Weyl invariance, non-compact duality and conformal higher-derivative sigma models

We study a system of $n$ Abelian vector fields coupled to $\frac{1}{2}n(n+1)$ complex scalars parametrising the Hermitian symmetric space $\mathsf{S}\mathsf{p}(2n,\mathbb{R})\mathrm{/}\mathsf{U}(n)$. This model is Weyl invariant and possesses the maximal non-compact duality group $\mathsf{S}\mathsf{p}(2n,\mathbb{R})$. Although both symmetries are anomalous in the quantum theory, they should be respected by the logarithmic divergent term (the ``induced action’') of the effective action obtained by integrating out the vector fields. We compute this induced action and demonstrate its Weyl and $\mathsf{S}\mathsf{p}(2n,\mathbb{R})$ invariance. The resulting conformal higher-derivative $\sigma$-model on $\mathsf{S}\mathsf{p}(2n,\mathbb{R})\mathrm{/}\mathsf{U}(n)$ is generalised to the cases where the fields take their values in (i) an arbitrary Kähler space; and (ii) an arbitrary Riemannian manifold. In both cases, the $\sigma$-model Lagrangian generates a Weyl anomaly satisfying the Wess-Zumino consistency condition.