Abstract
It is argued that 4d SU(N) \( \mathcal{N} \) = 4 SYM has an accumulation line of zero-temperature topologically ordered phases. Each of these phases corresponds to N bound states charged under electromagnetic \( {\mathbb{Z}}_N^{(1)} \) one-form symmetries. Each of the N bound states is made of two Dyonic flux components each of them extended over a two dimensional surface. They are localized at the fixed loci of a rotational action, and are argued to correspond to conformal blocks (or primaries) of an SU(N)1 WZNW model on a two-torus.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The Large N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transition, and confinement in gauge theories, Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, Microscopic origin of the Bekenstein-Hawking entropy of supersymmetric AdS5 black holes, JHEP 10 (2019) 062 [arXiv:1810.11442] [INSPIRE].
J.B. Gutowski and H.S. Reall, Supersymmetric AdS5 black holes, JHEP 02 (2004) 006 [hep-th/0401042] [INSPIRE].
M. Cvetič, H. Lü and C.N. Pope, Charged Kerr-de Sitter black holes in five dimensions, Phys. Lett. B 598 (2004) 273 [hep-th/0406196] [INSPIRE].
M. Cvetič, H. Lü and C.N. Pope, Charged rotating black holes in five dimensional U(1)3 gauged N = 2 supergravity, Phys. Rev. D 70 (2004) 081502 [hep-th/0407058] [INSPIRE].
D. Cassani and L. Papini, The BPS limit of rotating AdS black hole thermodynamics, JHEP 09 (2019) 079 [arXiv:1906.10148] [INSPIRE].
F. Larsen, J. Nian and Y. Zeng, AdS5 black hole entropy near the BPS limit, JHEP 06 (2020) 001 [arXiv:1907.02505] [INSPIRE].
G. Kántor, C. Papageorgakis and P. Richmond, AdS7 black-hole entropy and 5D \( \mathcal{N} \) = 2 Yang-Mills, JHEP 01 (2020) 017 [arXiv:1907.02923] [INSPIRE].
S. Choi, J. Kim, S. Kim and J. Nahmgoong, Large AdS black holes from QFT, arXiv:1810.12067 [INSPIRE].
F. Benini and E. Milan, Black Holes in 4D \( \mathcal{N} \) = 4 Super-Yang-Mills Field Theory, Phys. Rev. X 10 (2020) 021037 [arXiv:1812.09613] [INSPIRE].
S.M. Hosseini, K. Hristov and A. Zaffaroni, An extremization principle for the entropy of rotating BPS black holes in AdS5, JHEP 07 (2017) 106 [arXiv:1705.05383] [INSPIRE].
M. Honda, Quantum Black Hole Entropy from 4d Supersymmetric Cardy formula, Phys. Rev. D 100 (2019) 026008 [arXiv:1901.08091] [INSPIRE].
A. Arabi Ardehali, Cardy-like asymptotics of the 4d \( \mathcal{N} \) = 4 index and AdS5 blackholes, JHEP 06 (2019) 134 [arXiv:1902.06619] [INSPIRE].
J. Kim, S. Kim and J. Song, A 4d \( \mathcal{N} \) = 1 Cardy Formula, JHEP 01 (2021) 025 [arXiv:1904.03455] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, The asymptotic growth of states of the 4d \( \mathcal{N} \) = 1 superconformal index, JHEP 08 (2019) 120 [arXiv:1904.05865] [INSPIRE].
S. Murthy, The growth of the \( \frac{1}{16} \)-BPS index in 4d \( \mathcal{N} \) = 4 SYM, arXiv:2005.10843 [INSPIRE].
P. Agarwal, S. Choi, J. Kim, S. Kim and J. Nahmgoong, AdS black holes and finite N indices, Phys. Rev. D 103 (2021) 126006 [arXiv:2005.11240] [INSPIRE].
C. Romelsberger, Counting chiral primaries in N = 1, d = 4 superconformal field theories, Nucl. Phys. B 747 (2006) 329 [hep-th/0510060] [INSPIRE].
J. Kinney, J.M. Maldacena, S. Minwalla and S. Raju, An Index for 4 dimensional super conformal theories, Commun. Math. Phys. 275 (2007) 209 [hep-th/0510251] [INSPIRE].
F.A. Dolan and H. Osborn, Applications of the Superconformal Index for Protected Operators and q-Hypergeometric Identities to N = 1 Dual Theories, Nucl. Phys. B 818 (2009) 137 [arXiv:0801.4947] [INSPIRE].
A. Cabo-Bizet and S. Murthy, Supersymmetric phases of 4d \( \mathcal{N} \) = 4 SYM at large N, JHEP 09 (2020) 184 [arXiv:1909.09597] [INSPIRE].
