Target Space Structure of a Chiral Gauged

arXiv:hep-th/9204011v1 27 Mar 1992Target Space Structure of a Chiral GaugedWess-Zumino-Witten ModelSupriya K. Kar1 and Alok Kumar2Institute of Physics, Bhubaneswar-751005, INDIA.AbstractThe background for string propagation is obtained by a chiral gaug-ing of the SL(2, R) Wess-Zumino-Witten model. It is shown explicitlythat the resulting background fields satisfy the field equations of thethree dimensional string effective action and the target space has cur-vature singularity.Close connection of our solution with the threedimensional black string is demonstrated.IP/BBSR/92-18March ’92.1e-mail:supriya%iopb@shakti.ernet.in2e-mail:kumar%iopb@shakti.ernet.in

The target space structure of the gauged Wess-Zumino-Witten (WZW)model has been investigated by Witten [1] and others [2]. It was shown [1]that the vector or axial gauging of a U(1) subgroup of the SL(2, R) WZWmodel leads to the string propagation in a two dimensional target space withblack hole singularity.

There have been several generalizations of this result.In particular, two dimensional charged black hole [3], three dimensional blackstring [4] as well as four dimensional solutions with curvature singularities[5, 6] have been constucted as gauged WZW models. It has also been demon-strated that the two gaugings mentioned above are related by duality [7, 8]and they are anomaly free.In a recent paper [9], it has been shown that there is another gaugingfor the two dimensional WZW model, called ”chiral gauging”,which is alsoanomaly free.

Quantization of the chiral WZW model and their applicationsto the coset constructions have been done in ref. [9].In this paper, we investigate the target space structure of the SL(2, R)WZW model with a chiral U(1) gauging.It is shown that the resultingmodel describes a three dimesional target space with curvature singular-ity.

Furthermore, by an O(2, 2) transformation acting on background metric(Gµν), dilaton (φ) and antisymmetric tensor (Bµν), it is demonstrated explic-itly that our solution can be transformed to the three dimensional unchargedblack string of ref.[4]. The connection to the charged black string [4] is alsodiscussed.1

We start by writing down the action for the chiral gauged Wess-Zumino-Witten (CGWZW) model [9] :S = SW ZW + k2πZd2z Tr [ ARz U−1 ¯∂U + AL¯z ∂U U−1 + ARz U−1AL¯z U ](1)where,SW ZW =k4πZd2z tr [U−1∂U U−1 ¯∂U]+k12πZd2x dt ǫijkTr [U−1∂iU U−1∂jU U−1∂kU](2)and ARz (AL¯z ) in eq. (1) is the z(¯z) component of gauge field ARµ (ALµ).

Theclassical action S is invariant under the gauge transformation [9],δU = vL (¯z) U −U vR(z),δAL¯z = −¯∂vL (¯z)andδARz = ∂vR (z). (3)It has been shown in ref.

[9] that the theory is anomaly free. Therefore thegauge invariance is maintained at the quantum level as well.

For comparison,we also write down the action for the vector GWZW model [1, 10, 11] :SV =SW ZW+ k2πZd2z Tr [ A U−1 ¯∂U + ¯A ∂U U−1 + A U−1 ¯A U + ¯A A ](4)where in eq. (4) A and ¯A are the two light cone components of a gauge fieldAµ.2

From eqs. (1) and (4), it is observed that, except for the absence of one ofthe terms quadratic in gauge fields, the expression for the CGWZW model[9] is similar to that of the vector (or axial) one [1, 10, 11].

However it willbe seen that the absence of this term makes crucial difference in the targetspace structure.We now explicitly work out the case of the SL(2, R) WZW model when”chiral” gauging is done with respect to the U(1) subgroup generated by thecurrents JL¯z ≡−i2 Tr (σ2 U−1∂¯zU) and JRz ≡−i2 Tr (σ2 ∂zU U−1). Thenby representing the gauge fields as, AL¯z ≡( −i2AL¯z σ2), ARz ≡( −i2ARz σ2)and the sigma model field U as,U ≡exp ( i2φL σ2) exp (r2 σ1) exp ( i2φR σ2)(5)one obtains [10],SW ZW = k4πZd2z (−12∂φL ¯∂φL −12∂φR ¯∂φR + 12∂r ¯∂r −cosh r ∂φR ¯∂φL),(6)andS =SW ZW +k4πZd2z [ARz (¯∂φR + cosh r ¯∂φL)+ AL¯z (∂φL + cosh r ∂φR) −AL¯z ARz cosh r ].

(7)For comparison, we again write the GWZW action, eq. (4), for the vectorgauging in a similar form [10] :SV =SW ZW +k4πZd2z [ A(¯∂φR + cosh r ¯∂φL)+ ¯A(∂φL + cosh r ∂φR) −¯AA (cosh r + 1)](8)3

It is seen that the coefficients of the last term in the RHS. of eqs.

