The сigar soliton and the Ricci flows perturbation solutions in the two-dimensional string $\sigma$ model

Jun Yan

Teoretičeskaâ i matematičeskaâ fizika, Tome 215 (2023) no. 1, pp. 97-110

Résumé

The Ricci flows perturbation equations and the Weyl anomaly coefficients are derived in the two-dimensional bosonic string $\sigma$ model. These equations correspond to the two-loop flow equations for the graviton field $g_{\mu\nu}$ and the dilaton field $\Phi$. The one-loop perturbation solution of the cigar soliton can be expressed in terms of the hypergeometric functions. The two-loop asymptotic perturbation solution of the cigar soliton is reduced by using a small parameter expansion method. Moreover, analytic solutions of the second basic form $l$ and $n$ are obtained in accordance with the perturbation Gauss–Codazzi equations. The modified expression of the deformed principal curvatures of a two-dimensional surface can then be given in terms of $l$ and $n$. The influence of quantum Ricci flows on the space–time geometry is analyzed and discussed, and the physical meaning of the Weyl anomaly coefficients varying with the momentum scale $\lambda$ is also explained.

Détail

Mots-clés : cigar soliton, perturbation solutions
Keywords: Ricci flows, two-dimensional string $\sigma$ model.

Bibliographie

[1] R. S. Hamilton, “Three-manifolds with positive Ricci curvature”, J. Different. Geom., 17:2 (1982), 255–306 | DOI | MR

[2] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv: math/0211159

[3] S. Brendle, Ricci Flow and the Sphere Theorem, Graduate Studies in Mathematics, 111, Amer. Math. Soc., Providence, RI, 2010 | DOI | MR

[4] Y. Li, “Generalized Ricci flow I: Higher-derivative estimates for compact manifolds”, Anal. PDE, 5:4 (2012), 747–775 | DOI | MR

[5] Y. Li, “Generalized Ricci flow II: Existence for complete noncompact manifolds”, Differ. Geom. Appl., 66 (2019), 109–154 | DOI | MR

[6] D. Friedan, “Nonlinear models in $2+\varepsilon$ dimensions”, Phys. Rev. Lett., 45:13 (1980), 1057–1060 | DOI | MR

[7] D. Friedan, “Nonlinear models in $2+\varepsilon $ dimensions”, Ann. Phys., 163:2 (1985), 318–419 | DOI | MR

[8] L. Alvarez-Gaumé, D. Z. Freedman, S. Mukhi, “The background field method and the ultraviolet structure of the supersymmetry nonlinear $\sigma$-model”, Ann. Phys., 134:1 (1981), 85–109 | DOI

[9] E. Abdalla, M. C. B. Abdalla, K. D. Rothe, Nonperturbative Methods in Two-Dimensional Quantum Field Theory, World Sci., Singapore, 1991 | DOI | MR

[10] C. G. Callan, D. Friedan, E. J. Martinec, M. J. Perry, “Strings in back-ground fields”, Nucl. Phys. B, 262:4 (1985), 593–609 | DOI | MR

[11] J. Yan, B.-Y. Tao, S.-J. Wang, “Temperature duality on Riemann surface and string cosmological solutions for genus $g=1$ and $g=2$”, Chinese Phys. C, 23:7 (1999), 655–664

[12] R. R. Metsaev, A. A. Tseytlin, “Two-loop $\beta$-functions for the generalized bosonic sigma model”, Phys. Lett. B, 191:4 (1987), 354–362 | DOI

[13] R. R. Metsaev, A. A. Tseytlin, “Order $\alpha'$ (two-loop) equavalence of the string equations of motion and the $\sigma$-model Weyl invariance conditons: Dependence on the dilaton and the antisymmetric tensor”, Nucl. Phys. B, 293 (1987), 385–419 | DOI | MR

[14] A. A. Tseytlin, “Sigma model approach to string theory”, Internat. J. Modern Phys. A, 4:6 (1989), 1257–1318 | DOI | MR

[15] C. V. Johnson, D-Branes, Cambridge Monographs on Mathematical Physics, Cambridge Univ. Press, Cambridge, 2003 | MR

[16] N. D. Birrell, P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, 7, Cambridge Univ. Press, Cambridge, 1982 | MR

[17] P. Figueras, J. Lucietti, T. Wiseman, “Ricci solitons, Ricci flow and strongly coupled CFT in the Schwarzschild Unruh or Boulware vacua”, Class. Quantum Grav., 28:21 (2011), 215018, 40 pp. | DOI | MR

