Table Of ContentGalilean Conformal and Superconformal Symmetries
Jerzy Lukierski1
1
Institute for Theoretical Physics, University of Wrocl aw,
pl. Maxa Borna 9, 50-204 Wrocl aw, e-mail:[email protected]
Firstly we discuss briefly three different algebras named as nonrelativistic (NR)
1 conformal: Schro¨dinger, Galilean conformal and infinite algebra of local NR confor-
1
0 mal isometries. Further we shall consider in some detail Galilean conformal algebra
2
n (GCA) obtainedinthelimitc fromrelativistic conformalalgebra O(d+1,2) (d
→ ∞
a
J - number of space dimensions). Two different contraction limits providing GCA and
1
2 somerecently consideredrealizations willbebrieflydiscussed. Finally byconsidering
] NR contraction of D=4 superconformal algebra the Galilei conformal superalgebra
h
t
- (GCSA) is obtained, in the formulation using complex Weyl supercharges.
p
e
h
[ PACS numbers: 11.30.Ly, 11.30.Pb,11.25.Hf
1
v
2
I. INTRODUCTION
0
2
4
. The notion of nonrelativistic conformal symmetries was used at least in three-fold sense:
1
0
a) Schr¨odinger symmetries.
1
1
The d dimensional Galilei group G(d) can be represented as the following semidirect
:
v
i product
X
r G(d) = (O(d) R) B T2d, (1.1)
a
⊕
where R describes the time translations (generator H), O(d) - the d-dimensional space
rotations (generators J = J ) and the Abelian subgroups T2d represents the space trans-
ij ji
−
lations (generators P ) and Galilean boosts (generator B ). Almost forty years ago [1]–[4] to
i i
the corresponding Galilean algebra g(d) there were added two generators D and K, gener-
ating the dilatations (scale transformations) and the expansions (conformal transformations
of time). New generators are forming together with H the d = 0 conformed algebra O(2,1):
[D,H] = H , [K,H] = 2D, [D,K] = K. (1.2)
− −
In such a way we obtain the Schro¨dinger group Schr(d), which is obtained from (1.1) by
2
the enlargement of R O(1,1) to O(2,1) group
≃
Schr(d) = (SO(d) SO(2,1)) B T2d. (1.3)
⊕
The symmetries (1.3) can be introduced as the set of transformations preserving the
Schro¨dinger equation for free nonrelativistic massive particle (we put ~ = 1)
∂ ∆(d)
S (d)ψ = 0, S (d) i . (1.4)
m m ≡ ∂t − 2m
b b
Theparametermcanbeinterpretedingeometricwayasthecentral extension ofSchro¨dinger
algebra, which defines the “quantum” Schro¨dinger symmetries. The “quantum” Schro¨dinger
group can be obtained also as the enlargement by dilatations and expansions of the centrally
extended Galilean group, called “quantum” Galilean or Bargmann group.
b) Galilean conformal symmetries [5]–[13].
For d = 1 the Galilean conformal algebra is finite-dimensional and obtained by nonrel-
6
ativistic contraction c of D-dimensional (D = d + 1) relativistic conformal algebra
→ ∞
O(d+1,2). To two generators D, K which enlarge Galilei to Schro¨dinger algebra one adds d
Abelian generators F , describing Galilean accelerations defined by the nonrelativistic limit
i
of conformal translations. The structure of d-dimensional Galilean conformal algebra c(d),
denotedasGCA, isdescribed by thealgebrawiththefollowingsemi-direct productstructure
c(d) = (O(d) O(2,1)) A T3d, (1.5)
⊕
where T3d = (P ,B ,F ). It should be stressed that
i i i
i) From the contraction procedure follows that similarly as the pair of Galilean and
Poincar´e algebras, the Galileanand relativistic conformal algebras have the same dimension.
ii) GCA does not permit to introduce the mass-like parameter as its central extension,
i.e. it describes only the symmetries of massless NR dynamical systems1
iii) ByputtingF = 0in(1.5)wedonotarriveatthealgebra(1.3), becauseinSchro¨dinger
i
and Galilean conformal algebras the boosts B transforms differently under dilatations:
i
Schr(d) : [D,B ] = B c(d) : [D,B ] = 0. (1.6)
i i i
−
1 In principle however one can introduce the GCA - invariant model with continuous spectrum of nonrela-
tivistic masses.
