Five-dimensional N = 1 AdS superspace: Geometry, off-shell multiplets and dynamics

TL;DR

This paper develops the geometry and superspace techniques for five-dimensional N=1 anti-de Sitter superspace, constructing off-shell multiplets and supersymmetric actions, advancing supergravity formulations with eight supercharges.

Contribution

It introduces the differential geometry of AdS^{5|8}, defines off-shell supermultiplets, and constructs supersymmetric actions using harmonic and projective superspace methods.

Findings

Developed the geometry and isometries of AdS^{5|8}

Constructed various off-shell supermultiplets and actions

Presented new supersymmetric theories including sigma-models and Chern-Simons

Abstract

As a step towards formulating projective superspace techniques for supergravity theories with eight supercharges, this work is devoted to field theory in five-dimensional N = 1 anti-de Sitter superspace AdS^{5|8} = SU(2,2|1)/SO(4,1) x U(1) which is a maximally symmetric curved background. We develop the differential geometry of AdS^{5|8} and describe its isometries in terms of Killing supervectors. Various off-shell supermultiplets in AdS^{5|8} x S^2 are defined, and supersymmetric actions are constructed both in harmonic and projective superspace approaches. Several families of supersymmetric theories are presented including nonlinear sigma-models, Chern-Simons theories and vector-tensor dynamical systems. Using a suitable coset representative, we make use of the coset construction to develop an explicit realization for one half of the superspace AdS^{5|8} as a trivial fiber bundle with fibers isomorophic to four-dimensional Minkowski superspace.

Equations

$${\mathcal{D}}_{\hat{A}}=E_{\hat{A}}+\mathrm{i}{\Phi }_{\hat{A}}J+\frac{1}{2}{\Omega }_{\hat{A}}{}^{\hat{b}\hat{c}}{M}_{\hat{b}\hat{c}}=E_{\hat{A}}+\mathrm{i}{\Phi }_{\hat{A}}J+{\Omega }_{\hat{A}}{}^{\hat{\beta }\hat{\gamma }}{M}_{\hat{\beta }\hat{\gamma }}.$$

$$[J,{\mathcal{D}}_{\hat{\alpha }}^i]$$

$$[M_{\hat{\alpha }\hat{\beta }},{\mathcal{D}}_{\hat{\gamma }}^i]$$

$$\{{\mathcal{D}}_{\hat{\alpha }}^i,{\mathcal{D}}_{\hat{\beta }}^j\}$$

$$[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{\alpha }}^i]$$

$${\mathcal{D}}_{\hat{\alpha }\hat{\beta }}=({\Gamma }^{\hat{a}}{)}_{\hat{\alpha }\hat{\beta }}{\mathcal{D}}_{\hat{a}},$$

$$[{\mathcal{D}}_{\hat{\alpha }\hat{\beta }},{\mathcal{D}}_{\hat{\gamma }}^i]$$

$$J{f}^{ij}=(J_k^i{f}^{kj}+J_k^j{f}^{ik})=0⟺f^{ij}=m{J}^{ij},$$

$$({\mathcal{D}}_{\hat{\alpha }}^{i}F{)}^{\ast }=-(-1{)}^{ϵ(F)}{\mathcal{D}}_i^{\hat{\alpha }}{F}^{\ast },$$

$$x=\overline{x},m=\overline{m}.$$

$$C^{†}=-C,C=(C^i{}_j).$$

$$[{\mathcal{D}}_{\hat{\alpha }}^i,\{{\mathcal{D}}_{\hat{\beta }}^j,{\mathcal{D}}_{\hat{\gamma }}^k\}]+[{\mathcal{D}}_{\hat{\beta }}^j,\{{\mathcal{D}}_{\hat{\gamma }}^k,{\mathcal{D}}_{\hat{\alpha }}^i\}]+[{\mathcal{D}}_{\hat{\gamma }}^k,\{{\mathcal{D}}_{\hat{\alpha }}^i,{\mathcal{D}}_{\hat{\beta }}^j\}]=0$$

$$\displaystyle C^{i}_{~{}j}={{\rm i}\over 2}\omega J^{i}_{~{}j}~{},\qquad\omega=\Big{(}{m\over 2}-x\Big{)}~{}.$$

$$[{\mathcal{D}}_{\hat{a}},\{{\mathcal{D}}_{\hat{\alpha }}^i,{\mathcal{D}}_{\hat{\beta }}^j\}]+\{{\mathcal{D}}_{\hat{\alpha }}^i,[{\mathcal{D}}_{\hat{\beta }}^j,{\mathcal{D}}_{\hat{a}}]\}-\{{\mathcal{D}}_{\hat{\beta }}^j,[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{\alpha }}^i]\}=0$$

