Chern-Simons supergravity on supergroup manifolds
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Springer
Received: March 9, 2020 Accepted: May 5, 2020 Published: May 25, 2020
Chern-Simons supergravity on supergroup manifolds
a
Dipartimento di Scienze e Innovazione Tecnologica, Universit` a del Piemonte Orientale, viale T. Michel 11, 15121 Alessandria, Italy b INFN, Sezione di Torino, via P. Giuria 1, 10125 Torino, Italy c Arnold-Regge Center, via P. Giuria 1, 10125 Torino, Italy d Dipartimento di Scienze e Alta Tecnologia (DiSAT), Universit` a degli Studi dell’Insubria, via Valleggio 11, 22100 Como, Italy e INFN, Sezione di Milano, via G. Celoria 16, 20133 Milano, Italy
E-mail: [email protected], [email protected], [email protected] Abstract: We construct N=1 d=3 AdS supergravity within the group manifold approach and compare it with Achucarro-Townsend Chern-Simons formulation of the same theory. We clarify the relation between the off-shell super gauge transformations of the ChernSimons theory and the off-shell worldvolume supersymmetry transformations of the group manifold action. We formulate the Achucarro-Townsend model in a double supersymmetric action where the Chern-Simons theory with a supergroup gauge symmetry is constructed on a supergroup manifold. This framework is useful to establish a correspondence of degrees of freedom and auxiliary fields between the two descriptions of d=3 supergravity. Keywords: Chern-Simons Theories, Superspaces ArXiv ePrint: 2002.09400
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)110
JHEP05(2020)110
L. Castellani,a,b,c C.A. Cremoninid,e and P.A. Grassia,b,c
Contents 1
2 N = 1, d = 3 AdS supergravity as Chern-Simons
2
3 N = 1, d = 3 AdS supergravity in the group geometric approach 3.1 The Lagrangian 3.2 Action and symmetries 3.3 Curvature parametrizations and symmetries of the spacetime action
4 5 5 6
4 Off-shell N = 1, d = 3 AdS supergravity 4.1 Off-shell degrees of freedom 4.2 The extended superAdS algebra 4.3 Curvature parametrizations 4.4 The Lagrangian 4.5 Off-shell supersymmetry transformations 4.6 Field equations
8 8 8 8 9 9 10
5 Equivalence of transformations: trivial gauge transformations
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6 Supersymmetric Achucarro-Townsend model
13
7 Supersymmetry
16
8 Conclusions and outlook
18
A Gamma matrices in D = 3 A.1 Useful identities A.2 Fierz identity for two Majorana one-forms
18 18 19
1
Introduction
We consider the N = 1 anti-de Sitter supergravity action in d = 3, realized as the difference of two Chern-Simons actions [1], with respectively OSp(1|2) and Sp(2) connections. Starting from the Chern-Simons formulation, we derive the supergravity action following the steps of the Achucarro and Townsend construction. One obtains a theory whose fundamental 1-form fields are (after a simple redefinition) the dreibein V a , the spin connection ω ab and the Majorana gravitino ψ. The action is invariant by construction under the gauge transformations of OSp(1|2) ⊗ Sp(2). The transformations generated by the spinorial (Majorana) charge of the supergroup yield the N =
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