Invariants for minimal conformal supergravity in six dimensions

Published for SISSA by Springer Received: July 29, 2016 Revised: October 3, 2016 Accepted: November 29, 2016 Published: December 15, 2016 Daniel Butter,a Sergei M. Kuzenko,b Joseph Novakc and Stefan Theisenc a Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands b School of Physics M013, The University of Western Australia, 35 Stirling Highway, Crawley W.A. 6009, Australia c Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Germany E-mail: , , , Abstract: We develop a new off-shell formulation for six-dimensional conformal supergravity obtained by gauging the 6D N = (1, 0) superconformal algebra in superspace. This formulation is employed to construct two invariants for 6D N = (1, 0) conformal supergravity, which contain C 3 and CC terms at the component level. Using a conformal supercurrent analysis, we prove that these exhaust all such invariants in minimal conformal supergravity. Finally, we show how to construct the supersymmetric F F invariant in curved superspace. Keywords: Conformal Field Theory, Supergravity Models, Superspaces ArXiv ePrint: 1606.02921 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP12(2016)072 JHEP12(2016)072 Invariants for minimal conformal supergravity in six dimensions Contents 1 2 Conformal gravity in six dimensions 2.1 Conformal gravity in D > 3 spacetime dimensions 2.2 The C 3 invariants 2.3 The CC invariant 2.4 The Euler invariant 6 7 10 10 10 3 N = (1, 0) conformal superspace 3.1 The superconformal algebra 3.2 Gauging the superconformal algebra 3.3 Conformal supergravity 3.4 Introducing a compensator 12 12 13 14 17 4 An action principle for the supersymmetric C 3 invariant 4.1 Flat superspace actions and their generalization 4.2 Primary closed six-forms in superspace 4.3 The supersymmetric C 3 invariant 4.4 Other invariants 18 18 20 22 23 5 An action principle for the supersymmetric CC invariant 5.1 Non-primary closed forms in superspace 5.2 A non-primary six-form action principle 25 27 29 6 Discussion 31 A Notation and conventions 33 B The conformal Killing supervector fields of R6|8 36 C The Yang-Mills multiplet in conformal superspace 39 1 Introduction Conformal field theories (CFTs) play a distinguished role among relativistic quantum field theories. It has long been realized that they arise as fixed point theories of renormalization group flows and the study of their properties is clearly of interest. The enlarged symmetry group helps to constrain e.g. the general structure of correlation functions beyond what is already required by Poincaré invariance. Additional symmetries lead to further restrictions. –1– JHEP12(2016)072 1 Introduction One such symmetry which is very powerful in this respect is supersymmetry, in which case one deals with superconformal field theories (SCFTs). As mentioned before, symmetries in quantum field theories lead to restrictions on correlation functions which have to satisfy Ward identities. In correlation functions of conserved currents one finds, however, that the naive Ward identities which would follow from the symmetries cannot always be satisfied simultaneously. This happens in even dimensions and leads to (super)conformal anomalies which express the fact that imposing conservation and tracelessness of the energy-momentum tensor clashes in certain correlation functions. The general structure of these conformal or Weyl anomalies was analyzed by Deser and Schwimmer [14] who also introduced the classification into two types: type A and type B. In any even dimension there is always one type A anomaly and starting in four dimensions, an increasing number of type B anomalies. The easiest way to discuss them is to couple the conformal field theory to a metric background which serves as a source for the energy-momentum tensor. The anomalies then express the non-invariance of the effective action (generating functional) under a local Weyl rescaling of the metric. The anomalous variation of the non-local effective action results in anomalies which are local diffeomorphism invariant functions of the metric and its derivative, i.e. functions of the curvature and its covariant derivatives. The type A anomaly in any even dimension is given by the Euler density of that dimension; the type B anomalies are Weyl invariant expressions constructed from the curvature tensors and its covariant derivatives [14]. In four dimensions there is one such expression, the square of the Weyl tensor; in six dimensions there are two inequivalent contractions of three Weyl tensors and one Weyl invariant expression which involves two covariant derivatives. If we work in a topologically trivial background, only the type B anomalies contribute if one rescales the metric by a constant factor. In any dimension the possible Weyl anomalies can be found by imposing the WessZumino consistency condition [15], which expresses the obvious fact that two consecutive Weyl variations of the effective action must commute. Non-supersymmetric CFTs are then characterized by as many anomaly coefficients as there are solutions to the Wess-Zumino consistency condition: one in two, two in four and four in six dimensions, respectively. 1 Here we are concerned with unitary SCFTs. For an example of a higher-derivative classical SCFT, see [2]. –2– JHEP12(2016)072 It has been known since the early days of supersymmetry that superconformal theories can only exist in six or lower dimensions [1]. In six dimensions, where N = (p, q) Poincaré superalgebras exist for any integer p, q ≥ 0, superconformal algebras only exist for either p = 0 or q = 0. In fact, the only known non-trivial unitary CFTs in six dimensions are supersymmetric and arise as world-volume theories of appropriate brane configurations in string and M-theory and in F-theory, in the limit where gravity decouples. They realize either N = (2, 0) or N = (1, 0) superconformal symmetry. For these theories no Lagrangian description is known but they are believed to obey the axioms of quantum field theories. 1 They should, in particular, have local conserved current operators and among them a local conserved and traceless energy-momentum tensor [3, 4]. Evidence for the existence of N = (2, 0) theories was first given in [5–7]; for N = (1, 0) theories we refer to [3, 4, 8–13]. –3– JHEP12(2016)072 In SCFTs, the Weyl anomalies are accompanied by superconformal and R-symmetry anomalies; altogether they constitute the so-called super-Weyl anomalies. They are related by supersymmetry and various anomalies in bosonic and fermionic symmetry currents are packaged into anomaly supermultiplets. The most elegant way to exhibit this is using a manifestly supersymmetric formulation, i.e. superspace. In four dimensions, the super-Weyl anomalies were studied in [16, 17] in the N = 1 case and in [18] for N = 2. Furthermore, supersymmetry might also reduce the number of indepen (...truncated)