| Summary: | Abstract In both N $$ \mathcal{N} $$ = 1 and N $$ \mathcal{N} $$ = 2 supersymmetry, it is known that Sp(2n, ℝ) is the maximal duality group of n vector multiplets coupled to chiral scalar multiplets τ(x, θ) that parametrise the Hermitian symmetric space Sp(2n, ℝ)/U(n). If the coupling to τ 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 Γ[τ, τ ¯ $$ \overline{\tau} $$ ] 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 $$ \mathcal{N} $$ = 1 and N $$ \mathcal{N} $$ = 2 locally supersymmetric “induced actions” which determine the logarithmically divergent parts of the corresponding effective actions. In the N $$ \mathcal{N} $$ = 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 N $$ \mathcal{N} $$ = 1 super-Weyl anomalies that contain weight-zero chiral scalar multiplets Φ I taking values in a Kähler manifold. Explicit anomaly calculations are carried out in the n = 1 case.
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