Topology change in commuting saddles of thermal N=4 SYM theory

We study the large N saddle points of weakly coupled N=4 super Yang-Mills theory on S(1)xS(3) that are described by a commuting matrix model for the seven scalar fields {A(0), Phi(J)}. We show that at temperatures below the Hagedorn/'deconfinement' transition the joint eigenvalue distribution is S(1)xS(5). At high temperatures T >> 1/R-S3, the eigenvalues form an ellipsoid with topology S-6. We show how the deconfinement transition realises the topology change S(1)xS(5)-> S-6. Furthermore, we find compelling evidence that when the temperature is increased to T=1/(root lambda R-S3) the saddle with S-6 topology changes continuously to one with S-5 topology in a new second order quantum phase transition occurring in these saddles.