3
The subalgebra of c(d) obtained by putting F = 0 as well as Schr(d) belong to the one-
i
parameter family of generalized Schro¨dinger algebras Schr (d) with the following commuta-
z
tor of generators D and B 2
i
[D,B ] = (1 z)B , (1.7)
i i
−
where z is called a dynamical exponent and Schr(d) = Schr (d)(z = 2), c(d) Schr (d)(z =
2 1
⊃
1). The dilatations (scale transformations) described by Schr (d) transform the nonrela-
z
tivistic space-time coordinates (t,~x = (x ...x )) as follows
1 d
t′ = λzt ~x′ = λ~x. (1.8)
It is easy to see that the invariance of free Schro¨dinger equation (1.4) is achieved if z = 2.
c) Infinite-dimensional conformal extension of GCA [7–9, 12–15].
The degeneracy of nonrelativistic metric in space-time allows to introduce infinite-
dimensional Galileanconformalisometriesdescribed byvector fieldssatisfying NRconformal
Killing equations [13, 14]. Finite-dimensional GCA (1.5) is spanned by the vector fields
1 ∂ 1 ∂
X = kt2 +λt+ε + ω x +λx +ktx α t2 +β t+γ . (1.9)
(c(d)) (cid:18)2 (cid:19) ∂t (cid:18) ij j i i − 2 i i i(cid:19) ∂x
i
Amonginfinite-dimensional Galileanconformal isometries itisdistinguished thefollowing
set of NR Killing vector fields, denoted in [13] by cgal , generating reparametrization of
z
time t, local time-dependent NR space-time dilatations and translations (t ξ(t),x
i
→ →
(1+ 1ξ′(t))x +η (t)) as well as time-dependent O(d) rotations
z i i
∂ 1 ∂
X = ξ(t) + ω (t)x + ξ′(t)x +η (t) . (1.10)
∂t (cid:18) ij j z i i (cid:19) ∂x
i
Formula (1.10) contains as a subset the Galilean-conformal vector fields (1.9) as well as the
z-dependent Schro¨dinger algebra Schr (d).
z
It should be added that the infinite-dimensional NR conformal isometries do not have
their infinite-dimensional relativistic counterpart except if d = 1.
Further we shall restrict our considerations to the finite-dimensional GCA (see (1.5) and
(1.9)). InSect. 2 we shall discuss the derivationof GCA andwill consider two versions of NR
2 ThegeneralizedSchr¨odingeralgebrasSchrz(d)wereintroducedbyHenkel[8]. Ifwestillintroducearbitrary
parameter defining the scaling properties of H ([D,H] = (y 1)H) one obtains the Milne-conformal
−
algebra, with independent scaling of time and space (see also [13]).
4
contraction procedure leading to GCA. Subsequently we shall provide some recent results
about GLA realizations. In Sect. 3 we shall describe the supersymmetrization of D = 4
GCA [16]–[18]. In order to distinguish the presentation here from the one given in [16] with
real Majorana supercharges, we will formulate the Galilean conformal superalgebra in terms
of complex Weyl supercharges. In Sect. 4 we provide a brief outlook.
