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Geometric flows in Hořava-Lifshitz gravity gravity. (English) Zbl 1272.83026

Summary: We consider instanton solutions of Euclidean Hořava-Lifshitz gravity in four dimensions satisfying the detailed balance condition. They are described by geometric flows in three dimensions driven by certain combinations of the Cotton and Ricci tensors as well as the cosmological-constant term. The deformation curvature terms can have competing behavior leading to a variety of fixed points. The instantons interpolate between any two fixed points, which are vacua of topologically massive gravity with \(\Lambda > 0\), and their action is finite. Special emphasis is placed on configurations with SU(2) isometry associated with homogeneous but generally non-isotropic Bianchi IX model geometries. In this case, the combined Ricci-Cotton flow reduces to an autonomous system of ordinary differential equations whose properties are studied in detail for different couplings. The occurrence and stability of isotropic and anisotropic fixed points are investigated analytically and some exact solutions are obtained. The corresponding instantons are classified and they are all globally \( \mathbb{R} \times {S^3} \) and complete spaces. Generalizations to higher-dimensional gravities are also briey discussed.

MSC:

83C45 Quantization of the gravitational field
83C20 Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
35C08 Soliton solutions
14D21 Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory)
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[1] E.M. Lifshitz, On the theory of second order phase transitions I & II, Zh. Eksp. Teor. Fiz.11 (1941) 255 & 269.
[2] S. Chadha and H.B. Nielsen, Lorentz invariance as a low-energy phenomenon, Nucl. Phys.B 217 (1983) 125 [SPIRES]. · doi:10.1016/0550-3213(83)90081-0
[3] J. Iliopoulos, D.V. Nanopoulos and T.N. Tomaras, Infrared stability or anti grand unification, Phys. Lett.B 94 (1980) 141 [SPIRES].
[4] I. Antoniadis, J. Iliopoulos and T. Tomaras, On the infrared stability of gauge theories, Nucl. Phys.B 227 (1983) 447 [SPIRES]. · doi:10.1016/0550-3213(83)90568-0
[5] M. Petrini, Infrared stability of N = 4 super Yang-Mills theory, Phys. Lett.B 404 (1997) 66 [hep-th/9704004] [SPIRES].
[6] P. Hořava, Membranes at Quantum Criticality, JHEP03 (2009) 020 [arXiv:0812.4287] [SPIRES].
[7] P. Hořava, Quantum Gravity at a Lifshitz Point, Phys. Rev.D 79 (2009) 084008 [arXiv:0901.3775] [SPIRES].
[8] D. Orlando and S. Reffert, On the Renormalizability of Hořava-Lifshitz-type Gravities, Class. Quant. Grav.26 (2009) 155021 [arXiv:0905.0301] [SPIRES]. · Zbl 1172.83323 · doi:10.1088/0264-9381/26/15/155021
[9] F.-W. Shu and Y.-S. Wu, Stochastic Quantization of the Hořava Gravity, arXiv:0906.1645 [SPIRES].
[10] C. Charmousis, G. Niz, A. Padilla and P.M. Saffin, Strong coupling in Hořava gravity, JHEP08 (2009) 070 [arXiv:0905.2579] [SPIRES]. · doi:10.1088/1126-6708/2009/08/070
[11] M. Li and Y. Pang, A Trouble with Hořava-Lifshitz Gravity, JHEP08 (2009) 015 [arXiv:0905.2751] [SPIRES].
[12] D. Blas, O. Pujolàs and S. Sibiryakov, On the Extra Mode and Inconsistency of Hořava Gravity, JHEP10 (2009) 029 [arXiv:0906.3046] [SPIRES]. · doi:10.1088/1126-6708/2009/10/029
[13] D. Blas, O. Pujolàs and S. Sibiryakov, A healthy extension of Hořava gravity, arXiv:0909.3525 [SPIRES]. · Zbl 1250.83031
[14] D. Blas, O. Pujolàs and S. Sibiryakov, Comment on “Strong coupling in extended Hořava-Lifshitz gravity’, arXiv:0912.0550 <RefTarget Address=”http://arxiv.org/abs/0912.0550“ TargetType=”URL“/> [SPIRES <RefTarget Address=”http://www-spires.slac.stanford.edu/spires/find/hep/www?eprint=0912.0550“ TargetType=”URL”/> ]. · Zbl 1250.83031
[15] K. Koyama and F. Arroja, Pathological behaviour of the scalar graviton in Hořava-Lifshitz gravity, JHEP03 (2010) 061 [arXiv:0910.1998] [SPIRES]. · Zbl 1271.83033 · doi:10.1007/JHEP03(2010)061
[16] R. Hamilton, Three-manifolds with positive Ricci curvature, J. Diff. Geom.17 (1982) 255. · Zbl 0504.53034
[17] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math/0211159 [SPIRES]. · Zbl 1130.53001
[18] G. Perelman, Ricci flow with surgery on three-manifolds, math/0303109 [SPIRES]. · Zbl 1130.53002
[19] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, math/0307245 [SPIRES]. · Zbl 1130.53003
[20] H.-D. Cao, B. Chow, S.-C. Chu and S.-T. Yau eds., Collected Papers on Ricci Flow, Series in Geometry and Topology, 37, International Press, Somerville U.S.A. (2003). · Zbl 1108.53002
[21] J.W. Morgan and G. Tian, Ricci Flow and the Poincaré Conjecture, Clay Mathematics Monographs, Amer. Math. Soc., Cambridge U.S.A. (2007) [math.DG/0607607].
