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

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


Introduction
Recently, a closed non-relativistic (NR) string with a non-trivial spectrum of excitations was constructed [1,2]. The construction 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 analysis will 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 SuperPoincaré) /SO(9, 1) [8]. After treating the string case, we will then discuss the extension to non-relativistic d-branes.

Non-relativistic strings
The contraction of the Poincaré group associated with the NR string in n spacetime dimensions is obtained by letting c → ∞ after rescaling the coordinates as well as rescaling the Poincaré generators The contracted algebra is then 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 coset element 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 to rotations) by The Maurer-Cartan one-form Ω = −ig −1 dg is Ω = K a dv a + J a dθ a + P a (−v a dt + θ a dx + dX a ) − Hdt + P dx ≡ K a ω Ka + J a ω Ja + P a ω Pa − Hω H + P ω P From this, we construct the closed, invariant three-form Moreover, it is easy to obtain a two-form "potential" Φ 2 such that Ω 3 = dΦ 2 , namely modulo addition of any closed or exact 2-form. Φ 2 cannot be written in terms of the left-invariant one forms, so it is not 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 Note that the presence of the central charges Z and W break the SO(1, 1) invariance, since, for example, [K a , P b ] = iδ ab Z but [J a , P b ] = 0. We will explore this further elsewhere [11].
(II) We can construct a WZ action given by S W Z = T * Φ 2 where * Φ 2 is the pullback of the two-form on the world sheet and T = 1/ 2πl 2 s is the tension of the string. This WZ action takes the explicit form If we fix the static gauge, t = τ and x = ασ, α being a parameter, and we eliminate the nondynamical Goldstone fields, we get [12] The transverse coordinates are a collection of free, massless fields on the world-sheet.
If we compute the Noether charges associated with the transformations (5), we find for the central and topological elements in the gauge fixed form the formal expressions If Z should be different from zero we need our space to be homologically non-trivial, for example S 1 × R n−1 . For a closed string the coordinate x would then be wrapped [13] around S 1 , which implies x = 2πRkσ, σ ∈ [0, 1], where k is the winding number and R is the radius of S 1 . Similarly, if we have a homologically trivial transverse space N a is zero, but if we consider some directions of the transverse space as tori, say, we will have some of the N a = 0.

Non-relativistic d-branes
World-volume (longitudinal) dimensions are labeled by x α with α = 0, 1, · · · , d and Lorentzian metric η αβ = diag (−1, +1, · · · , +1), while the remaining (transverse) dimensions are labeled by X a with a = 1, · · · , D and Euclidean metric δ ab . The group elements in the coset are As before, the algebra, without central extensions 1 , is easily abstracted from the contracted (c → ∞) Galilean correspondence Thus the relevant non-vanishing commutators are as well as [K αa , m βγ ] = iη αγ K βa − iη αβ K γa , [K αa , M bc ] = iδ ac K αb − iδ ab K αc , etc. This leads to the one-form with component one-forms defined as 2 The Maurer-Cartan equations involving the non-vanishing structure constants, f Kαa,p β ,P b = η αβ δ ab , are given here by dω a = ω αa ∧ ω α . We now construct from the component one-forms a closed, invariant d + 2 form with ε 01···d ≡ +1 = −ε 01···d , to produce a Wess-Zumino brane action where T d is the tension of the d-brane. The boundary M d+1 = ∂M d+2 is the world-volume of the evolving brane. Straightforward calculation shows that an appropriate choice for the brane form potential is Recall the a, b, etc. indices are contracted with Euclidean metric, while α, β, etc. are handled with Lorentz metric. As a special case, reconsider the NR string, with d = 1. We have in agreement with (7) and (8). We note in passing that Ω 3 is an apparent modification of the usual torsion 3-form for non-linearly realized (compact) semi-simple Lie groups [15] when expressed in terms of the component forms (i.e. vielbeins) 3 .
2 The string case is obtained by writing v b = v 0b and θ b = v 1b . 3 Naively, this other 3-form, Ω3, would be built from the component forms by using the structure constants fK αa ,p β ,P b given earlier, and differs from Ω3 by replacement of ε αβ with η αβ . For an arbitrary d-brane, it would be Remarkably, this 3-form is not closed, for arbitrary d, but rather gives d Ω3 = dxα ∧ dv αb ∧ dx β ∧ dv βb .
If we fix the static gauge x α = σ α , the pullback to the d-brane world-volume goes as follows.
So the action is Eliminating the auxiliary Goldstone fields v γb using their equations of motion, v b γ = − ∂X b ∂x γ , gives the gauge fixed action So, in a straightforward generalization of the string case given previously, the X a considered as functions of the x α are free massless fields on the world-volume.
In terms of another world-volume parameterization, z α , α = 0, 1, · · · , d, and z β = η βα z α , Explicitly In this arbitrary parameterization The action in an arbitrary gauge is therefore Eliminating once again the Goldstone auxiliaries using their equations of motion, v γb = − M −1 γα ∂X b ∂zα , gives (for d = 2, see [12]) The quadratic in cofactors can be written more explicitly using (27) above.
Rather deceptively, (30) looks like the action for a nonlinear, interacting theory coupling the transverse dimensions to the world-volume variables. Of course, in the gauge x α = z α , we have M αβ = η αβ , and we recover a free-field action as in the string case -a simplification typical of re-parameterization invariance. The Noether charges for the d-brane, in the gauge fixed form (25), are given at any world-volume time as where a, b = 1, · · · , D, and α, β = 0, 1, · · · , d, but i = 1, · · · , d. Here, Π a is the canonically conjugate momentum associated with X a , so [X a (x) , . For the general d-brane, the commutation relations are just a straightforward generalization of the non-relativistic string case. However, a greater variety of central and topological charges may appear in the d-brane algebra, for example as These central and topological charges are given formally for the d-brane by 4 To avoid ambiguities in these formal expressions, we have checked the results against various Jacobi identities. A full discussion of all Jacobi identities is deferred [11].

Conclusion
We have shown how to construct non-relativistic string and brane actions from the structure of the underlying Galilei group. We will investigate quantum properties of these models and discuss the roles of their topological charges in a subsequent paper [11]. In certain cases, perhaps all, passive advections of non-relativistic strings and membranes also result from a Wess-Zumino action when expressed in the formalism of Nambu mechanics [16]. It would be interesting to combine fully that other formalism with the group theoretic approach of the present paper.