Abstract
We study the thermodynamic behaviour of the real β- and γ i -deformation of \( \mathcal{N}=4 \) Super Yang-Mills theory on \( \mathbb{R}\times {\mathrm{S}}^3 \) in the planar limit. These theories were shown to be the most general asymptotically integrable supersymmetric and non-supersymmetric field-theory deformations of \( \mathcal{N}=4 \) Super Yang-Mills theory, respectively. We calculate the first loop correction to their partition functions using an extension of the dilatation-operator and Pólya-counting approach. In particular, we account for the one-loop finite-size effects which occur for operators of length one and two. Remarkably, we find that the \( \mathcal{O}\left(\lambda \right) \) correction to the Hagedorn temperature is independent of the deformation parameters, although the partition function depends on them in a non-trivial way.
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, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 231] [hep-th/9711200] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.M. Polyakov, Gauge theory correlators from noncritical string theory, Phys. Lett. B 428 (1998) 105 [hep-th/9802109] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
N. Beisert et al., Review of AdS/CFT Integrability: An Overview, Lett. Math. Phys. 99 (2012) 3 [arXiv:1012.3982] [INSPIRE].
K. Zoubos, Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries, Lett. Math. Phys. 99 (2012) 375 [arXiv:1012.3998] [INSPIRE].
S.J. van Tongeren, Integrability of the AdS 5 × S 5 superstring and its deformations, J. Phys. A 47 (2014) 433001 [arXiv:1310.4854] [INSPIRE].
R.G. Leigh and M.J. Strassler, Exactly marginal operators and duality in four-dimensional \( \mathcal{N}=1 \) supersymmetric gauge theory, Nucl. Phys. B 447 (1995) 95 [hep-th/9503121] [INSPIRE].
S. Frolov, Lax pair for strings in Lunin-Maldacena background, JHEP 05 (2005) 069 [hep-th/0503201] [INSPIRE].
N. Beisert and R. Roiban, Beauty and the twist: The Bethe ansatz for twisted \( \mathcal{N}=4 \) SYM, JHEP 08 (2005) 039 [hep-th/0505187] [INSPIRE].
O. Lunin and J.M. Maldacena, Deforming field theories with U(1) × U(1) global symmetry and their gravity duals, JHEP 05 (2005) 033 [hep-th/0502086] [INSPIRE].
R. Hagedorn, Statistical thermodynamics of strong interactions at high-energies, Nuovo Cim. Suppl. 3 (1965) 147 [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, The Hagedorn/deconfinement phase transition in weakly coupled large-N gauge theories, Adv. Theor. Math. Phys. 8 (2004) 603 [hep-th/0310285] [INSPIRE].
O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas and M. Van Raamsdonk, A First order deconfinement transition in large-N Yang-Mills theory on a small S 3, Phys. Rev. D 71 (2005) 125018 [hep-th/0502149] [INSPIRE].
O. Aharony, J. Marsano and M. Van Raamsdonk, Two loop partition function for large-N pure Yang-Mills theory on a small S 3, Phys. Rev. D 74 (2006) 105012 [hep-th/0608156] [INSPIRE].
M. Mussel and R. Yacoby, The 2-loop partition function of large-N gauge theories with adjoint matter on S 3, JHEP 12 (2009) 005 [arXiv:0909.0407] [INSPIRE].
B. Sundborg, The Hagedorn transition, deconfinement and \( \mathcal{N}=4 \) SYM theory, Nucl. Phys. B 573 (2000) 349 [hep-th/9908001] [INSPIRE].
M. Spradlin and A. Volovich, A Pendant for Polya: The One-loop partition function of \( \mathcal{N}=4 \) SYM on R×S 3, Nucl. Phys. B 711 (2005) 199 [hep-th/0408178] [INSPIRE].
N. Beisert, The complete one loop dilatation operator of \( \mathcal{N}=4 \) super Yang-Mills theory, Nucl. Phys. B 676 (2004) 3 [hep-th/0307015] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, Non-conformality of γ i -deformed \( \mathcal{N}=4 \) SYM theory, J. Phys. A 47 (2014) 455401 [arXiv:1308.4420] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, The complete one-loop dilatation operator of planar real β-deformed \( \mathcal{N}=4 \) SYM theory, JHEP 07 (2014) 150 [arXiv:1312.2959] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, A piece of cake: the ground-state energies in γ i -deformed \( \mathcal{N}=4 \) SYM theory at leading wrapping order, JHEP 1409 (2014) 78 [arXiv:1405.6712] [INSPIRE].
