Galilean conformal and superconformal symmetries
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ELEMENTARY PARTICLES AND FIELDS Theory
Galilean Conformal and Superconformal Symmetries* J. Lukierski** Institute for Theoretical Physics, University of Wrocław, Poland Received May 5, 2011
Abstract—Firstly we discuss briefly three different algebras named as nonrelativistic (NR) conformal: ¨ Schrodinger, Galilean conformal, and infinite algebra of local NR conformal isometries. Further we shall consider in some detail Galilean conformal algebra (GCA) obtained in the limit c → ∞ from relativistic conformal algebra O(d + 1, 2) (d—number of space dimensions). Two different contraction limits providing GCA and some recently considered realizations will be briefly discussed. Finally by considering NR contraction of D = 4 superconformal algebra the Galilei conformal superalgebra (GCSA) is obtained, in the formulation using complex Weyl supercharges. DOI: 10.1134/S1063778812100134
1. INTRODUCTION The notion of nonrelativistic (NR) conformal symmetries was used at least in three-fold sense:
¨ a) Schrodinger Symmetries The d-dimensional Galilei group G(d) can be represented as the following semidirect product G(d) = (O(d) ⊕ R) T 2d ,
(1)
where R describes the time translations (generator H), O(d) stands for the d-dimensional space rotations (generators Jij = −Jji ), and the Abelian subgroups T 2d represent the space translations (generators Pi ) and Galilean boosts (generator Bi ). Almost forty years ago [1–4] to the corresponding Galilean algebra g(d) there were added two generators D and K, generating 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.
(2)
¨ In such a way we obtain the Schrodinger group Schr(d), which is obtained from (1) by the enlargement of R O(1, 1) to O(2, 1) group Schr(d) = (O(d) ⊕ O(2, 1)) T 2d . ∗ **
The text was submitted by the author in English. E-mail: [email protected]
(3)
The symmetries (3) can be introduced as the set of ¨ transformations preserving the Schrodinger equation for free NR massive particle (we put = 1) (d)
Δ ∂ . (4) Sm (d) ≡ i − ∂t 2m The parameter m can be interpreted in geometric ¨ way as the central extension of Schrodinger algebra, ¨ which defines the “quantum” Schrodinger symme¨ tries. The “quantum” Schrodinger group can be obtained also as the enlargement by dilatations and expansions of the centrally extended Galilean group, called “quantum” Galilean or Bargmann group. Sm (d)ψ = 0,
b) Galilean Conformal Symmetries [5–13] For d = 1 the Galilean conformal algebra (GCA) is finite-dimensional and obtained by NR contraction c → ∞ of D-dimensional (D = d + 1) relativistic conformal algebra O(d + 1, 2). To two generators ¨ D, K which enlarge Galilei to Schrodinger algebra one adds d Abelian generators Fi , describing Galilean accelerations defined by the NR limit of conformal translations. The structure of d-dimensional Galilean conformal algebra c(d), denoted as GCA, is described by the algebra
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