Some results on the representation theory of vertex operator algebras and integer partition identities

Integer partition identities such as the Rogers-Ramanujan identities have deep relations with the representation theory of vertex operator algebras, among many other fields of mathematics and physics. Such identities, when written in generating function form typically take the shape "product side'' = "sum side.'' In some vertex-operator-algebraic settings, the product sides arise naturally, and the problem is to explain, interpret and prove the sum sides, while some other settings pose an opposite problem. In this thesis, we provide some results on both types of problems. In Part I of this thesis, we interpret the sum sides of the Gollnitz-Gordon identities using Lepowsky-Wilson's Z-algebraic constructions applied to certain principally twisted level 2 standard modules for A52. In Part II, we give, following Dong-Lepowsky, explicit constructions for certain higher level twisted intertwining operators for sl2 these constructions are inspired by a desire to interpret Andrews-Baxter's q-series theoretic "motivated proof'' of the Rogers-Ramanujan identities and more generally, motivated proofs of the Gordon-Andrews and the Andrews-Bressoud identities given by Lepowsky-Zhu and Kanade-Lepowsky-Russell-Sills, respectively. These motived proofs are about explaining the "sum sides'' starting with the "product sides.'' In Part III, following an idea of J. Lepowsky, we introduce and analyze a Koszul complex related to the principal subspace of the level 1 vacuum module of sl2 this construction is expected to yield a "character formula'' for the principal subspaces, thereby explaining the emergence of "product sides.''.