Integrability of Nonlinear Equations of Motion on Two-Dimensional World Sheet Space-Time
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1 Commun. Theor. Phys. (Beijing, China 44 (005 pp c International Academic Publishers Vol. 44, No., August 5, 005 Integrability of Nonlinear Equations of Motion on Two-Dimensional World Sheet Space-Time YAN Jun Department of Physics, Sichuan Normal University, Chengdu 60066, China (Received November 30, 004 Abstract The integrability character of nonlinear equations of motion of two-dimensional gravity with dynamical torsion and bosonic string coupling is studied in this paper. The space-like and time-like first integrals of equations of motion are also found. PACS numbers: h Key words: two-dimensional gravity model, string coupling, integrability Introduction Two-dimensional gravity model is an interesting toy model for studying four-dimensional gravity and its quantization. In Ref. [], Katanaev and Volovich proposed a two-dimensional theory of gravity with dynamical metric and torsion. The instanton-like solutions and eigenmodes of the Katanaev Volovich model (K-V model have been found by Akdeniz et al. [] Moreover, the integrable and solvable characters of the equations of motion in K-V model are analyzed by Katanaev, [35] who also investigated the canonical quantization of two-dimensional gravity model. [6] The black-hole solutions in D gravity with torsion were examined by Solodukhin. [7] On the quantum level, it was shown that this model is renormalizable. [8] In our previous work, the stationary solutions of equation of motion in K-V model with string coupling were obtained by means of numerical integral method. [9] Bosonic String and Two-Dimensional Gravity Model with Dynamical Torsion The Lagrangian of two-dimensional gravity model with Nambu Goto bosonic string coupling is [] L = 4πα egαβ α X µ β X µ γ 4 er abcd β 4 et abc λe, ( where e is a zweibein field, g αβ (ξ is the metric in twodimensional manifold M with local coordinates ξ α (α = 0, = (τ, σ. M can be taken as a two-dimensional worldsheet spanned by the string along its evolution. α, γ, and β are coupling constants, λ is a cosmological constant. X µ (ξ is a string position variable (µ = 0,,..., d, the curvature tensor and torsion tensor can be written in terms of the metrical connection, R δ αβγ = α Γ δ βγ Γ ε αγγ δ βε (α β, ( T γ αβ = T γ βα = Γγ αβ Γγ βα, (3 which can also be expressed as another form in terms of Lorentz connection ωα ab = ωα ba, Rαβ ab = α ωβ ab ωα ac ωβc b (α β, T a αβ = α e α β ω ab α e βb (α β. (4 In the two-dimensional manifold M, the Lorentz connection and the curvature tensor can be expressed in terms of a pseudovector field B α, ω ab α = B α ε ab, R ab αβ = F αβ ε ab, F αβ = α B β β B α. (5 These formulations are very useful for analyzing the integrable character of the model. The Lagrangian ( yields the following second-order equations of motion: πα α α X µ = 0, (6 γ β F βα βt α abε ab = 0, (7 β β T βα a + βt abc T abc β 4 T bcde α a γf αβ + γ F bce α a λe α a 4πα βx µ β X µ e α a πα α X µ α X µ = 0. (8 The physical meaning of Eq. (8 can be explained as the interaction between the bosonic string and the worldsheet internal geometry. 3 Integrability Character of Equations of Motion In what follows, we investigate the integrable and solvable characters of the model within the framework of conformal gauge. By means of the general coordinate and local Lorentz transformations, the zweibein field can be transformed into the conformal gauge e a α = e ϕ δ a α, (9 The project supported by the Science Foundation of Sichuan Normal University
