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In QFT, the renormalization group is usually formulated in Euclidean signature. I will discuss time-dependent probes of the RG, in Lorentzian signature, and derive new dynamical constraints that govern the spread of local operators. Through a chain of Wick rotations and dualities, the same methods lead to new sum rules for inflationary correlators, which relate observable quantities like the inflationary speed of sound to properties of the UV.

 

We discuss the general structure of finite dimensional real Lie groups, which is called the Levi decomposition theorem. According to this classical result, any Lie group (up to global topology) splits as a semi-direct product of a non-semisimple part and of a semisimple part. In traditional gauge theory / field theory folklore mostly, only semisimple groups are considered. We will argue, that quite possibly non-semisimple, in particular so-called nilpotent Lie groups could also play a role for new kind of unification mechanisms.