Algebraic Structures Underlying Conformal Field Theory, and Logarithmic Modules for Vertex Operator Algebras
Conformal Field Theories, including string theory, are attempts in physics at describing all particle interactions but are also structures that arise naturally in mathematics through the study of representations of finite simple groups and Lie algebras and their connections to number theory and topology. More recently, both in physics and mathematics, logarithmic theories have garnered much attention. These theories correspond to allowing for spaces of particle states that include generalized eigenvectors for the quantized energy operator rather than just eigenvectors. We will give some motivation for the algebraic structures that arise from the geometry of propagating strings, such as vertex operator algebras and their modules, and present some aspects of logarithmic modules for a vertex operator algebra.