Nonrelativistic Maxwell ChernSimons gravity
HJE
Nonrelativistic Maxwell ChernSimons gravity
Jorge Zanelli 0 1 2 5 6 7 8
Symmetries, Gauge Symmetry
0 Casilla 160C , Concepci ́on , Chile
1 Av. Arturo Prat 514 , Valdivia , Chile
2 Luis Avil ́es
3 Departamento de F ́ısica, Universidad de Concepci ́on
4 Instituto de Ciencias F ́ısicas y Matem ́aticas, Universidad Austral de Chile
5 Joaquim Gomis
6 Mart ́ı i Franqu`es 1 , E08028 Barcelona , Spain
7 Universitat de Barcelona
8 Casilla 567 , Valdivia , Chile
We consider a nonrelativistic (NR) limit of (2 + 1)dimensional Maxwell ChernSimons (CS) gravity with gauge algebra [Maxwell] ⊕ u(1) ⊕ u(1). We obtain a finite NR CS gravity with a degenerate invariant bilinear form. We find two ways out of this difficulty: to consider i) [Maxwell] ⊕ u(1), which does not contain Extended Bargmann gravity (EBG); or, ii) the NR limit of [Maxwell] ⊕ u(1)⊕u(1)⊕u(1), which is a Maxwellian generalization of the EBG.
ChernSimons Theories; Field Theories in Lower Dimensions; SpaceTime

1 Introduction
2.1
2.2
3.1
3.2
3.3
2 2+1 relativistic gravity and Maxwell algebra
ChernSimons action
Relativistic symmetries and U(1) enlargements
3
Nonrelativistic Maxwell gravities
Exotic nonrelativistic Maxwell algebra
Bargmann NR Maxwell algebra
Maxwellian exotic Bargmann gravity
4
Discussion
A ChernSimons invariance
B Explicit ω and ωa connections
Therefore, the
study of nonrelativistic gravities [
3–21
] is a subject that could be useful to understand
nonrelativistic coupled systems in the boundary.
NR theories is a vast world characterized by giving up Lorentz symmetry as a
fundamental ingredient. This symmetry has prevailed as a key ingredient for fundamental
theories and therefore an elegant way to formulate a NR system is by considering a
limiting process from a relativistic one. In this sense a nonrelativistic system is a sector of a
more fundamental theory, but there are several different prescriptions for taking this limit.
One prescription could priviledge time together with one (or several) spatial directions, as
in theories for extended objects living in many dimensions such as string theories, where
these are natural options [22–24]. Each of those NR limits may have a physical interest of
its own. For each one the structure of the theory, and therefore its physical consequences,
change in different ways.
In this work, the NR limit is one in which the speed of light is taken to infinity. In
this process only time is a special direction and it is called the NR particle limit because
it preserves the rotation group of pointlike objects.
– 1 –
In flat spacetime the standard NR limit corresponds to the contraction of the Poincar´e
group into the Galilei group. Group deformations of this type can be systematized in
general with the Ino¨nu¨Wigner contractions of the group algebra.
A welldefined NR limit of a Lagrangian system can be framed as a regular contraction
of the relativistic symmetry algebra preserving the number of generators while keeping the
fields and the action finite. In the limit c → ∞ there might appear infinities in the expansion
of the original Lagrangian. An interesting aspect, recently found in the literature, is that a
Lagrangian system with a finite NR limit may require an enlargement of the field content
of the relativistic theory [18, 19, 22, 25]. Up to now a general method for the inclusion
of extra fields, algebra generators and the new pieces of the Lagrangian for the starting
HJEP05(218)47
theory, is not known.
An additional feature of the NR limit is that the symplectic form of the NR theory
might become degenerate, making some fields not determined by the field equations, thus
reducing the number of dynamical fields. In the case of a ChernSimons (CS) formulation
in three dimensions, the nondegeneracy of the bilinear invariant trace of gauge
generators, hGA, GBi, implies the nondegeneracy of the symplectic form, which would ensure
dynamically indeterminate fields in the NR theory.
A (2 + 1)−dimensional example of the potential degeneracy occurs in the contraction
to obtain the Bargmann algebra from the Poincar´e algebra with an extra Abelian
generator [15]. Neither the Galilei nor the Bargmann algebra admit a nondegenerate bilinear
form. However, in 2 + 1 dimensions with the use of a second central extension [26], in
what is called extended or exotic Bargmann algebra, a nondegenerate bilinear form can
be obtained and a NR CS formulation is found [19, 20]. Therefore, two Abelian generators
with their corresponding fields are required at the relativistic level1 and those fields are
crucial in order to obtain a finite Lagrangian in the NR limit.
Here we study the NR limit of a threedimensional gravity theory coupled to
electromagnetism using an extension of the Poincar´e algebra commonly known as the Maxwell
algebra [28, 29]. In this system, the generators of translations obey [PA, PB] = ZAB,
where the new generator ZAB transforms as an antisymmetric second rank Lorentz tensor
and commutes with spacetime translations. A particle realization of this algebra describes
the motion of a charge interacting with an electromagnetic field with group manifold
coordinates xA and θAB, conjugate to PA and ZAB, respectively. In the Maxwell particle
Lagrangian, the components of the constant electromagnetic field fAB are the canonical
momenta conjugate to θAB [30, 31]. This is the starting point of a rich family of extensions
allowed by the Poincar´e algebra, including higher rank tensors (for a recent classification
see [31]). The physical relevance of these algebras is related to the motion of a charge
distribution described by the coordinates of the center of mass and higher multipolar
moments. These moments can be identified with the duals to the generators of the extended
Poincar´e algebras [31].
