Abstract
When a (super) conformal field theory is placed on a non-trivial manifold, the (super) conformal symmetry is broken. However, it is still possible to derive broken Ward identities for these broken symmetries, which provide additional constraints on the theory. We derive and apply the broken Ward identities associated with the (super) conformal group on the thermal manifold \( {\mathcal{M}}_{\beta }={S}_{\beta}^1\times {\mathbb{R}}^{d-1} \) and \( \mathcal{M}={T}^2\times {\mathbb{R}}^{d-2} \). The novel constraints not only systematically reproduce known results, including an implicit formulation of the generalized Cardy formula, but also elegantly relate the thermal energy spectrum with the conformal spectrum.
Article PDF
Avoid common mistakes on your manuscript.
References
L. Iliesiu et al., The Conformal Bootstrap at Finite Temperature, JHEP 10 (2018) 070 [arXiv:1802.10266] [INSPIRE].
L. Iliesiu, M. Koloğlu and D. Simmons-Duffin, Bootstrapping the 3d Ising model at finite temperature, JHEP 12 (2019) 072 [arXiv:1811.05451] [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].
G.T. Horowitz and V.E. Hubeny, Quasinormal modes of AdS black holes and the approach to thermal equilibrium, Phys. Rev. D 62 (2000) 024027 [hep-th/9909056] [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].
D.T. Son and A.O. Starinets, Minkowski space correlators in AdS/CFT correspondence: Recipe and applications, JHEP 09 (2002) 042 [hep-th/0205051] [INSPIRE].
O. Aharony et al., The Hagedorn - deconfinement phase transition in weakly coupled large N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
J. Casalderrey-Solana et al., Gauge/String Duality, Hot QCD and Heavy Ion Collisions, Cambridge University Press (2014) [https://doi.org/10.1017/9781009403504] [INSPIRE].
M. Ammon and J. Erdmenger, Gauge/gravity duality: Foundations and applications, Cambridge University Press, Cambridge (2015) [https://doi.org/10.1017/CBO9780511846373] [INSPIRE].
H. Nastase, Introduction to the ADS/CFT Correspondence, Cambridge University Press (2015) [INSPIRE].
S. Caron-Huot, Holographic cameras: an eye for the bulk, JHEP 03 (2023) 047 [arXiv:2211.11791] [INSPIRE].
M. Dodelson, C. Iossa, R. Karlsson and A. Zhiboedov, A thermal product formula, arXiv:2304.12339 [INSPIRE].
M. Dodelson et al., Holographic thermal correlators from supersymmetric instantons, SciPost Phys. 14 (2023) 116 [arXiv:2206.07720] [INSPIRE].
Y. Gobeil, A. Maloney, G.S. Ng and J.-Q. Wu, Thermal Conformal Blocks, SciPost Phys. 7 (2019) 015 [arXiv:1802.10537] [INSPIRE].
L.F. Alday, M. Kologlu and A. Zhiboedov, Holographic correlators at finite temperature, JHEP 06 (2021) 082 [arXiv:2009.10062] [INSPIRE].
S. Caron-Huot, On supersymmetry at finite temperature, Phys. Rev. D 79 (2009) 125002 [arXiv:0808.0155] [INSPIRE].
N. Benjamin, J. Lee, H. Ooguri and D. Simmons-Duffin, Universal Asymptotics for High Energy CFT Data, arXiv:2306.08031 [INSPIRE].
C. Luo and Y. Wang, Casimir energy and modularity in higher-dimensional conformal field theories, JHEP 07 (2023) 028 [arXiv:2212.14866] [INSPIRE].
H. Osborn and A.C. Petkou, Implications of conformal invariance in field theories for general dimensions, Annals Phys. 231 (1994) 311 [hep-th/9307010] [INSPIRE].
S. Rychkov, EPFL Lectures on Conformal Field Theory in D≥3 Dimensions, arXiv:1601.05000 [https://doi.org/10.1007/978-3-319-43626-5] [INSPIRE].
