C[subscript 2]-cofiniteness of the vertex algebra V[subscript L] [superscript +] and classification of irreducible modules of the vertex operator algebra V[subscript L] [superscript +<r>]

Abstract

The first part of this thesis is the study of the representation theory of vertex algebras V[subscript L] [superscript +] when L are nondegenerate even lattices that are not positive de nite. In particular, we show that such vertex algebras and their irreducible weak modules satisfy the so-called C[subscript 2]-co niteness condition. For the second part of this thesis, we study the representation theory of a vertex operator algebra V[subscript L] [superscript +<r>]. Here L is a positive de nite even lattice of rank 2 and tau is an auto-morphism of V[subscript L] [superscript +]. We are able to classify some irreducible V[subscript L] [superscript +<r>] -modules that are contained in irreducible V[subscript L] [superscript +<r>] -modules. Furthermore, we construct and classify some tau-twisted modules of a vertex algebra M(1)[superscript +]. This vertex algebra is a vertex subalgebra of V[subscript L] [superscript +].