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Register a new userSymmetries of $mathcal{N} = (1,0)$ supergravity backgrounds in six dimensions
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General $mathcal{N}=(1,0)$ supergravity-matter systems in six dimensions may be described using one of the two fully fledged superspace formulations for conformal supergravity: (i) $mathsf{SU}(2)$ superspace; and (ii) conformal superspace. With motivation to develop rigid supersymmetric field theories in curved space, this paper is devoted to the study of the geometric symmetries of supergravity backgrounds. In particular, we introduce the notion of a conformal Killing spinor superfield $epsilon^alpha$, which proves to generate extended superconformal transformations. Among its cousins are the conformal Killing vector $xi^a$ and tensor $zeta^{a(n)}$ superfields. The former parametrise conformal isometries of supergravity backgrounds, which in turn yield symmetries of every superconformal field theory. Meanwhile, the conformal Killing tensors of a given background are associated with higher symmetries of the hypermultiplet. By studying the higher symmetries of a non-conformal vector multiplet we introduce the concept of a Killing tensor superfield. We also analyse the problem of computing higher symmetries for the conformal dAlembertian in curved space and demonstrate that, beyond the first-order case, these operators are defined only on conformally flat backgrounds.
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The supersymmetrization of curvature squared terms is important in the study of the low-energy limit of compactified superstrings where a distinguished role is played by the Gauss-Bonnet combination, which is ghost-free. In this letter, we construct its off-shell ${cal N} = (1, 0)$ supersymmetrization in six dimensions for the first time. By studying this invariant together with the supersymmetric Einstein-Hilbert term we confirm and extend known results of the $alpha$-corrected string theory compactified to six dimensions. Finally, we analyze the spectrum about the ${rm AdS}_3times{rm S}^3$ solution.
In four spacetime dimensions, all ${cal N} =1$ supergravity-matter systems can be formulated in the so-called $mathsf{U}(1)$ superspace proposed by Howe in 1981. This paper is devoted to the study of those geometric structures which characterise a background $mathsf{U}(1)$ superspace and are important in the context of supersymmetric field theory in curved space. We introduce (conformal) Killing tensor superfields $ell_{(alpha_1 dots alpha_m) ({dot alpha}_1 dots {dot alpha}_n)}$, with $m$ and $n$ non-negative integers, $m+n>0$, and elaborate on their significance in the following cases: (i) $m=n=1$; (ii) $m-1=n=0$; and (iii) $m=n>1$. The (conformal) Killing vector superfields $ell_{alpha dot alpha}$ generate the (conformal) isometries of curved superspace, which are symmetries of every (conformal) supersymmetric field theory. The (conformal) Killing spinor superfields $ell_{alpha }$ generate extended (conformal) supersymmetry transformations. The (conformal) Killing tensor superfields with $m=n>1$ prove to generate all higher symmetries of the (massless) massive Wess-Zumino operator.
We describe the supersymmetric completion of several curvature-squared invariants for ${cal N}=(1,0)$ supergravity in six dimensions. The construction of the invariants is based on a close interplay between superconformal tensor calculus and recently developed superspace techniques to study general off-shell supergravity-matter couplings. In the case of minimal off-shell Poincare supergravity based on the dilaton-Weyl multiplet coupled to a linear multiplet as a conformal compensator, we describe off-shell supersymmetric completions for all the three possible purely gravitational curvature-squared terms in six dimensions: Riemann, Ricci, and scalar curvature squared. A linear combination of these invariants describes the off-shell completion of the Gauss-Bonnet term, recently presented in arXiv:1706.09330. We study properties of the Einstein-Gauss-Bonnet supergravity, which plays a central role in the effective low-energy description of $alpha^prime$-corrected string theory compactified to six dimensions, including a detailed analysis of the spectrum about the ${rm AdS}_3times {rm S}^3$ solution. We also present a novel locally superconformal invariant based on a higher-derivative action for the linear multiplet. This invariant, which includes gravitational curvature-squared terms, can be defined both coupled to the standard-Weyl or dilaton-Weyl multiplet for conformal supergravity. In the first case, we show how the addition of this invariant to the supersymmetric Einstein-Hilbert term leads to a dynamically generated cosmological constant and non-supersymmetric (A)dS$_6$ solutions. In the dilaton-Weyl multiplet, the new off-shell invariant includes Ricci and scalar curvature-squared terms and possesses a nontrivial dependence on the dilaton field.
The ultrashort unitary (4,0) supermultiplet of 6d superconformal algebra OSp(8*|8) reduces to the CPT-self conjugate supermultiplet of 4d superconformal algebra SU(2,2|8) that represents the fields of maximal N=8 supergravity. The graviton in the (4,0) multiplet is described by a mixed tensor gauge field which can not be identified with the standard metric in 6d. Furthermore the (4,0) supermultiplet can be obtained as a double copy of (2,0) conformal supermultiplet whose interacting theories are non-Lagrangian. It had been suggested that an interacting non-metric (4,0) supergravity theory might describe the strongly coupled phase of 5d maximal supergravity. In this paper we study the implications of the existence of an interacting non-metric (4,0) supergravity in 6d. The (4,0) theory can be truncated to non-metric (1,0) supergravity coupled to 5,8 and 14 self-dual tensor multiplets that reduce to three of the unified magical supergravity theories in d=5. This implies that the three infinite families of unified N=2 , 5d Maxwell-Einstein supergravity theories (MESGTs) plus two sporadic ones must have uplifts to unified non-metric (1,0) tensor Einstein supergravity theories in d=6. These theories have non-compact global symmetry groups under which all the self-dual tensor fields including the gravitensor transform irreducibly. Four of these theories are uplifts of the magical supergravity theories whose scalar manifolds are symmetric spaces. The scalar manifolds of the other unified theories are not homogeneous spaces. We also discuss the exceptional field theoretic formulations of non-metric unified $(1,0)$ tensor-Einstein supergravity theories and conclude with speculations concerning the existence of higher dimensional non-metric supergravity theories that reduce to the $(4,0)$ theory in $d=6$.
We embed general solutions to 4D Einstein-Maxwell theory into $mathcal{N} geq 2$ supergravity and study quadratic fluctuations of the supergravity fields around the background. We compute one-loop quantum corrections for all fields and show that the $c$-anomaly vanishes for complete $mathcal{N}=2$ multiplets. Logarithmic corrections to the entropy of Kerr-Newman black holes are therefore universal and independent of black hole parameters.
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