Completing Verlinde Algebras

TL;DR

This paper calculates the completion of Verlinde algebras for certain Lie groups, linking it to twisted K-theory of free loop spaces, advancing understanding in algebraic topology and quantum field theory.

Contribution

It provides a new computation of Verlinde algebra completions at the augmentation ideal, connecting it to twisted K-theory of loop spaces for simply connected Lie groups.

Findings

Computed the completion of Verlinde algebras for simply connected Lie groups.

Established a link between Verlinde algebra completion and twisted K-theory of free loop spaces.

Extended previous results by Freed, Hopkins, Teleman, Dwyer, and Lahtinen.

Abstract

We compute the completion of the Verlinde algebra of a simply connected simple compact Lie group $G$ at the augmentation ideal of the representation ring. By results of Freed, Hopkins, Teleman and C.Dwyer and Lahtinen, this gives a computation of (non-equivariant) twisted $K$-theory of the free loop space of $BG$.