Non-relativistic three-dimensional supergravity theories and semigroup expansion method

Published for SISSA by Springer Received: October 13, 2020 Accepted: January 3, 2021 Published: February 11, 2021 Non-relativistic three-dimensional supergravity theories and semigroup expansion method a Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile b Instituto de Física, Pontificia Universidad Católica de Valparaíso, Casilla 4059, Valparaiso-Chile c DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy d INFN — Sezione di Torino, Via P. Giuria 1, 10125 Torino, Italy e Departamento de Ciencias, Facultad de Artes Liberales, Universidad Adolfo Ibáñez, Viña del Mar-Chile, Chile E-mail: , , , Abstract: In this work we present an alternative method to construct diverse nonrelativistic Chern-Simons supergravity theories in three spacetime dimensions. To this end, we apply the Lie algebra expansion method based on semigroups to a supersymmetric extension of the Nappi-Witten algebra. Two different families of non-relativistic superalgebras are obtained, corresponding to generalizations of the extended Bargmann superalgebra and extended Newton-Hooke superalgebra, respectively. The expansion method considered here allows to obtain known and new non-relativistic supergravity models in a systematic way. In particular, it immediately provides an invariant tensor for the expanded superalgebra, which is essential to construct the corresponding Chern-Simons supergravity action. We show that the extended Bargmann supergravity and its Maxwellian generalization appear as particular subcases of a generalized extended Bargmann supergravity theory. In addition, we demonstrate that the generalized extended Bargmann and generalized extended Newton-Hooke supergravity families are related through a contraction process. Keywords: Chern-Simons Theories, Supergravity Models, Gauge Symmetry, Classical Theories of Gravity ArXiv ePrint: 2010.01216 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP02(2021)094 JHEP02(2021)094 Patrick Concha,a Marcelo Ipinza,b Lucrezia Raverac,d and Evelyn Rodrígueze Contents 1 Introduction 1 2 Nappi-Witten superalgebra and Chern-Simons action 3 4 Generalized extended Newton-Hooke supergravity theory and semigroup expansion method 18 4.1 Extended Newton-Hooke supergravity 18 4.2 Generalized extended Newton-Hooke supergravity 23 5 Concluding remarks 1 27 Introduction The formulation of a non-relativistic (NR) three-dimensional supergravity theory has recently been approached in [1] and subsequently developed in [2, 3]. These last two years, the construction of NR supergravity actions has received a growing interest [4–8] considering different procedures.1 Such supergravity models correspond to supersymmetric extensions of diverse NR gravity theories. Unlike their bosonic counterparts, the construction of NR supergravity theories remains as a challenging task mainly motivated by the diverse applications of these models in the context of holography and relativistic field theory. The first proposed NR supergravity theory corresponds to a supersymmetric extension of the Newton-Cartan gravity which was obtained by gauging a Bargmann superalgebra [1]. At the bosonic level, the Newton-Cartan formalism allows to formulate in a geometric way a Newtonian gravity model that resembles General Relativity [10, 11]. Newton-Cartan gravity theories have been largely studied and extended with diverse purposes [12–27]. Remarkably, Newton-Cartan geometry has been useful to approach strongly coupled condensed matter systems [28–38] and NR effective field theories [39–43]. Nevertheless, an action principle for a NR supergravity theory requires to consider an approach different from the Newton-Cartan supergravity one. To this end, a Chern-Simons (CS) formalism 1 See [9] for a classification of N = 1 and N = 2 supersymmetric extensions of the Bargmann and extended Newton-Hooke algebras in (3 + 1) dimensions. –1– JHEP02(2021)094 3 Generalized extended Bargmann supergravity theory and semigroup expansion method 5 3.1 Extended Bargmann supergravity 6 3.2 Maxwellian extended Bargmann supergravity 8 3.3 Generalized Maxwellian extended Bargmann supergravity 11 3.4 Generalized extended Bargmann supergravity 15 was considered in [3] to construct a three-dimensional NR supergravity action invariant under an extended Bargmann superalgebra. Such superalgebra can be seen as the supersymmetric extension of the extended Bargmann algebra [44–48]. The extended Bargmann superalgebra admits an invariant bilinear form which ensures the proper construction of a well-defined CS action. Furthermore, the extended Bargmann gravity differs from the Newton-Cartan gravity at the matter coupling level, allowing all components of the Ricci tensor to be non-vanishing. On the other hand, the CS formalism has the advantage of offering a gauge-invariant action, this being an interesting three-dimensional toy model [49–51]. The Lie algebra expansion procedure has been introduced in [54] and subsequently developed by expanding Maurer-Cartan forms [52, 55]. An expansion method based on semigroups (S-expansion) has been then introduced in [53] and subsequently studied in [56–61]. Within the S-expansion procedure the expanded (super)algebra is obtained by combining the structure constants of a Lie (super)algebra with the multiplication law of a semigroup S. In addition, the S-expansion method provides us with the non-vanishing components of the invariant tensor for the expanded (super)algebra, which are crucial to construct CS actions. The S-expansion mechanism not only has been useful at the NR level2 [22, 27, 65–68] but also to obtain novel relativistic symmetries [69–73], superalgebras [74–79], and asymptotic symmetries [80–82], among others. In this work, we present an alternative procedure to construct various NR supergravity theories by considering the S-expansion method. We extend the results obtained in [22, 65] in which diverse NR symmetries are obtained by expanding the Nappi-Witten algebra. The Nappi-Witten symmetry was introduced in [83, 84] and can be seen as a central extension of the homogeneous part of the Galilei algebra. Here, we apply the S-expansion procedure to the Nappi-Witten superalgebra introduced in [8] to obtain known and new NR superalgebras. We get two families of NR superalgebras by considering two different semigroup families. In particular, we first show that the extended Bargmann superalgebra and its generalizations can be obtained as an S-expansion of the super Nappi-Witten algebra. Then, 2 Application at the NR level of the Lie algebra expansion method considering the Maurer-Cartan equations can be found in [5, 26, 62–64]. –2– JHEP02(2021)094 To go beyond Poincaré supergravity is a natural step to explore more general supergravity theories. Analogously, at the NR level, it is possible to extend the extended Bargmann su (...truncated)