K3--Fibrations and Heterotic-Type II String Duality

TL;DR

This paper explores the duality between heterotic and type II string theories with specific moduli, revealing how elliptic functions relate to geometric structures and matching perturbative results, thus deepening understanding of string dualities.

Contribution

It provides explicit mappings between heterotic and type II theories for certain moduli, connecting elliptic functions to geometric Picard-Fuchs equations and analyzing symmetries related to S-duality.

Findings

Elliptic j-functions appear in the duality map.

Mirror maps and Yukawa couplings match perturbative heterotic results.

Symmetries of instanton numbers relate to S-duality.

Abstract

We analyze the map between heterotic and type II N=2 supersymmetric string theories for certain two and three moduli examples found by Kachru and Vafa. The appearance of elliptic j-functions can be traced back to specializations of the Picard-Fuchs equations to systems for $K_3$ surfaces. For the three-moduli example we write the mirror maps and Yukawa couplings in the weak coupling limit in terms of j-functions; the expressions agree with those obtained in perturbative calculations in the heterotic string in an impressive way. We also discuss symmetries of the world-sheet instanton numbers in the type II theory, and interpret them in terms of S--duality of the non-perturbative heterotic string.