8.

We applied our results on vertex operator algebras, modules and intertwining operator algebras to obtain a nonmeromorphic extension of the moonshine module vertex operator algebra and to construct representations of braid groups from vertex operator algebras and their representations. The moonshine module is a vertex operator algebra whose automorphism group is the Monster. It was constructed by Frenkel, Lepowsky and Meurman. Borcherds proved the ``Monstrous Moonshine'' conjecture of Conway and Norton for Frenkel-Lepowsky-Meurman's structure. But this vertex operator algebra is only part of a so-called $Z_{2}$-orbifold conformal field theory. To understand the moonshine module and the Monster better and to obtain a more conceptual proof of the Monstrous Moonshine conjecture, one wants to study this orbifold conformal field theory in a mathematically rigorous way. We constructed the intertwining operator algebra of this conformal field theory (actually, this is an abelian intertwining operator algebra in the sense that its fusion algebra is the group algebra of an abelian group). In particular, we obtained a new and conceptual proof that the moonshine module is indeed a vertex operator algebra. We also constructed a twisted module for the moonshine module vertex operator algebra, a vertex operator super-subalgebra of this intertwining operator algebra and a mathematical construction of the superconformal structure associated to the moonshine module first observed by Dixon, Ginsparg and Harvey. It is easy to see that any genus-zero modular functor gives representations of braid groups. Since we constructed intertwining operator algebras from suitable vertex operator algebras and their representations and we constructed genus-zero modular functors from intertwining operator algebras, we obtained representations of braid groups from these vertex operator algebras and their representations.

9.

We further developed both algebraic and geometric techniques needed for our theory. The techniques we used are both algebraic and geometric. The algebraic method includes the formal variable approach to the theory of vertex operator algebras, algebraic Lie theory, algebraic representation theory of the Virasoro algebras and affine Lie algebras. The geometric method includes one and several complex variables, complex manifolds, holomorphic forms, holomorphic vector bundles, determinant lines and determinant line bundles. To apply the theory of determinant lines and determinant line bundles to our study, certain aspects of the theory are developed. In particular, we applied the theory of Sobolev spaces and elliptic boundary problems to construct the canonical isomorphisms associated to sewing of Riemann surfaces with boundaries and proved that these canonical isomorphisms are in fact holomorphic. Jointly with Lepowsky, we also proved that the ${\cal D}$-module approach of Beilinson and Drinfeld and the formal variable approach to vertex algebras are directly and precisely equivalent.