A. Cabo-Bizet, From multi-gravitons to Black holes: The role of complex saddles, arXiv:2012.04815 [INSPIRE].
K. Goldstein, V. Jejjala, Y. Lei, S. van Leuven and W. Li, Residues, modularity, and the Cardy limit of the 4d \( \mathcal{N} \) = 4 superconformal index, JHEP 04 (2021) 216 [arXiv:2011.06605] [INSPIRE].
V. Jejjala, Y. Lei, S. van Leuven and W. Li, SL(3, ℤ) Modularity and New Cardy limits of the \( \mathcal{N} \) = 4 superconformal index, JHEP 11 (2021) 047 [arXiv:2104.07030] [INSPIRE].
A. Arabi Ardehali and S. Murthy, The 4d superconformal index near roots of unity and 3d Chern-Simons theory, JHEP 10 (2021) 207 [arXiv:2104.02051] [INSPIRE].
A.A. Ardehali and J. Hong, Decomposition of BPS moduli spaces and asymptotics of supersymmetric partition functions, JHEP 01 (2022) 062 [arXiv:2110.01538] [INSPIRE].
A. Cabo-Bizet, On the 4d superconformal index near roots of unity: Bulk and Localized contributions, arXiv:2111.14941 [INSPIRE].
A.A. Gerasimov and S.L. Shatashvili, Higgs Bundles, Gauge Theories and Quantum Groups, Commun. Math. Phys. 277 (2008) 323 [hep-th/0609024] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Supersymmetric vacua and Bethe ansatz, Nucl. Phys. B Proc. Suppl. 192-193 (2009) 91 [arXiv:0901.4744] [INSPIRE].
N.A. Nekrasov and S.L. Shatashvili, Quantization of Integrable Systems and Four Dimensional Gauge Theories, in 16th International Congress on Mathematical Physics, Prague, Czechia (2009), pg. 265 [arXiv:0908.4052] [INSPIRE].
A. Antinucci, G. Galati and G. Rizi, On Continuous 2-Category Symmetries and Yang-Mills Theory, arXiv:2206.05646 [INSPIRE].
S. Nawata, Localization of N = 4 Superconformal Field Theory on S1 × S3 and Index, JHEP 11 (2011) 144 [arXiv:1104.4470] [INSPIRE].
E. Witten, Introduction to cohomological field theories, Int. J. Mod. Phys. A 6 (1991) 2775 [INSPIRE].
V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].
S. Cordes, G.W. Moore and S. Ramgoolam, Lectures on 2 − D Yang-Mills theory, equivariant cohomology and topological field theories, Nucl. Phys. B Proc. Suppl. 41 (1995) 184 [hep-th/9411210] [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].
F. Benini and E. Milan, A Bethe Ansatz type formula for the superconformal index, Commun. Math. Phys. 376 (2020) 1413 [arXiv:1811.04107] [INSPIRE].
L. Álvarez-Gaumé, C. Gomez and G. Sierra, Hidden Quantum Symmetries in Rational Conformal Field Theories, Nucl. Phys. B 319 (1989) 155 [INSPIRE].
A. Alekseev, L.D. Faddeev and M. Semenov-Tian-Shansky, Hidden quantum groups inside Kac-Moody algebra, Commun. Math. Phys. 149 (1992) 335 [INSPIRE].
E.P. Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B 300 (1988) 360 [INSPIRE].
G.W. Moore and N. Seiberg, Classical and Quantum Conformal Field Theory, Commun. Math. Phys. 123 (1989) 177 [INSPIRE].
G.W. Moore and N. Reshetikhin, A Comment on Quantum Group Symmetry in Conformal Field Theory, Nucl. Phys. B 328 (1989) 557 [INSPIRE].
D. Karabali and H.J. Schnitzer, BRST Quantization of the Gauged WZW Action and Coset Conformal Field Theories, Nucl. Phys. B 329 (1990) 649 [INSPIRE].
M. Spiegelglas and S. Yankielowicz, G/G topological field theories by cosetting G(k), Nucl. Phys. B 393 (1993) 301 [hep-th/9201036] [INSPIRE].
K. Gawędzki and A. Kupiainen, Coset Construction from Functional Integrals, Nucl. Phys. B 320 (1989) 625 [INSPIRE].
E. Witten, On Holomorphic factorization of WZW and coset models, Commun. Math. Phys. 144 (1992) 189 [INSPIRE].
E. Witten, Nonabelian Bosonization in Two-Dimensions, Commun. Math. Phys. 92 (1984) 455 [INSPIRE].
A.M. Polyakov and P.B. Wiegmann, Theory of Nonabelian Goldstone Bosons, Phys. Lett. B 131 (1983) 121 [INSPIRE].
D. Bernard, On the Wess-Zumino-Witten Models on the Torus, Nucl. Phys. B 303 (1988) 77 [INSPIRE].
G.W. Moore and N. Read, Nonabelions in the fractional quantum Hall effect, Nucl. Phys. B 360 (1991) 362 [INSPIRE].
