Persistent Spontaneous Symmetry Breaking at High Temperature
I will describe the results of a series of works in which we have found systems for which Spontaneous Symmetry Breaking(SSB) persists to very high temperatures. The systems involve Conformal Field Theories of scalars in fractional dimensions 3 < d < 4 and for Gauge Theories in d=4. The systems are studied in the limit of various large numbers of species. In the process we have found non supersymmetric conformal manifolds as well as cases of non supersymmetric effective potentials with flat directions. The SSB results in the formation of a dilaton. We also discuss some phase diagrams in the presence of relevant and marginally relevant operators made out of scalar fields in d=4.