A functional-analytic theory of vertex (operator) algebras, II
  For a finitely-generated vertex operator algebra of central charge c, a locally convex topological completion is constructed. We construct on the completion a structure of an algebra over the operad of the c/2-th power of the determinant line bundle over the moduli space of genus-zero Riemann surfaces with ordered analytically parametrized boundary components. In particular, the completion is a module for the semi-group of the c/2-th power of the determinant line bundle over the moduli space of conformal equivalence classes of annuli with analytically parametrized boundary components. The results in Part I for Z-graded vertex algebras are also reformulated in terms of the framed little disk operad. Using May's recognition principle for double loop spaces, one immediate consequence of such operadic formulations is that the compactly generated spaces corresponding to (or the k-ifications of) the locally convex completions constructed in Part I and in the present paper have the weak homotopy types of double loop spaces. We also generalize the results above to locally-grading-restricted conformal vertex algebras and to modules. 
Yi-Zhi Huang