  I briefly discuss effective Lagrangians for strong interactions while
concentrating on two specific lagrangians for QCD at large matter density. I
then introduce spectral duality in QCD a la Montonen and Olive. The latter is
already present in QCD in the hadronic phase. However it becomes transparent at
large chemical potential. Finally I show the relevance of having exact non
perturbative constraints such as t'Hooft anomaly conditions at zero and nonzero
quark chemical potential on the possible phases of strongly interacting matter.
An important outcome is that for three massless quarks at any chemical
potential the only non trivial solution of the constraints is chiral symmetry
breaking. This also shows that for three massless flavors at large quark
chemical potential CFL is the ground state.
