Abstract
In this paper, we initiate the study of holographic renormalization group flows for the metric of four-manifolds. In particular, we derive a set of equations which govern the evolution of a generic Kähler four-manifold along the renormalization group flow in seven-dimensional gauged supergravity. The physical eleven-dimensional M-theory setup is given by a stack of M5-branes wrapping a calibrated Kähler four-cycle inside a Calabi-Yau threefold. By topologically twisting the theory in the ultraviolet, we may choose an arbitrary Kähler metric on the four-cycle as an asymptotic boundary condition. We find that at the infrared fixed point, we reach a Kähler-Einstein metric, which can be interpreted as an indication of “uniformizing” behavior of the flow.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
M.H. Freedman, The topology of four-dimensional manifolds, J. Diff. Geom. 17 (1982) 357.
R.S. Hamilton, Three-manifolds with positive ricci curvature, J. Diff. Geom. 17 (1982) 255.
R.S. Hamilton, The formation of singularities in the Ricci flow, Surveys Diff. Geom. II (1995) 7.
W.P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc. 6 (1982) 357.
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math/0211159 [INSPIRE].
J. Streets and G. Tian, Hermitian Curvature Flow, arXiv:0804.4109.
J. Streets and G. Tian, A parabolic flow of pluriclosed metrics, Int. Math. Res. Not. 2010 (2010) 3101.
J. Streets and G. Tian, Regularity results for pluriclosed flow, arXiv:1008.2794.
D.H. Friedan, Nonlinear Models in Two + Epsilon Dimensions, Annals Phys. 163 (1985) 318 [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
N. Seiberg and E. Witten, Electric-magnetic duality, monopole condensation and confinement in N = 2 supersymmetric Yang-Mills theory, Nucl. Phys. B 426 (1994) 19 [Erratum ibid. B 430 (1994) 485] [hep-th/9407087] [INSPIRE].
N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [INSPIRE].
E. Witten, Monopoles and four manifolds, Math. Res. Lett. 1 (1994) 769 [hep-th/9411102] [INSPIRE].
J.M. Maldacena and C. Núñez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
J.P. Gauntlett, N. Kim and D. Waldram, M Five-branes wrapped on supersymmetric cycles, Phys. Rev. D 63 (2001) 126001 [hep-th/0012195] [INSPIRE].
J.P. Gauntlett and N. Kim, M five-branes wrapped on supersymmetric cycles. 2., Phys. Rev. D 65 (2002) 086003 [hep-th/0109039] [INSPIRE].
F. Benini and N. Bobev, Exact two-dimensional superconformal R-symmetry and c-extremization, Phys. Rev. Lett. 110 (2013) 061601 [arXiv:1211.4030] [INSPIRE].
F. Benini and N. Bobev, Two-dimensional SCFTs from wrapped branes and c-extremization, JHEP 06 (2013) 005 [arXiv:1302.4451] [INSPIRE].
P. Karndumri and E. Ó Colgáin, 3D supergravity from wrapped M5-branes, JHEP 03 (2016) 188 [arXiv:1508.00963] [INSPIRE].
I. Bah and V. Stylianou, Gravity duals of N = (0, 2) SCFTs from M5-branes, arXiv:1508.04135 [INSPIRE].
M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222] [INSPIRE].
M.T. Anderson, C. Beem, N. Bobev and L. Rastelli, Holographic Uniformization, Commun. Math. Phys. 318 (2013) 429 [arXiv:1109.3724] [INSPIRE].
D. Gaiotto, N = 2 dualities, JHEP 08 (2012) 034 [arXiv:0904.2715] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Four-dimensional wall-crossing via three-dimensional field theory, Commun. Math. Phys. 299 (2010) 163 [arXiv:0807.4723] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
N. Bobev and P.M. Crichigno, Universal RG Flows Across Dimensions and Holography, JHEP 12 (2017) 065 [arXiv:1708.05052] [INSPIRE].
K. Becker, M. Becker and A. Strominger, Five-branes, membranes and nonperturbative string theory, Nucl. Phys. B 456 (1995) 130 [hep-th/9507158] [INSPIRE].
K. Becker, M. Becker, D.R. Morrison, H. Ooguri, Y. Oz and Z. Yin, Supersymmetric cycles in exceptional holonomy manifolds and Calabi-Yau 4 folds, Nucl. Phys. B 480 (1996) 225 [hep-th/9608116] [INSPIRE].
J.P. Gauntlett, N.D. Lambert and P.C. West, Branes and calibrated geometries, Commun. Math. Phys. 202 (1999) 571 [hep-th/9803216] [INSPIRE].
G.W. Gibbons and G. Papadopoulos, Calibrations and intersecting branes, Commun. Math. Phys. 202 (1999) 593 [hep-th/9803163] [INSPIRE].
J.P. Gauntlett, Branes, calibrations and supergravity, in Strings and geometry. Proceedings, Summer School, Cambridge, U.K., March 24 - April 20, 2002, pp. 79-126, hep-th/0305074 [INSPIRE].
