The Worldsheet Geometry of Superconformal Field Theory and Vertex Operator Superalgebras

I will give an introduction to the worldsheet geometry underlying two-dimensional, holomorphic superconformal field theory, and indicate how the worldsheet geometry of propagating superstrings gives rise to the notion of vertex operator superalgebra. I will discuss some applications of this understanding of the relationship between the geometric and algebraic aspects of superconformal field theory, such as the construction of twisted modules for a vertex operator algebra (also known in physics as twisted sectors). Vertex operator algebras arose independently from conformal field theory in mathematics and have deep connections to the representation theory of finite groups and to number theory. I will discuss how these connections are strengthened by adding supersymmetry to the theory.