Teoria De Cuerdas

UB-ECM-PF-04/07

Non-relativistic strings and branes as non-linear realizations of Galilei groups

arXiv:hep-th/0404175v3 29 Jun 2004

Jan Brugues† , Thomas Curtright§ , Joaquim Gomis† , and Luca Mezincescu§
Departament ECM, Facultat de F´ ısica Institut de F´ ısica d’Altes Energies and CER for Astrophysics Particle Physics and Cosmology, Universitat de Barcelona Diagonal 647, E-08028Barcelona, Spain
§ †

Department of Physics, University of Miami Coral Gables, Florida 33124-8046 USA

February 1, 2008

Abstract We construct actions for non-relativistic strings and membranes purely as WessZumino terms of the underlying Galilei groups.

1

Introduction

Recently, a closed non-relativistic (NR) string with a non-trivial spectrum of excitations was constructed [1, 2]. Theconstruction was motivated by non-commutative open string (NCOS) theories [3, 4] in 1 + 1 dimensions [5]. Here we would like to elucidate the symmetries and the geometrical structure of the NR string. On the one hand, our goal will be to generalize the study of the free non-relativistic particle action as a Wess-Zumino (WZ) term of the ordinary Galilei group [6]. On the other hand, our analysiswill parallel the study of the relativistic Green-Schwarz string action which contains a WZ term [7] predicated on the non-trivial third cohomology group of (N = 2 SuperPoincar´) /SO(9, 1) [8]. After treating e the string case, we will then discuss the extension to non-relativistic d-branes.

2

Non-relativistic strings

The contraction of the Poincar´ group associated with the NR string in nspacetime dimene sions is obtained by letting c → ∞ after rescaling the coordinates x0 → cx0 ≡ ct , x1 → cx1 ≡ cx , xa → X a , a = {2, ..., n − 1} (1)

1

as well as rescaling the Poincar´ generators e P 0 → H/c , P 1 → P /c J0a → cJ0a ≡ cKa , Pa → Pa , The contracted algebra is then [Mab , Mcd ] = i (δac Mbd + δbd Mac − δad Mbc − δbc Mad ) [Ka , Mcd ] = i (δad Kc − δac Kd ) , [Ka , K] = iJa[Ja , Mcd ] = i (δad Jc − δac Jd ) , [Mab , P ] = i (δac Pb − δbc Pa ) [K, H] = iP , [Ka , H] = iPa , [K, P ] = iH [Ja , P ] = iPa (3)
c

J1a → cJ1a ≡ cJa Jab → Mab (2)

J01 → J01 ≡ K ,

[Ja , K] = iKa

Applying the general techniques [9] for non-linear realizations of spacetime symmetries, as have been applied to relativistic membranes and supermembranes [10], we consider the cosetelement (4) g = exp (−itH + ixP ) exp iX a Pa + iv b Kb + iθ c Jc where H, P are the unbroken translations, and X a (t, x) , v a (t, x) , θ a (t, x) are Goldstone fields associated with the broken generators P a , K a , J a , respectively. The stability group is generated by the transverse rotations J ab , and by K. The transformation properties of X a (t, x) , v a (t, x) , θ a (t, x) are given (up torotations) by t′ = t0 + t cosh v0 + x sinh v0 , x′ = x0 + t sinh v0 + x cosh v0
a v a ′ = v0 + v a cosh v0 + θ a sinh v0 , a θ a ′ = θ0 + v a sinh v0 + θ a cosh v0 a a a X a ′ = X a + X0 + (t0 + t cosh v0 + x sinh v0 ) v0 − (x0 + t sinh v0 + x cosh v0 ) θ0

(5)

The Maurer-Cartan one-form Ω = −ig−1 dg is Ω = Ka dv a + Ja dθ a + Pa (−v a dt + θ a dx + dX a ) − Hdt + P dx ≡ Ka ωKa + Ja ωJa + PaωPa − HωH + P ωP From this, we construct the closed, invariant three-form Ω 3 = ω K b ∧ ω Pb ∧ ω P − ω J b ∧ ω Pb ∧ ω H Moreover, it is easy to obtain a two-form “potential” Φ2 such that Ω3 = dΦ2 , namely Φ2 = 1 2 θ − v 2 dt ∧ dx + v a dX a ∧ dx − θ a dX a ∧ dt 2 (8) (7) (6)

modulo addition of any closed or exact 2-form. Φ2 cannot be written in terms of the left-invariant one forms, so it isnot invariant under the action of the group, and therefore the third cohomology group of the Galilei group as given by (3) is not trivial. There are two immediate developments. (I) We can construct an extended algebra with new non-trivial commutation relations. In part, these would be given by [Ka , Pb ] = iδab Z , [Ka , Jb ] = iδab W , 2 [P, Ka ] = iNa . (9)

Note that the presence of the...