Higher Derivative Couplings and Heterotic–Type I Duality in Eight Dimensions

Abstract We calculate F 4 and RT 4g−4 couplings in d = 8 heterotic and type I string vacua (with gauge and graviphoton field strengths F, T , and Riemann curvature R). The holomorphic piece Fg of the heterotic one–loop coupling R T 4g−4 is given by a polylogarithm of index 5 − 4g and encodes the counting of genus g curves with g nodes on the K3 of the dual F–theory side. We present closed expressions for world–sheet τ–integrals with an arbitrary number of lattice vector insertions. Furthermore we verify that the corresponding heterotic one–loop couplings sum up perturbative open string and non-perturbative D–string contributions on the type I side. Finally we discuss a type I one–loop correction to the R term.