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Modified Black Holes in Two Dimensional Gravity

arXiv:hep-th/9111044v1 22 Nov 1991IC/91/384November, 1991Modified Black Holes in Two Dimensional GravityNoureddine Mohammedi1International Centre for Theoretical PhysicsP. O.

Box 586, 34100 Trieste, Italy.AbstractThe SL(2, R)/U(1) gauged WZWN model is modified by a topological termand the accompanying change in the geometry of the two dimensional tar-get space is determined. The possibility of this additional term arises froma symmetry in the general formalism of gauging an isometry subgroup ofa non-linear sigma model with an antisymmetric tensor.It is shown, inparticular, that the space-time exhibits some general singularities for whichthe recently found black hole is just a special case.

From a conformal fieldtheory point of view and for special values of the unitary representations ofSL(2, R), this topological term can be interpreted as a small perturbationby a (1,1) conformal operator of the gauged WZWN action.1nour@itsictp.bitnet

1. IntroductionConformal field theories with central charge c = 1, when coupled to two-dimensional gravity provide interesting toy models for the study of stringtheories.

Perhaps the most appealing feature of these c = 1 theories is theirability to describe strings in non-critical dimensions in a non-perturbativeway [1-18].This is due to the recent success of the matrix model ap-proach to non-critical strings. Another property of these theories is theirtarget space-time interpretation as critical string theories propagating innon-trivial 1 + 1 dimensional backgrounds [6-9].

In particular, the metricof this two-dimensional target space exhibits a black hole singularity justas the Schwarzschild black hole in four dimensions [19,20]. Of great impor-tance, however, is the possibility of viewing these singularities as arising froma modular invariant SL(2, R)/U(1) coset WZWN model [19].In section two of this paper, we explore the change in the geometry of thetwo dimensional target space due a modification by a topological term of thegauged WZWN action.

The possibility of this additional term stems fromthe general formalism of gauging an arbitrary isometry subgoup of a non-linear sigma model with an antisymmetric tensor. We find that, in general,the scalar curvature is singular whenever the quantity116(1 −uv)2h4 + 2(uv −4)∂uX∂vX + 4u∂uX −4v∂vX + u2(∂uX)2 + v2(∂vX)2i(1.1)has zeros or is undefined.

Here (u, v) are the target space coordinates andX(u, v) is a scalar function introduced by the topological term.We also consider, in section three, the effects of the additional term on theconformal field theory of the black hole as described by the SL(2, R)/U(1)coset theory. We show that when the unitary representations |l, m > of theuniversal cover gSL(2, R) and the level k satisfy−l(l + 1)k −2 + m24k = 0 ,(1.2)the extra topological term is a conformal operator of dimension (1, 1).

There-fore it can be considered as a kind of perturbation of the ordinary SL(2, R)/U(1)coset model.1

1. The Topological TermThe non-linear sigma model with a Wess-Zumino term as given byS = 12πZd2x (√γγµνGij + ǫµνBij) ∂µφi∂νφj(2.1)is invariant under the global U(1) isometry transformations, (α is constant),δφi = αKi(φ) ,(2.2)provided that Ki is a Killing vector of the metric Gij,∇(iKj) = 0, and∂lHijkKl + Hljk∂iKl + Hilk∂jKl + Hijl∂kKl = 0 .

(2.3)The torsion is defined by Hijk = 32∂[iBjk]. It follows that the antisymmetrictensor Bij satisfies∂lBijKl + Blj∂iKl + Bil∂jKl = ∇iLj −∇jLi(2.4)for some vector Li [21].The above transformation can be made local, (α = α(x)), b y the intro-duction of a U(1) gauge field Aµ.

Then the gauged action takes the form[22,23]Sg = 12πZd2x{√γγµνGijDµφiDνφj + ǫµνBij∂µφi∂νφj −2ǫµνCiAµ∂νφi} ,(2.5)whereDµφi=∂µφi + AµKiCi=BijKj + Li . (2.6)The gauge field Aµ transforms as δAµ = −∂µα.

The action (2.5) is theninvariant under local gauge transformations if [22,23]∂jCiKj + Cj∂iKj = 0LiKi = 0. (2.7)The gauge field apprears only quadratically and with no derivatives.Hence, it can be eliminated via its equations of motion.

The resulting theory2

is again a non-linear sigma model with a new metric bGij and an antisymmet-ric tensor bBij given bybGij=Gij −1MGikGjlKkKl + CiCjbBij=Bij + 1MGikCjKk −GjkCiKk, ,(2.8)whereM = GijKiKj . (2.9)Notice that the new metric bGij would exhibit an explicit singular ity if Mhas zeros.

This is so if the old metric Gij is not positive definite as it is inthe case when the non-linear sigma model is defined on a non-compact groupmanifold. Using Eq.

