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[**Matan Mussel, Ran Yacoby[^1]** ]{}

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[*Weizmann Institute of Science,*]{}\
[*Rehovot 76100, Israel*]{}\
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[**ABSTRACT** ]{}

> We compute the 2-loop thermal partition function of Yang-Mills theory on a small 3-sphere, in the large $N$ limit with weak ’t Hooft coupling $\lambda=g_{YM}^2N$. We include $N_s$ scalars and $N_f$ chiral fermions in the adjoint representation of the gauge group $(S)U(N)$, with arbitrary Yukawa and quartic scalar couplings, assuming only commutator interactions. From this computation one can extract information on the perturbative corrections to the spectrum of the theory, and the correction to its Hagedorn temperature. Furthermore, the computation of the 2-loop partition function is a necessary step towards determining the order of the deconfinement phase transition at weak coupling, for which a 3-loop computation is needed.

Introduction {#sec:introduction}
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Overview and outline {#ssec:Overview}
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The interest in the thermodynamics of large $N$ Yang-Mills (**YM**) theories on a small 3-sphere $S^3$, originates in [@Witten:1998zw]. In that paper, the AdS/CFT correspondence [@Maldacena:1997re; @Witten:1998qj; @Gubser:1998bc] was used to study the thermal behavior of $\cal{N}=4$ supersymmetric Yang-Mills theory (**SYM**) on $S^3$, with gauge group $(S)U(N)$. As the correspondence dictates, in the large $N$ limit with strong ’t Hooft coupling $\lambda=g_{YM}^2N$, this behavior can be 