Arbitrary Spacetimes from the SL(2, R)/U(1) Coset Model

arXiv:hep-th/9212095v1 16 Dec 1992BONN-HE-92-38December, 1992Arbitrary Spacetimes from the SL(2, R)/U(1) Coset ModelNoureddine Mohammedi∗†Physikalisches Institutder Universit¨at BonnNussallee 12D-5300 Bonn 1, GermanyAbstractWe show that the gauged SL(2, R) WZWN model yields arbitrary spacetimes in two di-mensions. The c = 1 matter coupled to gravity and the black hole singularity are just twoparticular cases in these spacetimes.∗Work supported by the Alexander von Humboldt-Stiftung.†e-mail: nouri@avzw02.physik.uni-bonn.de

Since the discovery of the existence of a two dimensional black hole in the coset modelSL(2, R)/U(1) [1] and as a solution to the string beta functions [2,3], many efforts havebeen devoted to this subject. In particular, attempts have been directed towards under-standing the relationship between the black hole and the c = 1 matter field coupled totwo dimensional gravity [4-10].

All the evidences point to the fact that these two theoriesmight have the same origin.In this note, we show that by gauging the U(1) isometry subgroup of the SL(2, R)WZWN model one obtains an arbitrary geometry for the two dimensional target-space.The c = 1 matter coupled to gravity and the black hole are just two particular cases inthis arbitrary geometry.This arbitrariness in the geometry arises from the mathematical formalism of gaugingan isometry subgroup of a general non-linear sigma model [11,12]. This formalism was alsorecently used to identify the perturbations, by (1, 1) conformal operators, of the black holefound in the SL(2, R)/U(1) coset model [13].Our starting point is the ungauged SL(2, R) WZWN actionI(g)=k8πZΣ d2x√γγµνTrhg−1∂µg g−1∂νgi+k12πZB d3yǫµνρTrhg−1∂µg g−1∂νg g−1∂ρgi.

(1)Here B is a three dimensional manifold whose boundary is Σ and Tr is the trace in the twodimensional representation of SL(2, R). Let us parametrize the SL(2, R) group manifoldbyg = au−vb!, ab + uv = 1 .

(2)In this parametrization, the above action yieldsI(φ) = k4πZd2x (√γγµνGij + ǫµνBij) ∂µφi∂νφj ,(3)where the target-space metrice Gij and the antisymmetric tensor Bij areGij =1a2(1 −uv)12va12ua12va0−1212ua−120, Bij =00000−ln a0ln a0. (4)Here φ1 = a, φ2 = u, φ3 = v.1

A general non-linear sigma model as given in (3) possesses a global U(1) isometrysymmetry given byδφi = εKi(φ) ,(5)provided that Ki is a Killing vector of the metric Gij and the antisymmetric tensor Bijsatisfies∂lBijKl + Blj∂iKl + Bil∂jKl = ∇iLj −∇jLi ,(6)for some target-space vector Li [14].This global U(1) symmetry can be made local by introducing a U(1) gauge field Aµtransforming asδAµ = −∂µε . (7)The most general action involving the gauge field Aµ is written as [11,12]Igauged = k4πZd2x{√γγµνGijDµφiDνφj + ǫµνBij∂µφi∂νφj −2ǫµνCiAµ∂νφi} ,(8)whereDµφi = ∂µφi + AµKi(9)and the target-space function Ci(φ) is given byCi = BijKj + Li .

(10)Local gauge invariance implies then the following equations [11,12]∂jCiKj + Cj∂iKj = 0(11)LiKi = 0. (12)Eliminating the gauge fields from (8) results in a new non-linear sigma model of theform (3) with a new metric bGij and a new antisymmetric tensor bBij given bybGij=Gij −1MGikGjlKkKl −CiCjbBij=Bij + 1MGikCjKk −GjkCiKk,(13)whereM = GijKiKj .

(14)2

Notice that the new metric bGij would exhibit an explicit singularity if M has zeros. Thisis so if the old metric Gij is not positive definite as it is in the case when the non-linearsigma model is defined on a non-compact group manifold.

Using equation (13) we findbGijKj = 0 . (15)Therefore due to these null eigenvectors, the metric bGij cannot be inverted and we cannotanalyse the singularities of the gauged non-linear sigma model.

To overcome this difficulty,a gauge fixing term must be introduced.We would like now to apply this analyses to the non-linear sigma model of the SL(2, R)WZWN action. For this, we would like to gauge the non-compact one parameter symmetrygroup generated byδg = ε( 100−1!g + g 100−1!).