(2.8) we findbGijKj = 0 . (2.10)Therefore due to these null eigenvectors, the metric bGij cannot be invertedand we cannot analyse the singularities of the gauged non-linear sigma model.To overcome this difficulty, a gauge fixing term in the action (2.5) is necessary.The main remark in this note is that the defining equation for Li, in (2.4),is invariant under the shift [22]Li →Li + ∂iλ .

(2.11)To guarantee that the gauged action (2.5) is invariant under the above shift,we should modify our action in the following mannerSg →Sg + 12πZd2x ǫµνFµνX(φ) ,(2.12)where Fµν = ∂µAν −∂νAµ is the U(1) field strength. The additional term isgauge invariant if∂iXKi = 0 .

(2.13)The gauged action will be invariant under the shift (2.11) provided that X(φ)undergoes the transformationX →X + ∂iλ . (2.14)Furthermore, the term in Fµν is topological in nature.3

The possibility of the additional term in Eq. (2.12) may also be seen froma different point of view: In the normal coordinate expansion of the gaugedaction, when the gauge field is taken as a fixed background, terms propor-tional to Fµν are generated [22].

Indeed, in order to expand the action Sgaround some classical field φ in a covariant fashion [24], we introduce thequantum field ξi(x) which is the tangent at φi to the geodesic joining φi toφi + πi, where πi is a small perturbation around φi. We find that the firstterm in the expansion of the ǫµν term in Sg is given by [22]12πZd2x ǫµν HijkDµφiDνφj −CkFµνξk .

(2.15)It is clear, therefore, that divergences proportional to Fµν will appear uponquantisation. Hence, for a renormalisable theory we should add a term inFµν to the classical action.To see the consequences of this additional term on the black hole physics,let us apply our formalism to the SL(2, R) case.The group manifold isparametrised by 2g = au−v1a(1 −uv)!,(2.16)and the U(1) isometry group is generated by 3δa = 2αa , δu = δv = 0 .

(2.17)The gauged WZWN action is given by an expression of the form (2.5), (withthe factor12π in the front replaced by −k4π), and whereGij=−1a2 (1 −uv)−12va−12ua−12va012−12ua120Bij=00000−ln a0ln a0. (2.18)Here a = φ1 , u = φ2 , v = φ3, and Li and Ci are given byL1=C1 = 0L2=C2 = 2vL3=C3 = −2u .

(2.19)2In this section we are using the conventions of ref. [19].3Throughout this note we will restrict our analyses to gauging the non-compact U(1)subgroup.4

Adding the term proportional to Fµν, as given in (2.12) with X scaledby a factor k, and integrating out the gauge field leads to a non-linear sigmamodel with a metric bGijbGuu=12(1 −uv)h−v∂uX −(∂uX)2ibGvv=12(1 −uv)hu∂vX −(∂vX)2ibGuv=12(1 −uv)1 + 12u∂uX −12v∂vX −∂uX∂vX. (2.20)The coordinate a has been eliminated via the gauge choice a2 = 1 −uv [19].Notice that by setting X = 0 one gets the usual black hole metric [19,20].To illustrate the change in the geometry due to the Fµν term, let uscompute the scalar curvature for a particular case for the function X(u, v).First of all we find, using (2.13) and (2.17), that X is a function of thecoordinate u and v only.

Second of all, the only further restiction that onecan have on X is the requirement that it satisfies the equation of a scalarfield on the target space, namely∇2X(u, v) = 0= (−3u∂u −3v∂v + 2(2 −uv)∂u∂v −u2∂2u −v2∂2v) X,(2.21)where the covariant derivative ∇corresponds to the metric (2.18). The lastequation is also sufficient for the vanishing of the one-loop counterterm pro-portional to Fµν when the the gauge field is considered as a fixed background.This counterterm is calculated from the expansion in ξi(x) given by14πZd2x ǫµνFµν (∇i∇j −∇iCj) ξiξj .

(2.22)The equation in (2.21) has two particular solutionsX1(u, v)=xu−2 + yv−2X2(u, v)=z ln(uv ) ,(2.23)with x, y and z being some constants of integration. As an example, thescalar curvature corresponding to the second solution is given byR2 = −4 [u2v2(1 + 4z + 6z2 + 4z3) + z2(uv + 1)(1 + 2z + 2z2)](1 −uv)(uv + 2zuv + 2z2)2.

(2.24)5

In addition to the usual singularity at uv = 1, there appear another singular-ity at (uv+2zuv+2z2) = 0. In general the extra singularities are determinedby the curves in u and v for which the determinant of the metric bGij is zeroor undefined, as given in (1.1).1.The Conformal Field Theory Of The TopologicalTermIn what follows we will investigate the effects of the additional term involvingFµν on the conformal field theory of the black hole solution.

