We present two-parameter solutions of the low-energy four-dimensional
heterotic string which in the extremal limit reduce to supersymmetric monopole,
string and domain wall solutions. The effective scalar coupling to the Maxwell
field, $e^{-\alpha \phi} F_{\mu\nu} F^{\mu\nu}$, gives rise to a new string
black hole with $\alpha = \sqrt{3}$, in contrast to the pure dilaton black hole
solution which has $\alpha=1$. Implications of string/fivebrane duality in
$D=10$ to four-dimensional dualities are discussed.Comment: Based on talks given at MRST 94, McGill University and at
  Mini-workshop on Extended Objects, Newton Institute, March 1994, 4 pages, Te

Khuri, Ramzi R.

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We present two-parameter solutions of the low-energy four-dimensional heterotic string which in the extremal limit reduce to supersymmetric monopole, string and domain wall solutions. The effective scalar coupling to the Maxwell field, e^{-\alpha \phi} F_{\mu\nu} F^{\mu\nu}, gives rise to a new string black hole with \alpha = \sqrt{3}, in contrast to the pure dilaton black hole solution which has \alpha=1. Implications of string/fivebrane duality in D=10 to four-dimensional dualities are discussed.We present two-parameter solutions of the low-energy four-dimensional heterotic string which in the extremal limit reduce to supersymmetric monopole, string and domain wall solutions. The effective scalar coupling to the Maxwell field, $e~{-\alpha \phi} F_{\mu\nu} F~{\mu\nu}$, gives rise to a new string black hole with $\alpha = \sqrt{3}$, in contrast to the pure dilaton black hole solution which has $\alpha=1$. Implications of string/fivebrane duality in $D=10$ to four-dimensional dualities are discussed

