From Hagedorn to Lee-Yang: Partition functions of $\mathcal{N}=4$ SYM theory at finite $N$

Abstract: We study the thermodynamics of the maximally supersymmetric Yang-Mills theory with gauge group U(N) on R x S^3, dual to type IIB superstring theory on AdS_5 x S^5. While both theories are well-known to exhibit Hagedorn behavior at infinite N, we find evidence that this is replaced by Lee-Yang behavior at large but finite N: the zeros of the partition function condense into two arcs in the complex temperature plane that pinch the real axis at the temperature of the confinement-deconfinement transition. Concretely, we demonstrate this for the free theory via exact calculations of the (unrefined and refined) partition functions at N<=7 for the su(2) sector containing two complex scalars, as well as at N<=5 for the su(2|3) sector containing 3 complex scalars and 2 fermions. In order to obtain these explicit results, we use a Molien-Weyl formula for arbitrary field content, utilizing the equivalence of the partition function with what is known to mathematicians as the Poincare series of trace algebras of generic matrices. Via this Molien-Weyl formula, we also generate exact results for larger sectors.