T.J. Hollowood and S.P. Kumar, An \( \mathcal{N}=1 \) duality cascade from a deformation of \( \mathcal{N}=4 \) SUSY Yang-Mills theory, JHEP 12 (2004) 034 [hep-th/0407029] [INSPIRE].
D. Serban and M. Staudacher, Planar \( \mathcal{N}=4 \) gauge theory and the Inozemtsev long range spin chain, JHEP 06 (2004) 001 [hep-th/0401057] [INSPIRE].
N. Beisert, V. Dippel and M. Staudacher, A Novel long range spin chain and planar \( \mathcal{N}=4 \) super Yang-Mills, JHEP 07 (2004) 075 [hep-th/0405001] [INSPIRE].
C. Sieg and A. Torrielli, Wrapping interactions and the genus expansion of the 2-point function of composite operators, Nucl. Phys. B 723 (2005) 3 [hep-th/0505071] [INSPIRE].
A. Mauri, S. Penati, A. Santambrogio and D. Zanon, Exact results in planar \( \mathcal{N}=1 \) superconformal Yang-Mills theory, JHEP 11 (2005) 024 [hep-th/0507282] [INSPIRE].
S. Ananth, S. Kovacs and H. Shimada, Proof of all-order finiteness for planar β-deformed Yang-Mills, JHEP 01 (2007) 046 [hep-th/0609149] [INSPIRE].
S. Ananth, S. Kovacs and H. Shimada, Proof of ultra-violet finiteness for a planar non-supersymmetric Yang-Mills theory, Nucl. Phys. B 783 (2007) 227 [hep-th/0702020] [INSPIRE].
Q. Jin and R. Roiban, On the non-planar β-deformed \( \mathcal{N}=4 \) super-Yang-Mills theory, J. Phys. A 45 (2012) 295401 [arXiv:1201.5012] [INSPIRE].
D.Z. Freedman and U. Gürsoy, Comments on the β-deformed \( \mathcal{N}=4 \) SYM theory, JHEP 11 (2005) 042 [hep-th/0506128] [INSPIRE].
J. Fokken, C. Sieg and M. Wilhelm, unpublished.
M. Wilhelm, Amplitudes, Form Factors and the Dilatation Operator in \( \mathcal{N}=4 \) SYM Theory, arXiv:1410.6309 [INSPIRE].
G. Pólya, Kombinatorische Anzahlbestimmungen für Gruppen, Graphen und chemische Verbindungen, Acta Math. 68 (1937) 145.
B.I. Zwiebel, The psu(1, 1|2) Spin Chain of \( \mathcal{N}=4 \) Supersymmetric Yang-Mills Theory, Ph.D. Thesis (2007).
U. Gürsoy, Probing universality in the gravity duals of \( \mathcal{N}=1 \) SYM by gamma-deformations, JHEP 05 (2006) 014 [hep-th/0602215] [INSPIRE].
A. Hamilton, J. Murugan and A. Prinsloo, A note on the universality of the Hagedorn behavior of pp-wave strings, JHEP 02 (2008) 108 [arXiv:0712.3059] [INSPIRE].
M. Gomez-Reino, S.G. Naculich and H.J. Schnitzer, More pendants for Polya: Two loops in the SU(2) sector, JHEP 07 (2005) 055 [hep-th/0504222] [INSPIRE].
M. Günaydin and C. Saçlioglu, Oscillator Like Unitary Representations of Noncompact Groups With a Jordan Structure and the Noncompact Groups of Supergravity, Commun. Math. Phys. 87 (1982) 159 [INSPIRE].
M. Günaydin, D. Minic and M. Zagermann, 4D doubleton conformal theories, CPT and IIB string on AdS 5 × S 5, Nucl. Phys. B 534 (1998) 96 [Erratum ibid. B 538 (1999) 531] [hep-th/9806042] [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: 1411.7695
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
Fokken, J., Wilhelm, M. One-loop partition functions in deformed \( \mathcal{N}=4 \) SYM theory. J. High Energ. Phys. 2015, 18 (2015). https://doi.org/10.1007/JHEP03(2015)018
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2015)018