String Amplitudes and N = 2 , d = 4 Prepotential in Heterotic K 3 × T 2 Compactifications

For the gauge couplings, which arise after toroidal compactification of six–dimensional heterotic N=1 string theories from the T 2 torus, we calculate their one–loop corrections. This is performed by considering string amplitudes involving two gauge fields and moduli fields. We compare our results with the equations following from N=2 special geometry and the underlying prepotential of the theory. Moreover we find relations between derivatives of the N=2, d = 4 prepotential and world–sheet τ –integrals which appear in various string amplitudes of any T 2 –compactification. The vector multiplet sector of N=2 supergravity in four dimensions is governed by a holomorphic function, namely the prepotential F [1]. Up to second order derivatives the Lagrangian is constructed from it. The Kähler metric and the gauge couplings are expressible by derivatives of this prepotential. Effective N=2 supergravity theories in four dimensions arise e.g. after compactifications of heterotic string theories on K3×T 2 or type IIA or type IIB on a Calabi–Yau (CY) threefold. In the following we will concentrate on the first vacuum. In addition to the heterotic gauge group, which depends on the specific instanton embeddings and may be completely Higgsed away, a K3 × T 2 heterotic string compactification always possesses the U (1) 2 + × U (1) 2 − internal gauge symmetry. The first factor comes from the internal graviton and the last factor arises from the compactification of the six–dimensional tensor multiplet which describes the heterotic dilaton in six dimensions. The gauge fields of the left U (1)'s belong to vector multiplets whose scalars are the T and U moduli of the torus T 2. Their moduli space is described by special geometry [2][3]. Besides, there is a vector multiplet for the heterotic dilaton and the graviphoton, whereas the K3 moduli and the gauge bundle deformations come in scalars of hyper mulitplets. 2 L may be enhanced to SU (2) L × U (1) L for T = U , to SU (2) 2 L for T = U = i and SU (3) L at T = U = ρ [4]. Logarithmic singularities in the effective string action appear at these special points. This effect should be seen e.g. in the one–loop gauge couplings, where heavy strings have been integrated out. Modes, which have been integrated out and become light at these special points in the moduli space are responsible for these singularities. These one–loop couplings …