D. Simmons-Duffin, The Conformal Bootstrap, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and Strings, Boulder, U.S.A., June 01–26 (2015), p. 1–74 [https://doi.org/10.1142/9789813149441_0001] [arXiv:1602.07982] [INSPIRE].
R. Gopakumar, E. Perlmutter, S.S. Pufu and X. Yin, Snowmass White Paper: Bootstrapping String Theory, arXiv:2202.07163 [INSPIRE].
P. Di Francesco, P. Mathieu and D. Senechal, Conformal Field Theory, Springer-Verlag, New York (1997) [https://doi.org/10.1007/978-1-4612-2256-9] [INSPIRE].
K. Papadodimas, Topological Anti-Topological Fusion in Four-Dimensional Superconformal Field Theories, JHEP 08 (2010) 118 [arXiv:0910.4963] [INSPIRE].
A.B. Zamolodchikov, Integrable field theory from conformal field theory, Adv. Stud. Pure Math. 19 (1989) 641 [INSPIRE].
A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl. Phys. B 241 (1984) 333 [INSPIRE].
G. Mussardo, Statistical Field Theory, Oxford University Press (2020) [INSPIRE].
D. Bitko, T.F. Rosenbaum and G. Aeppli, Quantum critical behavior for a model magnet, Phys. Rev. Lett. 77 (1996) 940.
N. Chai et al., Thermal Order in Conformal Theories, Phys. Rev. D 102 (2020) 065014 [arXiv:2005.03676] [INSPIRE].
N. Chai et al., Symmetry Breaking at All Temperatures, Phys. Rev. Lett. 125 (2020) 131603 [INSPIRE].
M. Krech and D.P. Landau, Casimir effect in critical systems: A monte carlo simulation, Phys. Rev. E 53 (1996) 4414.
S. El-Showk and K. Papadodimas, Emergent Spacetime and Holographic CFTs, JHEP 10 (2012) 106 [arXiv:1101.4163] [INSPIRE].
J.L. Cardy, Conformal invariance and statistical mechanics, in the proceedings of the Les Houches Summer School in Theoretical Physics: Fields, Strings, Critical Phenomena, Les Houches, France, June 28 – August 05 (1988) [INSPIRE].
J.L. Cardy, Operator content and modular properties of higher dimensional conformal field theories, Nucl. Phys. B 366 (1991) 403 [INSPIRE].
M. Hogervorst and S. Rychkov, Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 (2013) 106004 [arXiv:1303.1111] [INSPIRE].
M. Hogervorst, S. Rychkov and B.C. van Rees, Truncated conformal space approach in d dimensions: A cheap alternative to lattice field theory?, Phys. Rev. D 91 (2015) 025005 [arXiv:1409.1581] [INSPIRE].
J.L. Cardy, Operator Content of Two-Dimensional Conformally Invariant Theories, Nucl. Phys. B 270 (1986) 186 [INSPIRE].
M. Berg, Manifest Modular Invariance in the Near-Critical Ising Model, arXiv:2302.01185 [INSPIRE].
M. Downing, S. Murthy and G.M.T. Watts, Modular symmetry of massive free fermions, arXiv:2302.01251 [INSPIRE].
L. Di Pietro and Z. Komargodski, Cardy formulae for SUSY theories in d = 4 and d = 6, JHEP 12 (2014) 031 [arXiv:1407.6061] [INSPIRE].
B. Assel et al., The Casimir Energy in Curved Space and its Supersymmetric Counterpart, JHEP 07 (2015) 043 [arXiv:1503.05537] [INSPIRE].
B. Mukhametzhanov and A. Zhiboedov, Modular invariance, tauberian theorems and microcanonical entropy, JHEP 10 (2019) 261 [arXiv:1904.06359] [INSPIRE].
S. Carlip, Logarithmic corrections to black hole entropy from the Cardy formula, Class. Quant. Grav. 17 (2000) 4175 [gr-qc/0005017] [INSPIRE].
I.H. Brevik, S. Nojiri, S.D. Odintsov and L. Vanzo, Entropy and universality of Cardy-Verlinde formula in dark energy universe, Phys. Rev. D 70 (2004) 043520 [hep-th/0401073] [INSPIRE].
