N = 3 conformal superspace in four dimensions

A bstract We develop a superspace formulation for N N = 3 conformal supergravity in four spacetime dimensions as a gauge theory of the superconformal group SU (2, 2 | 3). Upon imposing certain covariant constraints, the algebra of conformally covariant derivatives A= (ₐ, _ⁱ, ᵢ^ { }) ∇ A = ∇ a ∇ α i ∇ i α ⋅ is shown to be determined in terms of a single primary chiral spinor superfield, the super-Weyl spinor W α of dimension +1 / 2 and its conjugate. Associated with W α is its primary descendant B i j of dimension +2, the super-Bach tensor, which determines the equation of motion for conformal supergravity. As an application of this construction, we present two different but equivalent action principles for N N = 3 conformal supergravity. We describe the model for linearised N N = 3 conformal supergravity in an arbitrary conformally flat background and demonstrate that it possesses U (1) duality invariance. Additionally, upon degauging certain local symmetries, our superspace geometry is shown to reduce to the U (3) superspace constructed by Howe more than four decades ago. Further degauging proves to lead to a new superspace formalism, called SU (3) superspace, which can also be used to describe N N = 3 conformal supergravity. Our conformal superspace setting opens up the possibility to formulate the dynamics of the off-shell N N = 3 super Yang-Mills theory coupled to conformal supergravity.