| In the theory of vertex algebras, there is a construction known as the commutant, which associates to any vertex algebra V and any subalgebra A ⊂ V a new subalgebra Com( A,V ) ⊂ V . This construction was introduced by Frenkel-Zhu and is an abstraction of an earlier construction in conformal field theory due to Goddard-Kent-Olive. In this thesis, we interpret the commutant as a vertex algebra notion of invariant theory. We focus on a particular example, which is analogous to the ring DV g of invariant polynomial differential operators on V, where V is a module over a Lie algebra g . We present a strategy for reducing this commutant problem to a more classical problem. We give a complete description of a canonical subalgebra of this commutant algebra in the case where g = sl(2, C ) and V is the adjoint module. |
