Four-dimensional stringy black membrane.

An exact conformal field theory describing a four dimensional singular string background is obtained by chiral gauging a U(1) subgroup along with translations in R of an SL(2, R)×R Wess-Zumino-Witten model. It is shown that the target space-time describes a four dimensional black membrane. Furthermore various duality transformed solutions are constructed. These are also shown to correspond to various forms of four dimensional black membranes. IP/BBSR/92-35 May ’92. e-mail:supriya%[email protected] e-mail:pratik%[email protected] e-mail:sengupta%[email protected] In recent years the analogs of singular solutions in general theory of relativity have come to play an important role in the understanding of string theories in curved backgrounds. Intense investigation in this direction has established the existence of black holes and their higher dimensional generalizations (black strings and p-branes) as solutions to the low energy string equations of motion [1]. However the exact conformal field theories describing such singular string backgrounds were not understood. Very recently the existence of a graviton dilaton string background describing a two dimensional target space black hole was established in [2]. It was found immediately afterwards [3] that the exact conformal field theory describing this two dimensional black hole may be obtained by gauging the axial or the vector U(1) subgroup of an SL(2, R) Wess-Zumino-Witten (WZW) [4, 5] model. Considerable activity over the past year has unravelled the structure of this two dimensional black hole and its related generalization (black strings and p-branes) in string theories [6, 7, 8, 9, 10]. In an interesting article [11] Chung and Tye recently showed that an alternative anomaly free, left right asymmetric, chiral gauging of the WessZumino-Witten model was also possible. Recently one of us [SKK] in collaboration with A.Kumar has shown [12] that the target space-time of a chiral gauged SL(2,R) U(1) WZW model corresponds to a three dimensional black string similar to Horne and Horowitz [10]. In this article we construct an exact conformal field theory corresponding to a four dimensional black mem1 brane by chiral gauging the U(1) subgroup along with translations in R of an SL(2, R)×R WZWmodel. We also discuss various duality transformed solutions exploiting the isometries present. Furthermore we show that the target space-time structure leads to a product of a two dimensional black hole [3] and two non compact flat directions under certain simplifying assumptions. The action for the chiral gauged SL(2, R) WZW model is [11, 12] S = S + k 2π ∫ dz Tr [ ARz U ∂̄U + ALz̄ ∂U U −1 + ARz U A z̄U ]. (1) Where Az (A L z̄ ) in eq.(1) are the z(z̄) component of gauge field A R μ (A L μ) and U(z, z̄) ∈ SL(2, R). To obtain the conformal field theory describing a four dimensional string background we add a single non compact boson X representing the R direction to the WZW action. The coupling of the gauge field to the bosonic field X is given by a modification of the gauge invariant action of Isibashi et al [9]. Notice that the complete independence of the left and the right gauge fields in the chiral gauged model allows us the freedom of choosing different couplings of the bosonic field X to the gauge field. We have the following action; SG = 1 2π ∫ dz[∂X∂̄X + eLAz̄ ∂X + eRAz ∂̄X]. Adding this action to the WZW action with the gauge field assignments Az̄ ≡ ( − i 2Az̄ σ2), Az ≡ (− i 2Az σ2) we arrive at the following action. S = SWZW + 1 2π ∫ dz∂X∂̄X + k 2π ∫ dz [ − 1 2 ARz ( Tr[( 0 1 −1 0 )U ∂̄U ]− 2eR k ∂̄X )