2 84 YAN Jun Vol. 44 where ϕ(ξ is a scalar field. A solution of Eq. (7 is B α = ε αβ β χ, where χ is a scalar field. After fixing the gauge, the equations of motion and two constraints on -D space-time take the following form, [] ( 0 X µ = 0, (0 ( 0 χ + β γ (ϕ χ eϕ = 0, ( ( 0 ϕ γ β eϕ + β (ϕ χ eϕ γ + β γ (ϕ χ e ϕ = 0, ( β (ϕ χ + β ( 0ϕ + ϕ 0 χ χ λ e ϕ + β 4γ (ϕ χ eϕ 4πα 0X µ 0 X µ 4πα X µ X µ = 0, (3 β( 0 ϕ + 0 χ + 0 ϕ ϕ 0 χ χ πα 0X µ X µ = 0. (4 Proposition Equations of motion (0 (4 are equivalent to the following set of equations: ( 0 X µ = 0, (5 ( 0 f + (f Λ e ϕ = 0, (6 ( 0 ϕ + (f + f Λ e ϕ = 0, (7 f + f + f ϕ f ϕ f (f Λ e ϕ 4πα γ ẊµẊµ 4πα γ X µx µ = 0, (8 f + ff fϕ f ϕ 4πα γ ẊµX µ = 0. (9 Let us introduce new variable and define f = ϕ χ, (0 β = γ, Λ = λ γ. ( It is also convenient to introduce the following notations: ϕ = ϕ f = f, ϕ = ϕ, Ẋµ = Xµ f = f X µ = Xµ. ( Proof The proof is based on some algebraic manipulations. Equations (5 and (7 follow from Eqs. (0 and (. Equation (6 is a linear combination of Eqs. ( and (. Since f + f ϕ f ϕ f = τ χ ( χ + τ τ χ substituting Eq. (3 into Eq. (3 yields Eq. (8. Since ff fϕ f ϕ = τ = τ χ ( χ τ τ χ ϕ τ τ χ τ ϕ χ = 0 ϕ ϕ + 0 χ + χ, (3 ϕ τ τ χ τ ϕ τ χ = ( 0 ϕ ϕ 0 χ χ, (4 substituting Eq. (4 into Eq. (4 we obtain Eq. (9. Proposition In light cone coordinates ξ = σ τ, η = σ + τ, equations of motion (5 (9 take the following form: X µ ξη = 0, (5 4η + (f Λ e ϕ = 0, (6 4ϕ ξη + (f + f Λ e ϕ = 0, (7 η + fη ϕ η + 6πα γ (X ξµ X ηµ (X µ ξ Xµ η + 6πα γ (X ξµ + X ηµ (X µ ξ + Xµ η 8πα γ (X ξµ X ηµ (X µ ξ + Xµ η = 0, (8 ξ + fξ ϕ ξ + 6πα γ (X ξµ X ηµ (X µ ξ Xµ η + 6πα γ (X ξµ + X ηµ (X µ ξ + Xµ η where Proof 8πα γ (X ξµ X ηµ (X µ ξ + Xµ η = 0, (9 = f X µ ξ = Xµ = f η, Xµ η = Xµ η. ϕ ξ = ϕ In light cone coordinates, we have ( 0 4 η ϕ η = ϕ η, τ η
3 No. Integrability of Nonlinear Equations of Motion on Two-Dimensional World Sheet Space-Time 85 η + ξ. (30 Then, equations (5 (7 follow from Eqs. (5 (7 and Eq. (6 + Eq. (8 + Eq. (9 yields the following equation: ( 0 f + f + f + f + f + ff ( ϕ f + ϕf + ϕ f + ϕ f 4πα γ (ẊµẊµ + X µx µ πα γ ẊµX µ = 0. (3 Since ( 0 f = 4η, (3 f = f = ( + = η + η + ξ, (33 f = = (fξ + = fη + + fξ, (34 f = ( τ f = ( = fη + fξ, (35 f = τ ( + = η ξ, (36 ff = τ f f = ( ( + = fη fξ, (37 then ( 0 f + f + f + f + f + ff = 4η + (η + η + ξ + (f η + f ξ + (f η + + f ξ + (η ξ + (f η f ξ = 4η + 4f η, (38 ( ϕ f + ϕ f + ϕ f + ϕf = (ϕ η ϕ ξ ( + (ϕ η + ϕ ξ ( + + (ϕ η + ϕ ξ ( + (ϕ η ϕ ξ ( + = 4ϕ η. Substituting Eqs. (38 and (39 into Eq. (3, we obtain Eq. (8. Since f + ff fϕ f ϕ = (η ξ + (fη fξ (ϕ η + ϕ ξ ( (ϕ η ϕ ξ ( + = (η + f η ϕ η (ξ + f ξ ϕ ξ. (40 (39 Substituting Eqs. (40 and (9 into Eq. (8, we obtain Eq. (9. Lemma When X µ ξ = 0, Xµ η 0, equation (8 is one of the first integrals of Eqs. (6 and (7. Proof From Eq. (5, we have X µ ξ = Cµ (ξ, or Xµ η = C µ (η. Now we take Cµ (ξ = 0, Cµ (η 0, then equation (8 yields the following equation: η + f η ϕ η = 0. (4 Let us differentiate Eq. (6 with respect to η, 4ηη + f e ϕ + ϕ η e ϕ (f Λ = 0. (4 According to the linear combination of the following equations 4 Eq. (6 η Eq. (6 Eq. (7 ϕ η Eq. (6, we obtain So (43 ηη + η ϕ ξη η ϕ η = 0. (44 d (η + f η ϕ η = 0, (45 and integration over ξ yields the equation η + f η ϕ η = H(η, (46 where H(η is an arbitrary function o. Hence equation (8 is one of the first integral of equations (6 and (7, which corresponds to H(η = 0. Lemma When X µ ξ 0, Xµ η = 0, equation (9 is one of the first integrals of Eqs. (6 and (7. Proof Take C µ (ξ 0, Cµ (η = 0, and repeating the proof of Lemma with the change of coordinates ξ η, we succeed in proving that equation (9 is one of the first integrals of Eqs. (6 and (7, which corresponds to H(ξ = 0. Theorem On two-dimensional world sheet region D, let 0, 0, the space-like and time-like first integrals of equations of motion (5 (9 have the following form, (i Space-like first integrals: 4 = (f f + Λ exp f C (f f + Λ C