A realization of the Maxwell algebra in gravity theories has been studied in [32–35].
There, the ZAB extension was used also in an attempt to include the cosmological constant,
1The use of two U(1) factors in the symmetry group was also considered in [27] in relation to AdS3/CF T2.
– 2 –
something that we do not do in this work. Instead, we would like to see the effect of
including a covariantly constant electromagnetic field in the threedimensional CS gravity,
without introducing a cosmological constant, both in the relativistic and nonrelativistic
regimes.
Note that as an extension of Poincar´e symmetry, the Maxwell algebra is relativistic in
the sense that temporal and spatial directions are on equal footing. How are these algebras
modified in the limit c → ∞ was discussed in [36, 37]. Here we explore the NR limit
of gravitation theories that admit a CS formulation using the Maxwell algebra in 2 + 1
dimensions and we find several alternative CS theories for the NR Maxwell algebras.
A NR theory with a finite Lagrangian is found by following the approach that leads to
the exotic Bargmann algebra from the [Poincar´e] ⊕ u(1) ⊕ u(1) algebra. The addition of
the two extra Abelian generators in the relativistic algebra is motivated by the existence of
two central extensions of the Galilei algebra in 2+1 dimensions [26]. In the contraction of
the Lagrangian this procedure yields two central extensions, one corresponds to the mass
and the other to the noncommutativity of the boost generators [19].
In our case, we consider the NR contraction of the [Maxwell] ⊕ u(1)⊕u(1) algebra. The
presence of the two Abelian generators is enough to guarantee a finite Lagrangian in the NR
limit. In this case, however, the NR algebra has a degenerate bilinear form, which means
that at least one of the NR fields is dynamically indeterminate. One way to circunvent
this difficuty is to eliminate the Abelian generator associated to the noncommutativity of
the NR boosts. This gives rise to a fourparameter family of NR Lagrangians. There is a
particular choice of the free parameters in the NR limit of the theory that corresponds to
the contraction of the [Maxwell] ⊕ u(1) algebra, where the Abelian gauge field is related
to the mass central extension of the Galilei group. Both versions of NR Maxwell algebras
can be seen to fit in the family of Galilean extended algebras constructed in [36, 37].
An alternative way to circunvent the degeneracy difficulty found in the contraction
of [Maxwell]
⊕ u(1) ⊕ u(1) is to consider an extra U(1) field at the relativistic level.
Starting from the relativistic algebra [Maxwell] ⊕ u(1) ⊕ u(1) ⊕ u(1), we show that there is
a generalization of the trasformation used in [19] that includes the Maxwellian generators
such that the resulting bilinear form is non degenerate. The NR CS action obtained through
the contraction is what we call the Maxwellian Extended Bargmann Gravity, but it has
also the Extended Bargmann Gravity and the Exotic Gravity as subcases.
2
2+1 relativistic gravity and Maxwell algebra
2.1
ChernSimons action
In this section, we construct a gauge quasiinvariant gravity action under the Maxwell
algebra in 2 + 1 spacetime dimensions. The Maxwell algebra consists of the nine
generators: spacetime rotations JA, spacetime translations PA, and a new type of generators
ZA characterized and introduced in [28, 29]. The non vanishing commutators among these
– 3 –
HJEP05(218)47
generators are
[JA, JB] = ǫABC J C ,
[JA, ZB] = ǫABC ZC ,
[JA, PB] = ǫABC P C ,
[PA, PB] = ǫABC ZC ,
(2.1)
where the latin indeces are raised and lowered with the Minkowski metric, ηAB
=
diag(−1, 1, 1) = ηAB, and they are split as A = (0, a), with a = 1, 2. The conventions
for the LeviCivita symbol are ǫ012 = 1, ǫ012 = −1. The generators are in their dualized
form ZA = 21 ǫABC ZBC , and J A = 21 ǫABC JBC , and their inverse forms are JAB = ǫABC J C .
In [32–35] the ZAB was defined as ΛZ˜AB where Λ is the cosmological constant. Here,
instead, we would like to see the effect of including a covariant constant electromagnetic
by {PA, JA, ZA},
hJA, ZBi = hPA, PBi = α1ηAB, hJA, JBi = α2ηAB, hPA, JBi = α3ηAB,
(2.2)
where αi are real arbitrary constants. The invariance of this bilinear form under the action
of the Maxwell algebra requires that hJA, ZBi and hPA, PBi have the same global coefficient.
Hence, the most general quadratic Casimir invariant is C = α1(P APA + J AZA) + α2J AJA +
A gaugeinvariant gravity action with the 2 + 1 Maxwell algebra can be constructed
using the connection oneform A = Aµ dxµ taking values in the Maxwell algebra generated
A = EBPB + W BJB + KBZB,
where EB, W B, and KB are oneform fields. The CS form for this connection constructed
with the invariant bilinear form (2.2) defines an action for the relativistic gauge theory for
the Maxwell symmetry as
SR =
Z
hA ∧ dA +
A ∧ [A, A]i.
