Conformal Supergravity Research Articles

Higher-derivative supergravity, AdS4 holography, and black holes

We use conformal supergravity techniques to study four-derivative corrections in four-dimensional gauged supergravity. We show that the four-derivative Lagrangian for the propagating degrees of freedom of the mathcal{N} = 2 gravity multiplet is determined by two real dimensionless constants. We demonstrate that all solutions of the two-derivative equations of motion in the supergravity theory also solve the four-derivative equations of motion. These results are then applied to explicitly calculate the regularized on-shell action for any asymptotically locally AdS4 solution of the two-derivative equations of motion. The four-derivative terms in the supergravity Lagrangian modify the entropy and other thermodynamic observables for the black hole solutions of the theory. We calculate these corrections explicitly and demonstrate that the quantum statistical relation holds for general stationary black holes in the presence of the four-derivative corrections. Employing an embedding of this supergravity model in M-theory we show how to use supersymmetric localization results in the holographically dual three-dimensional SCFT to determine the unknown coefficients in the four-derivative supergravity action. This in turn leads to new detailed results for the first subleading {N}^{frac{1}{2}} correction to the large N partition function of a class of three-dimensional SCFTs on compact Euclidean manifolds. In addition, we calculate explicitly the first subleading correction to the Bekenstein-Hawking entropy of asymptotically AdS4 black holes in M-theory. We also discuss how to add matter multiplets to the supergravity theory in the presence of four-derivative terms and to generalize some of these results to six- and higher-derivative supergravity.

Superconformal geometries and local twistors

Superconformal geometries in spacetime dimensions D = 3, 4, 5 and 6 are discussed in terms of local supertwistor bundles over standard superspace. These natually admit superconformal connections as matrix-valued one-forms. In order to make contact with the standard superspace formalism it is shown that one can always choose gauges in which the scale parts of the connection and curvature vanish, in which case the conformal and S-supersymmetry transformations become subsumed into super-Weyl transformations. The number of component fields can be reduced to those of the minimal off-shell conformal supergravity multiplets by imposing constraints which in most cases simply consists of taking the even covariant torsion two-form to vanish. This must be supplemented by further dimension-one constraints for the maximal cases in D = 3, 4. The subject is also discussed from a minimal point of view in which only the dimension-zero torsion is introduced. Finally, we introduce a new class of supermanifolds, local super Grassmannians, which provide an alternative setting for superconformal theories.

Symmetries of mathcal{N} = (1, 0) supergravity backgrounds in six dimensions

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) 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 ϵα, which proves to generate extended superconformal transformations. Among its cousins are the conformal Killing vector ξa and tensor ζa(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 d’Alembertian in curved space and demonstrate that, beyond the first-order case, these operators are defined only on a limited class of backgrounds, including all conformally flat ones.

Polarization tensor in de Sitter gauge gravity

The gauge theory of the de Sitter group, [Formula: see text], in the ambient space formalism has been considered in this paper. This method is important to construction of the de Sitter super-conformal gravity and Quantum gravity. [Formula: see text] gauge vector fields are needed which correspond to [Formula: see text] generators of the de Sitter group. Using the gauge-invariant Lagrangian, the field equations of these vector fields have been obtained. The gauge vector field solutions are recalled. By using these solutions, the spin-[Formula: see text] gauge potentials has been constructed. There are two possibilities for presenting this tensor field: rank-[Formula: see text] symmetric and mixed symmetry rank-[Formula: see text] tensor fields. To preserve the conformal transformation, a spin-[Formula: see text] field must be represented by a mixed symmetry rank-[Formula: see text] tensor field, [Formula: see text]. This tensor field has been rewritten in terms of a generalized polarization tensor field and a de Sitter plane wave. This generalized polarization tensor field has been calculated as a combination of vector polarization, [Formula: see text], and tensor polarization of rank-2, [Formula: see text], which can be used in the gravitational wave consideration. There is a certain extent of arbitrariness in the choice of this tensor and we fix it in such a way that, in the limit, [Formula: see text], one obtains the polarization tensor in Minkowski spacetime. It has been shown that under some simple conditions, the spin-[Formula: see text] mixed symmetry rank-[Formula: see text] tensor field can be simultaneously transformed by unitary irreducible representation of de Sitter and conformal groups ([Formula: see text]).

