Abstract
The partition function of a three-dimensional \( \mathcal{N}=2 \) theory on the manifold ℳg,p, an S1 bundle of degree p over a closed Riemann surface Σg, was recently computed via supersymmetric localization. In this paper, we compute these partition functions at large N in a class of quiver gauge theories with holographic M-theory duals. We provide the supergravity bulk dual having as conformal boundary such three-dimensional circle bundles. These configurations are solutions to \( \mathcal{N}=2 \) minimal gauged supergravity and pertain to the class of Taub-NUT-AdS and Taub-Bolt-AdS preserving 1/4 of the supersymmetries. We discuss the conditions for the uplift of these solutions to M-theory, and compute the on-shell action via holographic renormalization. We show that the uplift condition and on-shell action for the Bolt solutions are correctly reproduced by the large N limit of the partition function of the dual superconformal field theory. In particular, the Σg × S1 = ℳg,0 partition function, which was recently shown to match the entropy of AdS4 black holes, and the S3 ≅ ℳ0,1 free energy, occur as special cases of our formalism, and we comment on relations between them.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Pestun et al., Localization techniques in quantum field theories, J. Phys. A 50 (2017) 440301 [arXiv:1608.02952] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Bethe/Gauge correspondence on curved spaces, JHEP 01 (2015) 100 [arXiv:1405.6046] [INSPIRE].
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
F. Benini and A. Zaffaroni, Supersymmetric partition functions on Riemann surfaces, Proc. Symp. Pure Math. 96 (2017) 13 [arXiv:1605.06120] [INSPIRE].
C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353 [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS 4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
S.L. Cacciatori and D. Klemm, Supersymmetric AdS 4 black holes and attractors, JHEP 01 (2010) 085 [arXiv:0911.4926] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS4, Phys. Lett. B 771 (2017) 462 [arXiv:1608.07294] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Passias, Holographic microstate counting for AdS 4 black holes in massive IIA supergravity, JHEP 10 (2017) 190 [arXiv:1707.06884] [INSPIRE].
F. Benini, H. Khachatryan and P. Milan, Black hole entropy in massive Type IIA, Class. Quant. Grav. 35 (2018) 035004 [arXiv:1707.06886] [INSPIRE].
A. Cabo-Bizet, V.I. Giraldo-Rivera and L.A. Pando Zayas, Microstate counting of AdS 4 hyperbolic black hole entropy via the topologically twisted index, JHEP 08 (2017) 023 [arXiv:1701.07893] [INSPIRE].
S.M. Hosseini and A. Zaffaroni, Large N matrix models for 3d \( \mathcal{N}=2 \) theories: twisted index, free energy and black holes, JHEP 08 (2016) 064 [arXiv:1604.03122] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
D.L. Jafferis, I.R. Klebanov, S.S. Pufu and B.R. Safdi, Towards the F-Theorem: N = 2 Field Theories on the Three-Sphere, JHEP 06 (2011) 102 [arXiv:1103.1181] [INSPIRE].
D.Z. Freedman and S.S. Pufu, The holography of F-maximization, JHEP 03 (2014) 135 [arXiv:1302.7310] [INSPIRE].
H. Casini and M. Huerta, On the RG running of the entanglement entropy of a circle, Phys. Rev. D 85 (2012) 125016 [arXiv:1202.5650] [INSPIRE].
C. Closset, H. Kim and B. Willett, Supersymmetric partition functions and the three-dimensional A-twist, JHEP 03 (2017) 074 [arXiv:1701.03171] [INSPIRE].
E. Witten, Topological σ-models, Commun. Math. Phys. 118 (1988) 411 [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].
D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
D. Martelli and J. Sparks, The gravity dual of supersymmetric gauge theories on a biaxially squashed three-sphere, Nucl. Phys. B 866 (2013) 72 [arXiv:1111.6930] [INSPIRE].
D. Martelli, A. Passias and J. Sparks, The supersymmetric NUTs and bolts of holography, Nucl. Phys. B 876 (2013) 810 [arXiv:1212.4618] [INSPIRE].
D. Martelli and A. Passias, The gravity dual of supersymmetric gauge theories on a two-parameter deformed three-sphere, Nucl. Phys. B 877 (2013) 51 [arXiv:1306.3893] [INSPIRE].
F. Azzurli, N. Bobev, P.M. Crichigno, V.S. Min and A. Zaffaroni, A universal counting of black hole microstates in AdS 4, JHEP 02 (2018) 054 [arXiv:1707.04257] [INSPIRE].
N. Halmagyi and S. Lal, On the on-shell: the action of AdS 4 black holes, JHEP 03 (2018) 146 [arXiv:1710.09580] [INSPIRE].
