Heterotic Gauge Couplings and their M – Theory Origin

We work out the relation between automorphic forms on SO(2 + s, 2, Z) and gauge one–loop corrections of heterotic K3×T 2 string compactifications for the cases s = 0, 1. We find that one–loop gauge corrections of any orbifold limit of K3 can always be expressed by their instanton numbers and generic automorphic forms. These functions classify also one– loop gauge thresholds of N=1 (0, 2) heterotic compactifications based on toroidal orbifolds T 6 /Z ν. We compare these results with the gauge couplings of M –theory compactified on S 1 /Z 2 × T 6 /Z ν using Witten's Calabi–Yau strong coupling expansion. Within the last years, many perturbative calculations have been accumulated in het-erotic K3 × T 2 compactifications, whose scalar field sector of N=2 vector multiplets contains , besides the generic S, T, U –moduli, which describe the dilaton, the size and shape of the torus T 2 , respectively, in addition Wilson line moduli. The latter parametrize non– trivial gauge background fields w.r.t. to the internal torus T 2. These results –summarized for the two derivative couplings in a holomorphic prepotential H and two other functions– lead to a contrôl of all perturbative one–loop corrections in the gauge sector [1][2][3][4][5], in the gravitational sector [3][4][6][7][8] and the one–loop Kähler corrections via the pre-potential [9][10][11]. N=2 heterotic–type-II duality [12] links the heterotic prepotential, given as sum over trilogarithms [3] with the type-II prepotential, given as weighted in-stanton expansions [13]. The main evidence for the equivalence of a pair with a rank three gauge group was the appearance of the j–function, the automorphic function of the perturbative duality group of the heterotic side, in the functional dependence of the CY couplings at a certain boundary of the CY moduli space, which has been identified with the weak coupling limit of the heterotic string. The appearance of automorphic functions of subgroups of SL(2, Z), the typical dependence of the perturbative heterotic couplings, is a general phenomenon in CY spaces of a special fibration structure which has been realized in [14]. Moreover it was demonstrated there, that this K3 fibration structure 1 implies the appearance of automorphic functions of modular groups of more variables, a mathematically surprising fact which was subsequently explored in [16]. In this way type-II heterotic duality imposes surprising relations of CY mirror maps to automorphic functions of heterotic duality groups, e.g. SO(2 + s, 2, …