Local supertwistors and conformal supergravity in six dimensions

The local supertwistor formalism, which involves a superconformal connection acting on the bundle of such objects over superspace, is used to investigate superconformal geometry in six dimensions. The geometry corresponding to (1, 0) and (2, 0) off-shell conformal supergravity multiplets, as well the associated finite super-Weyl transformations, are derived.

Relaxed hypermultiplet in four dimensional N=2 conformal supergravity

Superconformal matter multiplets play a crucial role in the construction of Poincare supergravity invariants. Off-shell multiplets allow for construction of general matter couplings in supergravity. In Nucl. Phys. B214 (1983) 519-531, relaxed hypermultiplet was constructed in rigid supersymmetry which on coupling with the real scalar multiplet allowed for an off-shell formulation of the rigid hypermultiplet. In this paper, we extend the relaxed hypermultiplet to conformal supergravity. For consistency with the superconformal algebra, we find that the fields have to be allowed to transform in a non-canonical way under SU(2) symmetry. We find suitable field redefinitions to obtain fields which are irreducible representations of SU(2) R-symmetry and present the full non-linear transformation rule.

New higher derivative action for tensor multiplet in mathcal{N} = 2 conformal supergravity in four dimensions

We will use the covariant superform approach to develop a new density formula for mathcal{N} = 2 conformal supergravity which is based on a fermionic multiplet whose lowest component is a dimension-5/2 spinor. We will show that this density formula admits an embedding of the real scalar multiplet of [1]. Upon using the embedding of the tensor multiplet into the real scalar multiplet, we will construct a new higher derivative action of the tensor multiplet in mathcal{N} = 2 conformal supergravity.

Mathcal{N} = 4 conformal supergravity: the complete actions

The most general class of 4D mathcal{N} = 4 conformal supergravity actions depends on a holomorphic function of the scalar fields that parametrize an SU(1, 1)/U(1) coset space. The bosonic sector of these actions was presented in a letter [1]. Here we provide the complete actions to all orders in the fermion fields. They rely upon a new mathcal{N} = 4 density formula, which permits a direct but involved construction. This density formula also recovers the on-shell action for vector multiplets coupled to conformal supergravity. Applications of these results in the context of Poincaré supergravity are briefly discussed.

Super heat kernel and one-loop divergence of super Yang–Mills theory in conformal supergravity

Abstract We consider super Yang–Mills (SYM) theory in $N=1$ conformal supergravity. Using the background field method and the Faddeev–Popov procedure, the quantized action of the theory is presented. Its one-loop effective action is studied using the heat kernel method. We shall develop a non-iterative scheme, generalizing the non-supersymmetric case, to obtain the super heat kernel coefficients. In particular, the first three coefficients, which govern the one-loop divergence, will be calculated. We shall also demonstrate how to schematically derive the higher-order coefficients. The method presented here can be readily applied to various quantum theories. We shall, as an application, derive the full one-loop divergence of SYM in conformal supergravity.

Super heat kernel of general second order operators in N=1 superspace and one-loop divergence of dilaton-coupled SYM theory

We shall develop a general technique to obtain the super heat kernel coefficients of an arbitrary second order operator in $N=1$ superspace. We focus on the space of conformal supergravity here but the method presented is equally applicable for other types of superspace. The first three coefficients which determine the one-loop divergence of the corresponding quantum theory will be calculated. As an application we shall present the one-loop logarithmic divergence of super Yang-Mills theory coupled to a string dilaton $S$. This is the first superfield calculation for SYM with a non-trivial gauge kinetic function, which generalize the previous result with a constant coupling strength. We also demonstrate that the method presented can be extended to the case of third order operators, with the restriction that its third order part is composed of only spinor derivatives.