4 86 YAN Jun Vol. 44 exp f C 4 = (f f + Λ exp (f f + Λ C exp f C + A, (47 f C + A, (48 where η = σ + constant, ξ = σ constant, while C, and A are integral constants. (ii Time-like first integrals: 4 = (f f + Λ exp f C (f f + Λ C exp f C 4 = (f f + Λ exp (f f + Λ C exp f + C + A, (49 f + C + A. (50 where η = constant τ, ξ = constant + τ, C, and A are integral constants. Proof Equation (5 yields the solutions X µ ξ = Cµ or X η µ = C µ (constant, then equations (8 and (9 become η + f η ϕ η = C 4πα γ, (5 ξ + f ξ ϕ ξ = C 4πα γ, (5 where C = C µ C µ and C = C µ C µ. (i Space-like first integrals: Let C = C, equations (5 and (5 take the following form: η + f η ϕ η = C 4πα γ, (53 ξ + fξ ϕ ξ = C 4πα γ. (54 Equations (53 and (54 can be integrated after dividing them by ( 0 and ( 0, respectively, C ln + f ϕ =, (55 C ln + f ϕ =, (56 where C = C /4πα γ. Let us consider τ = constant, then η = σ + constant and ξ = σ constant which leads to η = ξ + constant, and = =, (57 /f ( ξ = =. / (58 Substitution of = and Eqs. (55 and (56 into Eq. (6 yields the following equations: 4η + (f Λ exp f C = 0, (59 4ξ + (f Λ exp f C = 0. (60 The first integrals of the above equations are [9,0] 4 = (f f + Λ exp f C (f f + Λ C exp f C + A, (6 4 = (f f + Λ exp f C (f f + Λ C exp f C + A, (6 where C and A are integral constants. (ii Time-like first integrals: Letting C = C, equations (5 and (5 take the following form, η + f η ϕ η = C 4πα γ, (63 C ξ ξ + fξ ϕ ξ = 4πα γ. (64 Equations (53 and (54 can be integrated after dividing them by ( 0 and ( 0 respectively, C ln + f ϕ =, (65 C ln + f ϕ =, (66 where C = C/4πα γ. Let us consider σ = constant, then η = constant τ, ξ = constant + τ, which yield η + ξ = constant and = =, (67 / / = ( =. (68
5 No. Integrability of Nonlinear Equations of Motion on Two-Dimensional World Sheet Space-Time 87 Substitution of = and Eqs. (65 and (66 into Eq. (6 yields the following equations: 4η + (f Λ exp f C = 0, (69 4ξ + (f Λ exp f + C = 0. (70 The first integrals of the above equations are 4 = (f f + Λ exp f C (f f + Λ C exp f C 4 = (f f + Λ exp (f f + Λ C + A, (7 f + C exp f + C where C and A are integral constants. 4 Conclusion + A, (7 In summery, the integrability of nonlinear equations of motion in D gravity model with bosonic string coupling was investigated in this paper. The new spacelike and time-like first integrals of equations of motion on world sheet space-time were found. The velocity of a massive point-like particle can be expressed by these first integrals. [] The similar solvable characters in twodimensional gravity coupled to the nonlinear matter fields were analyzed and discussed, [3,4] an electric sine-gordon soliton solution in D dilaton gravity is obtained through coordinate transformation. [5] These results may be useful for studying the physical properties of two-dimensional dilaton gravity model. References [] M. Katanaev and I. Volovich, Ann. Phys. (NY 97 (990. [] K. Akdeniz, A. Kizilersu, and E. Rizaoglu, Phys. Lett. B 5 ( [3] M. Katanaev, J. Math. Phys. 3 ( [4] M. Katanaev, J.Math. Phys. 3 ( [5] M. Katanaev, J. Math. Phys. 34 ( [6] M. Katanaev, Nucl. Phys. B 46 ( [7] S. Solodukhin, Phys. Lett. B 39 ( [8] W. Kummerand and D. Schwarz, Nucl. Phys. B 38 (99 7. [9] J. Yan, B.Y. Tao, and S.K. Hu, High Energy Phys. and Nucl. Phys. 7 (993 3 (in Chinese. [0] X.M. Qiu, J. Yan, and D.Y. Peng, Gen. Rel. Grav. 9 ( [] D.Y. Peng, J. Yan, and X.M. Qiu, Chin. J. Comput. Phys. (in Chinese 5 ( [] J. Yan and B.Y. Tao, J. of Sichuan Normal University (Natural Science 7 ( (in Chinese. [3] J. Yan and X.M. Qiu, Gen. Rel. Grav. 30 ( [4] J. Yan, S.J. Wang, and B.Y. Tao, Commun. Theor. Phys. (Beijing, China 35 (00 9. [5] J. Yan and B.Y. Tao, High Energy Phys. and Nucl. Phys. 7 ( (in Chinese.
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