Abstract
When the cosmological constant is considered to be a thermodynamical variable in black hole thermodynamics, analogous to a pressure, its conjugate variable can be thought of as a thermodynamic volume for the black hole. In the AdS/CFT correspondence this interpretation cannot be applied to the CFT on the boundary but, from the point of view of the boundary SU(N) gauge theory, varying the cosmological constant in the bulk is equivalent to varying the number of colors in the gauge theory. This interpretation is examined in the case of AdS 5 × S 5, for \( \mathcal{N}=4 \) SUSY Yang-Mills at large N , and the variable thermodynamically conjugate to N , a chemical potential for color, is determined. It is shown that the chemical potential in the high temperature phase of the Yang-Mills theory is negative and decreases as temperature increases, as expected. For spherical black holes in the bulk the chemical potential approaches zero as the temperature is lowered below the Hawking-Page temperature and changes sign at a temperature that is within one part in a thousand of the temperature at which the heat capacity diverges.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D. Kastor, S. Ray and J. Traschen, Enthalpy and the mechanics of AdS black holes, Class. Quant. Grav. 26 (2009) 195011 [arXiv:0904.2765] [INSPIRE].
B.P. Dolan, The cosmological constant and the black hole equation of state, Class. Quant. Grav. 28 (2011) 125020 [arXiv:1008.5023] [INSPIRE].
M. Cvetič, G.W. Gibbons, D. Kubiznák and C.N. Pope, Black hole enthalpy and an entropy inequality for the thermodynamic volume, Phys. Rev. D 84 (2011) 024037 [arXiv:1012.2888] [INSPIRE].
N. Altamirano, D. Kubiznak, R.B. Mann and Z. Sherkatghanad, Thermodynamics of rotating black holes and black rings: phase transitions and thermodynamic volume, Galaxies 2 (2014) 89 [arXiv:1401.2586] [INSPIRE].
B.P. Dolan, D. Kastor, D. Kubiznák, R.B. Mann and J. Traschen, Thermodynamic volumes and isoperimetric inequalities for de Sitter black holes, Phys. Rev. D 87 (2013) 104017 [arXiv:1301.5926] [INSPIRE].
B.P. Dolan, The compressibility of rotating black holes in D-dimensions, Class. Quant. Grav. 31 (2014) 035022 [arXiv:1308.5403] [INSPIRE].
C.V. Johnson, Holographic heat engines, arXiv:1404.5982 [INSPIRE].
J. Maldacena, The large N-limit of superconformal field theories and supergravity,, Int. J. Theor. Phys. 38 (1999) 1113 [Adv. Theor. Math. Phys. 2 (1998) 252] [hep-th/9711200] [INSPIRE].
E. Witten, Anti-de Sitter space and holography, Adv. Theor. Math. Phys. 2 (1998) 253 [hep-th/9802150] [INSPIRE].
E. Witten, Anti-de Sitter space, thermal phase transitions and confinement in gauge theories Adv. Theor. Math. Phys. 2 (1998) 505 [hep-th/9803131].
H.B. Callen, Thermodynamics and an introduction to thermostatistics, Wiley, U.S.A: (2006).
D.Z. Freedman, S.S. Gubser, K. Pilch and N.P. Warner, Renormalization group flows from holography supersymmetry and a c theorem, Adv. Theor. Math. Phys. 3 (1999) 363 [hep-th/9904017] [INSPIRE].
R.C. Myers and A. Singh, Comments on holographic entanglement entropy and RG flows, JHEP 04 (2012) 122 [arXiv:1202.2068] [INSPIRE].
S.S. Gubser, I.R. Klebanov and A.W. Peet, Entropy and temperature of black 3-branes, Phys. Rev. D 54 (1996) 3915 [hep-th/9602135] [INSPIRE].
S.W. Hawking and D.N. Page, Thermodynamics of black holes in Anti-de Sitter space, Commun. Math. Phys. 87 (1983) 577 [INSPIRE].
J.M. Maldacena and C. Núñez, Towards the large-N limit of pure N =1 super Yang-Mills, Phys. Rev. Lett. 86 (2001) 588 [hep-th/0008001] [INSPIRE].
E. Conde, J. Gaillard and A.V. Ramallo, On the holographic dual of N =1 SQCD with massive flavors, JHEP 10 (2011) 023 [Erratum ibid. 1308 (2013) 082] [arXiv:1107.3803] [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: 1406.7267
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
Dolan, B.P. Bose condensation and branes. J. High Energ. Phys. 2014, 179 (2014). https://doi.org/10.1007/JHEP10(2014)179
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2014)179