N. Read and G.W. Moore, Fractional quantum Hall effect and nonAbelian statistics, Prog. Theor. Phys. Suppl. 107 (1992) 157 [hep-th/9202001] [INSPIRE].
C. Nayak, S.H. Simon, A. Stern, M. Freedman and S. Das Sarma, Non-Abelian anyons and topological quantum computation, Rev. Mod. Phys. 80 (2008) 1083 [arXiv:0707.1889] [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].
A. Cherman and A. Dhumuntarao, Confinement and graded partition functions for \( \mathcal{N} \) = 4 SYM, Phys. Rev. D 103 (2021) 066013 [arXiv:2012.12341] [INSPIRE].
M. Heydeman, L.V. Iliesiu, G.J. Turiaci and W. Zhao, The statistical mechanics of near-BPS black holes, J. Phys. A 55 (2022) 014004 [arXiv:2011.01953] [INSPIRE].
G. Festuccia and N. Seiberg, Rigid Supersymmetric Theories in Curved Superspace, JHEP 06 (2011) 114 [arXiv:1105.0689] [INSPIRE].
C. Closset and I. Shamir, The \( \mathcal{N} \) = 1 Chiral Multiplet on T2 × S2 and Supersymmetric Localization, JHEP 03 (2014) 040 [arXiv:1311.2430] [INSPIRE].
B. Assel, D. Cassani and D. Martelli, Localization on Hopf surfaces, JHEP 08 (2014) 123 [arXiv:1405.5144] [INSPIRE].
L.V. Iliesiu, M. Kologlu and G.J. Turiaci, Supersymmetric indices factorize, arXiv:2107.09062 [INSPIRE].
E. Witten, A Note On Complex Spacetime Metrics, arXiv:2111.06514 [INSPIRE].
M. Kontsevich and G. Segal, Wick Rotation and the Positivity of Energy in Quantum Field Theory, Quart. J. Math. Oxford Ser. 72 (2021) 673 [arXiv:2105.10161] [INSPIRE].
M.F. Atiyah and R. Bott, The Moment map and equivariant cohomology, Topology 23 (1984) 1 [INSPIRE].
A.G. Lezcano, J. Hong, J.T. Liu and L.A. Pando Zayas, The Bethe-Ansatz approach to the \( \mathcal{N} \) = 4 superconformal index at finite rank, JHEP 06 (2021) 126 [arXiv:2101.12233] [INSPIRE].
A. Cabo-Bizet, D. Cassani, D. Martelli and S. Murthy, The large-N limit of the 4d \( \mathcal{N} \) = 1 superconformal index, JHEP 11 (2020) 150 [arXiv:2005.10654] [INSPIRE].
J. Hong and J.T. Liu, The topologically twisted index of \( \mathcal{N} \) = 4 super-Yang-Mills on T2 × S2 and the elliptic genus, JHEP 07 (2018) 018 [arXiv:1804.04592] [INSPIRE].
S.M. Hosseini, A. Nedelin and A. Zaffaroni, The Cardy limit of the topologically twisted index and black strings in AdS5, JHEP 04 (2017) 014 [arXiv:1611.09374] [INSPIRE].
A. Arabi Ardehali, J. Hong and J.T. Liu, Asymptotic growth of the 4d \( \mathcal{N} \) = 4 index and partially deconfined phases, JHEP 07 (2020) 073 [arXiv:1912.04169] [INSPIRE].
A. González Lezcano, J. Hong, J.T. Liu and L.A. Pando Zayas, Sub-leading Structures in Superconformal Indices: Subdominant Saddles and Logarithmic Contributions, JHEP 01 (2021) 001 [arXiv:2007.12604] [INSPIRE].
A. Amariti, M. Fazzi and A. Segati, Expanding on the Cardy-like limit of the SCI of 4d \( \mathcal{N} \) = 1 ABCD SCFTs, JHEP 07 (2021) 141 [arXiv:2103.15853] [INSPIRE].
D. Cassani and Z. Komargodski, EFT and the SUSY Index on the 2nd Sheet, SciPost Phys. 11 (2021) 004 [arXiv:2104.01464] [INSPIRE].
F. Benini and G. Rizi, Superconformal index of low-rank gauge theories via the Bethe Ansatz, JHEP 05 (2021) 061 [arXiv:2102.03638] [INSPIRE].
D. Gaiotto, A. Kapustin, N. Seiberg and B. Willett, Generalized Global Symmetries, JHEP 02 (2015) 172 [arXiv:1412.5148] [INSPIRE].
S. Gukov and E. Witten, Gauge Theory, Ramification, And The Geometric Langlands Program, hep-th/0612073 [INSPIRE].
N. Seiberg, Dynamics of Exotic Theories, Spectrum and UV/IR Mixing, talk at Geometry of (S)QFT, Simons Center for Geometry and Physics, New York, U.S.A. (2021).
