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A note on supersymmetric Yang-Mills thermodynamics

The thermodynamics of supersymmetric Yang-Mills theories is studied by computing the two-loop correction to the canonical free energy and to the equation of state for theories with 16, 8 and 4 supercharges in any dimension $4\leq d\leq 10$, and in two dimensions at finite volume. In the four-dimensional case we also evaluate the first non-analytic contribution in the 't Hooft coupling to the free energy, arising from the resummation of ring diagrams. To conclude, we discuss some applications to the study of the Hagedorn transition in string theory in the context of Matrix strings and speculate on the possible physical meaning of the transition.