Holographic uniformization. (English) Zbl 1329.81299
Summary: We derive and study supergravity BPS flow equations for M5 or D3 branes wrapping a Riemann surface. They take the form of novel geometric flows intrinsically defined on the surface. Their dual field-theoretic interpretation suggests the existence of solutions interpolating between an arbitrary metric in the ultraviolet and the constant-curvature metric in the infrared. We confirm this conjecture with a rigorous global existence proof.
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
| 53C44 | Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) |
| 83E50 | Supergravity |
| 83E30 | String and superstring theories in gravitational theory |
| 81T60 | Supersymmetric field theories in quantum mechanics |
References:
| [1] | Chow, B., Knopf, D.: The Ricci flow: an introduction. Mathematical Surveys and Monographs, 110. Providence, RI: Amer. Math. Soc., 2004 · Zbl 1086.53085 |
| [2] | Hamilton, R.S.: The formation of singularities in the Ricci flow. In: Surveys in Differential Geometry Vol. II, Cambridge, MA: Int. Press, 1995, pp. 7–136 · Zbl 0867.53030 |
| [3] | Friedan D.H.: Nonlinear Models in Two + Epsilon Dimensions. Ann. Phys. 163, 318 (1985) · Zbl 0583.58010 · doi:10.1016/0003-4916(85)90384-7 |
| [4] | Hamilton R.S.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17(2), 255–306 (1982) · Zbl 0504.53034 |
| [5] | Anderson, M.T., Beem, C., Bobev, N., Rastelli L.: Work in progress |
| [6] | Gaiotto, D.: N = 2 dualities. http://arxiv.org/abs/0904.2715v2 [hep-th], 2009 |
| [7] | Gaiotto D., Moore G.W., Neitzke A.: Four-dimensional wall-crossing via three-dimensional field theory. Commun. Math. Phys. 299, 163–224 (2010) · Zbl 1225.81135 · doi:10.1007/s00220-010-1071-2 |
| [8] | Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin Systems, and the WKB Approximation. http://arxiv.org/abs/0907.3987v2 [hep-th], 2009 · Zbl 1358.81150 |
| [9] | Gaiotto, D., Maldacena, J.: The gravity duals of N = 2 superconformal field theories. http://arxiv.org/abs/0904.4466v4 [hep-th], 2009 · Zbl 1397.83038 |
| [10] | Maldacena J.M., Núñez C.: Supergravity description of field theories on curved manifolds and a no go theorem. Int. J. Mod. Phys. A16, 822–855 (2001) · Zbl 0984.83052 |
| [11] | Nirenberg, L.: Topics in Nonlinear Functional Analysis. Courant Lecture Series, New York: Courant Inst. Math. Sciences, 1974 · Zbl 0286.47037 |
| [12] | Kazdan J., Warner F.: Curvature functions for compact 2-manifolds. Ann. Math. 99, 14–47 (1974) · Zbl 0273.53034 · doi:10.2307/1971012 |
| [13] | Bah, I., Beem, C., Bobev, N., Wecht, B.: Work in progress |
| [14] | Fayyazuddin A., Smith D.J.: Warped AdS near horizon geometry of completely localized intersections of M5-branes. JHEP 0010, 023 (2000) · Zbl 0959.83051 · doi:10.1088/1126-6708/2000/10/023 |
| [15] | Brinne B., Fayyazuddin A., Mukhopadhyay S., Smith D.J.: Supergravity M5-branes wrapped on Riemann surfaces and their QFT duals. JHEP 0012, 013 (2000) · Zbl 0990.81609 · doi:10.1088/1126-6708/2000/12/013 |
| [16] | Witten E.: Topological Quantum Field Theory. Commun. Math. Phys. 117, 353 (1988) · Zbl 0656.53078 · doi:10.1007/BF01223371 |
| [17] | Bershadsky M., Vafa C., Sadov V.: D-branes and topological field theories. Nucl. Phys. B463, 420–434 (1996) · Zbl 1004.81560 · doi:10.1016/0550-3213(96)00026-0 |
| [18] | Benini F., Tachikawa Y., Wecht B.: Sicilian gauge theories and N = 1 dualities. JHEP 1001, 088 (2010) · Zbl 1269.81080 · doi:10.1007/JHEP01(2010)088 |
