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KEK preprint 91-99, KEK-TH 388

arXiv:hep-th/9108026v2 2 Sep 1991February 1, 2018FERMILAB-PUB-91/230-TKEK preprint 91-99, KEK-TH 388OCHA-PP-18Superstring in Two Dimensional Black HoleShin’ichi Nojiri⋆Theory Group, Fermi National Accelerator LaboratoryP.O.Box 500, Batavia, IL60510, USAabstractWe construct superstring theory in two dimensional black hole backgroundbased on supersymmetric SU(1, 1)/U(1) gauged Wess-Zumino-Witten model.⋆On leave of absence from National Laboratory for High Energy Physics (KEK),Oho 1-1,Tsukuba-shi,Ibaraki-ken 305,JAPAN and Ochanomizu University,1-1Otsuka 2, Bunkyo-ku, Tokyo 112, JAPAN. e-mail address :NOJIRI@JPNKEKVX,NOJIRI@JPNKEKVM

Recently it was shown that the SU(1, 1)/U(1) gauged Wess-Zumino-Witten(GWZW) model describes strings in a two dimensional black hole. [1]The stringpropagation and Hawking radiation in this black hole were discussed in Ref.2.When the level k of SU(1, 1) current algebra equals to 9/4, this model can beregarded as a two dimensional gravity coupled with c = 1 superconformal matter.We expect that this model could be one of toy models which provide a clue to solvethe dynamics of more “realistic” string models.The supersymmetric extension of this model appeared[3]as an exact solutionof ten-dimensional superstring theory corresponding to black fivebranes.

[4]In thispaper, we will consider the supersymmetric extension based on SU(1, 1)/U(1)supersymmetric GWZW (SGWZW) model. It has been shown[5]that supersym-metric SU(1, 1)/U(1) coset model has N = 2 supersymmetry due to Kazama-Suzuki[6]mechanizm and this model is equivalent to N = 2 superconformal modelsproposed by Dixon, Lykken and Peskin.

[7]The central charge c of this system isgiven by,c =3kk −2 . (1)Here k is the level of SU(1, 1) current algebra.

When k = 52, the central charge cequals to 15 and this conformal field theory describes a critical string theory. TheN = 1 supergravity coupled with c = 32ˆc = 32 superconformal matter would be de-scribed by this critical theory.

Furthermore the N = 2 superconformal symmetryof this model suggests that pure N = 2 supergravity would be also described bythis model when c = 6 (k = 4). Due to N = 2 superconformal symmetry, super-string theories in the two dimensional black hole background can be constructedby imposing GSO projection.

It is also expected that this superstring theory wouldbe equivalent to the matrix models which have space-time supersymmetry[8,9] andtopological superstring theories based on N = 2 superconformal topological fieldtheories. [10]The action SWZW(G) of N = 1 supersymmetric Wess-Zumino-Witten model2

is given by,[11]SWZW(G) = k2πZd2zd2θtrG−1DGG−1 ¯DG−k2πZdtd2zd2θ[trG−1DGG−1 ¯DGG−1∂tG + (D ↔¯Dterm)] . (2)Here we define covariant derivatives D and ¯D by using holomorphic and anti-holomorphic Grassmann coordinates θ and ¯θD ≡∂∂θ −θ ∂∂z ,¯D ≡∂∂¯θ −¯θ ∂∂¯z .

(3)The matrix superfield G, which is an element of a group G, is given byG = exp(iXaT aΦa) . (4)Here Φa is a superfield Φa = φa + θψa + ¯θ ¯ψa + θ¯θfa and T a is a generator ofthe algebra corresponding to G. The action (2) satisfies Polyakov-Wiegmann typeformula:SWZW(GH) = SWZW(G) + SWZW(H) + kπZd2zd2θtrG−1DG ¯DHH−1 .

