The vertex algebra with central charge may be defined as a module over the universal central extension of the Lie algebra of differential operators on the circle. For an integer , it was conjectured in the physics literature that should have a minimal strong generating set consisting of elements. Using a free field realization of due to Kac-Radul, together with a deformed version of Weyl's first and second fundamental theorems of invariant theory for the standard representation of , we prove this conjecture. A consequence is that the irreducible, highest-weight representations of are parametrized by a closed subvariety of .
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Andrew R. Linshaw, Invariant theory and the algebra with negative integral central charge. J. Eur. Math. Soc. 13 (2011), no. 6, pp. 1737–1768DOI 10.4171/JEMS/292