Strongly holomorphic c = 24 vertex operator algebras and modular forms

We study the graded traces (one-point correlation functions) of certain strongly holomorphic vertex operator algebras (VOAs) with central charge c = 24. Zhu proved that the graded traces are modular forms [Zhu96]. For each VOA, V, we want to know which modular forms are the graded trace of some (highest-weight) vector in V. In Part 1, we use the finite-dimensional reductive Lie algebra V 1 in V and its associated affine Lie algebra in End V to show that: If V1 is semisimple, then for any holomorphic modular form f, there exists an element of the VOA with graded trace equal to f. In Part 2, we define a family of highest-weight vectors in the moonshine module. Imitating calculations of Dong and Mason [DM00], we explicitly compute their graded traces and show them to be non-zero cusp forms of weight congruent to two modulo four.