B. Wang, E. Abdalla and R.-K. Su, Relating Friedmann equation to Cardy formula in universes with cosmological constant, Phys. Lett. B 503 (2001) 394 [hep-th/0101073] [INSPIRE].
R. Kubo, Statistical mechanical theory of irreversible processes. 1. General theory and simple applications in magnetic and conduction problems, J. Phys. Soc. Jap. 12 (1957) 570 [INSPIRE].
P.C. Martin and J.S. Schwinger, Theory of many particle systems. 1, Phys. Rev. 115 (1959) 1342 [INSPIRE].
S. Caron-Huot and G.D. Moore, Spacelike thermal correlators are almost time independent, Phys. Rev. D 106 (2022) 125015 [arXiv:2209.02641] [INSPIRE].
J. Fuchs, Thermal and Superthermal Properties of Supersymmetric Field Theories, Nucl. Phys. B 246 (1984) 279 [INSPIRE].
H. Aoyama and D. Boyanovsky, Goldstone Fermions in Supersymmetric Theories at Finite Temperature, Phys. Rev. D 30 (1984) 1356 [INSPIRE].
A.K. Das and M. Kaku, Supersymmetry at high temperatures, Phys. Rev. D 18 (1978) 4540 [INSPIRE].
L. Girardello, M.T. Grisaru and P. Salomonson, Temperature and Supersymmetry, Nucl. Phys. B 178 (1981) 331 [INSPIRE].
J.J. Friess, S.S. Gubser, G. Michalogiorgakis and S.S. Pufu, The stress tensor of a quark moving through N = 4 thermal plasma, Phys. Rev. D 75 (2007) 106003 [hep-th/0607022] [INSPIRE].
D. Berenstein and R. Mancilla, Aspects of thermal one-point functions and response functions in AdS black holes, Phys. Rev. D 107 (2023) 126010 [arXiv:2211.05144] [INSPIRE].
A.C. Petkou, Thermal one-point functions and single-valued polylogarithms, Phys. Lett. B 820 (2021) 136467 [arXiv:2105.03530] [INSPIRE].
R. Karlsson, M. Kulaxizi, A. Parnachev and P. Tadić, Black Holes and Conformal Regge Bootstrap, JHEP 10 (2019) 046 [arXiv:1904.00060] [INSPIRE].
R. Karlsson, A. Parnachev and P. Tadić, Thermalization in large-N CFTs, JHEP 09 (2021) 205 [arXiv:2102.04953] [INSPIRE].
R. Karlsson, A. Parnachev, V. Prilepina and S. Valach, Thermal stress tensor correlators, OPE and holography, JHEP 09 (2022) 234 [arXiv:2206.05544] [INSPIRE].
L. Eberhardt, Superconformal symmetry and representations, J. Phys. A 54 (2021) 063002 [arXiv:2006.13280] [INSPIRE].
F.A. Dolan and H. Osborn, Superconformal symmetry, correlation functions and the operator product expansion, Nucl. Phys. B 629 (2002) 3 [hep-th/0112251] [INSPIRE].
F.A. Dolan and H. Osborn, On short and semi-short representations for four-dimensional superconformal symmetry, Annals Phys. 307 (2003) 41 [hep-th/0209056] [INSPIRE].
J.I. Kapusta and C. Gale, Finite-temperature field theory: Principles and applications, Cambridge University Press (2011) [https://doi.org/10.1017/CBO9780511535130] [INSPIRE].
K.A. Intriligator, Bonus symmetries of N = 4 superYang-Mills correlation functions via AdS duality, Nucl. Phys. B 551 (1999) 575 [hep-th/9811047] [INSPIRE].
K.A. Intriligator and W. Skiba, Bonus symmetry and the operator product expansion of N = 4 SuperYang-Mills, Nucl. Phys. B 559 (1999) 165 [hep-th/9905020] [INSPIRE].