A. Cabo-Bizet, U. Kol, L.A. Pando Zayas, I. Papadimitriou and V. Rathee, Entropy functional and the holographic attractor mechanism, arXiv:1712.01849 [INSPIRE].
C. Klare, A. Tomasiello and A. Zaffaroni, Supersymmetry on Curved Spaces and Holography, JHEP 08 (2012) 061 [arXiv:1205.1062] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, Supersymmetric Field Theories on Three-Manifolds, JHEP 05 (2013) 017 [arXiv:1212.3388] [INSPIRE].
N.L. Balazs and A. Voros, Chaos on the pseudosphere, Phys. Rept. 143 (1986) 109 [INSPIRE].
L.C. Jeffrey and F.C. Kirwan, Localization for nonabelian group actions, Topology 34 (1995) 291 [alg-geom/9307001].
F. Benini, R. Eager, K. Hori and Y. Tachikawa, Elliptic Genera of 2d \( \mathcal{N}=2 \) Gauge Theories, Commun. Math. Phys. 333 (2015) 1241 [arXiv:1308.4896] [INSPIRE].
D.Z. Freedman and A.K. Das, Gauge Internal Symmetry in Extended Supergravity, Nucl. Phys. B 120 (1977) 221 [INSPIRE].
E.S. Fradkin and M.A. Vasiliev, Model of Supergravity with Minimal Electromagnetic Interaction, LEBEDEV-76-197 (1976) [INSPIRE].
A. Chamblin, R. Emparan, C.V. Johnson and R.C. Myers, Large N phases, gravitational instantons and the nuts and bolts of AdS holography, Phys. Rev. D 59 (1999) 064010 [hep-th/9808177] [INSPIRE].
S.W. Hawking, C.J. Hunter and D.N. Page, Nut charge, anti-de Sitter space and entropy, Phys. Rev. D 59 (1999) 044033 [hep-th/9809035] [INSPIRE].
A.H. Taub, Empty space-times admitting a three parameter group of motions, Annals Math. 53 (1951) 472 [INSPIRE].
E. Newman, L. Tamubrino and T. Unti, Empty space generalization of the Schwarzschild metric, J. Math. Phys. 4 (1963) 915 [INSPIRE].
C.W. Misner, The Flatter regions of Newman, Unti and Tamburino’s generalized Schwarzschild space, J. Math. Phys. 4 (1963) 924 [INSPIRE].
L.F. Alday, M. Fluder and J. Sparks, The large N limit of M2-branes on Lens spaces, JHEP 10 (2012) 057 [arXiv:1204.1280] [INSPIRE].
T. Eguchi and A.J. Hanson, Selfdual Solutions to Euclidean Gravity, Annals Phys. 120 (1979) 82 [INSPIRE].
N. Alonso-Alberca, P. Meessen and T. Ortín, Supersymmetry of topological Kerr-Newman-Taub-NUT-AdS space-times, Class. Quant. Grav. 17 (2000) 2783 [hep-th/0003071] [INSPIRE].
D. Klemm and M. Nozawa, Supersymmetry of the C-metric and the general Plebanski-Demianski solution, JHEP 05 (2013) 123 [arXiv:1303.3119] [INSPIRE].
M. Nozawa and T. Houri, Killing-Yano tensor and supersymmetry of the self-dual Plebanski-Demianski solution, Class. Quant. Grav. 33 (2016) 125008 [arXiv:1510.07470] [INSPIRE].
M. Nozawa, Euclidean supersymmetric solutions with the self-dual Weyl tensor, Phys. Lett. B 770 (2017) 166 [arXiv:1702.05210] [INSPIRE].
M.M. Caldarelli and D. Klemm, All supersymmetric solutions of N = 2, D = 4 gauged supergravity, JHEP 09 (2003) 019 [hep-th/0307022] [INSPIRE].
J.P. Gauntlett and O. Varela, Consistent Kaluza-Klein reductions for general supersymmetric AdS solutions, Phys. Rev. D 76 (2007) 126007 [arXiv:0707.2315] [INSPIRE].
S. de Haro, S.N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence, Commun. Math. Phys. 217 (2001) 595 [hep-th/0002230] [INSPIRE].
R. Emparan, C.V. Johnson and R.C. Myers, Surface terms as counterterms in the AdS/CFT correspondence, Phys. Rev. D 60 (1999) 104001 [hep-th/9903238] [INSPIRE].
K. Skenderis, Lecture notes on holographic renormalization, Class. Quant. Grav. 19 (2002) 5849 [hep-th/0209067] [INSPIRE].
I. Papadimitriou and K. Skenderis, Thermodynamics of asymptotically locally AdS spacetimes, JHEP 08 (2005) 004 [hep-th/0505190] [INSPIRE].
