On the thermodynamics of large N non-commutative super Yang-Mills theory

We study the thermodynamics of the large N non-commutative super Yang-Mills theory in the strong 't Hooft coupling limit in the spirit of AdS/CFT correspondence. Although the thermodynamic quantities of near-extremal D3-branes with NS B fields, which are dual gravity configurations of the non-commutative ${\cal N}=4$ super Yang-Mills theory, are the same as those without B fields, there are some differences in the thermodynamics of the ordinary and non-commutative super Yang-Mills theories. In particular we discuss the differences in three aspects. (1) We examine the $\alpha'^3 R^4$ corrections to the free energy and find that they give a negative entropy correction. This is consistent with the expectation that because of the non-commutative nature of the spacetime coordinates, there is a reduction of the degrees of freedom in the ultraviolet regime, compared to the ordinary case. (2) We find that the stress-energy tensor of the super Yang-Mills is not isotropic and its trace does not vanish, which confirms that the non-commutative super Yang-Mills theory is not conformal even in four dimensions. (3) We consider the thermodynamics of a probe D3-brane in the near-extremal D3-brane background. The entropy of the probe is increased relative to that in the background without B field, which suggests that in the low-temperature and weak coupling limit the one-loop contribution would not be exponentially suppressed for the non-commutative super Yang-Mills theory