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I. Masterov. Remark on N=2 supersymmetric extension of 1-conformal Galilei algebra UDC 530.1; 539.1
Remark on N=2 supersymmetric extension of 1-conformal Galilei algebra
I. Masterov
Laboratory of Mathematical Physics, Tomsk Polytechnic University, 634050 Tomsk, Lenin Ave. 30, Russian Federation
E-mail: masterov@mph.phtd.tpu.ru
i-conformal Galilei algebra is considered. N = 2 supersymmetric extension of this algebra is constructed. A relation between its representations in flat spacetime and in Newton-Hooke spacetime is discussed.
Keywords: conformai Galilei algebra, supersymmetrv.
1 Introduction
The non-relativistic version of the Anti-de Sitter /Conformai Field Theory (AdS/CFT) correspondence fl] stimulates extensive investigation of non-relativistic conformai algebras [2]- [4]. Algebras relevant for physical applications in flat non-relativistic spacetime as well as in Newton-Hooke spacetime (i.e. spacetime with cosmological repulsion/attraction [5]) belong to the family of the l-conformal Galilei algebras [7,16] l being a positive integer or half-integer.
Supersymmetric extension of the l-conformal Galilei algebras and their dynamical realizations have been studied in detail for l = 2 (the Schrödinger algebra) and l = 1 (the conformai Galilei algebra). In [8] an N =1 supersymmetric extension of the Schrödinger algebra was identified with the symmetry algebra of the non-relativistic spin-2 particle. In [9] it was shown that the non-relativistic limit of the Chern-Simons matter system in (2+1) dimensions is invariant under N =2 Schrödinger supersymmetry. Many-body quantum mechanics invariant under N =2 Schrödinger supersymmetry was studied in [10]. Relations between the Schrödinger superalgebra and relativistic super-conformal algebras were discussed in [11]- [12]. More recently, supersymmetric extensions of the conformai Galilei algebra were extensively investigated by applying various non-relativistic contractions [13]- [14].
N = 2
supersymmetric extension of the l-conformal Galilei algebra for the case of arbitrary l [15]. We do this in section 2. Representations of this superalgebra in flat and Newton-Hooke spacetimes are considered in section 3. We also give a coordinate transformation, which relates the representations.
2 N=2 supersymmetric extension of the l-conformal Galilei algebra
First let us recall the structure of the l-conformal Galilei algebra. It involves the generator of time trans-
lations H, the generator of dilatations D, the generator of special conformal transformations K, the generators of space rotations Mij, and a chain of vector generators C(n), n = 0,1,.., 21. In particular, for n = 0 one obtains the generator of space translations, n =1 gives
n
accelerations. The non-vanishing structure relations read [7]
[H, D] = H, [H,c(n)] = nC(n-1),
[H, K] = 2D, [D, c(n)] = (n - 1)c(n),
[D, K] = K, [K, c(n)] = (n - 21)c(n+1), (1)
[Mij ,C(n)] = -¿ifccjn) + j C(n),
[Mij , Mkl] = -Sik Mjl — SjlMik + SilMjk + Sjk Mil.
H D K one dimension so(2,1).
In order to construct an N =2 supersymmetric extension of this algebra, we introduce a pair of supersymmetry generators Q+ mid Q-, the super-conformal generators S+ mid S-, fermionic partners of the vector generators L(n)+ mid L(n) with n =
i i (n)
0,1,.., 21 — 1, extra bosonic vector generators Pi ) with n = 0,1,.., 21 — 2, and the bosonic generator J which corresponds to u(1)-R-symmetry. It is assumed that the odd generators are antihermitian conjugates of each other
(Q+)f = —Q-, (S+ )f = —S-,
(¿in)+)' = — L<nH (2)
The bosonic operators J Mid P(n) are taken to be antihermitian as well.
