Fock Space of Level infinity and Characters of Vertex Operators

Abstract

We present an extension of the trace of a vertex operator and explain a representation-theoretic interpretation of the trace. Specifically, we consider a twist of the vertex operator with infinitely many Casimir operators and compute its trace as a character formula. To do this, we define the Fock space of infinite level ${\mathfrak{F}}^{\infty }$. Then, we prove a duality between ${\mathfrak{gl}}_{\infty }$ and ${\mathfrak{a}}_{\infty }={\widehat{\mathfrak{gl}}}_{\infty }$ of Howe type, which provides a decomposition of ${\mathfrak{F}}^{\infty }$ into irreducible representations with joint highest weight vector for ${\mathfrak{gl}}_{\infty }$ and ${\mathfrak{a}}_{\infty }$. The decomposition of the Fock space ${\mathfrak{F}}^{\infty }$ into highest weight representations provides a method to calculate and interpret the extended trace.

References in corpus (2)

• The Refined Topological Vertex
• M-Strings