Thermodynamics of theories with sixteen supercharges in non-trivial vacua

Abstract

We study the thermodynamics of maximally supersymmetric U(N) Yang-Mills theory on $\mathds{R}\times S^2$ at large $N$. The model arises as a consistent truncation of ${\cal N}=4$ super Yang-Mills on $\mathds{R}\times S^3$ and as the continuum limit of the plane-wave matrix model expanded around the $N$ spherical membrane vacuum. The theory has an infinite number of classical BPS vacua, labeled by a set of monopole numbers, described by dual supergravity solutions. We first derive the Lagrangian and its supersymmetry transformations as a deformation of the usual dimensional reduction of ${\cal N}=1$ gauge theory in ten dimensions. Then we compute the partition function in the zero 't Hooft coupling limit in different monopole backgrounds and with chemical potentials for the $R$-charges. In the trivial vacuum we observe a first-order Hagedorn transition separating a phase in which the Polyakov loop has vanishing expectation value from a regime in which this order parameter is non-zero, in analogy with the four-dimensional case. The picture changes in the monopole vacua due to the structure of the fermionic effective action. Depending on the regularization procedure used in the path integral, we obtain two completely different behaviors, triggered by the absence or the appearance of a Chern-Simons term. In the first case we still observe a first-order phase transition, with Hagedorn temperature depending on the monopole charges. In the latter the large $N$ behavior is obtained by solving a unitary multi-matrix model with a peculiar logarithmic potential, the system does not present a phase transition and it always appears in a ``deconfined'' phase.