(5)Here H is also an element of G. This formula guarantees that the system describedby the action (2) has super Kac-Moody symmetry and N = 1 superconformalsymmetry.If F is a U(1) subgroup of G, we can gauge the global symmetry under thefollowing axial U(1) transformation, which is given by an element F of F,G →FGF . (6)The action of supersymmetric G/F gauged Wess-Zumino-Witten is given by,SG/F(G, A) =SWZW(FLGFR) −SWZW(F −1L F R)=SWZW(G)+ k2πZd2zd2θtr(A ¯A + AG ¯AG−1 + G−1DG ¯A + A ¯DGG−1) .

(7)3

Here FL and FR are elements of F and gauge fields A and ¯A are defined by,A = F −1L DFL,¯A = F −1R¯DFR . (8)The action (7) is invariant under the following U(1) gauge transformationG →FGF,A →A + F −1DF,¯A →¯A + F −1 ¯DF .

(FL,R →FL,RF)(9)A supersymmetric extension of string theory in a two dimensional black holebackground is given by setting G = SU(1, 1) in the action (7). We start withconsidering SU(1, 1) SWZW model.

By parametrizing G by,G = exp( i2ΦLσ2) exp(12Rσ1) exp( i2ΦRσ2),(10)with σi the Pauli matrices, we obtain the action SSU(1,1) of SU(1, 1) SWZW modelSSU(1,1) = k2πZd2zd2θ[−12DΦL ¯DΦL −12DΦR ¯DΦR−coshR DΦL ¯DΦR + 12DR ¯DR] . (11)The holomorphic (anti-holomorphic) conserved currents Ji ( ¯Ji) of this system aregiven by,2kG−1DG = J1σ1 + iJ2σ2 + J3σ3 ,2kG−1 ¯DG = ¯J1σ1 + i ¯J2σ2 + ¯J3σ3 ,(12)Ji = ji + θ ˜Ji + · · · ,¯Ji = ¯ji + ¯θ ¯˜Ji + · · · .

(13)Here · · · express the terms which vanish by using the equations of motion. If wedefine new currents ˆJi and ¯ˆJi by the following equationˆJi = ˜Ji −12kǫilmjljm,¯ˆJi = ¯˜Ji −12kǫilm¯jl¯jm,(14)These currents ˆJi and ¯ˆJi do not depend on the fermion currents ji and ¯ji.4

By expanding superfields ΦL,R and R into components,ΦL,R = φL,R + θψL,R + ¯θ ¯ψL,R + θ¯θfL,R,R2 = s + θη + ¯θ¯η + θ¯θg,(15)we can rewrite the SU(1, 1) SWZW action SSU(1,1) in Eq. (11) by a sum of non-supersymmetric SU(1, 1) WZW action ˜SSU(1,1) and free fermion actions:SSU(1,1) = ˜SSU(1,1) +14kπZd2z[j+ ¯∂j−−j2 ¯∂j2 + ¯j+∂¯j−−¯j2∂¯j2] ,(16)˜SSU(1,1) = k2πZd2z[−12(∂φL ¯∂φL + ∂φR ¯∂φR)−cosh(2s)∂φL ¯∂φR + 2∂s¯∂s] .

(17)Here j± and ¯j± are defined byj± ≡j1 ± ij3 ,¯j± ≡¯j1 ± i¯j3 . (18)The conserved currents corresponding to the non-supersymmetric SU(1, 1) WZWaction ˜SSU(1,1) (17) are given by ˆJi and ¯ˆJi in Eq.

(14).Fermionic currents j± and ¯j± can be written asj± =k2 exp(∓iφR)(η ± i2sinh(2s)ψL) ,¯j± =k2 exp(∓iφL)(¯η ± i2sinh(2s) ¯ψR) . (19)Note that there appear bosonic factors exp(∓iφR) and exp(∓iφL).

Due to thesefactors, the boundary conditions of j± and ¯j± are twisted although fermions η,¯η, ψL and ¯ψR, which is identified later with space-time fermionic coordinates, are5

periodic or anti-periodic. Therefore the eigenvalues of the zero modes of fermionnumber currents K and ¯K,K = 14k(j+j−−j−j+) ,¯K = 14k(¯j+¯j−−¯j−¯j+) ,(20)which satisfy the following operator product expansionsK(z)j±(w) ∼±1z −wj± ,¯K(¯z)¯j±( ¯w) ∼±1¯z −¯w¯j± ,(21)are not quantized.We now gauge the U(1) symmetry in the action (11) by following Eq.