N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
M.M. Caldarelli and D. Klemm, Supersymmetry of Anti-de Sitter black holes, Nucl. Phys. B 545 (1999) 434 [hep-th/9808097] [INSPIRE].
S.W. Hawking and S.F. Ross, Duality between electric and magnetic black holes, Phys. Rev. D 52 (1995) 5865 [hep-th/9504019] [INSPIRE].
S.M. Hosseini and N. Mekareeya, Large N topologically twisted index: necklace quivers, dualities and Sasaki-Einstein spaces, JHEP 08 (2016) 089 [arXiv:1604.03397] [INSPIRE].
C. Closset, T.T. Dumitrescu, G. Festuccia and Z. Komargodski, From Rigid Supersymmetry to Twisted Holomorphic Theories, Phys. Rev. D 90 (2014) 085006 [arXiv:1407.2598] [INSPIRE].
C. Closset, H. Kim and B. Willett, \( \mathcal{N}=1 \) supersymmetric indices and the four-dimensional A-model, JHEP 08 (2017) 090 [arXiv:1707.05774] [INSPIRE].
D. Martelli and J. Sparks, AdS(40/CFT(3) duals from M2-branes at hypersurface singularities and their deformations, JHEP 12 (2009) 017 [arXiv:0909.2036] [INSPIRE].
A. Bergman and C.P. Herzog, The Volume of some nonspherical horizons and the AdS/CFT correspondence, JHEP 01 (2002) 030 [hep-th/0108020] [INSPIRE].
D. Martelli and J. Sparks, The large N limit of quiver matrix models and Sasaki-Einstein manifolds, Phys. Rev. D 84 (2011) 046008 [arXiv:1102.5289] [INSPIRE].
D. Martelli and J. Sparks, Moduli spaces of Chern-Simons quiver gauge theories and AdS 4 /CFT 3, Phys. Rev. D 78 (2008) 126005 [arXiv:0808.0912] [INSPIRE].
F. Benini, C. Closset and S. Cremonesi, Quantum moduli space of Chern-Simons quivers, wrapped D6-branes and AdS4/CFT3, JHEP 09 (2011) 005 [arXiv:1105.2299] [INSPIRE].
A. Amariti, C. Klare and M. Siani, The Large N Limit of Toric Chern-Simons Matter Theories and Their Duals, JHEP 10 (2012) 019 [arXiv:1111.1723] [INSPIRE].
D. Gang, C. Hwang, S. Kim and J. Park, Tests of AdS 4 /CFT 3 correspondence for \( \mathcal{N}=2 \) chiral-like theory, JHEP 02 (2012) 079 [arXiv:1111.4529] [INSPIRE].
F. Benini, T. Nishioka and M. Yamazaki, 4d Index to 3d Index and 2d TQFT, Phys. Rev. D 86 (2012) 065015 [arXiv:1109.0283] [INSPIRE].
B. de Wit and H. Nicolai, N = 8 Supergravity, Nucl. Phys. B 208 (1982) 323 [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].
A. Azizi, H. Godazgar, M. Godazgar and C.N. Pope, Embedding of gauged STU supergravity in eleven dimensions, Phys. Rev. D 94 (2016) 066003 [arXiv:1606.06954] [INSPIRE].
M. Colleoni and D. Klemm, Nut-charged black holes in matter-coupled N = 2, D = 4 gauged supergravity, Phys. Rev. D 85 (2012) 126003 [arXiv:1203.6179] [INSPIRE].
H. Erbin and N. Halmagyi, Quarter-BPS Black Holes in AdS 4 -NUT from N = 2 Gauged Supergravity, JHEP 10 (2015) 081 [arXiv:1503.04686] [INSPIRE].
N. Bobev, T. Hertog and Y. Vreys, The NUTs and Bolts of Squashed Holography, JHEP 11 (2016) 140 [arXiv:1610.01497] [INSPIRE].
N. Bobev, P. Bueno and Y. Vreys, Comments on Squashed-sphere Partition Functions, JHEP 07 (2017) 093 [arXiv:1705.00292] [INSPIRE].
J.T. Liu, L.A. Pando Zayas, V. Rathee and W. Zhao, A One-loop Test of Quantum Black Holes in Anti de Sitter Space, arXiv:1711.01076 [INSPIRE].
L.J. Romans, Massive N = 2a Supergravity in Ten-Dimensions, Phys. Lett. B 169 (1986) 374 [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: 1712.08861
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
Toldo, C., Willett, B. Partition functions on 3d circle bundles and their gravity duals. J. High Energ. Phys. 2018, 116 (2018). https://doi.org/10.1007/JHEP05(2018)116
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2018)116