The component structure of conformal supergravity invariants in six dimensions

Published for SISSA by Springer Received: March 15, 2017 Accepted: April 20, 2017 Published: May 24, 2017 Daniel Butter,a,b Joseph Novakc and Gabriele Tartaglino-Mazzucchellid a Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands b George and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, College Station, TX 77843, U.S.A. c Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, Am Mühlenberg 1, D-14476 Golm, Germany d Instituut voor Theoretische Fysica, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven, Belgium E-mail: , , Abstract: In the recent paper arXiv:1606.02921, the two invariant actions for 6D N = (1, 0) conformal supergravity were constructed in superspace, corresponding to the supersymmetrization of C 3 and C2C. 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 2F action coupled to conformal supergravity. Exploiting the fact that the N = (2, 0) Weyl multiplet has a consistent truncation to N = (1, 0), we then verify that there is indeed only a single N = (2, 0) conformal supergravity invariant and reconstruct most of its bosonic terms by uplifting a certain linear combination of N = (1, 0) invariants. Keywords: Conformal Field Theory, Supergravity Models, Superspaces ArXiv ePrint: 1701.08163 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP05(2017)133 JHEP05(2017)133 The component structure of conformal supergravity invariants in six dimensions Contents 1 2 The 6D N = (1, 0) Weyl multiplet from superspace 2.1 Component fields and curvatures from superspace 2.2 Analysis of the curvature constraints 2.3 Different choices of conventional constraints 2.4 The supersymmetry transformations 4 5 8 10 14 3 The supersymmetric C 3 invariant 3.1 The A action principle 3.2 The supersymmetric C 3 invariant in components 14 15 17 4 The supersymmetric C2C invariant 4.1 The B action principle 4.2 The supersymmetric C2C invariant in components 4.3 The supersymmetric F 2F invariant in components 18 19 21 23 5 The N = (2, 0) conformal supergravity invariant 25 6 Discussion 28 A 6D N = (1, 0) conformal superspace A.1 The superconformal algebra A.2 Gauging the superconformal algebra in conformal superspace A.3 Different superspace frames 31 31 32 34 B Relating notation and conventions 39 1 Introduction The invariants for conformal gravity naturally arise in the study of conformal field theories on curved manifolds. Deser and Schwimmer divided the possible conformal anomalies into two families, type A and type B [1]. Type A anomalies are topological and always involve the Euler term, while type B anomalies are Weyl invariants built (in the purely gravitational case) from the Riemann tensor and its derivatives. In six dimensions, there are three independent conformal gravity invariants parametrizing the type B anomalies. –1– JHEP05(2017)133 1 Introduction Their Lagrangians are L1 = Cabcd C aef d Ce bc f , L2 = Cab cd Ccd ef Cef ab ,   6 a a a L3 = Cabcd δe 2 − 4Re + δe R C ebcd , 5 (1.1) 1 0 0 0 0 0 0 − εabcdef εa0b0c0d0e0f 0 Cab a b Ccd c d Cef e f = 8L1 + 4L2 , 8 (1.2) while the other was observed to contain L3 at the quadratic order. As we will see, it actually contains additional cubic terms in the Weyl tensor in the particular combination 4L1 − L2 + L3 . (1.3) We will refer to these two particular combinations as the C 3 and C2C invariants from now on. 1 Such an approach was advocated for constructing the 4D N = 4 conformal supergravity action [4] prior to its recent direct construction [5]. –2– JHEP05(2017)133 where Cabcd is the Weyl tensor and Rab is the Ricci tensor. When superconformal field theories are under consideration, the type B anomalies should correspond to conformal supergravity invariants and generally the number of such invariants decreases with more supersymmetry. In six dimensions, superconformal algebras only exist for N = (n, 0) (or N = (0, n)) [2], while the requirement that conformal supergravity does not contain higher spin fields limits n < 3. Besides general interest in superconformal field theories, N = (2, 0) models have been the focus of much interest due to their somewhat mysterious nature. Their existence was actually inferred by various arguments in string theory and they are believed to provide a description of the low-energy dynamics of multiple coincident M5-branes in M-theory. In regards to the type B anomalies, there are two obvious questions. First, how many supersymmetric invariants are permitted and what linear combinations of (1.1) do they correspond to? Second, what are their fully supersymmetric forms when couplings to the rest of the Weyl multiplet of conformal supergravity are included? While very strong evidence exists for the purely gravitational form of these anomalies in the supersymmetric cases — namely the existence of two invariants in (1, 0) and only one for (2, 0), which we discuss below — very little was known about their supersymmetric completions. In principle the answer to both questions could be investigated via indirect means by e.g. computing the conformal anomaly of various (1, 0) or (2, 0) matter multiplets coupled to (super)gravity, as advocated in [3].1 So far only the purely gravitational part of these computations have been performed in 6D, see e.g. [6–8]. Alternatively, one could construct the full supersymmetric invariants directly. Recently, the direct path was pursued in [9], where two (1, 0) Weyl invariants were built using superspace techniques, and a separate supercurrent analysis was given that established that there were at most two such invariants. One of them was observed to contain only C 3 terms in the particular combination 1 4L1 + L2 + L3 , 3 (1.4) and should be extendable to some (2, 0) conformal supergravity invariant containing additional terms involving other fields of the Weyl multiplet. To our knowledge, no analysis of the off-shell supersymmetric extension of this term has been performed. Another goal of this paper is to make a major step towards solving this problem. By analyzing the structure of the two (1, 0) conformal supergravity invariants, we will show that only a certain combination can be lifted to (2, 0) conformal supergravity; this combination has (1.4) as 2 In conventional superspace approaches, see e. g. [14–26], the structure group contains only the Lorentz and R-symmetry groups, and additional torsion superfields appear. –3– JHEP05(2017)133 The approach of [9] involved the direct construction of the two invariants from certain conformal primary superfields. These were composites built from the super-Weyl tensor and its derivatives within a novel superspace formulation of 6D N = (1, 0) conformal supergravity, (...truncated)