$$[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{b}}]$$

$$J^2\equiv -\frac{1}{2}{J}^{ij}{J}_{ij}.$$

$$\omega =-\frac{1}{4}m.$$

$$[{\mathcal{D}}_{\hat{a}},[{\mathcal{D}}_{\hat{b}},{\mathcal{D}}_{\hat{\alpha }}^i]]+[{\mathcal{D}}_{\hat{b}},[{\mathcal{D}}_{\hat{\alpha }}^i,{\mathcal{D}}_{\hat{a}}]]+[{\mathcal{D}}_{\hat{\alpha }}^i,[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{b}}]]=0,$$

$$[{\mathcal{D}}_{\hat{a}},[{\mathcal{D}}_{\hat{b}},{\mathcal{D}}_{\hat{c}}]]+[{\mathcal{D}}_{\hat{b}},[{\mathcal{D}}_{\hat{c}},{\mathcal{D}}_{\hat{a}}]]+[{\mathcal{D}}_{\hat{c}},[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{b}}]]=0,$$

$$\{{\mathcal{D}}_{\hat{\alpha }}^i,{\mathcal{D}}_{\hat{\beta }}^j\}$$

$$[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{\beta }}^i]$$

$$[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{b}}]$$

$$[{\mathcal{D}}_{\hat{\alpha }\hat{\beta }},{\mathcal{D}}_{\hat{\gamma }}^i]$$

$$J^i{}_j=J^I({\sigma }_I{)}_j^i$$

$$\{{\mathcal{D}}_{\hat{\alpha }}^{\underline{1}},{\mathcal{D}}_{\hat{\beta }}^{\underline{1}}\}$$

$$[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{\beta }}^{\underline{1}}]$$

$$[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{b}}]$$

$$u^{+i}{u}_i^{-}\equiv (u^{+}{u}^{-})\ne 0⟹{\delta }_j^i=\frac{1}{(u^{+}{u}^{-})}(u^{+i}{u}_j^{-}-u^{-i}{u}_j^{+}),$$

$${\mathcal{D}}_{\hat{\alpha }}^{\pm }$$

$$J^{++}\equiv J^{ij}{u}_i^{+}{u}_j^{+},J^{+-}$$

$$\{{\mathcal{D}}_{\hat{\alpha }}^{+},{\mathcal{D}}_{\hat{\beta }}^{+}\}$$

$$\{{\mathcal{D}}_{\hat{\alpha }}^{+},{\mathcal{D}}_{\hat{\beta }}^{-}\}$$

$$\{{\mathcal{D}}_{\hat{\alpha }}^{-},{\mathcal{D}}_{\hat{\beta }}^{-}\}$$

$$[{\mathcal{D}}_{\hat{a}},{\mathcal{D}}_{\hat{\alpha }}^{+}]$$

Full text

1. 1 Introduction

2. 2 Covariant derivatives

3. 3 Killing supervectors

4. 4 Harmonic superspace approach

5. 4.1 Analytic multiplets

6. 4.2 Harmonic action principle

7. 5 Projective superspace approach

8. 5.1 Projective multiplets

9. 5.2 Projective action principle

10. 6 Dynamical systems in projective superspace

11. 6.1 Projective multiplets revisited

12. 6.2 Projective action

13. 6.3 Nonlinear sigma-models

14. 6.4 Vector multiplet and Chern-Simons couplings

15. 6.5 Tensor multiplet and vector-tensor couplings

16. 7 Coset space realization

17. 7.1 Coset representative

18. 7.2 SO(4,1)$\times$U(1) covariance

19. 7.3 Representation of covariant derivatives

20. 7.4 Torsion and curvature

21. A 5D Conventions

1 Introduction

In four-dimensional ${\cal N}=2$ Poincaré supersymmetry, there exist two powerful formalisms to construct off-shell manifestly supersymmetic actions: harmonic superspace [1, 2] and projective superspace [3, 4, 5, 6]. Both approaches make use of the superspace ${\mathbb{R}}^{4|8}\times S^{2}$ and its supersymmetric subspaces, which were introduced for the first time by Rosly [7] who built on earlier ideas due to Witten [8]. Both approaches can naturally be extended to the case of $d$-dimensional supersymmetry with eight supercharges, for $d\leq 6$, where the appropriate flat superspace with auxiliary bosonic dimensions is ${\mathbb{R}}^{d|8}\times S^{2}$. Specifically, the harmonic superspace formulations were developed in [9, 10] for $d=5$, and in [11] for $d=6$. The projective superspace formulations were developed in [10] for $d=5$, and in [12, 13] for $d=6$.

