Superconformal Tensor Calculus Research Articles

Components of curvature-squared invariants of minimal supergravity in five dimensions

We present for the first time the component structure of the supersymmetric completions for all curvature-squared invariants of five-dimensional, off-shell (gauged) minimal supergravity, including all fermions. This is achieved by using an interplay between superspace and superconformal tensor calculus techniques, and by employing results from arXiv:1410.8682 and arXiv:2302.14295. Our analysis is based on using a standard Weyl multiplet of conformal supergravity coupled to a vector and a linear multiplet compensator to engineer off-shell Poincaré supergravity. We compute all the descendants of the composite linear multiplets that describe gauged supergravity together with the three independent four-derivative invariants. These are the building blocks of the locally superconformal invariant actions. A derivation of the primary equations of motion for minimal gauged off-shell supergravity deformed by an arbitrary combination of these three locally superconformal invariants, is then provided. Finally, all the covariant descendants in the multiplets of equations of motion are obtained by applying a series of Q-supersymmetry transformations, equivalent to successively applying superspace spinor derivatives to the primary equations of motion.

Component actions of liberated mathcal{N} = 1 supergravity and new Fayet-Iliopoulos terms in superconformal tensor calculus

We explicitly compute the component action of certain recently discovered new mathcal{N} = 1 supergravity actions which enlarge the space of scalar potentials allowed by supersymmetry and also contain fermionic interaction terms that become singular when supersymmetry is unbroken. They are the “Liberated Supergravity” introduced by Farakos, Kehagias and Riotto, and supergravities with a new Kähler-invariant Fayet-Iliopoulos term proposed by Antoniadis, Chatrabhuti, Isono, and Knoops. This paper is complementary to our previous papers [Phys. Rev. D103 (2021) 025008 and 105006], in which new constraints on the coupling constants of those new theories were found. In this paper we spell out many details that were left out of our previous papers.

Magnetic deformation of super-Maxwell theory in supergravity

A necessary condition for partial breaking of mathcal{N} = 2 global supersymmetry is the presence of nonlinear deformations of the field transformations which cannot be generated by background values of auxiliary fields. This work studies the simplest of these deformations which already occurs in mathcal{N} = 1 global supersymmetry, and its coupling to supergravity. It can be viewed as an imaginary constant shift of the D-auxiliary real field of an abelian gauge multiplet. We show how this deformation describes the magnetic dual of a Fayet-Iliopoulos term, a result that remains valid in supergravity, using its new-minimal formulation. Local supersymmetry and the deformation induce a positive cosmological constant. Moreover, the deformed U(1) Maxwell theory coupled to supergravity describes upon elimination of the auxiliary fields the gauging of R-symmetry, realised by the Freedman model of 1976. To this end, we construct the chiral spinor multiplet in superconformal tensor calculus by working out explicitly its transformation rules and use it for an alternative description of the new-minimal supergravity coupled to a U(1) multiplet. We also discuss the deformed Maxwell theory in curved superspace.

Curvature squared invariants in six-dimensional mathcal{N} = (1, 0) supergravity

We describe the supersymmetric completion of several curvature-squared invariants for mathcal{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 Poincaré 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 super-gravity, which plays a central role in the effective low-energy description of α′-corrected string theory compactified to six dimensions, including a detailed analysis of the spectrum about the AdS3 × S3 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)dS6 solutions. In the dilaton-Weyl multiplet, the new off-shell invariant includes Ricci and scalar curvaturesquared terms and possesses a nontrivial dependence on the dilaton field.

The component structure of conformal supergravity invariants in six dimensions

In the recent paper arXiv:1606.02921, the two invariant actions for 6D mathcal{N}=left(1,0right) conformal supergravity were constructed in superspace, corresponding to the supersymmetrization of C3 and C□C. In this paper, we provide the translation from superspace to the component formulation of superconformal tensor calculus, and we give the full component actions of these two invariants. As a second application, we build the component form for the supersymmetric F□F action coupled to conformal supergravity. Exploiting the fact that the mathcal{N}=left(2,0right) Weyl multiplet has a consistent truncation to mathcal{N}=left(1,0right) , we then verify that there is indeed only a single mathcal{N}=left(2,0right) conformal supergravity invariant and reconstruct most of its bosonic terms by uplifting a certain linear combination of mathcal{N}=left(1,0right) invariants.

Off-shell N $$ \mathcal{N} $$ = 2 linear multiplets in five dimensions

We present a superconformal tensor calculus for an arbitrary number of five dimensional N=2 linear multiplets. We also demonstrate how to construct higher derivative invariants and higher order supersymmetric off-diagonal models. Finally, we show the procedure required for the derivation of the supersymmetric completion of the non-Abelian $F^4$ action.