(7).We consider the case that the U(1) symmetry is generated by σ2. Since the U(1)symmetry is compact, the resulting conformal field theory describes the Euclideanblack hole.

The theory of the Lorentzian black hole can be obtained by replacingσ2 by σ3 or simply by analytic continuating ΦL,R →iΦL,R.By the parametrization (10) the SU(1, 1)/U(1) gauged SWZW action takesthe formSSU(1,1)/U(1) =SSU(1,1) + k2πZd2zd2θ[4(1 + coshR)A ¯A+ 2iA( ¯DΦL + coshR ¯DΦR) + 2i(DΦL + coshR DΦR) ¯A] . (22)Here SSU(1,1) is SU(1, 1) SWZW action in Eq.(11).

By using the following redefi-nitions,Φ ≡ΦL −ΦR ,A′ ≡A + i2coshR DΦL + DΦR1 + coshR,¯A′ ≡¯A + i2coshR ¯DΦR + ¯DΦL1 + coshR,(23)the action (22) can be rewritten as follows,SSU(1,1)/U(1) = k2πZd2zd2θ[12tanh2R2 DΦ ¯DΦ+ 12DR ¯DR + 4(1 + coshR)A′ ¯A′ . (24)The action (22) and (24) are invariant under the following infinitesimal gauge6

transformation corresponding to Eq. (9),δΦL = δΦR = Λ ,δA = −i2DΛ ,δ ¯A = −i2¯DΛ .

(25)We fix this gauge symmetry by imposing the gauge conditionΦL = −ΦR = ˜Φ . (26)By integrating gauge fields A and ¯A in the action (22) or (24),⋆and by integratingauxiliary fields, we obtain the following action,S(1) = kπZd2z[tanh2s(∂φ¯∂φ −∂¯ψ ¯ψ + ψ ¯∂ψ)−2 sinhscosh3s(ηψ ¯∂φ + ¯η ¯ψ∂φ) + 4 tanh2s η¯ηψ ¯ψ+ ∂s¯∂s −∂¯η ¯η + η ¯∂η] .

(27)Here we write superfields ˜Φ and R in terms of components:˜Φ = φ + θψ + ¯θ ¯ψ + θ¯θf ,R2 = s + θη + ¯θ¯η + θ¯θg. (28)This system has N = 1 supersymmetry since the starting action (22) and gaugecondition (25) are manifestly supersymmetric.

In fact, this action is nothing butthe action of (1,1) supersymmetric σ model[13]in two dimensional black hole back-ground.⋆The integration of the gauge fields induces the dilaton term in the action but we nowneglect this term. The gauge fixed action which is correct at the quantum level is givenlater in this paper.7

The N = 1 supersymmetry in the action (27) is extended to N = 2 super-symmetry since this action is invariant under the following holomorphic (anti-holomorphic) U(1) symmetry:δψ = −u(z)tanhsηδη =u(z)tanhs ψ ,(29)δ ¯ψ = −¯u(¯z)tanhs ¯ηδ¯η =¯u(¯z)tanhs ¯ψ . (30)Here u(z) (¯u(¯z)) is a holomorphic (anti-holomorphic) parameters of the transfor-mation.The transformations (29) and (30) tell that the currents of this U(1)symmetry can be regarded as fermion number currents with respect to space-timefermion coordinates, η, ψ, ¯η and ¯ψ.

By commuting this U(1) symmetry transfor-mation with the original N = 1 supersymmetry transformation, we obtain anothersupersymmetry transformation and we find that the action has N = 2 supersymme-try. On the other hand, in case of the Lorentzian black hole, the obtained algebra isnot exactly N = 2 superconformal algebra.† Usual N = 2 superconformal algebrais given by{G+n , G−m} =4Ln+m + 2(m −n)Jn+m + c12m(m2 −1)δm+n,0 ,[Jn, G±m] = ± Gn+m ,etc.