(16)This transformation is of the form (5) where the corresponding Killing vectors are givenbyK1 = 2a , K2 = 0 , K3 = 0 . (17)We deduce from equations (4) and (10) thatC1 = L1 , C2 = L2 , C3 = L3 .

(18)Furthermore, equation (12) leads toC1 = L1 = 0(19)while equation (11) is solved byC2 = f(u, v) , C3 = h(u, v) ,(20)where f(u, v) and h(u, v) are two arbitrary functions. These two functions are, however,not independent.

They are related by the defining equation for Li. Indeed, equation (6)leads to2 = ∂vf(u, v) −∂uh(u, v) .

(21)This differential equation has the following general solutionf(u, v)=v −∂uX(u, v)h(u, v)=−u −∂vX(u, v) ,(22)3

where X(u, v) is an arbitrary function. Therefore the quantities C1, C2 and C3 have beencompletely determined.

Consequently, the non-vanishing components of the new metricbGij are listed belowbG22=−14(1 −uv)h2v (∂uX) −(∂uX)2ibG23=−14(1 −uv) [2 −u (∂uX) + v (∂vX) −(∂uX) (∂vX)]bG33=−14(1 −uv)h−2u (∂vX) −(∂vX)2i. (23)The scalar curvature corresponding to this metric is found to beR=ABA=−64 + 96hv (∂uX) (∂vX)2 −u (∂vX) (∂uX)2i+32h2 + 5uv −u2v2i(∂uX) (∂vX)−8h4 −2uv + u2v2i(∂uX)2 (∂vX)2+8 (3 −uv)hu2 (∂uX)3 (∂vX) + v2 (∂vX)3 (∂uX)i+32h1 −3uv + 3u2v2 −u3v3i h∂2uX ∂2vX−(∂v∂uX)2i+4u (∂uX)h32 −24u (∂uX) + 8u2 (∂uX)2 −u3 (∂uX)3i−4v (∂vX)h32 + 24v (∂vX) + 8v2 (∂vX)2 + v3 (∂vX)3i+(1 −uv)2 h32 (∂uX)2 ∂2vX−32 (∂vX)2 ∂2uXi+(1 −uv)2 h−8u (∂uX)3 ∂2vX−8v (∂vX)3 ∂2uXi+(1 −uv)2 h−32u∂2uX(∂vX) −32v∂2vX(∂uX)i+(1 −uv)2 h−8u∂2uX(∂uX) (∂vX)2 −8v∂2vX(∂vX) (∂uX)2i+(1 −uv)2 h16u (∂v∂uX) (∂uX)2 (∂vX) + 16v (∂v∂uX) (∂vX)2 (∂uX)iB=−(1 −uv)h4 −4u (∂uX) + 4v (∂vX) + u2 (∂uX)2 + v2 (∂vX)2−2 (2 −uv) (∂uX) (∂vX)]2 .

(24)Therefore, due to the arbitrariness of the function X(u, v), we can obtain any geometrywe like from gauging the U(1) isometry subgroup of the SL(2, R) WZWN model.Inparticular if we chooseX(u, v) = constant(25)4

then the metric bGij exhibits the black hole geometry found in [1]. On the other hand,the c = 1 matter field coupled to two dimensional gravity is realized by all the functionsX(u, v) which are solutions to the differential equationA = 0 .

(26)This arbitrariness in the geometry is better understood in the language of the SL(2, R)group manifold. It turns out that the metric bGij in (23) arises, upon eliminating the gaugefield, from the following actionI(g, A)=I(g) + k4πZd2x√γγµνTr(Aµ 100−1!g−1∂νg + Aµ 100−1!∂νgg−1+AµAν" 100−1!

100−1!+ 100−1!g 100−1!g−1#)+k4πZd2xǫµνTr"Aµ 100−1!g−1∂νg −Aµ 100−1!∂νgg−1#+k4πZd2xǫµνTr"∂µAν 100−1! X00−X!#.

(27)Without the last term, this action is what is commonly written down for the SL(2, R)/U(1)WZWN model [15,16] when gauging the U(1) subgroup as given in (16). The last term,however, is a topological termk4πZd2xǫµνFµνX (u, v)(28)whose presence is required by the mathematical structure arising from gauging the U(1)isometry of a general non-linera sigma model.

Therefore, there is no reason for ignoringsuch a term when dealing with gauged WZWN models. Furthermore, this is the same termwhich was interpreted in ref.

[13] as generating perturbations, by (1, 1) conformal operators,of the SL(2, R)/U(1) black hole.Acknowledgements:I would like to thank W. Nahm for showing his interest in thesubject. The financial support from the Alexander von Humboldt-Stiftung is also herebyacknowledged.5

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