II. GALILEAN CONFORMAL SYMMETRIES
Following the derivation of Galilei algebra by NR contraction c of Poincar´e algebra,
→ ∞
one can as well perform the contraction c of (d+2)(d+3) generators of relativistic con-
→ ∞ 2
formal algebra O(d+1,2). Denoting by P = (P ,P ), M = (M = 1ǫ M ), M = N )
µ 0 i µν ij 2 ijk k i0 i
(µ,ν = 0,1,...d; i,j = 1,2...d) the Poincar´e generators, we get D = (d+1)-dimensional
relativistic conformal algebraby addingthedilatationgeneratorD andconformal generators
K = (K ,K ). The “natural” NR contraction c leads to Galilean conformal algebra
µ 0 i
→ ∞
c(d) = (H,P ,M ,B ,D,K,F ) if we rescale the relativistic generators as follows [10]
i k k i
P = H , N = cB ,
0 c i i
K = cK, K = c2F , (2.1)
0 i i
(P ,M ,D remain unchanged). One obtains besides the O(2,1) algebra (see (1.2)) and O(d)
i i
generators M the following O(2,1) O(d) covariance relations (i,j,k = 1...d)
ij
⊗
[H,P ] = 0, [H,B ] = P , [H,F ] = 2B ,
i i i i i
[K,P ] = 2B , [K,B ] = F , [K,F ] = 0, (2.2)
i i i i i
−
[D,P ] = P , [D,B ] = 0, [D,F ] = F ,
i i i i i
−
and (l = 1,2,3)
[M ,A ] = δ A δ A , (2.3)
ij k;l jk i;l ik j;l
−
where A = (P ,B ,F ) describe the maximal Abelian subalgebra T3d (see (1.5)).
i;l i i i
We shall consider the various contraction procedures leading to GCA and present some
results on GCA realizations.
a) Nonuniqueness of contraction procedure.
Besides the “natural” or “physical” contraction limit (2.1) one can use as well other
contraction O(d+ 1,2) c(d) if we observe that the semidirect structure of c(d) given
−−λ→−−∞→
5
by(1.5)canbeobtainedbythecanonical contractionfromthefollowing coset decomposition
of the relativistic conformal group
O(d+1,2) = , (2.4)
H·K
where = O(d) O(2,1) and = O(d+1,2) . The corresponding generators of relativistic
H ⊗ K O(d)⊗O(2,1)
conformal algebra c (d) = h k (h = h , k = k ), describe (pseudo)Riemannian
rel i r
⊕ { } { }
symmetric pair. If we performbthebresbcaling b
h′ = h k′ = λk (2.5)
i i r r
we get in the limit λ exactly the algebraic structure (1.5), with the O(2,1) generators
→ ∞
described by the relativistic conformal generators P , K and D [11, 17]. The rescaling (2.5)
0 0
in comparison with (2.1) is simpler, but the contraction parameter λ can not be related in
universal way (for all indices r in (2.5)) with light velocity c. In order to find the relation of
two contraction procedures let us observe that the choice of generators describing O(d+1,2)
conformal algebra is not unique. In particular one can introduce the following new basis
P = λP K = λ−1K , (2.6)
µ µ µ µ
e e e e
with generators M and D unchanged (M = M , D = D). It is easy to check that
µν µν µν
composing the rescalings (2.1) and (2.6) onfe obtains e
e
P = λ H, K = c K,
0 c 0 λe (2.7)
Pe = cP , Ke = c2 F ,
i i i e i
λ
e e
and
N = cB , M = M , D = D. (2.8)
i i i i
e f e e
If we put λ = c the relations (2.7–2.8) become identical to (2.5) provided that λ = c. It
should beeadded that in such a contraction scheme the dimensional analysis is consistent
only if we identify the dimensionality of space and time, what implies that the rescaling
parameter c = λ = λ should be considered as dimensionless.
The rescaling (2.e6) is the only one which preserves the relativistic conformal algebra. For
GCA the class of rescalings preserving the algebraic structure is larger. One can show the
invariance of c(d) under the following rescaling depending on the parameter k
H′ = ρ−k−1H, K′ = ρk+1K,
6
P′ = ρ−kP , B′ = ρB , F′ = ρk+2F ,
i i i i i i
M′ = M , D′ = D. (2.9)
ij ij
If k = 0 in such a way one gets the renormalization (c′ = ρc) of the rescaling (2.1), and if
k = 1 we obtain the modification (λ′ = ρλ) of the contraction parameter in (2.5).
