Abstract
A class of 3d \( \mathcal{N}=2 \) supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction [1] in 3d-3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class, the massive supersymmetric vacua of the theory contain the classical geometries on a 4d simplicial complex. The corresponding 4d simplicial geometries are locally constant curvature (either dS or AdS), in the sense that they are made by gluing geometrical 4-simplices of the same constant curvature. When the simplicial complex is sufficiently refined, the simplicial geometries can approximate all possible smooth geometries on 4-manifold. At the quantum level, we propose that a class of holomorphic blocks defined in [2] from the 3d \( \mathcal{N}=2 \) gauge theories are wave functions of quantum 4d simplicial geometries. In the semiclassical limit, the asymptotic behavior of holomorphic block reproduces the classical action of 4d Einstein-Hilbert gravity in the simplicial context.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
T. Dimofte, D. Gaiotto and S. Gukov, Gauge Theories Labelled by Three-Manifolds, Commun. Math. Phys. 325 (2014) 367 [arXiv:1108.4389] [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].
H.-J. Chung, T. Dimofte, S. Gukov and P. Sulkowski, 3d-3d Correspondence Revisited, arXiv:1405.3663 [INSPIRE].
T. Dimofte, 3d Superconformal Theories from Three-Manifolds, arXiv:1412.7129 [INSPIRE].
T. Dimofte, D. Gaiotto and S. Gukov, 3-Manifolds and 3d Indices, Adv. Theor. Math. Phys. 17 (2013) 975 [arXiv:1112.5179] [INSPIRE].
T. Dimofte and S. Gukov, Chern-Simons Theory and S-duality, JHEP 05 (2013) 109 [arXiv:1106.4550] [INSPIRE].
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].
A. Gadde, L. Rastelli, S.S. Razamat and W. Yan, The 4d Superconformal Index from q-deformed 2d Yang-Mills, Phys. Rev. Lett. 106 (2011) 241602 [arXiv:1104.3850] [INSPIRE].
A. Gadde, S. Gukov and P. Putrov, Fivebranes and 4-manifolds, arXiv:1306.4320 [INSPIRE].
J. Yagi, 3d TQFT from 6d SCFT, JHEP 08 (2013) 017 [arXiv:1305.0291] [INSPIRE].
C. Cordova and D.L. Jafferis, Complex Chern-Simons from M5-branes on the Squashed Three-Sphere, arXiv:1305.2891 [INSPIRE].
S. Lee and M. Yamazaki, 3d Chern-Simons Theory from M5-branes, JHEP 12 (2013) 035 [arXiv:1305.2429] [INSPIRE].
E. Witten, Fivebranes and Knots, arXiv:1101.3216 [INSPIRE].
D. Gaiotto, G.W. Moore and Y. Tachikawa, On 6d \( \mathcal{N}=\left(2,0\right) \) theory compactified on a Riemann surface with finite area, PTEP 2013 (2013) 013B03 [arXiv:1110.2657] [INSPIRE].
Y. Luo, M.-C. Tan, J. Yagi and Q. Zhao, Ω-deformation of B-twisted gauge theories and the 3d-3d correspondence, JHEP 02 (2015) 047 [arXiv:1410.1538] [INSPIRE].
M.-C. Tan, M-Theoretic Derivations of 4d-2d Dualities: From a Geometric Langlands Duality for Surfaces, to the AGT Correspondence, to Integrable Systems, JHEP 07 (2013) 171 [arXiv:1301.1977] [INSPIRE].
M.-C. Tan, An M-Theoretic Derivation of a 5d and 6d AGT Correspondence and Relativistic and Elliptized Integrable Systems, JHEP 12 (2013) 031 [arXiv:1309.4775] [INSPIRE].
S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP 11 (2015) 155 [arXiv:1507.00261] [INSPIRE].
T. Regge, General relativity without coordinates, Nuovo Cim. 19 (1961) 558 [INSPIRE].
B. Bahr and B. Dittrich, Regge calculus from a new angle, New J. Phys. 12 (2010) 033010 [arXiv:0907.4325] [INSPIRE].
T. Dimofte, Quantum Riemann Surfaces in Chern-Simons Theory, Adv. Theor. Math. Phys. 17 (2013) 479 [arXiv:1102.4847] [INSPIRE].
T. Dimofte, D. Gaiotto and R. van der Veen, RG Domain Walls and Hybrid Triangulations, Adv. Theor. Math. Phys. 19 (2015) 137 [arXiv:1304.6721] [INSPIRE].