[18] E. Woolgar, “Some applications of Ricci flow in physics”, Can. J. Phys., 86:4 (2008), 645–651 | DOI

[19] M. Nitta, “Conformal sigma models with anomalous dimensions and Ricci solitons”, Modern Phys. Lett. A, 20:8 (2005), 577–584 | DOI | MR

[20] C. Lambert, V. Suneeta, “Stability analysis of the Witten black hole (cigar soliton) under world-sheet renoralization group flow”, Phys. Rev. D, 86:8 (2012), 084041, 7 pp. | DOI

[21] V. Husain, S. Seahra, “Ricci flows, wormholes and critical phenomena”, Class. Quantum Grav., 25:22 (2008), 222002, 9 pp. | DOI | MR

[22] J. Samwel, S. R. Chowdhury, “Energy, entropy, and the Ricci flow”, Class. Quant. Grav., 25:3 (2008), 035012, 17 pp. | DOI | MR

[23] M. Headrick, T. Wiseman, “Ricci flow and black holes”, Class. Quantum Grav., 23:23 (2006), 6683–6707 | DOI | MR

[24] A. A. Tseytlin, “Sigma model renormaliztion group flow, “central charge” actoin, and Perelman's entropy”, Phys. Rev. D, 75:6 (2007), 064024, 6 pp. | DOI | MR

[25] V. Ruchin, O. Vacaru, S. I. Vacaru, “On relativistic generalization of Perelman's W-entropy and thermodynamic description of gravitational fields and cosmology”, Eur. Phys. J. C, 77:3 (2017), 184, 27 pp. | DOI

[26] S. I. Vacaru, “Nonholonomic Ricci flows. II. Evolution equations and dynamics”, J. Math. Phys., 49:4 (2008), 043504, 27 pp. | DOI | MR

[27] J. Samuel, S. R. Chowdhury, “Geometric flows and black hole entropy”, Class. Quantum Grav., 24:11 (2007), F47–F54 | DOI | MR

[28] I. Bakas, D. Orlando, P. M. Petropoulos, “Ricci flows and expansion in axion-dilaton cosmology”, JHEP, 01 (2007), 040, 27 pp. | DOI | MR

[29] T. Oliynyk, V. Suneeta, E. Woolgar, “Metric for gradient renormalizatoin group flow of the worldsheet sigma model beyond first order”, Phys. Rev. D, 76:4 (2007), 045001, 7 pp. | DOI | MR

[30] T. Oliynyk, V. Suneeta, E. Woolgar, “A gradient flow for worldsheet nonlinear sigma models”, Nucl. Phys. B, 739:3 (2006), 441–458 | DOI | MR

[31] T. A. Oliynyk, “The second-order renormalization group flow for nonlinear sigma models in two dimensions”, Class. Quantum Grav., 26:10 (2009), 105020, 8 pp. | DOI | MR

[32] J. R. Mureika, P. Nicolini, “Aspects of noncommutative $(1+1)$-dimensional black holes”, Phys. Rev. D, 84:4 (2011), 044020, 12 pp. | DOI

[33] J.-G. Peng, Q. Chen, Differential Geometry, Higher Education Press, Beijing, 2002 (in Chinese)

[34] A. Popov, Lobachevsky Geometry and Modern Nonlinear Problems, Birkäuser, Cham, 2014 | DOI | MR

[35] A. A. Slavnov, L. D. Faddeev, Vvedenie v kvantovuyu teoriyu kalibrovochnykh polei, Nauka, M., 1988 | MR

[36] S. Das, K. Prabhu, S. Kar, “Higher order geometric flows on three-dimensional locally homogeneous spaces”, J. Math. Phys., 54:1 (2013), 013509, 29 pp. | DOI | MR

[37] J. Gegenberg, G. Kunstatter, “The generalized Ricci flow for three-dimensional manifolds with one Killing vector”, J. Math. Phys., 47:3 (2006), 032304, 6 pp. | DOI | MR

[38] E. Witten, “String theory and black holes”, Phys. Rev. D, 44:2 (1991), 314–324 | DOI | MR

[39] R. Dijkgraaf, H. L. Verlinde, E. P. Verlinde, “String propagation in a black hole geometry”, Nucl. Phys. B, 371:1–2 (1992), 269–314 | DOI | MR

[40] I. Bars, K. Sfetsos, “Conformally exact metric and dilaton in string theory on curved spacetime”, Phys. Rev. D, 46:10 (1992), 4510–4519 | DOI | MR