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

Abstract

In both $\mathcal{N}=1$ and $\mathcal{N}=2$ 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 $\mathcal{N}=1$ and $\mathcal{N}=2$ locally supersymmetric "induced actions" which determine the logarithmically divergent parts of the corresponding effective actions. In the $\mathcal{N}=1$ 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 $\mathcal{N}=1$ super-Weyl anomalies that contain weight-zero chiral scalar multiplets ${\Phi }^I$ taking values in a Kähler manifold. Explicit anomaly calculations are carried out in the $n=1$ case.

arXiv comment31 pages. V4: published version

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