[22] D. Friedan, Nonlinear Models in Two Epsilon Dimensions, Phys. Rev. Lett.45 (1980) 1057 [SPIRES]. · doi:10.1103/PhysRevLett.45.1057
[23] D.H. Friedan, Nonlinear Models in Two + Epsilon Dimensions, Ann. Phys.163 (1985) 318 [SPIRES]. · Zbl 0583.58010 · doi:10.1016/0003-4916(85)90384-7
[24] C. Schmidhuber and A.A. Tseytlin, On string cosmology and the RG flow in 2 - D field theory, Nucl. Phys.B 426 (1994) 187 [hep-th/9404180] [SPIRES]. · Zbl 1049.81052 · doi:10.1016/0550-3213(94)90131-7
[25] I. Bakas, D. Orlando and P.M. Petropoulos, Ricci flows and expansion in axion-dilaton cosmology, JHEP01 (2007) 040 [hep-th/0610281] [SPIRES]. · doi:10.1088/1126-6708/2007/01/040
[26] M. Cvetič, G.W. Gibbons, H. Lü and C.N. Pope, Cohomogeneity one manifolds of Spin(7) and G2holonomy, Phys. Rev.D 65 (2002) 106004 [hep-th/0108245] [SPIRES].
[27] I. Bakas and K. Sfetsos, unpublished work (2004).
[28] G.W. Gibbons, private communication (2005).
[29] F. Bourliot, J. Estes, P.M. Petropoulos and P. Spindel, Gravitational instantons, self-duality and geometric flows, arXiv:0906.4558 [SPIRES]. · Zbl 1190.83020
[30] J.B. Hartle and S.W. Hawking, Wave function of the universe, Phys. Rev.D28 (1983) 2960 [SPIRES]. · Zbl 1370.83118
[31] C.W. Misner, Mixmaster universe, Phys. Rev. Lett.22 (1969) 1071 [SPIRES]. · Zbl 0177.28701 · doi:10.1103/PhysRevLett.22.1071
[32] C.W. Misner, Quantum cosmology. 1, Phys. Rev.186 (1969) 1319 [SPIRES]. · Zbl 0186.28604 · doi:10.1103/PhysRev.186.1319
[33] V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys.19 (1970) 525 [SPIRES]. · doi:10.1080/00018737000101171
[34] V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz, A General Solution of the Einstein Equations with a Time Singularity, Adv. Phys.31 (1982) 639 [SPIRES]. · doi:10.1080/00018738200101428
[35] J.D. Barrow, Chaotic behavior in general relativity, Phys. Rept.85 (1982) 1 [SPIRES]. · doi:10.1016/0370-1573(82)90171-5
[36] I. Bakas, F. Bourliot, D. Lüst and M. Petropoulos, Mixmaster universe in Hořava-Lifshitz gravity, Class. Quant. Grav.27 (2010) 045013 [arXiv:0911.2665] [SPIRES]. · Zbl 1186.83116 · doi:10.1088/0264-9381/27/4/045013
[37] Y.S. Myung, Y.-W. Kim, W.-S. Son and Y.-J. Park, Chaotic universe in the z = 2 Hovava-Lifshitz gravity, arXiv:0911.2525 [SPIRES].
[38] L. Carroll, Alice’s Adventures in Wonderland, MacMillan, London U.K. (1865).
[39] C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, Freeman, San Francisco U.S.A. (1973).
[40] M. Henneaux, A. Kleinschmidt and G.L. Gomez, A dynamical inconsistency of Hořava gravity, Phys. Rev.D 81 (2010) 064002 [arXiv:0912.0399] [SPIRES].
[41] S. Deser, R. Jackiw and S. Templeton, Topologically massive gauge theories, Ann. Phys.140 (1982) 372 [Erratum ibid.185 (1988) 406] [SPIRES]. · doi:10.1016/0003-4916(82)90164-6
[42] S. Deser, R. Jackiw and S. Templeton, Three-Dimensional Massive Gauge Theories, Phys. Rev. Lett.48 (1982) 975 [SPIRES]. · doi:10.1103/PhysRevLett.48.975
[43] T. Eguchi, P.B. Gilkey and A.J. Hanson, Gravitation, Gauge Theories and Differential Geometry, Phys. Rept.66 (1980) 213 [SPIRES]. · doi:10.1016/0370-1573(80)90130-1
[44] G.W. Gibbons and C.N. Pope, The Positive Action Conjecture and Asymptotically Euclidean Metrics in Quantum Gravity, Commun. Math. Phys.66 (1979) 267 [SPIRES]. · doi:10.1007/BF01197188
[45] Hamilton, RS, The Ricci flow on surfaces, No. 71, 237-262 (1988), Providence U.S.A.