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arXiv:hep-th/9412150v1  16 Dec 1994CERN-TH.7467/94McGill/94–51Black Hole Solutions in String TheoryRamzi R. Khuri†CERN, Theory DivisionCH-1211 Geneva 23, SwitzerlandandMcGill University, Physics DepartmentMontreal, PQ, H3A 2T8, CanadaABSTRACTWe present two-parameter solutions of the low-energy four-dimensional heteroticstring which in the extremal limit reduce to supersymmetric monopole, stringand domain wall solutions. The effective scalar coupling to the Maxwell field,e−αφFµνFµν , gives rise to a new string black hole with α =√3, in contrast to thepure dilaton black hole solution which has α = 1. Implications of string/fivebraneduality in D = 10 to four-dimensional dualities are discussed.In recent work1, supersymmetric soliton solutions of the four-dimensional het-erotic string were presented, describing monopoles, strings and domain walls. Thesesolutions admit the D = 10 interpretation of a fivebrane wrapped around 5, 4 or 3of the 6 compactified dimensions and are arguably exact to all orders in α′. Inthis talk, we extend all three solutions to two-parameter solutions of the low-energyequations of the four-dimensional heterotic string2. The two-parameter solution ex-tending the supersymmetric monopole corresponds to a magnetically charged blackhole, while the solution extending the supersymmetric domain wall corresponds toa black membrane. By contrast, the two-parameter string solution does not possessa finite horizon and corresponds to a naked singularity.All three solutions involve both the dilaton and the modulus fields, and are thusto be contrasted with pure dilaton solutions3. In particular, the effective scalarcoupling to the Maxwell field, e−αφFµνFµν , gives rise to a new string black hole withα =√3, in contrast to the pure dilaton solution of the heterotic string which has α = 13. It thus resembles the black hole arising from Kaluza-Klein theories which also hasα =√3, and which reduces to the Pollard-Gross-Perry-Sorkin4 magnetic monopolein the extremal limit. The fact that the heterotic string admits α =√3 black holesalso has implications for string/fivebrane duality5. Both electric/magnetic dualityand string/string duality in D = 4 may be seen as a consequence of string/fivebraneduality in D = 10.We begin with the two-parameter black hole. Inspired by the wrapping of aBased on talks given at MRST 94, McGill University, May 1994 and at Mini-workshop onExtended Objects, Newton Institute, March 1994 and condensed from work already published1,2.† Supported by a World Laboratory Fellowship.1fivebrane around five of the six compactified dimensions (x5, x6, x7, x8, x9), it wasshown1 that the tree-level effective action for the D = 10 heterotic string may bereduced to the following four-dimensional formS1 =12κ2∫d4x√−ge−2Φ−σ1(R+ 4(∂Φ)2 + 4∂σ1 · ∂Φ− 14e2σ1FµνFµν), (1)where µ, ν = 0, 1, 2, 3. Here gµν is the string sigma-model metric and Φ is the dilaton.In the case of toroidal compactification, with N = 4 supersymmetry in D = 4, σ1is a modulus field, g44 = e−2σ1 , and Fµν = Hµν4 where H = dB and B is the stringantisymmetric tensor. However, actions of this type also appear in a large class ofN = 1 supergravity theories6. The solution is given by2e−2Φ = e2σ1 =(1− r−r),ds2 = −(1− r+r)(1− r−r)−1dt2 +(1− r+r)−1dr2 + r2(1− r−r)dΩ22,Fθϕ =√r+r− sin θ, (2)where here, and throughout this paper, we set the dilaton vev Φ0 equal to zero.This represents a magnetically charged black hole with event horizon at r = r+ andinner horizon at r = r−. The magnetic charge and mass of the black hole are givenbyg1 =4pi√2κ(r+r−)12 ,M1 = 2piκ2(2r+ − r−). (3)Changing coordinates via y = r − r− and taking the extremal limit r+ = r− yields:e2Φ = e−2σ1 =(1 +r−y),ds2 = −dt2 + e2Φ (dy2 + y2dΩ22) ,Fθϕ = r− sin θ, (4)which is just the tree-level supersymmetric monopole solution without a Yang-Millsfield which saturates the Bogomol’nyi bound√2κM1 ≥ g1. Note that the monopolearises in the gravitational sector of the string, as can be seen from an earlier solutionfound in purely bosonic string theory7. Supersymmetric extensions of this solutionboth with and without gauge fields were found in 8.Next we derive a two-parameter string solution which, however, does not possessa finite event horizon and consequently cannot be interpreted as a black string.This is inspired by the wrapping of the fivebrane around four of the compactifieddimensions (x6, x7, x8, x9). The action is given byS2 =12κ2∫d4x√−ge−2Φ−2σ2(R + 4(∂Φ)2 + 8∂σ2 · ∂Φ+ 2(∂σ2)2 − 12e4σ2FµFµ), (5)2In the case of the torus, σ2 is the modulus field g44 = g55 = e−2σ2 and Fµ = Hµ45. Atwo-parameter family of solutions is now given by2e2Φ = e−2σ2 = (1 + k/2− λ ln y),ds2 = −(1 + k)dt2 + (1 + k)−1(1 + k/2− λ ln y)dy2 + y2(1 + k/2− λ ln y)dθ2 + dx23,Fθ = λ√1 + k, (6)where for k = 0 we recover the supersymmetric string soliton solution1 which is dualto the elementary string solution of Dabholkar et al9. The solution shown in Eq.