Non-compact duality, super-Weyl invariance and effective actions

In both N = 1 and N = 2 supersymmetry, it is known that Sp(2n;R) is the maximal duality group of n vector multiplets coupled to chiral scalar multiplets tau(x, theta) that parametrise the Hermitian symmetric space Sp(2n;R)=U(n). If the coupling to is introduced for n superconformal gauge multiplets in a supergravity background, the action is also invariant under super-Weyl transformations. Computing the path integral over the gauge prepotentials in curved superspace leads to an effective action Gamma[tau, (tau) over bar] with the following properties: (i) its logarithmically divergent part is invariant under super-Weyl and rigid Sp(2n;R) transformations; (ii) the super-Weyl transformations are anomalous upon renormalisation. In this paper we describe the N = 1 and N = 2 locally supersymmetric \induced actions" which determine the logarithmically divergent parts of the corresponding effective actions. In the N = 1 case, superfield heat kernel techniques are used to compute the induced action of a single vector multiplet (n = 1) coupled to a chiral dilaton-axion multiplet. We also describe the general structure of N = 1 super-Weyl anomalies that contain weight-zero chiral scalar multiplets Phi(I) taking values in a Kahler manifold. Explicit anomaly calculations are carried out in the n = 1 case.