Topology Change and New Phases in Thermal N=4 SYM Theory

We characterise the ground state of weakly coupled N=4 super Yang-Mills theory on S^1 x S^3 by a joint eigenvalue distribution for seven scalar fields, {A_0, Phi_J}. We show that at temperatures below the Hagedorn/`deconfinement' transition the distribution is S^1 x S^5. At high temperatures T >> 1/R_{S^3}, the eigenvalues form an ellipsoid with topology S^6. We show how the deconfinement transition realises the topology change S^1 x S^5 -> S^6. Furthermore, we find compelling evidence that when the temperature is increased to T = 1/(\lambda^{1/2} R_{S^3}) the phase with S^6 topology changes continuously to one with S^5 topology in a new second order quantum phase transition, separating the weak and strong coupling regimes at high temperature