Abstract
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.
Highlights
/ or supersymmetry of the fixed points TUV and TIR, we can choose T to be the space of all theories with such properties, e.g. the space of 4d N = 1 theories
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 dowain walls? Analyzing these questions in a wide variety of contexts — from counting RG walls to AdS/CFT correspondence — will provide favorable answers, but will lead us to a unified general framework that is powerful enough to account for peculiar RG flows and predict new physical phenomena
Summary
We discuss deformation-equivalence of RG flows. In particular, our motivation comes from such questions as “Are there distinct topological sectors for RG flows?” and, if so, “What invariants can distinguish different flows?” Since RG flow is a continuous flow in the theory space T , π0(T ) gives rise to different homotopy types of RG flows. The main theorem of Morse theory gives information about how these pieces fit together: T has the homotopy type of a cell complex, with one cell of dimension μ for each critical point of index μ. It can be convenient to combine the second sum into a polynomial (or power series) with nonnegative coefficients Q(q) = k qkQk. Setting q = −1 in (2.16) gives the Morse theorem: χ(T ) = (−1)kNk. The beauty of the relations (2.11)–(2.17) between RG flows and topology of T is that, while they ought to hold true if the strongest version of the C-theorem holds, they do not explicitly refer to the C-function.
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