Linearised actions for mathcal{N} -extended (higher-spin) superconformal gravity

The off-shell actions for mathcal{N} -extended conformal supergravity theories in three dimensions were formulated in [1, 2] for 1 ≤ mathcal{N} ≤ 6 using a universal approach. Each action is generated by a closed super three-form which is constructed in terms of the constrained geometry of mathcal{N} -extended conformal superspace. In this paper we initiate a program to recast these actions (and to formulate their higher-spin counterparts) in terms of unconstrained gauge prepotentials as integrals over the full superspace. We derive transverse projection operators in mathcal{N} -extended Minkowski superspace and then use them to construct linearised rank-n super-Cotton tensors and off-shell mathcal{N} -extended superconformal actions. We also propose off-shell gauge-invariant actions to describe massive higher-spin super-multiplets in mathcal{N} -extended supersymmetry. Our analysis leads to general expressions for identically conserved higher-spin current multiplets in mathcal{N} -extended supersymmetry.

Superconformal vector multiplet self-couplings and generalised Fayet-Iliopoulos terms

As an extension of the recent construction of generalised Fayet-Iliopoulos terms in N=1 supergravity given in [1], we present self-interactions for a vector multiplet coupled to conformal supergravity. They are used to construct new models for spontaneously broken local supersymmetry.

Off-shell N = 2 \u2192 N = 1 reduction in 4D conformal supergravity

We discuss N = 2 → N = 1 reduction in four dimensional conformal supergravity. In particular, we keep the off-shell structure of supermultiplets (except hypermultiplets). As we will show, starting with (almost) off-shell conformal supergravity makes the procedure simpler than that from N = 2 Poincaré supergravity, which makes it easier to show the correspondence to the standard N = 1 conformal supergravity. We find that the N = 1 superconformal symmetry is simply realized by truncating the gravitino multiplet. We also discuss the consistency with the original N = 2 system and show the reduced N = 1 conformal supergravity action.

Bimetric, conformal supergravity and its superstring embedding

We discuss the connection between Weyl2 supergravity and superstrings and further discuss holography between 4-dimensional, mathcal{N} = 4 superconformal Weyl2 supergravity and mathcal{N} = 8, higher spin-four theory on AdS5. The Weyl2 plus Einstein supergravity theory is a special kind of a bimetric gravity theory and consists of a massless graviton multiplet plus an additional massive spin-two supermultiplet. Here, we argue that the additional spin-two field and its superpartners originate from massive excitations in the open string sector; just like the mathcal{N} = 4 super Yang-Mills gauge fields, they are localized on the world volume of D3-branes. The ghost structure of the Weyl action should be considered as an artifact of the truncation of the infinitely many higher derivative terms underlying the massive spin 2 action. In field theory, mathcal{N} = 4 Weyl2 supergravity exhibits superconformal invariance in the limit of vanishing Planck mass. In string theory the additional spin-two fields become massless in the tensionless limit. Therefore low string scale scenarios with large extra dimensions provide (almost) superconformal field theories with almost massless open string spin-two fields. The full mathcal{N} = 4 scalar potential including the Yang-Mills matter multiplets is presented and the supersymmetric vacua of Einstein Supergravity are shown, as expected, to be vacua of massive Weyl supergravity. Other vacua are expected to exist which are not vacua of Einstein supergravity. Finally, we identify certain spin-four operators on the 4-dimensional boundary theory that could be the holographic duals of spin-four fields in the bulk.

Supersymmetry anomalies in mathcal{N}=1 conformal supergravity

We solve the Wess-Zumino consistency conditions of mathcal{N}=1 off-shell conformal supergravity in four dimensions and determine the general form of the superconformal anomalies for arbitrary a and c anomaly coefficients to leading non trivial order in the gravitino. Besides the well known Weyl and R-symmetry anomalies, we compute explicitly the fermionic mathcal{Q} - and mathcal{S} -supersymmetry anomalies. In particular, we show that mathcal{Q} -supersymmetry is anomalous if and only if R-symmetry is anomalous. The mathcal{Q} - and mathcal{S} -supersymmetry anomalies give rise to an anomalous supersymmetry transformation for the supercurrent on curved backgrounds admitting Killing spinors, resulting in a deformed rigid supersymmetry algebra. Our results may have implications for supersymmetric localization and supersymmetry phenomenology. Analogous results are expected to hold in dimensions two and six and for other supergravity theories. The present analysis of the Wess-Zumino consistency conditions reproduces the holographic result of arXiv:1703.0429 and generalizes it to arbitrary a and c anomaly coefficients.

