Linearized $\mathcal{N}=2$ conformal supergravity in the harmonic approach

Abstract

Using the harmonic superspace approach, we construct the superconformal harmonic action for $\mathcal{N}=2$ Weyl supermultiplet. The fundamental objects of the theory are unconstrained analytic potentials $h^{++\alpha\dot{\alpha}}, h^{++\alpha+}, h^{++\dot{\alpha}+}, h^{(+4)}$, which distinguishes our construction among the previously known ones. An important role is played by the ``half-analyticity'' conditions introduced in arXiv:2407.08524 [hep-th]. The structure of the harmonic linearized $\mathcal{N}=2$ Weyl action to large extent repeats the structure of the $\mathcal{N}=2$ Maxwell action, which suggests a conjecture on the possible structure of the complete nonlinear $\mathcal{N}=2$ Weyl theory action in the harmonic superspace. We provide a detailed study of the rigid superconformal properties of the proposed action and prove its invariance in both harmonic and chiral superspaces. First steps are also undertaken towards constructing the nonlinear $\mathcal{N}=2$ Weyl action, based on an analogy with $\mathcal{N}=2$ Maxwell action just mentioned and a generalization of the concept of half-analyticity to curved harmonic superspace.