(7) and (8)are different. As pointed out earlier, the difference is due to different gaugechoices in the two cases.

Now, as in ref. [11], we decouple the gauge fields bya field redefinition :ARz≡ˆARz + (∂φL + cosh r ∂φR)cosh rAL¯z ≡ˆAL¯z + (¯∂φR + cosh r ¯∂φL)cosh r(9)and obtain the action for the CGWZW model as,S =k4πZd2z (12∂φL ¯∂φL + 12∂φR ¯∂φR+ 12∂r ¯∂r +1cosh r∂φL ¯∂φR −ˆAL¯z ˆARz cosh r ).

(10)By comparing this action with the following one, describing the string prop-agation :S = 12πZd2z [(Gµν + Bµν)∂xµ ¯∂xν ],(11)one obtains the background metric (Gµν) and antisymmetric tensor (Bµν).For example from eqs. (10) and (11), by defining an invariant distance :ds2 =−k4h−dr2 −dφL2 −dφR2 −2cosh rdφLdφRi,(12)we getG =−1000−1−1cosh r0−1cosh r−1,(13)and similarlyB =00000−1cosh r01cosh r0.

(14)4

As in ref. [1], given Gµν and Bµν, eqs.

(13) and (14), the background value forthe dilaton ’φ’ can be determined by solving the field equations of the stringeffective action. This procedure is followed later in the paper.

The resultingdilaton background is,φ = −ln cosh r + const. (15)It can also be shown that the background dilaton of eq.

(15) can be generatedfrom eq. (10) by doing the gauge field integration in the corresponding pathintegral and by regularizing a det ( −2π2k cosh r) factor as in ref.

[7].We now show that Gµν, Bµν and φ fields obtained above are in factthe solutions of the string effective action. A characteristic feature of thebackgrounds in eqs.

(13) - (15) is that they are dependent only on a singlecoordinate.String effective action with such background fields has beenstudied extensively by Meissner and Veneziano [12]. They worked with timedependent backgrounds, however their formalism can be applied to our caseby replacing t with r in their paper.

Following ref. [12], it can be shown thatfor the backgrounds of the type discussed in the previous paragraph, G andB can alaways be brought to the form,G ≡ −100G!

;B ≡ 000B! (16)where the first entry in a row or column of matrices G and B in eq.

(16)denotes the r coordinate. To the lowest order in α′, the string tension, thecomplete set of field equations following from the string effective action for5

this case are :( ˙Φ)2 −14Tr[(G−1 ˙G)(G−1 ˙G)] + 14Tr[(G−1 ˙B)(G−1 ˙B)] −V = 0 ,(17)( ˙Φ)2 −2¨Φ+ 14Tr[(G−1 ˙G)(G−1 ˙G)]−14Tr[(G−1 ˙B)(G−1 ˙B)]−V + ∂V∂Φ = 0 , (18)−˙Φ ˙G + ¨G −˙GG−1 ˙G −˙BG−1 ˙B = 0,(19)and−˙Φ ˙B + ¨B −˙BG−1 ˙G −˙GG−1 ˙B = 0,(20)whereΦ ≡φ −ln√detG(21)and dots denote the derivative with respect to the r coordinate. By compar-ing eqs.

(13) and (14) with eq. (16) we have, for our solutions,G = − 11cosh r1cosh r1!,B = − 01cosh r−1cosh r0!.

(22)This implies,G−1 = −coth2r 1−1cosh r−1cosh r1!,(23)Tr (G−1 ˙G)2 = 21sinh2r cosh2r +1sinh2r,(24)Tr (G−1 ˙B)2 = 21sinh2r cosh2r −1sinh2r(25)and˙Φ ≡˙φ −12 det ˙Gdet G= −coth r.(26)Using eqs. (24) - (26) it is observed that the field eq.

(17) is satisfied for aconstant potential V = 1. It can be checked that this is the same value as6

for the ungauged WZW model. Also, since ¨Φ =1sinh2r , eq.

(18) is satisfiedfor the same value V = 1. Eqs.

(19) and (20) can be satisfied by using thefollowing expressions :−˙Φ ˙G =1cosh r 0110!,−˙Φ ˙B =1cosh r 01−10!,(27)¨G =2cosh3r −1cosh r 0110!,(28)¨B =2cosh3r −1cosh r 01−10!,(29)˙GG−1 ˙G =1cosh3r −cosh r11−cosh r!,(30)˙BG−1 ˙B =1cosh3r cosh r11cosh r! (31)and˙BG−1 ˙G =1cosh3r −cosh r1−1cosh r!= − ˙GG−1 ˙BT .