1
3
The explicit form in terms of the oneform fields reads2
SR =
Z
α1 2KARA(W ) + EATA + α2
W AdWA +
1
+α3EARA(W ) ,
where each term accompanying αi in the Lagrangian is quasiinvariant under the Maxwell
symmetry. The Lorentz curvature, the torsion, and the KA curvature are given by
2Here, the wedge product ∧ between differential forms is understood, i.e., W AEB = W A ∧ EB.
RA(W ) = dW A −
RA(E) = DW EA = T A,
RA(K) = DW KA −
2
1 ǫABC WBWC ,
2
1 ǫABC EBEC ,
– 4 –
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
where the covariant derivative is DW ΦA := dΦA − ǫABC WBΦC .
The field equations derived from (2.5) are
δEA :
δW A :
δKA :
2α1TA + α3RA(W ) = 0,
2α1RA(K) + 2α2RA(W ) + α3TA = 0,
2α1RA(W ) = 0.
(2.9)
(2.10)
(2.11)
Clearly, these equations dynamically determine every field by the vanishing of every
curvature RA(W ) = 0, TA = 0, and RA(K) = 0, provided α1 6= 0 (regardless of the choices for
α2 and α3), otherwise KA would be a redundant field of the theory, i.e., and the bilinear
form (2.2) would be degenerate.
The vanishing of the RA(K) can be rephrased as the constancy of the covariant
derivative of KA: 2DW KA = ǫABC EBEC . This is analogous to the constancy of the background
electromagnetic field in flat spacetime, considered to define a system invariant under the
Maxwell symmetry.
A second order formulation of this system could be considered by postulating T A = 0
and algebraicly solving for W A as a function of EA and ∂EA, so that (2.9) becomes a set
of second order equations for the metric gµν = ηABEµAEνB and KA. This procedure would
give rise to new degrees of freedom, analogous to the topologically massive gravity [38–40].
2.2
Relativistic symmetries and U(1) enlargements
An infinitesimal gauge transformation for the Maxwell algebravalued oneform connection
A, is given by δΛA = dΛ + [A, Λ], where Λ = ρAPA + λAJA + ΘAZA is an algebravalued
gauge parameter. The infinitesimal transformation on the field components is
δΛEA = DW ρA − ǫABC EBλC ,
δΛW A = DW λA,
δΛKA = DW ΘA − ǫABC EBρC − ǫABC KBλC .
(2.12)
(2.13)
(2.14)
The relativistic CS action (2.5) is invariant up to a boundary term under the
infinitesimal transformations (2.12)–(2.14).
As explained in the introduction, a straightforward limit of the relativistic action (2.5)
gives an infinite result. In order to cancel the divergence, one can include extra auxiliary
Abelian fields. This choice is directly inspired by the [Poincar´e] ⊕ u(1) ⊕ u(1) algebra
that allows us to obtain the Extended Bargmann algebra and a CS theory for it [19].
Following this pattern, we include two extra U(1) oneform gauge fields, M and S in the
connection (2.3) as
A = EBPB + W BJB + KBZB + M Y1 + SY2.
(2.15)
The invariant bilinear form for the new algebra, [Maxwell]⊕u(1) ⊕ u(1), is a simple
extension of the bilinear form of the original Maxwell algebra. The new elements can
always be brought to satisfy hY1, Y1i = α4, hY1, Y2i = α5, and hY2, Y2i = 0; with α4 and
– 5 –
α5 arbitrary real constants. With this new connection and invariant bilinear form, the
relativistic CS action (2.5) becomes
SR =
Z
the divergences that arise in the contraction procedure.
3
Nonrelativistic Maxwell gravities
We now consider Ino¨nu¨Wigner contractions for the extended Maxwell algebra. In order
to carry out the contractions we express the relativistic algebra generators with a linear
combination of new generators that involves a dimensionless parameter ξ. By taking the
limit ξ → ∞ one obtains a NR version of the Maxwell algebra. As stated in the
introduction, there are several inequivalent Ino¨nu¨Wigner contractions that may define different
NR algebras. These new algebras can be used to construct new (2 + 1)dimensional CS
theories, the NR Maxwell gravities.
The inclusion of the extra U(1) fields is essential for the limiting process, and it is
through a suitable choice of the constants αi that we obtain different meaningful NR
versions of the Maxwell relativistic action.
In subsection 3.1 we build a first nonrelativistic Maxwell gravity as a CS theory for
a connection valued on a NR version of the [Maxwell] ⊕ u(1) ⊕ u(1) algebra. Because
it is built in a similar way as the socalled Exotic Galilei algebra, we call it the Exotic
NonRelativistic Maxwell algebra (ENRM). It turns our that the dynamics of the theory
is not fully determined because this algebra has a degenerate bilinear form.
In subsection 3.2, we consider a contraction from the [Maxwell] ⊕ u(1) algebra. The
resulting NR algebra admits a nondegenerate bilinear form, and consequently a
dynamically welldefined CS theory. Because it requires the addition of a central extension, as in
the Bargmann algebra with respect to the Galilei algebra, we call this second new algebra
as the Bargmann NonRelativistic Maxwell algebra (BNRM).