| [19] | Bershadsky M., Johansen A., Sadov V., Vafa C.: Topological reduction of 4-d SYM to 2-d sigma models. Nucl. Phys. B448, 166–186 (1995) · Zbl 1009.58502 · doi:10.1016/0550-3213(95)00242-K |
| [20] | Nastase H., Vaman D., van Nieuwenhuizen P.: Consistent nonlinear KK reduction of 11-d supergravity on AdS(7) x S(4) and selfduality in odd dimensions. Phys. Lett. B469, 96–102 (1999) · Zbl 0987.81570 |
| [21] | Nastase H., Vaman D., van Nieuwenhuizen P.: Consistency of the AdS(7) x S(4) reduction and the origin of selfduality in odd dimensions. Nucl. Phys. B581, 179–239 (2000) · Zbl 0985.83026 · doi:10.1016/S0550-3213(00)00193-0 |
| [22] | Cvetic M., Duff M.J., Hoxha P., Liu J.T., Lu H., Lu J.X., Martinez-Acosta R., Pope C.N. et al.: Embedding AdS black holes in ten-dimensions and eleven-dimensions. Nucl. Phys. B558, 96–126 (1999) · Zbl 0951.83033 · doi:10.1016/S0550-3213(99)00419-8 |
| [23] | Gauntlett J.P., Mac Conamhna O.A.P., Mateos T., Waldram D.: AdS spacetimes from wrapped M5 branes. JHEP 0611, 053 (2006) · doi:10.1088/1126-6708/2006/11/053 |
| [24] | Pernici M., Pilch K., van Nieuwenhuizen P.: Gauged Maximally Extended Supergravity in Seven-dimensions. Phys. Lett. B143, 103 (1984) |
| [25] | Liu J.T., Minasian R.: Black holes and membranes in AdS(7). Phys. Lett. B457, 39–46 (1999) · Zbl 0992.83075 |
| [26] | Gunaydin M., Romans L.J., Warner N.P.: Gauged N = 8 Supergravity in Five-Dimensions. Phys. Lett. B154, 268 (1985) |
| [27] | Pernici M., Pilch K., van Nieuwenhuizen P.: Gauged N = 8 D = 5 Supergravity. Nucl. Phys. B259, 460 (1985) · doi:10.1016/0550-3213(85)90645-5 |
| [28] | Gunaydin M., Romans L.J., Warner N.P.: Compact and Noncompact Gauged Supergravity Theories in Five-Dimensions. Nucl. Phys. B272, 598 (1986) · doi:10.1016/0550-3213(86)90237-3 |
| [29] | Bobev N., Kundu A., Pilch K., Warner N.P.: Supersymmetric Charged Clouds in AdS 5. JHEP 1103, 070 (2011) · Zbl 1301.81106 · doi:10.1007/JHEP03(2011)070 |
| [30] | Gauntlett J.P., Mac Conamhna O.A.P.: AdS spacetimes from wrapped D3-branes. Class. Quant. Grav. 24, 6267 (2007) · Zbl 1133.83012 · doi:10.1088/0264-9381/24/24/009 |
| [31] | Boyer C.P., Finley J.D.: Killing vectors in self-dual, Euclidean Einstein spaces. J. Math. Phys. 23, 1126 (1982) · Zbl 0484.53051 · doi:10.1063/1.525479 |
| [32] | Bakas, I.: Area Preserving Diffeomorphisms And Higher Spin Fields In Two-dimensions. Supermembranes and Physics in (2+1) Dimensions: proceedings. Edited by M. Duff, C.N. Pope, E. Sezgin. Teaneck, N.J.: World Scientific, 1990 |
| [33] | Saveliev M.V.: On the integrability problem of a continuous Toda system. Theor. and Math. Phys. 3, 1024–1031 (1993) |
| [34] | Lin H., Lunin O., Maldacena J.M.: Bubbling AdS space and 1/2 BPS geometries. JHEP 0410, 025 (2004) · doi:10.1088/1126-6708/2004/10/025 |
| [35] | Bianchi M., Freedman D.Z., Skenderis K.: Holographic renormalization. Nucl. Phys. B 631, 159 (2002) · Zbl 0995.81075 · doi:10.1016/S0550-3213(02)00179-7 |
| [36] | Gubser S.S.: Curvature singularities: The good, the bad, and the naked. Adv. Theor. Math. Phys. 4, 679–745 (2000) · Zbl 0984.83036 |
| [37] | Behrndt K., Chamseddine A.H., Sabra W.A.: BPS black holes in N = ,2 five-dimensional AdS supergravity. Phys. Lett. B442, 97–101 (1998) · Zbl 1002.83517 |
| [38] | Guillemin, V., Pollack, A.: Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974 · Zbl 0361.57001 |