(32)Therefore we have explicitly shown that the solution, eqs. (13) - (15),satisfy the field equations of the three dimensional string effective action.The scalar curvature R can be computed as [12],R = 2 [ ∂2∂r2(ln√det G)] + [ ∂∂r(ln√det G)]2 + 14 Tr (G−1 ˙G)2(33)and we find for our caseR = −721cosh2r.

Therefore the target space hascurvature singularity. However it does not occur for any real value of r inthe coordinate system of our choice.7

Next we show that the solutions, eqs. (13) - (15), can be transformed to athree dimensional uncharged black string by an O(2, 2) transformation actingon G, B and Φ.It was demonstrated in ref.

[12] that the string effective action, as well asthe equations of motion, (17) - (20), in space-time dimension D (= d + 1)have an invariance under an O(d, d) transformation acting on a matrix,M ≡ G−1−G−1BBG−1G −BG−1B! (34)as M →˜M = ΩM ΩT and Φ →Φ, where Ωis an O(d, d) matrix satisfyingΩT η Ω= η and η ≡ 0II0.

Therefore, new solutions ( ˜M) can be generatedfrom the old ones (M) through this transformation. To show the relation tothe uncharged black string, we apply this procedure to our case.

We findthat for G and B as in eqs. (22 ) and for an O(2, 2) matrix Ω:Ω≡1√200−1√201√21√200−1√21√201√2001√2(35)we have,˜M = ˜G−100˜G!

(36)where˜G = −(cosh r + 1)sinh2r cosh r11cosh r!. (37)By comparing eqs.

(34) and (36) it is noticed that in the O(2, 2) trans-formed coordinate system, the antisymmetric tensor B is absent. Since Φ8

remains unchanged under this transformation, therefore we get˙˜φ −12 det ˙˜Gdet ˜G= −coth rwhich implies, using (det ˜G) = coth2 r2,˜φ = −ln sinh2r2+ const. (38)To show that ˜G and ˜φ in eqs.

(37) and (38) are in fact the three dimensionaluncharged black string, we now diagonalize ˜G by an orthogonal transforma-tion :˜G′ ≡O ˜G OT = −coth2( r2)00−1! (39)where O =1√2 111−1.From the above results we see that the three dimensional metric G ofeq.

(13) is transformed, under combined O(2, 2) (Ω) and orthogonal trans-formations (O), to˜G′ =−1000−coth2( r2)000−1(40)The invariant distance corresponding to the metric ˜G′ is therefore,ds2 =k4 hdr2 + coth2(r2) dx2 + dy2i(41)where x and y are the corresponding transformed coordinates.It is now recognised that the metric in eq. (41) represents a target spacewhich is the cross product of the Euclidean two dimensional ”dual” black hole9

[1, 10] with a flat one dimensional coordinate. The solution for the dilatonin eq.

(38) is also the one corresponding to the ”dual” two dimensional blackhole. From the three dimensional point of view, as pointed out in ref.

[4] , thissolution can be interpreted as the uncharged ”dual” black string solution.In fact our original solution, eqs. (13) and (14), for G and B, is also relatedto the charged black string even prior to the O(2, 2) transformation.

If wejust diagonalize the G in eq. (13) we get the invariant distance :ds2 =k4hdr2 + (1cosh r + 1) dx2 + (1cosh r −1)dy2i(42)and the antisymmetric tensor :B =1cosh r 0000010−10!.

(43)Then by defining ρ = cosh r and y = i t we have,ds2 =k4h−1 −1ρdt2 +1 + 1ρdx2 +1 −1ρ−11 + 1ρ−1dρ2ρ2i, (44)B = iρ 0000010−10! (45)andφ = −ln ρ + const.

(46)This solutions matches with the one in ref. [4] for the charged black string ifone takes λ = −12 and defines ρ = 2 ˆr in eqs.

(12) and (13) of this reference.However the B field in this case is imaginary. This field can be made real by10

keeping t space-like. The value λ = −12 in ref.

[4] implies that the free boson’x’ of the [SL(2, R) × R] WZW model is time like.To conclude, in this paper the background graviton, dilaton and antisym-metric tensor fields resulting from the chiral U(1) gauging of the SL(2, R)WZW model were obtained. It was shown that the backgrounds satisfy thefield equations of the string effective action and target space has curvaturesingularity.

Relation of our solution with the three dimensional black stringhas been shown. An interesting aspect of the chiral gauging is that the result-ing model has the same target space dimension as the original one.

Whereasdue to the vector or axial gauging the target space dimension is reduced.It will be interesting to examine the duality properties of the CGWZWmodels as in the vector and axial cases. Also, since the chiral gauging is left-right assymmetric in nature, it should be possible to obtain exact conformalfield theory solutions for the heterotic strings in nontrivial background usingCGWZW models.11

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