J.M. Maldacena, The large N limit of superconformal field theories and supergravity, Int. J. Theor. Phys. 38 (1999) 1113 [hep-th/9711200] [INSPIRE].
J.P. Gauntlett, O.A.P. Mac Conamhna, T. Mateos and D. Waldram, AdS spacetimes from wrapped M5 branes, JHEP 11 (2006) 053 [hep-th/0605146] [INSPIRE].
P. Figueras, O.A.P. Mac Conamhna and E. Ó Colgáin, Global geometry of the supersymmetric AdS 3 /CF T 2 correspondence in M-theory, Phys. Rev. D 76 (2007) 046007 [hep-th/0703275] [INSPIRE].
E. Witten, Some comments on string dynamics, in Future perspectives in string theory. Proceedings, Conference, Strings’95, Los Angeles, U.S.A., March 13-18, 1995, pp. 501-523, hep-th/9507121 [INSPIRE].
N. Seiberg and E. Witten, Comments on string dynamics in six-dimensions, Nucl. Phys. B 471 (1996) 121 [hep-th/9603003] [INSPIRE].
M. Berkooz, A supergravity dual of a (1,0) field theory in six-dimensions, Phys. Lett. B 437 (1998) 315 [hep-th/9802195] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
T. Dimofte, 3d Superconformal Theories from Three-Manifolds, in New Dualities of Supersymmetric Gauge Theories, J. Teschner ed., (2016) pp. 339, [arXiv:1412.7129].
M. Pernici, K. Pilch and P. van Nieuwenhuizen, Gauged Maximally Extended Supergravity in Seven-dimensions, Phys. Lett. B 143 (1984) 103 [INSPIRE].
H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistent nonlinear K K reduction of 11-d supergravity on AdS 7 × S 4 and selfduality in odd dimensions, Phys. Lett. B 469 (1999) 96 [hep-th/9905075] [INSPIRE].
H. Nastase, D. Vaman and P. van Nieuwenhuizen, Consistency of the AdS 7 × S 4 reduction and the origin of selfduality in odd dimensions, Nucl. Phys. B 581 (2000) 179 [hep-th/9911238] [INSPIRE].
M. Cvetič et al., Embedding AdS black holes in ten-dimensions and eleven-dimensions, Nucl. Phys. B 558 (1999) 96 [hep-th/9903214] [INSPIRE].
M. Fluder, work in progress.
B.S. Acharya, J.P. Gauntlett and N. Kim, Five-branes wrapped on associative three cycles, Phys. Rev. D 63 (2001) 106003 [hep-th/0011190] [INSPIRE].
I. Bah, M. Gabella and N. Halmagyi, BPS M5-branes as Defects for the 3d-3d Correspondence, JHEP 11 (2014) 112 [arXiv:1407.0403] [INSPIRE].
M. Fluder, unpublished notes.
J. Song and B. Weinkove, Lecture notes on the Kähler-Ricci flow, arXiv:1212.3653.
H.-D. Cao, Deformation of kähler matrics to kähler-einstein metrics on compact kähler manifolds, Invent. Math. 81 (1985) 359.
S.-T. Yau, On the ricci curvature of a compact kähler manifold and the complex monge-ampére equation, I, Commun. Pure Appl. Math. 31 (1978) 339.
A. Futaki, An Obstruction to the Existence of Einstein Kähler Metrics, Invent. Math. 73 (1983) 437.
G. Tian, Kähler-einstein metrics with positive scalar curvature, Invent. Math. 130 (1997) 1.
B.S. Acharya and S. Gukov, M theory and singularities of exceptional holonomy manifolds, Phys. Rept. 392 (2004) 121 [hep-th/0409191] [INSPIRE].
M. Itoh, Moduli of half conformally flat structures, Math. Ann. 296 (1993) 687.
L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [INSPIRE].
M. Fluder, \( 4d\mathcal{N}=1/2d \) Yang-Mills Duality in Holography, arXiv:1712.06596 [INSPIRE].
I. Bah, C. Beem, N. Bobev and B. Wecht, AdS/CFT Dual Pairs from M5-Branes on Riemann Surfaces, Phys. Rev. D 85 (2012) 121901 [arXiv:1112.5487] [INSPIRE].
I. Bah, C. Beem, N. Bobev and B. Wecht, Four-Dimensional SCFTs from M5-Branes, JHEP 06 (2012) 005 [arXiv:1203.0303] [INSPIRE].
M.F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. Roy. Soc. Lond. A 308 (1982) 523 [INSPIRE].
C. Beem and A. Gadde, The N = 1 superconformal index for class S fixed points, JHEP 04 (2014) 036 [arXiv:1212.1467] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1710.09479
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Fluder, M. Kähler uniformization from holographic renormalization group flows of M5-branes. J. High Energ. Phys. 2018, 46 (2018). https://doi.org/10.1007/JHEP08(2018)046
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)046