a novel conjecture linking all genus topological amplitudes to coupled Chern-Simons theories.

Findings

Exact computation of open string amplitudes using enumerative techniques and Chern-Simons theory.

Prediction of a nontrivial relation between open and closed string amplitudes.

Conjecture relating all genus topological amplitudes to coupled Chern-Simons systems.

Abstract

We continue the study of a class of geometric transitions proposed by Aganagic and Vafa which exhibit open string instanton corrections to Chern-Simons theory. In this paper we consider an extremal transition for a local del Pezzo model which predicts a highly nontrivial relation between topological open and closed string amplitudes. We show that the open string amplitudes can be computed exactly using a combination of enumerative techniques and Chern-Simons theory proposed by Witten some time ago. This yields a striking conjecture relating the topological amplitudes of all genus of the local del Pezzo model to a system of coupled Chern-Simons theories.