Open string instantons and relative stable morphisms

TL;DR

This paper demonstrates how to compute topological open string amplitudes using algebraic geometry techniques, specifically relative stable morphisms, and successfully reproduces known physics results through explicit calculations.

Contribution

It introduces a method employing relative stable morphisms and virtual localization to compute open string amplitudes, extending previous physics results to algebraic geometry context.

Findings

Reproduces Ooguri and Vafa's results for multiple covers of holomorphic discs

Shows no open string instantons with more than one boundary component in the considered case

Validates the algebraic approach for open string amplitude calculations

Abstract

We show how topological open string theory amplitudes can be computed by using relative stable morphisms in the algebraic category. We achieve our goal by explicitly working through an example which has been previously considered by Ooguri and Vafa from the point of view of physics. By using the method of virtual localization, we successfully reproduce their results for multiple covers of a holomorphic disc, whose boundary lies in a Lagrangian submanifold of a Calabi-Yau 3-fold, by Riemann surfaces with arbitrary genera and number of boundary components. In particular we show that in the case we consider there are no open string instantons with more than one boundary component ending on the Lagrangian submanifold.