D. Gaiotto, G.W. Moore and A. Neitzke, Wall-crossing, Hitchin Systems and the WKB Approximation, arXiv:0907.3987 [INSPIRE].
V.V. Fock and A.B. Goncharov, Moduli spaces of local systems and higher Teichmüller theory, math/0311149.
Y. Kabaya, Parametrization of PSL(2, C)-representations of surface groups, arXiv:1110.6674 [INSPIRE].
T. Dimofte and R. van der Veen, A Spectral Perspective on Neumann-Zagier, arXiv:1403.5215 [INSPIRE].
O. Aharony, A. Hanany, K.A. Intriligator, N. Seiberg and M.J. Strassler, Aspects of N = 2 supersymmetric gauge theories in three-dimensions, Nucl. Phys. B 499 (1997) 67 [hep-th/9703110] [INSPIRE].
L.K. Hua and I. Reiner, On the Generators of the Symplectic Modular Group, Trans. Am. Math. Soc. 65 (1949) 415.
T.D. Dimofte and S. Garoufalidis, The Quantum content of the gluing equations, arXiv:1202.6268 [INSPIRE].
E. Witten, \( \mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{Z}}\right) \) action on three-dimensional conformal field theories with Abelian symmetry, hep-th/0307041 [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].
N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
L.D. Faddeev, Discrete Heisenberg-Weyl group and modular group, Lett. Math. Phys. 34 (1995) 249 [hep-th/9504111] [INSPIRE].
Y. Yoshida and K. Sugiyama, Localization of 3d \( \mathcal{N}=2 \) Supersymmetric Theories on S 1 × D 2, arXiv:1409.6713 [INSPIRE].
E. Witten, Topological Quantum Field Theory, Commun. Math. Phys. 117 (1988) 353.
E. Witten, Phases of N = 2 theories in two-dimensions, Nucl. Phys. B 403 (1993) 159 [hep-th/9301042] [INSPIRE].
S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys. B 367 (1991) 359 [INSPIRE].
N. Nekrasov and E. Witten, The Omega Deformation, Branes, Integrability and Liouville Theory, JHEP 09 (2010) 092 [arXiv:1002.0888] [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].
E. Witten, A New Look At The Path Integral Of Quantum Mechanics, arXiv:1009.6032 [INSPIRE].
E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].
T. Dimofte, S. Gukov, J. Lenells and D. Zagier, Exact results for perturbative Chern-Simons theory with complex gauge group, Commun. Num. Theor. Phys. 3 (2009) 363.
D. Cooper, M. Culler, H. Gillet, D.D. Long and P.B. Shalen, Plane curves associated to character varieties of 3-manifolds, Invent. Math. 118 (1994) 47.
S. Gukov and I. Saberi, Lectures on Knot Homology and Quantum Curves, arXiv:1211.6075 [INSPIRE].
S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory and the A polynomial, Commun. Math. Phys. 255 (2005) 577 [hep-th/0306165] [INSPIRE].
S. Garoufalidis, On the characteristic and deformation varieties of ab knot, Geom. Topol. Monogr. 7 (2004) 291 [math/0306230].
R. Gelca, On the relation between the A-polynomial and the Jones polynomial, Proc. Am. Math. Soc. 130 (2002) 1235. [math/0004158].
C. Frohman, R. Gelca and W. Lofaro, The A-polynomial from the noncommutative viewpoint, Trans. Am. Math. Soc. 354 (2002) 735 [math/9812048].
S. Nawata, P. Ramadevi and Zodinmawia, Trivalent graphs, volume conjectures and character varieties, Lett. Math. Phys. 104 (2014) 1303 [arXiv:1404.5119] [INSPIRE].
H.M. Haggard, M. Han and A. Riello, Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Maps, arXiv:1506.03053 [INSPIRE].
R. Brown, Topology and Groupoids, www.groupoids.org (2006).
H.M. Haggard, M. Han, W. Kaminski and A. Riello, \( \mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{C}}\right) \) Chern-Simons Theory, a non-Planar Graph Operator and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry, Nucl. Phys. B 900 (2015) 1 [arXiv:1412.7546] [INSPIRE].
H.M. Haggard, M. Han, W. Kaminski and A. Riello, \( \mathrm{S}\mathrm{L}\left(2,\mathrm{\mathbb{C}}\right) \) Chern-Simons theory, flat connection, and 4-dimensional quantum geometry, to appear.
