（A）dS和Chern-Simons引力的Newton-Hooke/Carrollian展开资源-CSDN文库

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JHEP02(2020)009

Published for SISSA by Springer

Received: January 9, 2020

Accepted: January 15, 2020

Published: February 3, 2020

Newton-Hooke/Carrollian expansions of (A)dS and

Chern-Simons gravity

Joaquim Gomis,

Axel Kleinschmidt,

b,c

Jakob Palmkvist

and

Patricio Salgado-Rebolledo

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

Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut),

Am M¨uhlenberg 1, DE-14476 Potsdam, Germany

International Solvay Institutes,

ULB-Campus Plaine CP231, BE-1050 Brussels, Belgium

Department of Mathematical Sciences,

Chalmers University of Technology and University of Gothenburg,

SE-412 96 G¨oteborg, Sweden

Instituto de F´ısica, Pontiﬁcia Universidad Cat´olica de Valpara´ıso,

Casilla 4059, Valparaiso, Chile

E-mail: joaquim.gomis@ub.edu, axel.kleinschmidt@aei.mpg.de,

jakob.palmkvist@chalmers.se, patricio.salgado@pucv.cl

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

Open Access,

 The Authors.

Article funded by SCOAP

https://doi.org/10.1007/JHEP02(2020)009

JHEP02(2020)009

and, upon taking quotients, has a connection to Lie algebra expansions and Kac-Moody

algebras. One interesting conceptual point made in [12] is that the sequence of Lie algebras

naturally comes with a generalisation of Minkowski space with more coordinates and an

extension of the Minkowski metric. Using this generalised Minkowski space it is possible to

deﬁne particle actions invariant under these extended algebras that naturally incorporate

post-Newtonian corrections in the non-relativistic limit.

In the present article, we study the non-relativistic symmetries obtained by Lie algebra

expansion of the AdS or dS algebra in D space-time dimensions, i.e. so(D−1, 2) or so(D, 1).

This generalises previous constructions to include a cosmological constant and generates

an inﬁnite family of algebras of Newton-Hooke [19, 20] type. By taking the limit of the

cosmological constant to zero one recovers the non-relativistic algebras introduced in [10]

and further studied in [8, 12, 18].

Besides the non-relativistic limit related to Galilean symmetries (c → ∞) we also

consider the case of ultra-relativistic Carrollian limit (c → 0).

As is known from [24] the

Carroll algebra can be understood by very speciﬁc changes in the commutation relations

and associated changes in the starting point of the algebra expansion procedure. Applying

the expansion procedure then produces an inﬁnite family of Lie algebras associated with

the Carrollian limit. We shall also show how our construction in both cases is related

to aﬃne Kac-Moody algebras, where the role of the expansion parameter is taken by the

spectral (or loop) parameter of the aﬃne algebra.

After the introduction of the inﬁnite family of algebras we consider gravitational models

based on them by focussing on the case of 2+1 space-time dimensions. We construct Chern-

Simons theories based on these algebras since they admit non-degenerate bilinear forms. We

show that the family of algebras systematically generates non-relativistic gravity theories

extended by a cosmological constant, such as extended Newton-Hooke gravity [25, 26], that

reduces to extended Bargmann gravity for Λ → 0 [27, 28], and post-Newtonian gravity [11].

Our construction also connects to recent discussions of non-relativistic expansions of

the metric as described in [9, 10, 29–33]. This can be made very precise in the context of

the Chern-Simons formulation and we shall elaborate on this connection in the conclusions.

In an appendix, we generalise the expansion procedure to obtain an inﬁnite family of

extensions of the non-relativistic Maxwell algebra. Moreover, we outline that our method

is not only applicable non-relativistic symmetries corresponding to point particle limits,

but also to extended objects where the D covariant dimensions are split into p + 1 and

D − p − 1 directions for a p-brane [34, 35]. Similar constructions in the case without a

cosmological constant have been considered previously in [8, 24, 36, 37].

Note added. While this manuscript was being ﬁnalised, the preprints [38, 39] appeared

on the arXiv that have some overlap with some of our results.

Work on Carroll symmetries in the context of electrodynamics and brane dynamics can be found for

instance in [21–23].

In [11] this was called ‘extended gravity’ but in the light of the results of [12] this is better understood

as a post-Newtonian correction.

– 2 –

JHEP02(2020)009

2 Expansions of the (A)dS algebra

In this section, we will construct an inﬁnite family of non-relativistic algebras of the

Newton-Hooke and Carroll AdS type. The ﬁrst members of this family will be given

by the Newton-Hooke algebra in the former case and the Carroll AdS algebra in the latter

(without extensions) [19, 20]. We also obtain an inﬁnite-dimensional algebra that contains

these and other intermediate cases, generalises the one obtained in [8, 10, 18], as quotients.

All our algebras contain a cosmological constant as we start from the AdS algebra.

We will construct the series of non-relativistic algebras by means of Lie algebra ex-

pansions [5–7]. More precisely, we will use the semigroup expansion technique [7] with

semigroup S

(N)

, which will be deﬁned below.

Our starting point is the (A)dS algebra in D dimensions:

[

] = 2η

C[B

, (2.1a)

[

] = 4η

[A[C

D]B]

, (2.1b)

[

] = −Λ

, (2.1c)

which denotes in a uniﬁed manner so(D −1, 2) (for Λ < 0) and so(D, 1) (for Λ > 0). Here,

and

are the generators of spacetime translations and Lorentz transformations,

respectively. Capital indices run over A = 0, . . . , D −1 and the Minkowski metric has been

chosen to have mostly plus signature.

In order to perform a non-relativistic expansion of the (A)dS algebra, it is convenient

to decompose the relativistic indices in the time and space components, A = (0, a), where

a = 1, . . . , D − 1, and relabel the Lie algebra generators

→ {

≡

→ {

≡

H ,

}. (2.2)

Then, the commutation relations (2.1) can be rewritten in the form

[

H] =

, (2.3a)

[

] = δ

H , (2.3b)

[

] = 2δ

c[b

, (2.3c)

[

] =

, (2.3d)

[

] = 2δ

c[b

, (2.3e)

[

] = 4δ

[a[c

d]b]

, (2.3f)

[

H] = Λ

, (2.3g)

[

] = −Λ

. (2.3h)

We note that the decomposition (2.2) is adapted to point particles in the sense that one

direction — that can be thought of as the world-line direction of the particle — is singled

out. In appendix A.2, we also consider the case of extended objects.

2.1 Extended Newton-Hooke algebras

In order to perform expansions of (A)dS of Newton-Hooke type, we ﬁrst note that (2.3)

allows for the following subspace decomposition V = V

⊕ V

[8]

= {

}, V

= {

}, (2.4)

Notice this convention is diﬀerent from the one in [12] where

≡

was used.

– 3 –

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