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Abstract:   Holomorphic vertex operator algebras at central charges up to 24 have been almost fully classified and it appears that they can all be constructed as cyclic orbifolds of lattice vertex operator algebras. At the same time, very little is known about the situation at higher central charge. Intuition from physics tells us that higher central charge analogues of the moonshine vertex operator algebra may exist, but so far all attempts at their construction have failed. The goal of this work is to explore the set of holomorphic vertex operator algebras at higher central charge using non-abelian orbifold theory.

I will begin the talk with a review of the orbifold theory of strongly rational vertex operator algebras. Then I will develop a theory of holomorphic extensions of metacyclic orbifolds as a generalisation of the cyclic theory.

Finally, I will prove the existence of a holomorphic vertex operator algebra at central charge 72 that cannot be constructed as a cyclic orbifold of a lattice vertex operator algebra. If there is time I will discuss some of the challenges arising in trying to construct analogues of the moonshine module.