Toward the group completion of the Burau representation

TL;DR

This paper develops a topological framework for the Burau representation, connecting it to topological quantum field theories, homology models, and string theory, offering new insights into knot theory and algebraic topology.

Contribution

It constructs a monoidal topological groupoid for finite subsets of the plane and interprets the Burau representation as a topological quantum field theory, linking it to homology and string theory models.

Findings

Burau representation as a braided monoidal TQFT

Construction of an infinite cyclic cover for finite subsets of 

Relation between Burau TQFT and SU(2) Wess-Zumino-Witten model

Abstract

Following Boardman-Vogt, McDuff, Segal, and others, we construct a monoidal topological groupoid or space of finite subsets of the plane, and interpret the Burau representation of knot theory as a topological quantum field theory defined on it. Its determinant or {\bf writhe} is an invertible braided monoidal TQFT which group completes to define a Hopkins-Mahowald model for integral homology as an $E_2$ Thom spectrum. We use these ideas to construct an infinite cyclic (Alexander) cover for the space of finite subsets of $\C$, and we argue that the TQFT defined by Burau is closely related to the SU(2)-valued Wess-Zumino-Witten model for string theory on $\R^3_+$.