Minimal \( \mathcal{N}=4 \) topologically massive supergravity

Published for SISSA by Springer Received: November 30, 2016 Revised: March 2, 2017 Accepted: March 8, 2017 Published: March 21, 2017 Sergei M. Kuzenko,a Joseph Novakb and Ivo Sachsc a School of Physics M013, The University of Western Australia, 35 Stirling Highway, Crawley W.A. 6009, Australia b Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Germany c Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität, Theresienstraße 37, D-80333 München, Germany E-mail: , , Abstract: Using the superconformal framework, we construct a new off-shell model for N = 4 topologically massive supergravity which is minimal in the sense that it makes use of a single compensating vector multiplet and involves no free parameter. As such, it provides a counterexample to the common lore that two compensating multiplets are required within the conformal approach to supergravity with eight supercharges in diverse dimensions. This theory is an off-shell N = 4 supersymmetric extension of chiral gravity. All of its solutions correspond to non-conformally flat superspaces. Its maximally supersymmetric solutions include the so-called critical (4,0) anti-de Sitter superspace introduced in [25], and well as warped critical (4,0) anti-de Sitter superspaces. We also propose a dual formulation for the theory in which the vector multiplet is replaced with an off-shell hypermultiplet. Upon elimination of the auxiliary fields belonging to the hypermultiplet and imposing certain gauge conditions, the dual action reduces to the one introduced in [20]. Keywords: Extended Supersymmetry, Supergravity Models, Superspaces ArXiv ePrint: 1610.09895 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP03(2017)109 JHEP03(2017)109 Minimal N = 4 topologically massive supergravity Contents 1 2 The N = 4 vector multiplets 2.1 Kinematics 2.2 Dynamics 3 3 7 3 Minimal topologically massive supergravity 3.1 Action principle and equations of motion 3.2 Analysing the equations of motion 9 9 10 4 Component actions 4.1 The component conformal supergravity action 4.2 The component vector multiplet actions 4.3 N = 4 topologically massive supergravity in components 17 17 18 20 5 Discussion 24 A The geometry of N = 4 conformal superspace 25 B The geometry of SO(4) superspace 28 C Super-Weyl gauge conditions 30 1 Introduction A unique feature of three spacetime dimensions (3D) is the existence of topologically massive Yang-Mills and gravity theories. They are obtained by augmenting the usual YangMills action or the gravitational action by a gauge-invariant topological mass term. Such a mass term coincides with a non-Abelian Chern-Simons action in the Yang-Mills case [1–4] and with a Lorentzian Chern-Simons term in the case of gravity [3, 4]. Without adding the Lorentzian Chern-Simons term, the pure gravity action propagates no local degrees of freedom. The Lorentzian Chern-Simons term can be interpreted as the action for conformal gravity in three dimensions [3, 5, 6].1 Topologically massive theories of gravity possess supersymmetric extensions. In particular, N = 1 topologically massive supergravity was introduced in [9] and its cosmological extension followed in [10]. The off-shell formulations for N -extended topologically massive supergravity theories were presented in [11] for N = 2 and in [12] for N = 3 and N = 4. In 1 The usual Einstein-Hilbert action for 3D gravity with a cosmological term can also be interpreted as the Chern-Simons action for the anti-de Sitter group [7, 8]. –1– JHEP03(2017)109 1 Introduction all of these theories, the action functional is a sum of two terms, one of which is the action for pure N -extended supergravity (Poincaré or anti-de Sitter) and the other is the action for N -extended conformal supergravity. The off-shell actions for N -extended supergravity theories in three dimensions were given in [13] for N = 1, [14, 15] for N = 2, and [14] for the cases N = 3, 4. The off-shell actions for N -extended conformal supergravity were given in [5] for N = 1, [16] for N = 2, and [17] for N = 3, 4. The latter work made use of the formulation for N -extended conformal supergravity presented in [18]. In [20] a supergravity action functional was also postulated to generate the dynamical equations given. This action was claimed to be off-shell without giving technical details. In this paper we propose a new off-shell model for N = 4 topologically massive supergravity which is minimal in the sense that it makes use of a single compensating vector multiplet. The theory is consistent only if the term corresponding to N = 4 conformal supergravity is turned on. An important maximally supersymmetric solution for this theory is the socalled critical (4,0) AdS superspace introduced in [25]. Our supergravity theory is first presented in a manifestly supersymmetric form, and then its action functional is reduced to components. By choosing appropriate gauge conditions at the component level and performing a duality transformation, we show how to reduce our off-shell supergravity action to the one postulated in [20]. This paper is organised as follows. In section 2 we recall the superspace geometry of the two N = 4 vector multiplets and the corresponding locally supersymmetric actions. In section 3 we present two models for minimal N = 4 topologically massive supergravity, analyse their equations of motion and give a brief discussion of the maximally supersymmetric solutions. Section 4 is devoted to the component structure of minimal N = 4 topologically massive supergravity. Concluding comments are given in section 5. The main body of the paper is accompanied with three technical appendices. The essential details 2 The only known models which pick precisely this value are the topologically gauged ABJ(M) models of [21–23]. –2– JHEP03(2017)109 The off-shell structure of 3D N = 4 supergravity [14] is analogous to that of 4D N = 2 supergravity (see, e.g., [19] for a pedagogical review) in the sense that two superconformal compensators are required (for instance, two off-shell vector multiplets, one of which is selfdual and the other anti-self-dual) in order to realise pure Poincaré or anti-de Sitter (AdS) supergravity theories. We recall that the equations of motion for pure N = 4 Poincaré or AdS supergravity are inconsistent if one makes use of a single compensator [12]. By construction, the off-shell N = 4 topologically massive supergravity theory of [12] makes use of two compensators. However, in [20] the consistent system of dynamical equations was proposed for N = 4 topologically massive AdS supergravity with a single compensating hypermultiplet, following earlier work in [21–23] on ABJ(M) models. A peculiar feature of this model, like those considered in [21–23], is that it has no free parameter. Consequently the dimensionless combination, µ`, of mass µ and AdS radius ` takes a f (...truncated)