Finally, in subsection 3.3, we consider a contraction from the [Maxwell] ⊕ u(1) ⊕ u(1) ⊕
u(1) algebra. As the previous case, the resulting NR algebra admits a non degenerate
bilinear form, and consequently a dynamically welldefined CS theory. The difference is
that in this case the NR CS action has the Exotic Bargmann gravity as a subcase. We
name the resulting NR theory as the Maxwellian Exotic Bargman gravity.
3.1
Exotic nonrelativistic Maxwell algebra
A NR version of the Maxwell algebra can be obtained from an Ino¨nu¨Wigner contraction
of the relativistic[ [Maxwell]⊕u(1) ⊕ u(1) algebra. The contraction can be motivated by
considering the NR limit for the action of a massive Maxwell particle [30] and the
transformations used to obtain the extended Bargmann algebra in [19]. The explicit relation
– 6 –
HJEP05(218)47
between the relativistic generators with the new NR generators {H˜ , P˜a, M˜ , J˜, Z˜, Z˜a, G˜a, S˜}
with a = 1, 2, ǫab ≡ ǫ0ab, ǫab ≡ ǫ0ab such that ǫabǫac = −δcb. The dimensionless parameter ξ
introduced to perform the contraction. We use a tilde for the nonrelativistic generators. In
the limit ξ → ∞, the contracted algebra from (2.1) has the following nonzero commutators
This algebra has three central extensions given by the generators M˜ , S˜, and Z˜. Two
of the central extensions are related to the two extra U(1) generators, but the third comes
directly from the Maxwell generator Z0. Note that the Exotic Bargmann algebra in [19]
is the subalgebra obtained by suppressing the Maxwell generators Z˜a and Z˜ (they appear
only on the right hand side).
The following nonrelativistic (degenerate) invariant bilinear form is obtained directly
from the contraction (3.1) of the relativistic bilinear form (2.2)
hM˜ , H˜ i = hJ˜, Z˜i = −α˜1,
hP˜a, P˜bi = hG˜a, Z˜bi = α˜1δab,
hJ˜, J˜i = −α˜2,
hH˜ , S˜i = hM˜ , J˜i = −α˜3,
hG˜a, P˜bi = α˜3δab,
where α˜1 = α1, α˜2 = α2, and α˜3 = α3/ξ are taken as constants in the NR theory.
The coefficients α4 and α5 are also present in the previous bilinear form, but we need to
choose them as α4 = α1 α5 = α3. As we will see in the following, these particular values
guarantee finiteness of the NR Lagrangian. While invariant under the NR algebra, the
bilinear form (3.3) is degenerate. In fact, the ENRM algebra (3.2) can not be equipped
with a nondegenerate invariant bilinear form. To prove this one may solve the most general
bilinear form that is invariant under the nonrelativistic algebra (3.2) and check that its
determinant vanishes.
we consider the oneform connection
In order to build a CS action invariant up to a surface term under the algebra (3.2),
A˜ = τ H˜ + eaP˜a + ωJ˜ + ωaG˜a + kZ˜ + kaZ˜a + mM˜ + sS˜.
(3.4)
– 7 –
Time translations
Space translations
Boosts
Spatial rotations
Maxwell central charge
Maxwell spatial field
First central charge
Second central charge
˜
H
P˜a
G˜a
˜
J
˜
Z
Z˜a
˜
M
˜
S
Gauge
fields
τ
e
a
ωa
ω
k
k
m
s
a
R(τ )
Ra(eb)
Ra(ωb)
R(ω)
R(k)
Ra(kb)
R(m)
R(s)
(3.5)
(3.6)
nonrelativistic notations in the following table:
Nonrelativistic elements Symmetry
Generators
Curvature
The NR fields are related to the relativistic ones by the inverse of the
transformation (3.1) in order to ensure that A = A˜ [18]. The curvature associated to this connection
where we have written it in terms of the field curvatures
can be written as
SNR =
Z
The NR CS action (2.4) built from the connection (3.4) and the bilinear form (3.3)
α˜1 h−2kR(ω) + 2kaRa(ωb) + eaRa(eb) − τ R(m) − mR(τ )i − α˜2ωR(ω)
+α˜3 heaRa(ωb) − τ R(s) − mR(ω)i .
(3.7)
It is worth noting that, as expected, for the choices α˜1 = 0 = α˜2 one obtains the same
NR action derived in [19], which is a CS action for the Exotic Bargmann algebra.
– 8 –
The field equations are
δτ :
δea :
δω :
δωa :
δk :
δka :
δm :
δs :
which implies R(τ ) = R(ω) = 0 = Ra(ωb) = Ra(eb) = Ra(kb), and
α˜1R(m) + α˜3R(s) = 0,
α˜1Ra(eb) + α˜3Ra(ωb) = 0,
α˜1Ra(kb) + α˜3Ra(eb) = 0,
α˜1R(ω) = 0,
α˜1Ra(ωb) = 0,
α˜3R(τ ) = 0,
α˜1R(τ ) + α˜3R(ω) = 0,
R(k) is arbitrary, which is a consequence of the degenerate bilinear form (see appendix A).
On the other hand, since A = A˜, the components of the relativistic gauge fields in
terms of the NR ones can be expressed as follows
E0 = ξτ +
M = ξτ −
1
2ξ
1
2ξ
m,
m,
W 0 = ω +
S = ω −
1
1
2ξ2 s,
2ξ2 s,
W a =
Ka = ξka,
1 a
w ,
ξ
Ea = ea,
K0 = k.
