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Constraining RG flow in three-dimensional field theory

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The entanglement entropy S(R) across a circle of radius R
has been invoked recently in deriving general constraints on renormalization
group flow in three-dimensional field theory. 
At conformal fixed points, the negative of the finite part of the
entanglement entropy, which is called F, is equal to the free energy on the
round three-sphere. The F-theorem states that F decreases under RG flow.

Along the RG flow it has recently been shown that the
renormalized entanglement entropy {\cal F}(R) = -S(R) + R S'(R), which is equal
to F at the fixed points, is a monotonically decreasing function.  I will review various three-dimensional field
theories where we can calculate F on the three-sphere and compute its change
under RG flow, including free field theories, perturbative fixed points, large
N field theories with double trace deformations, gauge theories with large
numbers of flavors, and supersymmetric theories with at least {\cal N} = 2
supersymmetry.  I will also present
calculations of the renormalized entanglement entropy along the RG flow in free
massive field theory and in holographic examples.