  Maximally supersymmetric (p+1)-dimensional Yang-Mills theory at large N and
finite temperature, with possibly compact spatial directions, has a rich phase
structure. Strongly coupled phases may have holographic descriptions as black
branes in various string duality frames, or there may be no gravity dual. In
this paper we provide tools in the gauge theory which give a simple and unified
picture of the various strongly coupled phases, and transitions between them.
Building on our previous work we consider the effective theory describing the
moduli of the gauge theory, which can be computed precisely when it is weakly
coupled far out on the Coulomb branch. Whilst for perturbation theory naive
extrapolation from weak coupling to strong gives little information, for this
moduli theory naive extrapolation from its weakly to its strongly coupled
regime appears to encode a surprising amount of information about the various
strongly coupled phases. We argue it encodes not only the parametric form of
thermodynamic quantities for these strongly coupled phases, but also certain
transcendental factors with a geometric origin, and allows one to deduce
transitions between the phases. We emphasise it also gives predictions for the
behaviour of a large class of local operators in these phases.