M. Brigante, G. Festuccia and H. Liu, Inheritance principle and non-renormalization theorems at finite temperature, Phys. Lett. B 638 (2006) 538 [hep-th/0509117] [INSPIRE].
K. Furuuchi, From free fields to AdS: Thermal case, Phys. Rev. D 72 (2005) 066009 [hep-th/0505148] [INSPIRE].
N. Arkani-Hamed et al., Deconstructing (2,0) and little string theories, JHEP 01 (2003) 083 [hep-th/0110146] [INSPIRE].
J. Hayling, C. Papageorgakis, E. Pomoni and D. Rodríguez-Gómez, Exact Deconstruction of the 6D (2,0) Theory, JHEP 06 (2017) 072 [arXiv:1704.02986] [INSPIRE].
V. Niarchos, C. Papageorgakis and E. Pomoni, Type-B Anomaly Matching and the 6D (2,0) Theory, JHEP 04 (2020) 048 [arXiv:1911.05827] [INSPIRE].
V. Niarchos, C. Papageorgakis, A. Pini and E. Pomoni, (Mis-)Matching Type-B Anomalies on the Higgs Branch, JHEP 01 (2021) 106 [arXiv:2009.08375] [INSPIRE].
T. Bourton, A. Pini and E. Pomoni, The Coulomb and Higgs branches of \( \mathcal{N} \) = 1 theories of Class \( {\mathcal{S}}_k \), JHEP 02 (2021) 137 [arXiv:2011.01587] [INSPIRE].
J. Giedt, E. Poppitz and M. Rozali, Deconstruction, lattice supersymmetry, anomalies and branes, JHEP 03 (2003) 035 [hep-th/0301048] [INSPIRE].
H.-C. Kim, S.S. Razamat, C. Vafa and G. Zafrir, Compactifications of ADE conformal matter on a torus, JHEP 09 (2018) 110 [arXiv:1806.07620] [INSPIRE].
E. Parisini, K. Skenderis and B. Withers, Embedding formalism for CFTs in general states on curved backgrounds, Phys. Rev. D 107 (2023) 066022 [arXiv:2209.09250] [INSPIRE].
E. Parisini, K. Skenderis and B. Withers, The Ambient Space Formalism, arXiv:2312.03820 [INSPIRE].
L.V. Delacretaz, A.L. Fitzpatrick, E. Katz and M.T. Walters, Thermalization and chaos in a 1+1d QFT, JHEP 02 (2023) 045 [arXiv:2207.11261] [INSPIRE].
L.V. Delacretaz, A.L. Fitzpatrick, E. Katz and M.T. Walters, Thermalization and hydrodynamics of two-dimensional quantum field theories, SciPost Phys. 12 (2022) 119 [arXiv:2105.02229] [INSPIRE].
O. Fukushima and R. Hamazaki, Violation of Eigenstate Thermalization Hypothesis in Quantum Field Theories with Higher-Form Symmetry, Phys. Rev. Lett. 131 (2023) 131602 [arXiv:2305.04984] [INSPIRE].
S. Datta, P. Kraus and B. Michel, Typicality and thermality in 2d CFT, JHEP 07 (2019) 143 [arXiv:1904.00668] [INSPIRE].
D. Rodriguez-Gomez and J.G. Russo, Correlation functions in finite temperature CFT and black hole singularities, JHEP 06 (2021) 048 [arXiv:2102.11891] [INSPIRE].
Acknowledgments
It is a pleasure to thank Antonio Antunes, Davide Cassani, Alejandra Castro, Pietro Ferrero, Stefano Giusto, Apratim Kaviraj and Vasilis Niarchos for very useful discussions at different stages of the work. AM and EP have benefited from the German Research Foundation DFG under Germany’s Excellence Strategy – EXC 2121 Quantum Universe – 390833306.
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: 2306.12417
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
Marchetto, E., Miscioscia, A. & Pomoni, E. Broken (super) conformal Ward identities at finite temperature. J. High Energ. Phys. 2023, 186 (2023). https://doi.org/10.1007/JHEP12(2023)186
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP12(2023)186