−
b) The representations of GCA.
The contractions (2.1) and (2.5) can be also applied to the space-time realization of
relativistic conformal algebra3
∂ ∂
P = ∂ , K = x2 2x x ,
µ −∂xµ µ ∂xµ − µ ν∂xν
∂
M = x ∂ x ∂ , D = x . (2.10)
µν µ∂xν − ν∂xµ µ∂xµ
After substituting x = ct and performing the rescaling (2.1) one gets the vector fields
0
realization of GCA in nonrelativistic space-time (x ,t)
i
H = ∂ , K = t2 ∂ 2tx ∂ ,
∂t ∂t − i ∂xi
P = ∂ , B = t ∂ , F = t2 ∂ ,
i −∂xi i − ∂xi i − ∂xi
M = x ∂ x ∂ , D = x ∂ t∂ . (2.11)
ij i∂xj − j∂xi i∂xi − ∂t
In order to obtain the rescaling (2.5) one should rescale in (2.10) the space coordinates
x x′ = λx and remain time coordinate unchanged. Further, if we introduce in (2.11)
i → i i
the rescaling of nonrelativistic space-time
x′ = ρkx , t′ = ρk+1t, (2.12)
i i
we obtain the rescaling (2.9) of GCA.
One can introduce as well the finite-dimensional spinors of GCA defined as the spinorial
representations of the compact part O(d) O(2,1) of GCA (for that purpose we should
⊕
consider the double coverings SO(d) SU(1,1)). Further we shall consider d = 3 and use
⊗
SO(3) = SU(2).
3 We consider here the spinless realization of relativistic conformal algebra.
7
The fundamental spinor representation of d = 3 GCA which can be called the nonrel-
ativistic d = 3 twistor is identified with the double spinor t C4 (α = 1,2;A = 1,2)
α;A
∈
transforming as follows under SU(2) SU(1,1) [19]
⊗
′ = A BT A SU(2) B SU(1,1), (2.13)
t t ∈ ∈
where = t is a 2 2 complex matrix. The general nonrelativistic conformal spinors
α;A
t { } ×
of rank (n,m) are described by the set of 2n+m complex variables transforming as follows
t′α1...αn;A1...Am = Aα1α′1 ...Aαnα′n ·tα′1...α′n;A′1...A′m ·(BT)A′1A1...(BT)A′mAm. (2.14)
The fundamental (n = m = 1) NR conformal spinors (twistors) leave invariant the following
SU(2) SU(1,1) norm
×
(1), (2) = t¯(1) (σ ) t(2) , (2.15)
t t α˙;A˙ 3 A˙A α;A
(cid:10) (cid:11)
where t¯(1) = (t )⋆, and Pauli matrix σ describes the U(1,1) metric.
α˙;A˙ α;A 3
An interesting question arises whether on NR level one can repeat the constructions
known from the Penrose formalism of relativistic twistors (see e.g. [20, 21]) which are de-
fined as fundamental representations of SU(2,2) = SO(4,2). In such geometric framework
twistor components are primary, and space-time as well as relativistic phase space coor-
dinates are composite. First step in searching of such NR counterpart is to look for NR
twistor realizations of GCA. Because GCA has nonsemisimple structure, it contains Abelian
subalgebra (see T3d in (1.5)), and that implies the following results [19].
i) The NR twistorial realizations of GCA exist only in the space of N twistors with
N 2.
≥
ii) Contrary to the twistor realizations of relativistic conformal algebra the NR twistor
realizations of GCA for any N do not provide the positive-definite Hamiltonian H.