[46] A.U.O. Kisisel, O. Sarioglu and B. Tekin, Cotton flow, Class. Quant. Grav.25 (2008) 165019 [arXiv:0803.1603] [SPIRES]. · Zbl 1147.83025 · doi:10.1088/0264-9381/25/16/165019
[47] J. Isenberg and M. Jackson, Ricci flow of locally homogeneous geometries on closed manifolds, J. Diff. Geom.35 (1992) 723. · Zbl 0808.53044
[48] G. Darboux, Mémoire sur la théorie des coordonnées curvilignes et des systèmes orthogonaux, Ann. Ec. Normale Supér.7 (1878) 101. · JFM 10.0500.04
[49] G.-H. Halphen, Sur un système d’équations différentielles, C.R. Acad. Sc. Paris92 (1881) 1001.
[50] G.-H. Halphen, Sur certains système d’équations différentielles, C.R. Acad. Sc. Paris92 (1881) 1004.
[51] L.A. Takhtajan, A Simple example of modular forms as tau functions for integrable equations, Theor. Math. Phys.93 (1992) 1308 [SPIRES]. · doi:10.1007/BF01083528
[52] M.F. Atiyah and N.J. Hitchin, Low-Energy Scattering of Nonabelian Monopoles, Phys. Lett.A 107 (1985) 21 [SPIRES]. · Zbl 1177.53069
[53] M.F. Atiyah and N.J. Hitchin, The Geometry and Dynamics of Magnetic Monopoles, Porter Lectures, Princeton University Press, Princeton U.S.A. (1988). · Zbl 0671.53001
[54] G.W. Gibbons and N.S. Manton, Classical and Quantum Dynamics of BPS Monopoles, Nucl. Phys.B 274 (1986) 183 [SPIRES]. · doi:10.1016/0550-3213(86)90624-3
[55] Y. Nutku and P. Bakler, Homogeneous, anisotropic three manifolds of topologically massive gravity, Annals Phys.195 (1989) 16 [SPIRES]. · Zbl 0875.53036 · doi:10.1016/0003-4916(89)90094-8
[56] G. Calcagni, Cosmology of the Lifshitz universe, JHEP09 (2009) 112 [arXiv:0904.0829] [SPIRES]. · doi:10.1088/1126-6708/2009/09/112
[57] E. Kiritsis and G. Kofinas, Hořava-Lifshitz Cosmology, Nucl. Phys.B 821 (2009) 467 [arXiv:0904.1334] [SPIRES]. · Zbl 1203.83068 · doi:10.1016/j.nuclphysb.2009.05.005
[58] R. Brandenberger, Matter Bounce in Hořava-Lifshitz Cosmology, Phys. Rev.D 80 (2009) 043516 [arXiv:0904.2835] [SPIRES].
[59] G.W. Gibbons and S.W. Hawking, Classification of Gravitational Instanton Symmetries, Commun. Math. Phys.66 (1979) 291 [SPIRES]. · doi:10.1007/BF01197189
[60] G.W. Gibbons, S.W. Hawking and M.J. Perry, Path Integrals and the Indefiniteness of the Gravitational Action, Nucl. Phys.B 138 (1978) 141 [SPIRES]. · doi:10.1016/0550-3213(78)90161-X
[61] A. Kehagias and K. Sfetsos, The black hole and FRW geometries of non-relativistic gravity, Phys. Lett.B 678 (2009) 123 [arXiv:0905.0477] [SPIRES].
[62] H. Lü, J. Mei and C.N. Pope, Solutions to Hořava Gravity, Phys. Rev. Lett.103 (2009) 091301 [arXiv:0904.1595] [SPIRES]. · doi:10.1103/PhysRevLett.103.091301
[63] E. Kiritsis and G. Kofinas, On Hořava-Lifshitz ‘Black Holes’, JHEP01 (2010) 122 [arXiv:0910.5487] [SPIRES]. · Zbl 1269.83033 · doi:10.1007/JHEP01(2010)122
[64] M. Gurses, Killing Vector Fields in Three Dimensions: A Method to Solve Massive Gravity Field Equations, arXiv:1001.1039 [SPIRES]. · Zbl 1237.83031
[65] D.D.K. Chow, C.N. Pope and E. Sezgin, Classification of solutions in topologically massive gravity, arXiv:0906.3559 [SPIRES]. · Zbl 1190.83077
[66] R. Bach, Zur Weylschen Relativitätstheorie und der Weylschen Erweiterung des Krümmungsbegriffs, Math. Zeitschr.9 (1921) 110. · JFM 48.1035.01 · doi:10.1007/BF01378338
[67] Calabi, E.; Yau, S-T (ed.), Extremal Kähler metric, No. 102 (1982), Princeton U.S.A.
[68] E.A. Bergshoeff, O. Hohm and P.K. Townsend, Massive Gravity in Three Dimensions, Phys. Rev. Lett.102 (2009) 201301 [arXiv:0901.1766] [SPIRES]. · doi:10.1103/PhysRevLett.102.201301
[69] I. Bakas and C. Sourdis, work in progress.
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