Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity
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Springer
Received: January 9, 2020 Accepted: January 15, 2020 Published: February 3, 2020
Joaquim Gomis,a Axel Kleinschmidt,b,c Jakob Palmkvistd and Patricio Salgado-Rebolledoe a
Departament de F´ısica Qu` antica i Astrof´ısica and Institut de Ci`encies del Cosmos (ICCUB), Universitat de Barcelona, Mart´ı i Franqu`es, ES-08028 Barcelona, Spain b Max-Planck-Institut f¨ ur Gravitationsphysik (Albert-Einstein-Institut), Am M¨ uhlenberg 1, DE-14476 Potsdam, Germany c International Solvay Institutes, ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium d Department of Mathematical Sciences, Chalmers University of Technology and University of Gothenburg, SE-412 96 G¨ oteborg, Sweden e Instituto de F´ısica, Pontificia Universidad Cat´ olica de Valpara´ıso, Casilla 4059, Valparaiso, Chile
E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: We construct finite- and infinite-dimensional non-relativistic extensions of the Newton-Hooke and Carroll (A)dS algebras using the algebra expansion method, starting from the (anti-)de Sitter relativistic algebra in D dimensions. These algebras are also shown to be embedded in different affine Kac-Moody algebras. In the three-dimensional case, we construct Chern-Simons actions invariant under these symmetries. This leads to a sequence of non-relativistic gravity theories, where the simplest examples correspond to extended Newton-Hooke and extended (post-)Newtonian gravity together with their Carrollian counterparts. Keywords: Classical Theories of Gravity, Space-Time Symmetries, Global Symmetries, p-branes ArXiv ePrint: 1912.07564
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP02(2020)009
JHEP02(2020)009
Newton-Hooke/Carrollian expansions of (A)dS and Chern-Simons gravity
Contents 1 Introduction
1
2 Expansions of the (A)dS algebra 2.1 Extended Newton-Hooke algebras 2.2 Extended Carroll (A)dS algebras
3 3 7 9 10 13
4 Chern-Simons non-relativistic gravities 4.1 Extended Newton-Hooke gravities 4.2 Extended Carrollian gravities
14 15 20
5 Conclusions and outlook
22
A Further generalisations A.1 Non-relativistic expansions of the Maxwell algebra A.2 Non-relativistic brane expansions of (A)dS
24 25 26
1
Introduction
Non-relativistic symmetries are usually obtained from relativistic ones by means of Lie algebra contractions. The most well-known example is the Wigner-In¨on¨ u contraction of the Poincar´e algebra that leads to the Galilei algebra [1] by sending the speed of light to infinity. Lie algebra contractions necessarily preserve the number of generators but can alter the cohomology of the Lie algebra, thus allowing for different central extensions like the Bargmann extension of the Galilei algebra [2], which is crucial when taking the limit of the relativistic particle action to the non-relativistic one [3, 4]. In recent years, more general constructions of non-relativistic symmetry algebras from relativistic ones have been explored. One example is given by the
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