Higher derivative couplings and heterotic-Type I duality in eight dimensions

Abstract

We calculate F 4 and R 4 T 4 g−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 F g of the heterotic one-loop coupling R 4 T 4 g−4 is given by a polylogarithm of index 5 − 4 g 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 2 term.