Irreversibility of renormalization group flows

5 problems

• Irreversibility of RG flows on three-dimensional defects

  Open

  Defects such as impurity lines or surfaces inside a critical material carry their own count of degrees of freedom. This count is proven to drop under coarse-graining for line, surface and four-dimensional defects, but not for three-dimensional ones.

• Does the sphere free energy decrease along five-dimensional RG flows?

  Open

  In three dimensions, a number computed by placing the theory on a sphere is proven to drop under coarse-graining (the F-theorem). Whether the analogous number drops in five dimensions is unknown.

• Does scale invariance imply conformal invariance in four dimensions?

  Partially resolved

  A theory that looks the same at all magnifications usually has a larger symmetry, conformal symmetry. Whether this always holds in four dimensions is not proven.

• Does the a-anomaly decrease along all six-dimensional RG flows?

  Partially resolved

  In two and four dimensions a number counting effective degrees of freedom is proven to drop under coarse-graining. In six dimensions this is proven only for special supersymmetric cases.

• Is four-dimensional RG flow a gradient flow of the a-function?

  Partially resolved

  In four dimensions a certain count of degrees of freedom is proven to be lower at long distances than at short ones. A stronger claim, that the couplings always move steadily downhill along a height function given by this count, is checked only in perturbation theory.