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

In both ${\cal N}=1$ and ${\cal N}=2$ supersymmetry, it is known that $\mathsf{Sp}(2n, {\mathbb R})$ is the maximal duality group of $n$ vector multiplets coupled to chiral scalar multiplets $\tau (x,\theta)$ that parametrise the Hermitian symmetric space $\mathsf{Sp}(2n, {\mathbb R})/ \mathsf{U}(n)$. If the coupling to $\tau$ 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 $\Gamma [\tau, \bar \tau]$ with the following properties: (i) its logarithmically divergent part is invariant under super-Weyl and rigid $\mathsf{Sp}(2n, {\mathbb R})$ transformations; (ii) the super-Weyl transformations are anomalous upon renormalisation. In this paper we describe the ${\cal N}=1$ and ${\cal N}=2$ locally supersymmetric "induced actions" which determine the logarithmically divergent parts of the corresponding effective actions. In the ${\cal N}=1$ 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 ${\cal N}=1$ super-Weyl anomalies that contain weight-zero chiral scalar multiplets $\Phi^I$ taking values in a K\"ahler manifold. Explicit anomaly calculations are carried out in the $n=1$ case.