Using these last expressions on the action (2.5) and then taking the limit ξ → ∞, the
action (3.7) is also obtained. This procedure gives in general a divergent piece for the NR
action. Note however, that in our case there is a delicate balance between the extra U(1)
gauge fields and the coefficients αi that exactly compensates the two divergences coming
from the terms −ξ2τ dτ and −ξτ dω, this is the main reason for using the extra fields.
Now, in spite of the introduction of the two U(1) fields to eliminate the divergences
in the Lagrangian, the contraction process gave rise to an algebra with a degenerate
bilinear form that does not produce a CS action with fully determinate dynamics. Then, a
preliminary lesson is that guaranteeing finiteness of the action in the contraction process
does not guarantee a nondegenerate bilinear form. Therefore a natural question emerges:
is it possible to have a different contraction yielding a NR algebra with a nondegenerate
bilinear form?
From (3.9), it is natural to expect that a solution of the degeneracy problem could
come from eliminating one of the three oneform fields m, k or s, corresponding to the
three central extension of the algebra (3.2). However, eliminating m or k by dropping from
the beginning the associated central charges M˜ or Z˜ does not solves the problem because
the resulting algebra does not admit a nondegenerate invariant bilinear form. On the other
hand if one eliminates S˜ the resulting algebra admits a nondegenerate invariant bilinear
form. This later option has a CS formulation and it is explored on the next subsection.
α˜1(2R(ω) + R(k)) + α˜3R(m) + α˜2R(ω) = 0,
(3.8)
(3.9)
(3.10)
An alternative point of view would be to consider the NR Lagrangian (3.7) as the
starting point and drop directly from it the fields m or k. This would give rise to equations
that determine all the remaining fields dynamically. However, there is not guarantee that
the action would be gauge invariant under the resulting nonrelativistic gauge symmetry
because the Lagrangian would not necessarily be a CS form.
At nonrelativistic level, let us consider putting to zero the central generator S˜. Following
the general steps shown in appendix A we find the resulting algebra admits an invariant
HJEP05(218)47
nondegenerate bilinear form
hM˜ , H˜ i = hJ˜, Z˜i = −α˜1,
with α˜1 6= 0. The corresponding CS action is given by
α˜1 h−2kR(ω) + 2kaRa(ωb) + eaRa(eb) − τ R(m) − mR(τ )i − α˜2ωR(ω)
+α˜0τ R(τ ) + 2α˜6τ R(ω).
The field equations are R(τ ) = R(ω) = Ra(ωb) = Ra(eb) = Ra(kb) = R(k) = R(m) =
0. In this way we have obtained an action invariant under the Bargmann nonrelativistic
Maxwell algebra that is also dynamically determinate.
The previous theory is built out of the NR algebra (3.2) (with S˜ = 0). A remaining
question is weather this theory can be obtained as a limiting process from a relativistic
theory. We do not have a general answer. However, by considering α2 = 0 = α3 from the
beginning (thus α˜2 = 0 as well as α˜0 = α˜6=0), this theory is obtained from a contraction
of a [Maxwell]⊗U(1) relativistic theory. Because α3 = 0, there is not a divergence of the
form −ξτ dω in the limiting process, and therefore it is enough an unique U(1) gauge field
in order to eliminate the divergence coming from ξ2τ dτ .
Then, we may take a minimalistic starting point at the very beginning by considering
at the relativistic level only the term that guarantee a nondegenerated Maxwell CS gravity.
That corresponds to use constants α2 = 0 = α3 in the relativistic bilinear form (2.2). The
EinsteinHilbert and the Exotic gravity Lagrangian are absent from the beginning.
Then, we start with a relativistic algebra [Maxwell]⊗u(1) and consider the following
transformation for the relativistic generators
˜
H
2ξ
P0 =
Pa = P˜a,
(3.14)
Using these transformations we can obtain the Bargmann NR Maxwell algebra
(BNRM) that has only two central extensions: M˜ and Z˜. The only change with respect
to the algebra (3.2) is the vanishing of [G˜a, G˜b] = 0.
The BNRM algebra can be equipped with a nondegenerate invariant bilinear form
Finally, the CS action invariant under the BNRM algebra is
A˜ = τ H˜ + eaP˜a + ωJ˜ + ωaG˜a + kZ˜ + kaZ˜a + mM˜ .
SNR =
Z
hA˜dA˜ + 1 [A˜, A˜], A˜i,
3
= α˜1
Z
[−2kR(ω) + 2kaRa(ωb) + eaRa(eb) − τ R(m) − mR(τ )] .
With this formulation the dynamics of the system is totally determined, that is F (A˜) =
0, or equivalently, the field equations imply the vanishing of all the curvatures associated
to the fields. That is, every curvature in (3.5) vanishes and R(s) is absent in this set up.
The field equations for a CS theory imply that the connection is locally flat, A˜ = g−1dg,
with g any element of the gauge group (the connection is “pure gauge”). Nevertheless, it
is wellknown that there are solutions which are not globally flat due to nontrivial features
of the spacetime topology. Therefore, in looking for those solutions it is worth to analyze
the equations of motion more carefully, component by component.
