Superspace Approaches to $$\mathscr {N} = \text{1}$$ Supergravity

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We develop a superspace formulation for $$ \mathcal{N} $$ 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 $$ {\nabla}_A=\left({\nabla}_a,{\nabla}_{\alpha}^i,{\nabla}_i^{\overset{\cdot }{\alpha }}\right) $$ ∇ A = ∇ a ∇ α i ∇ i α ⋅ is shown to be determined in terms of a single primary chiral spinor superfield, the super-Weyl spinor Wα of dimension +1/2 and its conjugate. Associated with Wα is its primary descendant Bij 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 $$ \mathcal{N} $$ N = 3 conformal supergravity. We describe the model for linearised $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ N = 3 conformal supergravity. Our conformal superspace setting opens up the possibility to formulate the dynamics of the off-shell $$ \mathcal{N} $$ N = 3 super Yang-Mills theory coupled to conformal supergravity.

$$ \mathcal{N} $$ = 3 conformal superspace in four dimensions

Kuzenko,

Raptakis

2024

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Effective actions in supersymmetric gauge theories: heat kernels for non-minimal operators

Grasso

Kuzenko

2023

We study the quantum dynamics of a system of n Abelian $$ \mathcal{N} $$ N = 1 vector multiplets coupled to $$ \frac{1}{2}n\left(n+1\right) $$ 1 2 n n + 1 chiral multiplets which parametrise the Hermitian symmetric space Sp(2n, ℝ)/U(n). In the presence of supergravity, this model is super-Weyl invariant and possesses the maximal non-compact duality group Sp(2n, ℝ) at the classical level. These symmetries should be respected by the logarithmically divergent term (the “induced action”) of the effective action obtained by integrating out the vector multiplets. In computing the effective action, one has to deal with non-minimal operators for which the known heat kernel techniques are not directly applicable, even in flat (super)space. In this paper we develop a method to compute the induced action in Minkowski superspace. The induced action is derived in closed form and has a simple structure. It is a higher-derivative superconformal sigma model on Sp(2n, ℝ)/U(n). The obtained $$ \mathcal{N} $$ N = 1 results are generalised to the case of $$ \mathcal{N} $$ N = 2 local supersymmetry: a system of n Abelian $$ \mathcal{N} $$ N = 2 vector multiplets coupled to $$ \mathcal{N} $$ N = 2 chiral multiplets XI parametrising Sp(2n, ℝ)/U(n). The induced action is shown to be proportional to $$ \int {\textrm{d}}^4x{\textrm{d}}^4\theta {\textrm{d}}^4\overline{\theta}E\mathfrak{K}\left(X,\overline{X}\right) $$ ∫ d 4 x d 4 θ d 4 θ ¯ E K X X ¯ , where $$ \mathfrak{K}\left(X,\overline{X}\right) $$ K X X ¯ is the Kähler potential for Sp(2n, ℝ)/U(n). We also apply our method to compute DeWitt’s a2 coefficients in some non-supersymmetric theories with non-minimal operators.

Higher-derivative deformations of the ModMax theory

Kuzenko,

Raptakis

2024