Abelian tensor hierarchy in 4D N $$ \mathcal{N} $$ = 1 conformal supergravity

We consider Abelian tensor hierarchy in four-dimensional ${\cal N}=1$ supergravity in the conformal superspace formalism, where the so-called covariant approach is used to antisymmetric tensor fields. We introduce $p$-form gauge superfields as superforms in the conformal superspace. We solve the Bianchi identities under the constraints for the superforms. As a result, each of form fields is expressed by a single gauge invariant superfield. The action of superforms is shown with the invariant superfields. We also show the relation between the superspace formalism and the superconformal tensor calculus.

Component versus superspace approaches to D = 4, N = 1 conformal supergravity

The superspace formulation of N=1 conformal supergravity in four dimensions is demonstrated to be equivalent to the conventional component field approach based on the superconformal tensor calculus. The detailed correspondence between two approaches is explicitly given for various quantities; superconformal gauge fields, curvatures and curvature constraints, general conformal multiplets and their transformation laws, and so on. In particular, we carefully analyze the curvature constraints leading to the superconformal algebra and also the superconformal gauge fixing leading to Poincare supergravity since they look rather different between two approaches.

Newton-Cartan supergravity with torsion and Schrödinger supergravity

We derive a torsionfull version of three-dimensional $$ \mathcal{N}=2 $$ Newton-Cartan supergravity using a non-relativistic notion of the superconformal tensor calculus. The “superconformal” theory that we start with is Schrodinger supergravity which we obtain by gauging the Schrodinger superalgebra. We present two non-relativistic $$ \mathcal{N}=2 $$ matter multiplets that can be used as compensators in the superconformal calculus. They lead to two different off-shell formulations which, in analogy with the relativistic case, we call “old minimal” and “new minimal” Newton-Cartan supergravity. We find similarities but also point out some differences with respect to the relativistic case.

Conformal supergravity in five dimensions: new approach and applications

We develop a new off-shell formulation for five-dimensional (5D) conformal supergravity obtained by gauging the 5D superconformal algebra in superspace. An important property of the conformal superspace introduced is that it reduces to the superconformal tensor calculus (formulated in the early 2000's) upon gauging away a number of superfluous fields. On the other hand, a different gauge fixing reduces our formulation to the SU(2) superspace of arXiv:0802.3953, which is suitable to describe the most general off-shell supergravity-matter couplings. Using the conformal superspace approach, we show how to reproduce practically all off-shell constructions derived so far, including the supersymmetric extensions of $R^2$ terms, thus demonstrating the power of our formulation. Furthermore, we construct for the first time a supersymmetric completion of the Ricci tensor squared term using the standard Weyl multiplet coupled to an off-shell vector multiplet. In addition, we present several procedures to generate higher-order off-shell invariants in supergravity, including higher-derivative ones. The covariant projective multiplets proposed in arXiv:0802.3953 are lifted to conformal superspace, and a manifestly superconformal action principle is given. We also introduce unconstrained prepotentials for the vector multiplet, the ${\cal{O}}(2)$ multiplet (i.e., the linear multiplet without central charge) and ${\cal{O}}(4+n)$ multiplets, with $n=0,1,\dots$ Superform formulations are given for the BF action and the non-abelian Chern-Simons action. Finally, we describe locally supersymmetric theories with gauged central charge in conformal superspace.

Supersymmetric completion of Gauss-Bonnet combination in five dimensions

Based on superconformal tensor calculus in five dimensions, we construct the supersymmetric completion of Gauss-Bonnet combination. We study the vacuum solutions with AdS 2 × S 3 and AdS 3 × S 2 structures. We also analyze the spectrum around a maximally supersymmetric Minkowski5.

The linear multiplet and ectoplasm

In the framework of the superconformal tensor calculus for 4D N=2 supergravity, locally supersymmetric actions are often constructed using the linear multiplet. We provide a superform formulation for the linear multiplet and derive the corresponding action functional using the ectoplasm method (also known as the superform approach to the construction of supersymmetric invariants). We propose a new locally supersymmetric action which makes use of a deformed linear multiplet. The novel feature of this multiplet is that it corresponds to the case of a gauged central charge using a one-form potential not annihilated by the central charge (unlike the standard N=2 vector multiplet). Such a gauge one-form can be chosen to describe a variant nonlinear vector-tensor multiplet. As a byproduct of our construction, we also find a variant realization of the tensor multiplet in supergravity where one of the auxiliaries is replaced by the field strength of a gauge three-form.