• Equation R(τ ) = dτ = 0 implies that the spacetime can be foliated in an absolute
time direction.
• At the relativistic level the vanishing of torsion, T A = 0, allows to solve algebraically
for the connection in terms of the vielbein, its inverse and its first derivatives,
W A = W A(EB, ∂EB). This is what allows to pass from the first to the second
order formulation of gravity. However, in the NR case the situation is more subtle.
The relativistic vielbein EA is identified with the NR fields τ and ea, while the
relativistic connection W A is identified with the NR fields ω and ωa. However, by simply
counting equations, the vanishing of Ra(eb) is not enough to solve for ωa and ω in terms of
the NR fields (τ, ea). This is a known issue in the NewtonCartan theory: the NR
torsionfree equation only allows to solve the connection up to an arbitrary oneform. A natural
way to fix the indeterminancy is to impose an extra equation for m¯, a curvature equation
R(m¯ ). In the present framework, the issue is automatically solved because the vielbein is
actually identified with τ, ea and m, see (3.10). And, in the NR theory m satisfies the
equation R(m) = 0 [13]. Then, the system
(3.15)
Ra(eb) = dea + ǫabτ ωb − ǫabebω = 0,
R(m) = dm + ǫabeaωb = 0,
(3.18)
(3.19)
is a linear system that can be algebraically solved for (ωa, ω) in terms of (τ, ea, m). The
explicit solutions are (see appendix B)
1
2
ω = ωµ dxµ =
ǫab∂[ν ecρ]ecµ eaν ebρ − 2ǫab∂[µ eνa]ebν − τµ ωbν ebν dxµ ,
ωa = ωaν dxν = ǫab∂[µ mν]e
bµ + 2ǫcdeaν ∂[µ eξa]eξbτ µ
dxν .
(3.20)
(3.21)
Note that ωaν should be replaced in the r.h.s. of (3.20) to get the full expression for ω.
• Equation R(ω) = dω = 0 is trivially solved too. It means that the expression (3.20)
is locally a gradient of a scalar function. For example, if there is one missing point
in the spatial section this allows for ω = dθ, which corresponds to the geometry of a
cone in 2+1 dimensions.
• Once the previous solutions are replaced, equation Ra(ωb) = 0 is a second order
differential equation for the fields τ, ea and m. It corresponds to some of equations
of the 2+1 NewtonCartan equations of motion, i.e., a NR version of the Einstein
field equations.
Maxwell fields.
• The remaining equations, Ra(kb) = 0, R(k) = 0 determine dynamically the NR
One interesting problem for future work would be to study the dynamical contents, and
in particular the classical solutions of these theories both in the relativistic and NR regimes.
3.3
Maxwellian exotic Bargmann gravity
An alternative way to circunvent the degeneracy of the bilinear form in the
[Maxwell] ⊕ u(1) ⊕ u(1) system, is to add one more u(1) gauge field. In this way one
finds that there is a NR contraction such that each term in the relativistic action has a
NR counterpart. In other words, we do not need to assume the vanishing of α2 or α3 at
the relativistic level. In particular the EinsteinCartan term leads in the NR limit to a
NewtonCartan term. We observe that the final NR action contains three pieces: 1) the
CS action for the exotic Bargmann algebra [19, 20]; 2) the CS action for a new NR Maxwell
algebra; and 3) a CS action for the NR exotic Gravity. The differences with respect to
the previous NR Maxwell algebra presented in (3.2), are the addition of the commutator
[Z˜, G˜a] = ǫabZ˜b plus a new central generator T˜ (in the rˆole played previously by Z˜) in
[G˜a, Z˜b] = −ǫabT˜ and [P˜a, P˜b] = −ǫabT˜. Then, at the relativistic level we start with three
U(1) fields: Y1, Y2, and Y3
A = EBPB + W BJB + KBZB + M Y1 + SY2 + T Y3.
(3.22)
The non zero entries for the bilinear form of the new Abelian generators are
hY1, Y1i = hY2, Y3i = α1,
hY2, Y2i = α2,
hY1, Y2i = α3.
The contraction is defined through the following identifications
P0 =
J0 =
Z0 =
˜
H
2ξ
˜
J
2
˜
Z
,
Note that the identifications for the Poincar´e algebra generators (first two lines), the
contracted algebra is the Exotic Bargmann algebra.
In terms of the NR generators and fields, the connection is
A˜ = τ H˜ + eaP˜a + ωJ˜ + ωaG˜a + kZ˜ + kaZ˜a + mM˜ + sS˜ + tT˜.
(3.24)
The contracted new NR Maxwell algebra, that we shall call Maxwellian Exotic Bargmann
(MEB) algebra , is
It is interesting to note that in contrast with the algebras found in 3.1 and 3.2, this is not
an extension of the Galilean algebra in the sense of [37]. The technical point is that the Z˜
generator does not appear as a central extension in any of the levels defined in [37]. Thus
the expectation is that this algebra may be obtained by enlarging an algebra different from
the Galilean one.
