I will review some recent progress on six-dimensional conformal field theories, highlighting applications to the
dynamics of M5-branes. On an orbifold singularity, M5s can fractionate, and they can recombine by sometimes
leaving behind a 'frozen' version of the singularity. The CFTs describing these processes also happen to be
ubiquitous building blocks for more complicated ones. There is also evidence for other processes where branes
recombine in patterns associated to nilpotent orbits of Lie groups, with precise checks obtained by anomalies and
moduli space dimensions.

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