Heterotic/Type II Triality and Instantons on $K_3$

TL;DR

The work investigates instanton effects for half-BPS $F^4$-type couplings in 16-supercharge theories by leveraging heterotic on $T^4$ and Type IIA on $K_3$ at the $T^4/\mathbb{Z}_2$ orbifold point, with extensions to other dimensions. It delivers a precise test of heterotic–Type IIA duality in six dimensions by equating a one-loop heterotic amplitude to a tree-level Type II result, using $SO(4,4)$ triality and Hecke identities to relate lattice sums, and showing that D-instanton contributions in Type II arise naturally in the Type II picture. The paper also provides exact non-perturbative results for Type I', F on $K_3$, M on $K_3$, and IIB on $K_3$, and derives the general D-instanton structure for Type II on $T^4/\mathbb{Z}_2$, along with NS5-brane corrections in $K_3\times T^2$ and the bulk index $\mu(N)=\sum_{d|N} 1/d^{3}$. Together, these results deepen understanding of instanton calculus in 16-supersymmetry settings, establish concrete duality checks, and reveal structured non-perturbative contributions across M-, II-, and IIB-theory descriptions.

Abstract

A detailed understanding of instanton effects for half-BPS couplings is pursued in theories with 16 supersymmetries. In particular, we investigate the duality between heterotic string on $T^4$ and type IIA on $K_3$ at the $T^4/Z_2$ orbifold point, as well as their higher and lower dimensional versions. We present a remarkably clean quantitative test of the duality at the level of $F^4$ couplings, by completely matching a purely one-loop heterotic amplitude to a purely tree-level type II result. The triality of SO(4,4) and several other miracles are shown to be crucial for the duality to hold. Exact non-perturbative new results for type I', F on $K_3$, M on $K_3$, and IIB on $K_3$ are found, and the general form of D-instanton contributions in type IIA or B on $T^4/Z_2$ is obtained. We also analyze the NS5-brane contributions in type II on $K_3\times T^2$, and predict the value $μ(N)=\sum_{d|N} (1/d^3)$ for the bulk contribution to the index of the NS5-brane world-volume theory on $K_3 \times T^2$.