R-symmetry and the topological twist of N=2 effective supergravities of heterotic strings

We discuss R-symmetries in locally supersymmetric N=2 gauge theories coupled to hypermultiplets which can be thought of as effective theories of heterotic superstring models. In this type of supergravities a suitable off-shell R-symmetry exists which can be used to topologically twist the theory: the vector multiplet containing the dilaton-axion field has different R-charge assignments with respect to the other vector multiplets. Correspondingly a system of coupled instanton equations emerges mixing gravitational and Yang--Mills instantons with triholomorphic hyperinstantons and axion-instantons. For the tree-level classical special manifolds ST(n)=SU(1,1)/U(1)\times SO(2,n)/SO(2) \times SO(n) R-symmetry with the specified properties is a continuous symmetry, but for the quantum corrected manifolds {\hat {ST}}(n) a discrete R-symmetry is sufficient and we argue that it exists.