(6) infact represents a naked singularity, since the event horizon is pushed out to r+ =∞,which agrees with the Horowitz-Strominger “no-4D-black-string” theorem10.Finally, we consider the two-parameter black membrane solution. In this case,we wrap the fivebrane around three of the compactified dimensions (x7, x8, x9). How-ever, the four-dimensional action necessary to yield membrane solutions is not ob-tained by a simple dimensional reduction of the ten-dimensional action because ofthe non-vanishing of F = H456. Instead, the effective action is obtained by treatingF 2 as a cosmological constant and is given byS3 =12κ2∫d4x√−ge−2Φ−3σ3(R+ 4(∂Φ)2 + 12∂σ3 · ∂Φ+ 6(∂σ3)2 − e6σ3 12F 2), (7)In the case of the torus, σ3 is the modulus field g44 = g55 = g66 = e−2σ3 . The two-parameter black membrane solution is then2e−2Φ = e2σ3 =(1− rr−),ds2 = −(1− rr+)(1− rr−)−1dt2 +(1− rr+)−1(1− rr−)−4dr2 + dx22 + dx23,F = −(r+r−)−1/2. (8)This solution represents a black membrane with event horizon at r = r+ and innerhorizon at r = r−. Changing coordinates via y−1 = r−1 − r−1− and taking the extremallimit yieldse2Φ = e−2σ3 =(1 +yr−),ds2 = −dt2 + dx22 + dx23 + e2Φdy2,F = − 1r−. (9)which is just the supersymmetric domain wall solution1.Consider the generic toroidal compactification of the heterotic string, where thefour-dimensional theory is given by N = 4 supergravity coupled to 22 N = 4 vectormultiplets. Then the Maxwell field Fµν in D = 4 (or its dual F˜µν) and the scalar fieldφ come from the D = 10 3-form (or 7-form) and dilaton plus modulus field of theheterotic string (or heterotic fivebrane). Thus, the D = 4 electric/magnetic dualitycan be interpreted as a reduction of D = 10 string/fivebrane duality.3Another reduction of string/fivebrane duality is D = 4 string/string duality1.The compactified heterotic string displays a target space duality O(6, 22, Z). It isalso conjectured to display the strong/weak coupling SL(2, Z) S-duality relating thedilaton and the axion, which is certainly there in the field theory limit. The “dualityof dualities” suggestion11,1 is that, under string/fivebrane duality, the roles of S andT dualities are interchanged. The picture that emerges is one in which the massivestates of the string correspond to extreme black holes.We have shown only that these two-parameter configurations are solutions of thefield theory limit of the heterotic string. Although the extreme one-parameter solu-tions are expected to be exact to all orders in α′, the same reasoning does not carryover to the new two-parameter solutions. It would be also interesting to see whetherthe generalization of the one-parameter solutions to the two-parameter solutions canbe carried out when we include the Yang-Mills coupling. This would necessarily in-volve giving up the self-duality condition on the Yang-Mills field strength, however,since the self-duality condition is tied to the extreme,√2κMp+1 = gp+1, supersymmet-ric solutions. Finally, there is the question of whether these solutions are peculiarto the toroidal compactification or whether they survive in more realistic orbifoldor Calabi-Yau models. Although the actions S1, S2 and S3 were originally derivedin the context of the torus1, they also appear in a large class of N = 1 supergravitytheories.References1. M. J. Duff and R. R. Khuri, Nucl. Phys. B411 (1994) 473.2. M. J. Duff, R. R. Khuri, R. Minasian and J. Rahmfeld, Nucl. Phys. B418(1994) 195.3. G. W. Gibbons, Nucl. Phys. B207 (1982) 337; G. W. Gibbons and K. Maeda,Nucl. Phys. B298 (1988) 741; D. Garfinkle, G. T. Horowitz and A. Strominger,Phys Rev. D43 (1991) 3140; R. Kallosh, A. Linde, T. Ortin, A. Peet and A. VanProeyen, Phys Rev. D46 (1992) 5278.4. D. Pollard, J. Phys. A16 (1983) 565; D. J. Gross and M. J. Perry, Nucl. Phys.B226 (1983) 29; R. D. Sorkin, Phys. Rev. Lett. 51 (1983) 87.5. M. J. Duff, Class. Quantum Grav. 5 (1988) 189; A. Strominger, Nucl. Phys.B343 (1990) 167; M. J. Duff and J. X. Lu, Nucl. Phys. B354 (1991) 129.6. E. Cremmer, S. Ferrara, L. Girardello and A. Van Proeyen, Nucl. Phys. B212(1983) 413.7. R. R. Khuri, Phys. Lett. B259 (1991) 261.8. R. R. Khuri, Phys. Lett. B294 (1992) 325; Nucl. Phys. B387 (1992) 315.9. A. Dabholkar, G. Gibbons, J. A. Harvey and F. R. Ruiz, Nucl. Phys. B340(1990) 33.10. G. T. Horowitz and A. Strominger, Nucl. Phys. B360 (1991) 2930.11. J. H. Schwarz and A. Sen, Phys. Lett. B312 (1993) 105; P. Bine´truy, Phys.Lett. B315 (1993) 80; A. Sen Int. J. Mod. Phys. A9 (1994) 3707.4

Black Hole Solutions in String Theory

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