  Interpreting renormalization group flows as solitons interpolating between
different fixed points, we ask various questions that are normally asked in
soliton physics but not in renormalization theory. Can one count RG flows? Are
there different "topological sectors" for RG flows? What is the moduli space of
an RG flow, and how does it compare to familiar moduli spaces of
(supersymmetric) dowain walls? Analyzing these questions in a wide variety of
contexts --- from counting RG walls to AdS/CFT correspondence --- will not only
provide favorable answers, but will also lead us to a unified general framework
that is powerful enough to account for peculiar RG flows and predict new
physical phenomena. Namely, using Bott's version of Morse theory we relate the
topology of conformal manifolds to certain properties of RG flows that can be
used as precise diagnostics and "topological obstructions" for the strong form
of the C-theorem in any dimension. Moreover, this framework suggests a precise
mechanism for how the violation of the strong C-theorem happens and predicts
"phase transitions" along the RG flow when the topological obstruction is
non-trivial. Along the way, we also find new conformal manifolds in well-known
4d CFT's and point out connections with the superconformal index and
classifying spaces of global symmetry groups.