Other GCA realizations which lead to GCA-covariant classical mechanics models is pro-
vided by the application of the geometric techniques of nonlinear realizations [22] to various
cosets of Galilean conformal group. By using the inverse Higgs method [23, 24] applied to
the Cartan-Maurer one-forms spanned by GCA one gets the extension of standard confor-
mal mechanics of de Alfaro, Fubini and Furlan [25] in the presence of nonvanishing space
dimensions as well as the geometric derivation in D = 2 + 1 of the GCA model presented
8
in [26] and its generalization4.
Finally let us recall that dynamical realizations of GCA were obtained [13] in the model
describing Souriau “Galilean photons” [27] and as well in the magnetic-like nonrelativistic
limit of Maxwell electrodynamics [28].
III. EXTENDED D=4 RELATIVISTIC CONFORMAL SUPERALGEBRA
SU(2,2,N) AND ITS NONRELATIVISTIC CONTRACTION
The D = 4 N-extended relativistic conformal algebra [29, 30] is supersymmetrized by
supplementing the N complex Weyl Poincar´e supercharges Q Q¯ i = (Q )⋆ satisfying the
αi α˙ αi
relations5
Q ,Q¯ j = 2(σ ) Pµδ j, (3.1)
{ αi β } µ αβ˙ i
Q ,Q = Q¯ i,Q¯ j = 0, (3.2)
{ αi βj} { α˙ β˙ }
by N complex conformal Weyl supercharges Si,S¯i = (S )⋆, representing “supersymmetric
α α˙ αi
roots” of conformal momenta K
µ
S ,S¯ j = 2(σ ) Kµδ j, (3.3)
{ αi β˙ } − µ αβ˙ i
S ,S = S¯ i,S¯ j = 0. (3.4)
{ αi βj} { α˙ β˙ }
Besides we have the relations
Q ,S = δ [(σ ) Mµν 2ǫ (D+iA)] 2iǫ T , (3.5a)
αi βj ij µν αβ αβ αβ ij
{ } − −
Q¯ i,S¯ j = δij[(σ¯ ) Mµν 2ǫ (D iA)]+2iǫ T¯ij, (3.5b)
{ α˙ β } µν α˙β˙ − α˙β˙ − α˙β˙
Q ,S¯ j = Q¯ i,S = 0, (3.5c)
{ αi β˙ } { α˙ βj}
S ,S = S¯ i,S¯ j = 0, (3.5d)
{ αi βj} { α˙ β˙ }
4 The standard conformal mechanics [25] employs as symmetry c(0)= O(2,1), with only conformal trans-
formations of the time coordinate.
5 In the relation(3.2) we didput the Poincar´esuperalgebracentral chargesequalto zero. It canbe further
shown that this requirement follows from the Jacobi identities for the extended conformal superalgebra
[29].
9
where (T )⋆ = T¯ij,
ij
i i
(σ ) = (σ ) (σ¯ ) (σ¯ ) = (σ¯ ) (σ ) , (3.6)
µν αβ [µ αγ˙ ν] γ˙β µν α˙γ˙ [µ α˙β ν] βγ˙
2 2
and the complex generators T of SU(N) algebra satisfy the Hermicity condition
ij
T = T¯ji T = T(S) +iT(A), (3.7)
ij ⇒ ij ij ij
where T(S) (T(A)) are real and symmetric (antisymmetric)
ij ij
T(S) = T(S) T(A) = T(A) T(S) = 0. (3.8)
ij ji ij − ji ii
The trace of operator-valued matrix generators T has been separated and denoted by A
ij
(axial charge).
Under theD=4conformalgenerators(M ,P ,K ,D)thesupercharges Qi ,Si transform
µν µ µ α α
as follows
[M ,Q ] = 1(σ ) βQ , [P ,Qi ] = 0,
µν αi −2 µν α βi µ α
[M ,S¯ i] = 1(σ¯ ) β˙ S¯ i, [P ,S¯i] = (σ¯ ) β˙ S¯i ,
µν α˙ −2 µν α˙ β˙ µ α˙ − µ α˙ β˙ (3.9)
[K ,Q ] = (σ ) β˙ S¯ i, [D,Q ] = iQ ,
µ αi − µ α β˙ αi −2 αi
[K ,S¯i] = 0, [D,S¯i] = iS¯i .