H.M. Haggard, M. Han, W. Kaminski and A. Riello, Four-dimensional Quantum Gravity with a Cosmological Constant from Three-dimensional Holomorphic Blocks, Phys. Lett. B 752 (2016) 258 [arXiv:1509.00458] [INSPIRE].
J.W. Barrett, R.J. Dowdall, W.J. Fairbairn, F. Hellmann and R. Pereira, Lorentzian spin foam amplitudes: Graphical calculus and asymptotics, Class. Quant. Grav. 27 (2010) 165009 [arXiv:0907.2440] [INSPIRE].
J.W. Barrett, R.J. Dowdall, W.J. Fairbairn, H. Gomes and F. Hellmann, Asymptotic analysis of the EPRL four-simplex amplitude, J. Math. Phys. 50 (2009) 112504 [arXiv:0902.1170] [INSPIRE].
M. Han and M. Zhang, Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Lorentzian Theory, Class. Quant. Grav. 30 (2013) 165012 [arXiv:1109.0499] [INSPIRE].
M.-X. Han and M. Zhang, Asymptotics of Spinfoam Amplitude on Simplicial Manifold: Euclidean Theory, Class. Quant. Grav. 29 (2012) 165004 [arXiv:1109.0500] [INSPIRE].
M. Han and T. Krajewski, Path Integral Representation of Lorentzian Spinfoam Model, Asymptotics and Simplicial Geometries, Class. Quant. Grav. 31 (2014) 015009 [arXiv:1304.5626] [INSPIRE].
J.W. Barrett and T.J. Foxon, Semiclassical limits of simplicial quantum gravity, Class. Quant. Grav. 11 (1994) 543 [gr-qc/9310016] [INSPIRE].
F. Conrady and L. Freidel, On the semiclassical limit of 4d spin foam models, Phys. Rev. D 78 (2008) 104023 [arXiv:0809.2280] [INSPIRE].
E. Suárez-Peiró, A Schläfli differential formula for simplices in semi-riemannian hyperquadrics, Gauss-Bonnet formulas for simplices in the de Sitter sphere and the dual volume of a hyperbolic simplex, Pacific J. Math. 194 (2000) 229.
B. Bahr and B. Dittrich, Improved and Perfect Actions in Discrete Gravity, Phys. Rev. D 80 (2009) 124030 [arXiv:0907.4323] [INSPIRE].
J.B. Hartle and R. Sorkin, Boundary Terms in the Action for the Regge Calculus, Gen. Rel. Grav. 13 (1981) 541 [INSPIRE].
R. Sorkin, Time Evolution Problem in Regge Calculus, Phys. Rev. D 12 (1975) 385 [Erratum ibid. D 23 (1981) 565] [INSPIRE].
S. Chun, S. Gukov and D. Roggenkamp, Junctions of surface operators and categorification of quantum groups, arXiv:1507.06318 [INSPIRE].
M. Han, Five branes, junctions, and quantum gravity, in preparation.
E. Witten, Five-brane effective action in M-theory, J. Geom. Phys. 22 (1997) 103 [hep-th/9610234] [INSPIRE].
E. Witten, AdS/CFT correspondence and topological field theory, JHEP 12 (1998) 012 [hep-th/9812012] [INSPIRE].
E. D’Hoker, J. Estes, M. Gutperle and D. Krym, Exact Half-BPS Flux Solutions in M-theory II: Global solutions asymptotic to AdS 7 × S 4, JHEP 12 (2008) 044 [arXiv:0810.4647] [INSPIRE].
D. Fiorenza, H. Sati and U. Schreiber, Multiple M5-branes, String 2-connections and 7d nonabelian Chern-Simons theory, Adv. Theor. Math. Phys. 18 (2014) 229 [arXiv:1201.5277] [INSPIRE].
D. Gang, N. Kim and S. Lee, Holography of 3d-3d correspondence at Large-N, JHEP 04 (2015) 091 [arXiv:1409.6206] [INSPIRE].
D.-S. Li, Z.-W. Liu, J.-B. Wu and B. Chen, M5-branes in AdS 4 × Q 1,1,1 spacetime, Phys. Rev. D 90 (2014) 066005 [arXiv:1406.1892] [INSPIRE].
T. Thiemann, Modern Canonical Quantum General Relativity, Cambridge University Press, Cambridge U.K. (2007).
C. Rovelli, Quantum Gravity, Cambridge University Press, Cambridge U.K. (2004).
C. Rovelli, and F. Vidotto, Covariant Loop Quantum Gravity: An Elementary Introduction to Quantum Gravity and Spinfoam Theory, Cambridge University Press, Cambridge U.K. (2014).