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.

N=2 supersymmetric higher spin gauge theories and current multiplets in three dimensions

We describe several families of primary linear supermultiplets coupled to three-dimensional ${\cal N}=2$ conformal supergravity and use them to construct topological $BF$-type terms. We introduce conformal higher-spin gauge superfields and associate with them Chern-Simons-type actions that are constructed as an extension of the linearised action for ${\cal N}=2$ conformal supergravity. These actions possess gauge and super-Weyl invariance in any conformally flat superspace and involve a higher-spin generalisation of the linearised ${\cal N}=2$ super-Cotton tensor. For massless higher-spin supermultiplets in (1,1) anti-de Sitter (AdS) superspace, we propose two off-shell Lagrangian gauge formulations, which are related to each other by a dually transformation. Making use of these massless theories allows us to formulate consistent higher-spin supercurrent multiplets in (1,1) AdS superspace. Explicit examples of such supercurrent multiplets are provided for models of massive chiral supermultiplets. Off-shell formulations for massive higher-spin supermultiplets in (1,1) AdS superspace are proposed.

Conformal Symmetry in Field Theory and in Quantum Gravity

Conformal symmetry always played an important role in field theory (both quantum and classical) and in gravity. We present construction of quantum conformal gravity and discuss its features regarding scattering amplitudes and quantum effective action. First, the long and complicated story of UV-divergences is recalled. With the development of UV-finite higher derivative (or non-local) gravitational theory, all problems with infinities and spacetime singularities might be completely solved. Moreover, the non-local quantum conformal theory reveals itself to be ghost-free, so the unitarity of the theory should be safe. After the construction of UV-finite theory, we focused on making it manifestly conformally invariant using the dilaton trick. We also argue that in this class of theories conformal anomaly can be taken to vanish by fine-tuning the couplings. As applications of this theory, the constraints of the conformal symmetry on the form of the effective action and on the scattering amplitudes are shown. We also remark about the preservation of the unitarity bound for scattering. Finally, the old model of conformal supergravity by Fradkin and Tseytlin is briefly presented.

Superconformal index of higher derivative mathcal{N}=1 multiplets in four dimensions

Supersymmetric partition function of mathcal{N}=1 superconformal theories on Sβ1 × S3 is related to the superconformal index receiving contributions from short representations. The leading coefficients in the small β (high “temperature”) expansion of the index were previously related to the conformal anomaly coefficients of the theory. Assumptions underlying universality of these relations were tested only for simplest low-spin unitary multiplets. Here we consider examples of higher derivative non-unitary mathcal{N}=1 multiplets that naturally appear in the context of extended conformal supergravities and compute their superconformal index. We compare the coefficients in the small β expansion of the index with those proposed earlier for unitary multiplets and suggest some modifications that should apply universally to all types of theories. We also comment on the structure of subleading terms and the case of mathcal{N}=4 conformal supergravity.

Heterotic and bosonic string amplitudes via field theory

Previous work has shown that massless tree amplitudes of the type I and IIA/B superstrings can be dramatically simplified by expressing them as double copies between field-theory amplitudes and scalar disk/sphere integrals, the latter containing all the α′-corrections. In this work, we pinpoint similar double-copy constructions for the heterotic and bosonic string theories using an α′-dependent field theory and the same disk/sphere integrals. Surprisingly, this field theory, built out of dimension-six operators such as (DμFμν)2, has previously appeared in the double-copy construction of conformal supergravity. We elaborate on the α′ → ∞ limit in this picture and derive new amplitude relations for various gauge-gravity theories from those of the heterotic string.