µ α α˙ −2 α˙
Further we obtain in consistency with the relation T = 0 that
ii
1
[T ,Q ] = δ Q δ Q ,
ij αk ik αj ij αk
− 4
1
[T ,S ] = δ S δ S , (3.10)
ij αk ik αj jk αi
− 4
where T satisfy the SU(N) algebra relations:
ij
[T ,T ] = δ T δ T . (3.11)
ij kl il jk ik jl
−
The U(1) axial charge A satisfies the relations
i 4
[A,Q ] = 1 Q ,
αk 4 (cid:18) − N(cid:19) αk
i 4
[A,S ] = 1 S , (3.12)
αk αk
4 (cid:18) − N(cid:19)
and commutes with all other bosonic generators forming O(2,4) SU(N) algebra. We see
⊕
from (3.13) that for N = 4 the axial charge A becomes a central charge.
10
We shall further assume that N = 2k and rewrite the D=4 relativistic superconformal
algebra in different fermionic basis
1
Q± = Q ǫ Ω Q¯j ,
αi 2 αi ± αβ ij β˙
(cid:16) (cid:17)
1
Q¯±i = Q¯i ǫ Ωij Q , (3.13)
α˙ 2 α˙ ± α˙β˙ βj
(cid:0) (cid:1)
where the real matrix Ω (Ωij (Ω )⋆ = Ω ) is a 2k 2k symplectic metric (Ω2 = 1,
ij ij
≡ × −
ΩT = Ω). We choose for simplicity that
−
0
k
Ω = −1 (3.14)
0
k
1
The Weyl supercharges (3.13) satisfy the subsidiary symplectic-Majorana conditions
Q¯±i = ǫ Ωij Q± . (3.15)
α˙ ± αβ βj
Similarly one can introduce the supercharges S±i, S¯±i, with the same subsidiary condition
α α
S¯±i = ǫ Ωij S± . (3.16)
α˙ ± αβ βj
The relations (3.1–3.4) can be represented now as follows (r = 1,2,3)
Q±,Q¯±j = δj δ P ,
{ αi β˙ } i αβ˙ 0
Q±,Q¯∓j = δj (σ P ) ,
{ αi β˙ } i r r αβ˙
Q±,Q± = Q±,Q∓ = Q¯±i,Q¯±j = Q¯±i,Q¯∓j = 0 , (3.17)
{ αi βj} { αi β˙j} { α˙ β˙ } { α˙ β˙ }
and
S±i,S¯±j = δj δ K ,
{ α β˙ } − i αβ˙ 0
S±i,S¯∓j = δj (σ K ) ,
{ α β˙ } − i r r αβ˙
S±,S± = S±,S∓ = S¯±i,S¯±j = S¯±i,S∓j = 0. (3.18)
{ αi βj} { αi βj} { α˙ β˙ } { α˙ β˙ }
Further from (3.5a–3.5d), (3.6) one obtains
1 1
Q±,S± = δ (σ ) M ǫ D + ǫ T(−)S +iT(+)A , (3.19a)
{ αi βj} 2 ij rs αβ rs − αβ 2 αβ ij ij
(cid:16) (cid:17)
(cid:2) (cid:3)
1 1
Q+,S− = δ (σ ) M ǫ A ǫ T(+)A iT(+)S , (3.19b)
{ αi βj} 2 ij r0 αβ r0 − αβ − 2 αβ ij − ij
(cid:16) (cid:17)
(cid:2) (cid:3)
Q±,S¯±j = Q±,S¯∓j = 0, (3.19c)
{ αi β˙ } { αi β˙ }