F. Benini and A. Zaffaroni, A topologically twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
K. Costello, Supersymmetric gauge theory and the Yangian, arXiv:1303.2632 [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, II, ICCM Not. 06 (2018) 120 [arXiv:1802.01579] [INSPIRE].
K. Costello, E. Witten and M. Yamazaki, Gauge Theory and Integrability, I, ICCM Not. 06 (2018) 46 [arXiv:1709.09993] [INSPIRE].
E. Witten, Integrable Lattice Models From Gauge Theory, Adv. Theor. Math. Phys. 21 (2017) 1819 [arXiv:1611.00592] [INSPIRE].
O. Aharony, F. Benini, O. Mamroud and E. Milan, A gravity interpretation for the Bethe Ansatz expansion of the \( \mathcal{N} \) = 4 SYM index, Phys. Rev. D 104 (2021) 086026 [arXiv:2104.13932] [INSPIRE].
A. Kapustin and E. Witten, Electric-Magnetic Duality And The Geometric Langlands Program, Commun. Num. Theor. Phys. 1 (2007) 1 [hep-th/0604151] [INSPIRE].
E. Frenkel, Affine algebras, Langlands duality and Bethe ansatz, in 11th International Conference on Mathematical Physics (ICMP-11), Paris, France (1994) [q-alg/9506003].
Y. Yoshida, Factorization of 4d N = 1 superconformal index, arXiv:1403.0891 [INSPIRE].
W. Peelaers, Higgs branch localization of \( \mathcal{N} \) = 1 theories on S3 × S1, JHEP 08 (2014) 060 [arXiv:1403.2711] [INSPIRE].
F. Nieri and S. Pasquetti, Factorisation and holomorphic blocks in 4d, JHEP 11 (2015) 155 [arXiv:1507.00261] [INSPIRE].
G. ’t Hooft, Topology of the Gauge Condition and New Confinement Phases in Nonabelian Gauge Theories, Nucl. Phys. B 190 (1981) 455 [INSPIRE].
J.L. Cardy and E. Rabinovici, Phase Structure of Z(p) Models in the Presence of a Theta Parameter, Nucl. Phys. B 205 (1982) 1 [INSPIRE].
J.L. Cardy, Duality and the Theta Parameter in Abelian Lattice Models, Nucl. Phys. B 205 (1982) 17 [INSPIRE].
A.D. Shapere and F. Wilczek, Selfdual Models with Theta Terms, Nucl. Phys. B 320 (1989) 669 [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. 430 (1994) 485] [hep-th/9407087] [INSPIRE].
K.A. Intriligator and N. Seiberg, Duality, monopoles, dyons, confinement and oblique confinement in supersymmetric SO(Nc) gauge theories, Nucl. Phys. B 444 (1995) 125 [hep-th/9503179] [INSPIRE].
E. Witten, The N matrix model and gauged WZW models, Nucl. Phys. B 371 (1992) 191 [INSPIRE].
M. Blau and G. Thompson, Lectures on 2 − D gauge theories: Topological aspects and path integral techniques, in Summer School in High-energy Physics and Cosmology, Trieste, Italy (1993), pg. 0175 [hep-th/9310144] [INSPIRE].
E. Witten, Quantum Field Theory and the Jones Polynomial, Commun. Math. Phys. 121 (1989) 351 [INSPIRE].
M. Blau and G. Thompson, Derivation of the Verlinde formula from Chern-Simons theory and the G/G model, Nucl. Phys. B 408 (1993) 345 [hep-th/9305010] [INSPIRE].
S. Okuda and Y. Yoshida, G/G gauged WZW model and Bethe Ansatz for the phase model, JHEP 11 (2012) 146 [arXiv:1209.3800] [INSPIRE].
S.G. Naculich and H.J. Schnitzer, Level-rank duality of the U(N) WZW model, Chern-Simons theory, and 2 − D qYM theory, JHEP 06 (2007) 023 [hep-th/0703089] [INSPIRE].
C. Korff and C. Stroppel, The \( \hat{s} \)l(n)k-WZNW fusion ring: A combinatorial construction and a realisation as quotient of quantum cohomology, Adv. Math. 225 (2010) 200 [arXiv:0909.2347].
N.M. Bogoliubov, A.G. Izergin and N.A. Kitanine, Correlation functions for a strongly correlated boson system, Nucl. Phys. B 516 (1998) 501 [solv-int/9710002] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2111.14942
Rights and permissions
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.
About this article
Cite this article
Cabo-Bizet, A. Quantum phases of 4d SU(N) \( \mathcal{N} \) = 4 SYM. J. High Energ. Phys. 2022, 52 (2022). https://doi.org/10.1007/JHEP10(2022)052
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2022)052