| [39] | Smale S.: An infinite dimensional version of Sard’s theorem. Amer. J. Math. 87, 861–866 (1965) · Zbl 0143.35301 · doi:10.2307/2373250 |
| [40] | Nirenberg L.: Variational and topological methods in non linear problems. Bull. Amer. Math. Soc. 4, 267–302 (1981) · Zbl 0468.47040 · doi:10.1090/S0273-0979-1981-14888-6 |
| [41] | Eells J.: A setting for global analysis. Bull. Amer. Math. Soc. 72, 751–807 (1966) · doi:10.1090/S0002-9904-1966-11558-6 |
| [42] | Anderson M.: Einstein metrics with prescribed conformal infinity on 4-manifolds. Geom. & Funct. Anal. 18, 305–366 (2008) · Zbl 1148.53033 · doi:10.1007/s00039-008-0668-5 |
| [43] | Yau S.-T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampere equation I. Comm. Pure Appl. Math. 31, 339–411 (1978) · Zbl 0369.53059 · doi:10.1002/cpa.3160310304 |
| [44] | Andersson, L., Chruściel, P.: Solutions of the constraint equations in general relativity satisfying ’hyperboloidal boundary conditions’. Dissertationes Math. 355, (1996) · Zbl 0873.35101 |
| [45] | Mazzeo R.: Elliptic theory of differential edge operators, I. Comm. PDE 16, 1615–1644 (1991) · Zbl 0745.58045 · doi:10.1080/03605309108820815 |
| [46] | Melrose, R.: Geometric Scattering Theory. Cambridge: Cambridge Univ. Press, 1995 · Zbl 0849.58071 |
| [47] | Mazzeo R., Melrose R.: Pseudodifferential operators on manifolds with fibered boundaries. Asian J. Math. 2, 833–866 (1998) · Zbl 1125.58304 |
| [48] | Calderon, A.: Uniqueness in the Cauchy problem for partial differential equations. Amer. J. Math. 80, 16–36 (1958) · Zbl 0080.30302 |
| [49] | Gilbarg, D., Trudinger, N.: Elliptic Partial Differential Equations of Second Order. Grundlehren Series, Vol. 224, Berlin: Springer Verlag, 1983 · Zbl 0562.35001 |
| [50] | Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Grundlehren Series, Vol. 130, Berlin: Springer Verlag, 1966 · Zbl 0142.38701 |
| [51] | Girardello L., Petrini M., Porrati M., Zaffaroni A.: Novel local CFT and exact results on perturbations of N = 4 superYang Mills from AdS dynamics. JHEP 9812, 022 (1998) · Zbl 0949.81053 · doi:10.1088/1126-6708/1998/12/022 |
| [52] | Freedman D.Z., Gubser S.S., Pilch K., Warner N.P.: Renormalization group flows from holography supersymmetry and a c theorem. Adv. Theor. Math. Phys. 3, 363–417 (1999) · Zbl 0976.83067 |
| [53] | Heemskerk I., Polchinski J.: Holographic and Wilsonian Renormalization Groups. JHEP 1106, 031 (2011) · Zbl 1298.81181 · doi:10.1007/JHEP06(2011)031 |
| [54] | Faulkner T., Liu H., Rangamani M.: Integrating out geometry: Holographic Wilsonian RG and the membrane paradigm. JHEP 1108, 051 (2011) · Zbl 1298.81173 |
| [55] | Acharya B.S., Gauntlett J.P., Kim N.: Five-branes wrapped on associative three cycles. Phys. Rev. D63, 106003 (2001) |
| [56] | Gauntlett J.P., Kim N., Waldram D.: M Five-branes wrapped on supersymmetric cycles. Phys. Rev. D63, 126001 (2001) |
| [57] | Perelman, G.: The Entropy formula for the Ricci flow and its geometric applications. http://arxiv.org/abs/math/0211159v1 [math-dg], 2002 · Zbl 1130.53001 |
| [58] | Gauntlett J.P., Kim N., Pakis S., Waldram D.: Membranes wrapped on holomorphic curves. Phys. Rev. D65, 026003 (2002) |
| [59] | Naka, M.: Various wrapped branes from gauged supergravities. http://arxiv.org/abs/hep-th/0206141v2 , 2002 |
| [60] | Lu H., Pope C.N., Townsend P.K.: Domain walls from anti-de Sitter space-time. Phys. Lett. B391, 39–46 (1997) · Zbl 0956.83072 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