The MEB algebra admits a non degenerate bilinear form obtained from the relativistic
bilinear form (2.2) in the limit ξ → ∞. To keep the action finite there is a rescaling of the
relativistic action parameters that can be absorved on each non relativistic parameter as
α1 = α˜1, α2 = ξ2α˜2, and α3 = ξα˜3.
which is non degenerate if α˜1 6= 0, in analogy with the relativistic case.
hH˜ , M˜ i = hJ˜, T˜i = −α˜1,
hP˜a, P˜bi = hG˜a, Z˜bi = α˜1δab,
hJ˜, S˜i = −α˜2,
hG˜a, G˜bi = α˜2δab,
hH˜ , S˜i = hJ˜, M˜ i = −α˜3,
hG˜a, P˜bi = α˜3δab,
(3.25)
(3.26)
The NR CS curvature along the generators of the MEB algebra is
α˜1 h2kaRa(ωb) + eaRa(eb) − 2mR(τ ) − 2sR(k) − 2tR(t)i
−α˜2 h−2sR(ω) + ωaRa(ωb)i + 2α˜3 heaRa(ωb) − τ R(s) − mR(ω)i . (3.28)
Because the bilinear form does not become degenerate in the contraction process the CS
formulation guarantees that the equation of motion from this NR action are the independent
vanishing of all the curvatures (3.27). Thus, the dynamics is not redundant.
4
Discussion
We have examined a class of NR CS Maxwell gravities in 2+1 dimensions obtained as
the NR particle limit of relativistic CS Maxwell gravity. At the Lagrangian level, the
relativistic action written as a series expansion in terms of the dimensionless parameter
ξ in general contains infinities. The coefficient of the most divergent term is always NR
invariant [24] while the remaining terms, in particular the finite one, are in general not
invariant under the NR symmetry group. Following the idea of [18, 22] we consider the
addition of an extra piece such that the total relativistic Lagrangian has a NR expansion
in which the first term is finite. Inspired by [19], we consider the CS action associated
with [Maxwell] ⊕ u(1) ⊕ u(1) algebra in threedimensional spacetime. In this way, a
finite NR CS Lagrangian is obtained, with an invariant bilinear form which is generically
degenerate. Therefore the NR Maxwell gravity has field equations that do not determine
all the dynamical fields. We found two ways to cure this difficulty.
First, the field equations (3.8) themselves suggest various ways to cure the dynamical
indeterminacy. One possibility that gives a Lagrangian invariant under NR transformations
with a nondegenerate bilinear form corresponds to the vanishing of the central charge
associated to the non commutativity of the Galilean boost, i.e. [G˜a, G˜b] = 0. The Lagrangian
depends on four parameters, α˜1, α˜2, α˜0, α˜6 and when the last three are put to zero, the
theory becomes the NR limit of a CS relativistic system with [Maxwell] ⊕ u(1) gauge algebra.
This case is studied in subsection 3.2.
Second, starting with [Maxwell] ⊕ u(1) ⊕ u(1) ⊕ u(1) at the relativistic level it is
shown in section 3.3 that there is a generalization of the trasformation used in [19] for this
relativistic algebra that leads to a NR algebra admitting a non degenerate bilinear form.
The NR CS action obtained through the contraction contains the Extended Bargmann
Gravity and the Exotic Gravity as subcases.
One of the points that needs further study is to analyze in detail the action (3.28) and
its dynamical contents. In particular to construct the second order metric formulation, i. e.,
the NR theory obtained by substituting ω and ωa into the NR action (3.28). Note however
that the advances in the dynamical analysis performed for the system of subsection 3.2
apply for the system of section 3.3 (vanishing of curvatures in (3.27)). In paticular the explicit
HJEP05(218)47
solution for the connection fields ω and ωa worked out in the appendix B are the same.
The torsion equations Ra(eb) = 0 can be algebraically solved for ωa and it allows us to
write the field equations as a second order system. However, if the torsion equation is not
obtained as the variation of the action with respect to ωa, the second order system obtained
by substituting ωa in the action needs not to be equivalent to the first order one and in
general possesses different degrees of freedom. In particular, this might be the case when
the torsion equation is enforced by a Lagrange mutiplier, in which case the resulting second
order system may have propagating degrees of freedom while the system with torsion not
forced to vanish may have no propagating degrees of freedom at all [38–41].
Another interesting problem is the construction of a supergravity theory associated to
super Maxwell algebra [42] in 2+1 dimensions.
Also in the context of supersymmetry, note that in this work we have reinforced the
idea that adding bosonic fields at the relativistic level helps in having a welldefined NR
limit. As shown for instance in [42–45], supersymmetry may require additional bosonic
fields to be welldefined too. If the number and nature of the extra fields coincide in
formulating the supersymmetric version for the theory and in having a NR limit it would
suggest that there is a deeper connection. Then, this would be a criteria to select classical
relativistic theories before addressing their quantization.
Acknowledgments
We are grateful to Eric Bergshoeff, Niels Obers, and Jan Rosseel for insightful discussions
and in particular, for pointing out to us the issue discussed in section 3.3. We are also
grateful to Axel Kleinschmidt and Oscar Fuentealba for enlightening discussions. JG has been
supported in part by FPA201346570C21P and Consolider CPAN and by the Spanish
goverment (MINECO/FEDER) under project MDM20140369 of ICCUB (Unidad de
Excelencia Mar´ıa de Maeztu). JG has also been visiting professor to the Universidad Austral
de Chile under grant PAI801620047 from CONICYT. EF is partially funded by
Fondecyt grant 11150467. DH and LA are partially founded by Conicyt grants 21160649 and
21160827, respectively. JZ has been partially funded by Fondecyt grants numbers 1140155
and 1180368. The Centro de Estudios Cient´ıficos (CECs) is funded by the Chilean
Government through the Centers of Excellence Base Financing Program of Conicyt.