In addition to (5) we impose the following structure
TSPU Bulletin. 2012. 13 (128)
relations [15]
{Q+, Q-} = 2iH, {Q±, ST} = 2iD ± J,
{S+, S-} = 2iK, [Q±, C(n)] = nL,(n-1)±,
[H,S ±] = Q±, [Q±,p(n)] = iL,(n)±,
[K, Q±] = -S±, [S±, C(n)] = (n - 21)L,(n)±
[D,Q±] = -1 Q±, [S±,p(n)] = iL(n+1)±,
[D, S±] = 2S±, [H, L(n)±] = nL(n-1)±,
[J, Q±] = ±iQ±, [H,Pj(n)] = nPj(n-1),
[J, S±] = ±iS±, [J, L(n)±] = ±iL(n)±,
{Q±,l(”)t} = *c(n) t nPf-1),
{S±, l(”)t} = ¿c(”+1) t (n - 21 + 1)Pj(n), [D,L(n)±] = (n - 1 + 1/2)L((n)±,
[D,Pj(n)] = (n - 1 + 1)Pj(n),
[K,L,(n)±] = (n - 21 + 1)L,(n+1)±,
[K,Pj(n)] = (n - 21 + 2)Pj(n+1),
M ,Lkn)±] = -*fc 4n)± + j L(n)±,
M, Pkn)] = -¿¿j + ¿¿fcP(n). (3)
Note that 1 = 1/2 reproduces the well known N =2 Schrôdinger superalgebra [10].
3 Realizations in superspace
First let us construct a realization of the superalgebra (3) in flat superspace. Introducing two Grassmann variables 0+ and 0-, which are complex conjugates of each other (0+)^ = 0-, one finds [15] (see also a related work [16])
D = tdt + 1xjdj + 20 dg- + ^0+dg+,
K = t2dt + 21txjdj + t0-dg- + t0+dg+,
S± = t0±dt + itdgT + 210± Xjdj + 0±0^dgT Q± = idgT + 0± dt,
H = dt, J = i0+dg+ - i0-dg-, c(n) = tnd n = 0,.., 21,
P(n) = 0-0+t”dj n = 0,.., 21 - 2,
L(n)± = 0±tn3i, n = 0,.., 21 - 1,
Mj = Xjdj - Xj dj.
Discarding the fermions one reproduces a realiza-1
In order to construct a realization of the superalgebra (3) in Newton-Hooke spacetime extended by fermionic variables, we introduce an analogue of Niederer’s transformation. Guided by the analysis in
[10], we first consider a coordinate transformation [15]
t' = R tan(t/R), t' = R tanh(t/R),
xi = (cos (t/R))-21 xi, xi = (cosh (t/R))-2lxi,
(0±)' = (cos (t/R))-10±,
(0±)' = (cosh (t/R))-10±, (5)
where the prime denotes coordinates parameterizing flat superspace. Here the left/right column corresponds to Newton-Hooke spacetime with negative/positive cosmological constant. Then we consider
1
algebra
1 1 i
H ^ H ± RK t R J, Q± ^ Q± ± RS±, (6)
where the upper/lower sign in the generator of time translations corresponds to negative/positive cosmological constant. The operators of realization (3) in Newton-Hooke superspacetime are constructed by using (5) in paper [15].
4 Conclusion
To summarize, in this paper we have constructed an N = 2 supersymmetric extension of the 1-conformal Galilei algebra and its realizations in flat spacetime and in Newton-Hooke spacetime. A coordinate transformation which links the realizations was given. An infinite-dimensional extension was proposed.
Acknowledgement
This research has been supported by the grant for LRSS, project No 224.2012.2, by the FTP “Research and Pedagogical Cadre for Innovative Russia”, contract (4) No. 14.B37.21.1298, RFBR grant 12-02-31591.
References
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Received 01.10.2012
if. В. Мастеров
ЗАМЕТКА ОБ N=2 СУПЕРСИММЕТРИЧНОМ РАСШИРЕНИИ 1-КОНФОРМНОЙ
АЛЕЕБРЫ ЕАЛИЛЕЯ
Построено N=2 суперснмметрнчное расширение 1-конформной алгебры Галилея и ее представления в плоском суперпространстве и суперпространстве Ньютона-Гука. Установлены координатные преобразования, связывающие построенные представления.
Ключевые слова: алгебра Галилея, конформная симметрия, N=2 суперсимметрия.
Мастеров И. В., аспирант.
Томский политехнический университет.
Пр. Ленина, 30, Томск, Россия, 634050.
E-mail: masterov@tpu.ru