(31)and the hermiticities of the operators are assigned by(G+n )† = G−−n ,J†n = J−n . (32)The algebra which appears in the Lorentzian case is identical with Eq.

(31), but†Note that any Lorentzian manifold is not K¨ahler.8

the assignment of the hermiticities is different from Eq. (32):(G+n )† = G+−n ,(G−n )† = G−−n ,J†n = −J−n .

(33)This is not so surprizing since this algebra also appears in flat two dimensionalLorentzian space-time which is a subspace of flat ten dimensional space-time inusual Neveu-Schwarz-Ramond model. Even in Lorentzian case, we have a U(1)current and superstring theories can be constructed by imposing GSO projection.In order to consider the spectrum of this theory, we choose the following gaugecondition instead of Eq.

(25) ,¯DA −D ¯A = 0 . (34)This gauge conditon allows us to parametrize the gauge fields A and ¯A asA = DΠ ,¯A = −¯DΠ .

(35)By shifting the fields ΦL,R,ΦL →ΦL + 2iΠ ,ΦR →ΦR −2iΠ ,(36)the gauge fixed action S(2) is given by a sum of SU(1, 1) SWZW action SSU(1,1)in Eq. (11), free field action SΠ and (free) ghost action SFP.S(2) =SSU(1,1) + SΠ + SFP ,SΠ = −4kπZd2zd2θDΠ ¯DΠ ,SFP = k2πZd2zd2θBD ¯DC .

(37)Here B and C are anti-ghost and ghost superfields.9

The BRS charge QB which defines the physical states is given byQB =IdzC(DΠ −i4kJ2) +Id¯zC( ¯DΠ + i4k¯J2) . (38)This BRS charge gives constraints on the physical states,DΠ −i2J2 = ¯DΠ + i2¯J2 = 0 ,(39)which tell that B, C, Π and J2 (or ¯J2) make so-called “quartet” structure[13]similarto the structure which appeared in the quantization of Neveu-Schwarz-Ramondmodel based on BRS symmetry.

[14,15]The action which describes superstring theory in the two dimensional black holeis simply given by a sum of SU(1, 1) WZW action (17), free fermion actions (16)and the actions of free superfield and free ghost and anti-ghost superfields (37).Furthermore the constraints (39) imposed by th BRS charge (38) can be easilysolved with respect to free superfields Π. Therefore if we can find the spectrumof the bosonic string in the two dimensional black hole,[1,2] we can also find thespectrum of this string theory.The U(1) current, which corresponds to the transformations (29) and (30) aregiven by,[5]J = −2ik −2ˆJ2 +kk −2K ,¯J = −2ik −2¯ˆJ2 +kk −2¯K .

(40)Here ˆJ2 and ¯ˆJ2 are defined by Eq. (14) and fermion number currents K and ¯K aredefined by Eq.(20).

These U(1) currents commute with the BRS charge (38) andwe can impose GSO projection consistently. Note that GSO projection does notgive any constraint on the representations of SU(1, 1) current algebra since theeigenvalues of the zero modes in the currents K and ¯K are not quantized althoughthose in J and ¯J are quantized.Recently the string model based on SU(1,1)×U(1)U(1)coset model was discussed.

[16,17]This model describes the strings in two[16]or three dimensional[17]charged black10

holes. By adjusting the radius of the U(1) boson, we will obtain N = 2 super-conformal theory with c > 3[7]in the same way as N = 2 minimal model wasconstructed from SU(2)×U(1)U(1)[18].

The obtained model should be equivalent to themodel discussed here.I would like to acknowledge discussions with N. Ishibashi, M. Li, J. Lykken andA. Strominger.

I am also indebted to M. Kato, E. Kiritsis, A. Sugamoto, T. Uchinoand S.-K. Yang for the discussion at the early stage. I wish to thank the theorygroups of SLAC, UC Santa Barbara and Fermilab, where this work was done, fortheir hospitalities.

I am grateful to Soryuushi Shougakkai for the financial support.11

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