For this purposethe SL(2, R) WZWN model is parametrised by 4g = ei2 θLσ2e12 rσ1ei2θLσ3 ,(3.1)where (r, θL, θR) are real Euler coordinates and σi are the Pauli matrices.The local gauge transformations correspond toδθL = δθR = α , δr = 0 . (3.2)The gauged SL(2, R) action takes the formS = Swzwn[r, θL, θR]+k2πZd2zhA¯∂θR + cosh r ¯∂θL+¯A (∂θL + cosh r∂θR) −¯AA (cosh r + 1)i, (3.3)where the action Swzwn is given bySwzwn = k4πZd2z¯∂r∂r −¯∂θL∂θL −¯∂θR∂θR −2 cosh r ¯∂θL∂θR.

(3.4)The gauge field A = (A, ¯A) is traded for two complex scalars φL and φR(φL = φ∗R) through [25]A = ∂φL ,¯A = ¯∂θR . (3.5)By redefining the fields as [26]θL →θL + φL, , θR →θR + φR(3.6)4In this section we will follow the notation of ref.

[26].6

one finds that the gauge fixed action has a dependence on φL and φR throughtheir difference φ = φL −φR only, and is given bySgf = Swzwn[r, θL, θR] + S[φ] + S[b, c] ,(3.7)where S[φ] describes a free scalar fieldS [φ] = −k4πZd2z∂φ¯∂φ . (3.8)The Jacobian of the gauge fixing is given by a (1, 0) ghost system representedby the actionS [b, c] =Zd2zb¯∂c + ¯b∂¯c.

(3.9)The energy momentum tensor corresponding to this action is given byT(z) =1k −2ηabJaJb + k4∂φ∂φ + b∂c ,(3.10)and the SL(2, R) currents are given byJ3(z)=k (∂θL + cosh r∂θR)J±(z)=ke±iθL (∂r ± i sinh r∂θR) . (3.11)The Virasoro algebra generated by T(z) has a central chargec =3kk −2 −1 .

(3.12)The primary fields of this Euclidean coset theory have the form [26]T lmn(r, θL, θR) = P lωLωR(cosh r)eiωLθL+iωRθR ,(3.13)where the quantum numbers ωL and ωR are the eigenvalues of J30 and ¯J30,respectively and l labels the SL(2, R) isospin. The functions P lωLωR are theJacobi functions, and ωL and ωR take their values on the m × n latticeωL = 12(m + nk) ,ωR = −12(m −nk) .

(3.14)The integers m and n are interpreted, respectively, as the discrete momentumand the winding number of the string in the θ = 12(θL −θR) direction [26].7

The vertex operators T lmn are eigenfunctions of the Virasoro operators L0and ¯L0, which are represented by the differential operatorsL0=−△0k −2 −1k∂∂θ2L¯L0=−△0k −2 −1k∂∂θ2R,(3.15)with △0 being the Casimir (or the Laplacian) on the group manifold and isgiven by△0 = ∂2∂r2 + cosh r ∂∂r +1sinh2 r ∂2∂θ2L−2 cosh r∂2∂θL∂θR+ ∂2∂θ2R!. (3.16)The eigenvalues of L0 and ¯L0 are the conformal weights of the primary fieldsand are expressed ashlmn=−l(l + 1)k −2 + (m + nk)24k¯hlmn=−l(l + 1)k −2 + (m −nk)24k.

(3.17)The term in Fµν that we want to add to our gauged action is given bySadd = kπZd2z X(r, θL, θR)¯∂∂φ . (3.18)On the other hand equation (2.13) yields(∂θL + ∂θR) X = 0 .

(3.19)ThereforeX(r, θL, θR) = X(r, θL −θR) . (3.20)This additional term may be regarded as a perturbation to the conformalfield theory of the coset model.

Hence, we require that the operator ¯∂∂φXhas conformal dimension (1, 1) with respect to the energy momentum tensorof the unperturbed theory in (2.10). Since ¯∂∂φ is already of dimenion (1, 1)we must haveL0X = ¯L0X = 0 .

(3.21)Therefore X is a conformal operator of the form (3.13) and of dimension(0, 0), and since it depends only on the difference θL −θR we deduce that8

the winding number n must be zero. Hence X(r, θ) is given by all the vertexoperatorsX(r, θ) = P l12m,−12 m(cosh r)ei2 m(θL−θR) ,(3.22)for which the representations and the level k have to satisfy−l(l + 1)k −2 + m24k = 0 .

(3.23)To summarise, we have modified the gauged WZWN model by a terminvolving the field strengh Fµν of the U(1) gauge group. We found that theaddition of this term changes completely the singularity structure of the two-dimensional target space.

We have calculated explicitly the scalar curvatureand found the singularity for a particular case of the additional term. Theconformal field theory of the black hole in the persence of the Fµν term wasalso analysed.

In particular, this term can be treated as a perturbation by a(1, 1) conformal operator of the SL(2, R)/U(1) coset model.Acknowledgements : I would like to thank K. S. Narain, E. Gava, H.Sarmadi, S. Panda and B. Rai for many useful discussions. The financialsupport from IAEA and UNESCO is also hereby acknowledged.9

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