$ \mathcal{N} = 2 $ conformal superspace in four dimensions

We develop the geometry of four dimensional N=2 superspace where the entire conformal algebra of SU(2,2|2) is realized linearly in the structure group rather than just the SL(2,C) x U(2)_R subgroup of Lorentz and R-symmetries, extending to N=2 our prior result for N=1 superspace. This formulation explicitly lifts to superspace the existing methods of the N=2 superconformal tensor calculus; at the same time the geometry, when degauged to SL(2,C) x U(2)_R, reproduces the existing formulation of N=2 conformal supergravity constructed by Howe.

New higher-derivative couplings in 4D $ \mathcal{N} = 2 $ supergravity

Using the off-shell formulation for general N = 2 supergravity-matter systems developed in arXiv:0905.0063, we propose a construction to generate a restricted chiral superfield from any real weight-zero projective multiplet L. One can choose L to be composed of tensor multiplets, O(2n) multiplets, with n=2,3,..., and polar hypermultiplets. In conjunction with the standard procedure to induce a N = 2 linear multiplet from any chiral weight-one scalar, we obtain a powerful mechanism to generate higher-derivative couplings in supergravity. In the case that L is a homogeneous function of n tensor multiplets of degree zero, we show that our construction is equivalent to that developed by de Wit and Saueressig in arXiv:hep-th/0606148 using the component superconformal tensor calculus. We also work out nontrivial examples with L composed of O(2n)and tensor multiplets.

Anomaly-mediation and sequestering from a higher-dimensional viewpoint

We study a five-dimensional supergravity model with boundary-localized visible sector exhibiting anomaly-mediated supersymmetry breaking, in which the central requirements of sequestering and radius stabilization are achieved perturbatively. This makes it possible to understand these various mechanisms in a more integrated and transparent fashion, mostly from the higher-dimensional viewpoint. Local supersymmetry, in the presence of visible sector quantum effects, is enforced by the formalism of the five-dimensional superconformal tensor calculus. The construction results in only mild warping, which allows a natural supersymmetry-breaking mediation mechanism of (finite) boundary-to-boundary gravity loops to co-dominate with anomaly-mediation, thereby solving the latter's tachyonic slepton problem. We make the non-trivial check that this can occur while dangerous loops of stabilizing fields remain highly suppressed. Our discussion is a well-controlled starting point for considering other generalizations of anomaly-mediation, or for string theory realizations.

Hypermultiplets and hypercomplex geometry from six to three dimensions

The formulation of hypermultiplets that has been developed for five-dimensional matter multiplets is by dimensional reductions translated into the appropriate spinor language for six and four dimensions. We also treat the theories without actions that have the geometrical structure of hypercomplex geometry. The latter is the generalization of hyper-Kähler geometry that does not require a Hermitian metric and hence corresponds to field equations without action. The translation tables of this paper allow the direct application of superconformal tensor calculus for the hypermultiplets using the available Weyl multiplets in six and four dimensions. Furthermore, the hypermultiplets in three dimensions that result from reduction of vector multiplets in four dimensions are considered, leading to a superconformal formulation of the c-map and an expression for the main geometric quantities of the hyper-Kähler manifolds in the image of this map.

Gauge and Non-Gauge Tensor Multiplets in 5D Conformal Supergravity

An off-shell formulation of two distinct tensor multiplets,a massive tensor multiplet and a tensor gauge multiplet, is presented in superconformal tensor calculus in five-dimensional space-time. Both contain a rank 2 antisymmetric tensor field, but there is no gauge symmetry in the former, while it is a gauge field in the latter. Both multiplets commonly have 4 bosonic and 4 fermionic {\em on-shell} modes, but the former consists of 16(boson)+16(fermion) component fields while the latter consists of 8(boson)+8(fermion) component fields.

Superconformal Tensor Calculus on an Orbifold in 5D

Superconformal tensor calculus on an orbifold S1/Z2 is given in five-dimensional (5D) spacetime. The four-dimensional superconformal Weyl multiplet and various matter multiplets are induced on the boundary planes from the 5D supermultiplets in the bulk. We identify those induced 4D supermultiplets and clarify a general method for coupling the bulk fields to the matter fields on the boundaries in a superconformal invariant manner.

Superconformal Tensor Calculus in Five Dimensions

We present a full superconformal tensor calculus in five spacetime dimensions in which the Weyl multiplet has 32 Bose plus 32 Fermi degrees of freedom. It is derived by the dimensional reduction from the 6D superconformal tensor calculus. We present two types of 32+32 Weyl multiplets, vector multiplet, linear multiplet, hypermultiplet and nonlinear multiplet. Their superconformal transformation laws and the embedding and invariant action formulas are given.

Erratum to “Superconformal tensor calculus and matter couplings in six dimensions”

Erratum to “Superconformal tensor calculus and matter couplings in six dimensions”