ChernSimons invariance
Let us recall two important properties of CS actions [46]. First, we show that the
quasiinvariance under the symmetry group is guaranteed. The infinitesimal variation of the
Lagrangian under the gauge symmetry, δλA = Dλ = dλ + [A, λ], is a total derivative
1
3
δλhAdA +
A[A, A]i = dh2F (A)λ − AδλAi,
where we used the Bianchi identity DF = 0. Note that the CS Lagrangian requires an
invariant bilinear form.
Second, we show that if the invariant bilinear form, gAB ≡ hXA, XBi, is
nondegenerated then the CS curvature vanishes. And therefore, all curvature components
which are themselves curvatures for each field, vanish too. The CS equation of motion is
0 = hF (A)δAi = F AδABgAB.
Thus, because δAB is arbitrary if gAB is invertible we have F A = 0. Note however that
the contrary is not neccesarily true: the vanishing of the curvatures does not guarantee the
existence of a nondegenerated invariant bilinear form.
Note that to find an invariant bilinear form for a given algebra is a straightforward
linear problem. The algebra is encoded in the structure constants [XA, XB] = fABC XC
and the invariance of a generic bilinear form gAB = hXA, XBi is the statement hXA, XBi =
hXA + δXA, XB + δXBi or equivalently
(A.1)
(A.2)
(A.3)
(A.4)
(B.1)
(B.2)
(B.3)
(B.4)
hXA, XBi − hXA + [XA, XC ], XB + [XB, XC ]i = 0
fBCDgAD + fACDgDB = 0
Then, given the structure constants the most general invariant bilinear form is
determined. Weather that solution for gAB is degenerate or not is a different question. Note
further that if gAB is nondegenerate the most general invariant bilinear allows us to have
all the quadratic Casimir objects. In fact, the equation [gABXAXB, XC ] = 0 is equivalent
to (A.4).
B
Explicit ω and ωa connections
To compute the nonrelativistic components of the connections, ω and ωa, we follow the
standard strategy (see [47]) but using our notation.
Computation of ωμ. Consider
21 Rµνa (ec) = ∂[µ eνa] + ǫabτ
[µ ωbν] + ǫabω[µ ebν] = 0,
then we multiply by eaρ, i.e., 12 Rµνa (ec)eaρ = 0 and write its cyclic permutations
∂[µ eνa]eaρ + ǫabτ
[µ ωbν]eaρ + ωρ[νµ ] = 0,
∂[ρeµa]eaν + ǫabτ[ρωbµ ]eaν + ων[µρ ] = 0,
∂[ν eρa]eaµ + ǫabτ[ν ωbρ]eaµ + ωµ [ρν] = 0,
where ωρνµ ≡ ǫabω[µ ebν]eaρ is antisymmetric in the first two and the last two indexes. We
sum the first two equations and subtract the third one to get
∂[µ eνa]eaρ+ǫabτ
[µ ωbν]eaρ+∂[ρeµa]eaν +ǫabτ[ρωbµ ]eaν −∂[ν eρa]eaµ −ǫabτ[ν ωbρ]eaµ +ωρνµ = 0. (B.5)
To invert the relation we have ωρνµ ebν eaρ = ǫabωµ , then
−ǫabω
µ =
(∂µ eνaebν − ∂ν eµa ebν ) +
2
1 ǫabτµ ωbν ebν
1
2
1
2
+ (∂ρeµb eaρ − ∂µ ebρeaρ) −
2
1 ǫbcτµ ωcρeaρ − ∂[ν ecρ]ecµ ebν eaρ,
we used τ µ eµa = 0. Now, multiplying by −ǫab/2 (remember ǫabǫab = −2), it yields
ωµ = ǫab∂[ν eµa]ebν +
1
2 ǫab∂[ν ecρ]ecµ eaν ebρ −
1
2
τµ ωbν ebν .
First, we replace the expression (B.7) into the curvature Rµνa (eb)
Computation of ωa.
μ
and compute the expression
this gives the symmetric part
antisymmetric part
eµc τ ν Rµνa (eb) + eµa τ ν Rµνc (eb) = 0,
c)
ǫ(abeµ c)ωbµ = 2eµ (aτ ν ∂[µ eν].
ǫ[abeµ c]ωbµ = −e[aµ ec]ν ∂[µ mν],
ǫabτ µ ωbµ = −2eaµ τ ν ∂[µ mν].
Now we use the R(m) = 0 equation. By computing 2Rµν (m)eaµ ebν = 0 we obtain the
From the curvature equation we also need the contraction 2Rµν (m)eaµ τ ν = 0, it yields
Summing (B.9) and (B.10), and then multiplying by eσc and − ǫa2b , it yields
2
ωσa =
ǫab τ µ ∂[σeµb ] + ebµ ecστ ν ∂[µ ecν] + ebµ ∂[σmµ ] + τσebν τ µ ∂[µ mν] ,
we used the relation eµc eσc = δσµ − τ µ τσ and the (B.11).
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