Hopf-Algebraic Structure of String Renormalization

Abstract

The formalization of renormalization in quantum field theory was a conceptual milestone that enabled the precise calculation of higher-order perturbative processes. It was later revealed that the underlying combinatorics of this procedure are governed by the algebraic structure of a Hopf algebra. This senior thesis aims to extend this algebraic framework to processes in string theory. Unlike in QFT, in string theory, ultraviolet (UV) divergences arise from geometric degenerations at the boundaries of the worldsheet moduli space, specifically when closed loops shrink to zero length. These short-distance UV divergences can be equivalently interpreted via open/closed string duality as an infrared (IR) phenomenon caused by a closed string propagating over a long distance and going on-shell. Our goal is to not just better understand these divergences, but also to better understand how Hopf algebras are somehow universal in describing renormalization procedures.