Instanton Numbers and Exchange Symmetries in $N=2$ Dual String Pairs

TL;DR

The paper examines $N=2$, $D=4$ string vacua with dual heterotic and type II descriptions, focusing on the elliptically fibered Calabi-Yau $WP_{1,1,2,8,12}(24)$ with $h^{1,1}=3$ and $h^{2,1}=243$. It shows that genus-0 and genus-1 instanton numbers on the type II side are encoded in the coefficients of modular forms, namely $E_4E_6/\eta^{24}$ and $E_2E_4E_6/\eta^{24}$, in the heterotic weak coupling limit, and relates the non-perturbative $S$-$T$ exchange to six-dimensional heterotic/heterotic duality via a moduli map between K"ahler and heterotic moduli. It derives explicit relations $n^r_{d_1,0,d_3}= -2 c_1(kl)$ with $d_1=k+l$, $d_3=k$, and demonstrates reflection and exchange symmetries of the rational instanton numbers, together with a simple $4\pi S\leftrightarrow T$ exchange for suitable parameters. The discussion extends to higher-genus couplings $F_g$ and to other CYs with the same Hodge numbers, highlighting how modular structure and duality constrain non-perturbative data across dimensions.

Abstract

In this note, we comment on Calabi-Yau spaces with Hodge numbers $h_{1,1}=3$ and $h_{2,1}=243$. We focus on the Calabi-Yau space $WP_{1,1,2,8,12}(24)$ and show how some of its instanton numbers are related to coefficients of certain modular forms. We also comment on the relation of four dimensional exchange symmetries in certain $N=2$ dual models to six dimensional heterotic/heterotic string duality.