In projective superspace, off-shell multiplets are reasonably short and can readily be expressed in terms of 4D ${\cal N}=1$ superfields. The latter property is very appealing from the point of view of brane(-world) models. It is also expected [14, 15] that projective superspace should be relevant in the context of hybrid string theory [16]. For these and similar possible applications, one actually needs projective superspace techniques for supergravity. So far, to the best of our knowledge, the projective superspace approach has been mastered only in the flat case.

In harmonic superspace, the prepotential structure of 4D ${\cal N}=2$ supergravity is well understood [17, 2], and similar constructions are clearly applicable in five and six dimensions, see [18] for the six-dimensional case. What is still missing here, in our opinion, is a properly incorporated covariant formalism of differential geometry for superfield supergravity, which should be similar in spirit to the famous Wess-Zumino approach to (the old minimal formulation for) 4D ${\cal N}=1$ supergravity reviewed in [19]. In four-dimensional ${\cal N}=1$ supergravity, it has been recognized for a long time that the most efficient approach to superfield supergravity occurs if one merges together and uses, depending on a concrete application, both the covariant and prepotential techniques [20, 21].

Unlike the purely prepotential approach pursued in [17, 2], this paper is targeted at (making the first step towards) developing covariant superfield techniques for supergravity theories with eight supercharges. Our point of departure is as follows. It is known that all information about off-shell supergravity formulations (including the structure of possible matter multiplets) is encoded in the corresponding algebra of covariant derivatives. We would like to use only this input and try to develop techniques to construct supersymmetric actions both in the harmonic and projective settings. In this paper we consider one particular supergravity background – five-dimensional ${\cal N}=1$ anti-de Sitter superspace, AdS5|8, and explicitly develop harmonic and projective formulations in a covariant fashion using only the language of differential geometry. We believe that similar ideas should be applicable for a general supergravity background, as well as in four and six space-time dimensions. In particular, the case of 4D ${\cal N}=2$ anti-de Sitter superspace111The 4D ${\cal N}=2$ anti-de Sitter superspace was studied in detail in [22] where a manifestly supersymmetric formulation for the off-shell 4D ${\cal N}=2$ anti-de Sitter higher spin supermultiplets [23] was given. A few years later, some formal aspects of this superspace were also discussed in [24]. can be treated similarly.

This paper is organized as follows. In section 2 we derive the algebra of the covariant derivatives for 5D ${\cal N}=1$ anti-de Sitter superspace by solving the Bianchi identities. In section 3 the isometries of AdS5|8 are realized in terms of Killing supervectors. In section 4 we introduce analytic multiplets over the harmonic superspace AdS${}^{5|8}\times S^{2}$ and formulate the harmonic superspace action. Various projective multiplets are defined in section 5, as well as the projective superspace action is formulated. A remarkable feature of this supersymmetric action is that it is uniquely determined by two independent requirements: (i) projective invariance; (ii) invariance under the isometry group SU$(2,2|1)$. Some important examples of dynamical systems in the AdS projective superspace are given in section 6. An explicit coset construction for one half of AdS5|8 (Poincaré chart) is elaborated in section 7. Our 5D notation and conventions are collected in Appendix A.

2 Covariant derivatives

In this section, we develop the differential geometry of five-dimensional ${\cal N}=1$ anti-de Sitter superspace, AdS5|8. This is a supersymmetric version of spaces of constant curvature and, similar to all symmetric spaces, it can be realized as a coset space, specifically AdS${}^{5|8}={\rm SU}$(2,2$|$1)/SO(4,1)$\times$U(1). Group-theoretical aspects of AdS5|8 will be discussed in section 7.

Let $z^{\hat{M}}=(x^{\hat{m}},\theta^{\hat{\mu}}_{i})$ be local bosonic ($x$) and fermionic ($\theta$) coordinates parametrizing AdS5|8, where $\hat{m}=0,1,\cdots,4$, $\hat{\mu}=1,\cdots,4$, and $i=\underline{1},\underline{2}$. The Grassmann variables $\theta^{\hat{\mu}}_{i}$ are assumed to obey a standard pseudo-Majorana reality condition. Since the holonomy group of AdS5|8 is ${\rm SO}(4,1)\times{\rm U}(1)$, the superspace covariant derivative ${\cal D}_{\hat{A}}=({\cal D}_{\hat{a}},{\cal D}_{\hat{\alpha}}^{i})$ can be chosen to have the form

[TABLE]

Here $E_{\hat{A}}=E_{\hat{A}}{}^{\hat{M}}(z)\partial_{\hat{M}}$