Rational Curves on Calabi-Yau Threefolds

TL;DR

This paper discusses mathematical methods for counting rational curves on Calabi-Yau threefolds, inspired by mirror symmetry and string theory, aiming to resolve the challenge of calculating instanton corrections in these geometries.

Contribution

It introduces mathematical techniques to compute finite rational curves on Calabi-Yau threefolds, aligning with physical path integral results and addressing open problems in instanton correction calculations.

Findings

Mathematical techniques for counting rational curves

Alignment with physical path integral calculations

Potential solutions for instanton correction problems

Abstract

By considering mirror symmetry applied to conformal field theories corresponding to strings propagating in quintic hypersurfaces in projective 4-space, Candelas, de la Ossa, Green and Parkes calculated the ``number of rational curves on the hypersurface'' by comparing three point functions. Actually, the number of curves may be infinite for special examples; what is really being calculated is a path integral. The point of this talk is to give mathematical techniques and examples for computing the finite number that ``should'' correspond to an infinite family of curves (which coincides with that given by the path integral in every known instance), and to suggest that these techniques should provide the answer to the not yet solved problem of how to calculate instanton corrections to the three point function in general.