M. Han, W. Huang and Y. Ma, Fundamental structure of loop quantum gravity, Int. J. Mod. Phys. D 16 (2007) 1397 [gr-qc/0509064] [INSPIRE].
A. Ashtekar and J. Lewandowski, Background independent quantum gravity: A Status report, Class. Quant. Grav. 21 (2004) R53 [gr-qc/0404018] [INSPIRE].
A. Perez, The Spin Foam Approach to Quantum Gravity, Living Rev. Rel. 16 (2013) 3 [arXiv:1205.2019] [INSPIRE].
C. Rovelli, Zakopane lectures on loop gravity, PoS(QGQGS 2011)003 [arXiv:1102.3660] [INSPIRE].
J. Engle, E. Livine, R. Pereira and C. Rovelli, LQG vertex with finite Immirzi parameter, Nucl. Phys. B 799 (2008) 136 [arXiv:0711.0146] [INSPIRE].
L. Freidel and K. Krasnov, A New Spin Foam Model for 4d Gravity, Class. Quant. Grav. 25 (2008) 125018 [arXiv:0708.1595] [INSPIRE].
M. Han, 4-dimensional Spin-foam Model with Quantum Lorentz Group, J. Math. Phys. 52 (2011) 072501 [arXiv:1012.4216] [INSPIRE].
W.J. Fairbairn and C. Meusburger, Quantum deformation of two four-dimensional spin foam models, J. Math. Phys. 53 (2012) 022501 [arXiv:1012.4784] [INSPIRE].
M. Han, Cosmological Constant in LQG Vertex Amplitude, Phys. Rev. D 84 (2011) 064010 [arXiv:1105.2212] [INSPIRE].
M. Han, Covariant Loop Quantum Gravity, Low Energy Perturbation Theory and Einstein Gravity with High Curvature UV Corrections, Phys. Rev. D 89 (2014) 124001 [arXiv:1308.4063] [INSPIRE].
M. Han, On Spinfoam Models in Large Spin Regime, Class. Quant. Grav. 31 (2014) 015004 [arXiv:1304.5627] [INSPIRE].
M. Han, Semiclassical Analysis of Spinfoam Model with a Small Barbero-Immirzi Parameter, Phys. Rev. D 88 (2013) 044051 [arXiv:1304.5628] [INSPIRE].
A. Banburski, L.-Q. Chen, L. Freidel and J. Hnybida, Pachner moves in a 4d Riemannian holomorphic Spin Foam model, Phys. Rev. D 92 (2015) 124014 [arXiv:1412.8247] [INSPIRE].
T. Thiemann, Quantum spin dynamics (QSD), Class. Quant. Grav. 15 (1998) 839 [gr-qc/9606089] [INSPIRE].
T. Thiemann, The Phoenix Project: master constraint programme for loop quantum gravity, Class. Quant. Grav. 23 (2006) 2211 [gr-qc/0305080].
T. Thiemann, Quantum spin dynamics: VIII. The master constraint, Class. Quant. Grav. 23 (2006) 2249 [gr-qc/0510011].
M.-x. Han and Y.-g. Ma, Master constraint operator in loop quantum gravity, Phys. Lett. B 635 (2006) 225 [gr-qc/0510014] [INSPIRE].
M. Han and T. Thiemann, On the Relation between Operator Constraint –, Master Constraint –, Reduced Phase Space – and Path Integral Quantisation, Class. Quant. Grav. 27 (2010) 225019 [arXiv:0911.3428] [INSPIRE].
M. Han and T. Thiemann, On the Relation between Rigging Inner Product and Master Constraint Direct Integral Decomposition, J. Math. Phys. 51 (2010) 092501 [arXiv:0911.3431] [INSPIRE].
M. Han, Path-integral for the Master Constraint of Loop Quantum Gravity, Class. Quant. Grav. 27 (2010) 215009 [arXiv:0911.3432] [INSPIRE].
V. Bonzom and L. Freidel, The Hamiltonian constraint in 3d Riemannian loop quantum gravity, Class. Quant. Grav. 28 (2011) 195006 [arXiv:1101.3524] [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: 1509.00466
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, 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 license, and indicate if changes were made.
About this article
Cite this article
Han, M. 4d quantum geometry from 3d supersymmetric gauge theory and holomorphic block. J. High Energ. Phys. 2016, 65 (2016). https://doi.org/10.1007/JHEP01(2016)065
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2016)065