The symplectic Verlinde algebras and string K-theory

TL;DR

This paper develops string topology operations in twisted K-theory, computes related Verlinde algebra completions for symplectic groups, and links these to twisted K-theory of certain pro-spectra, advancing the understanding of symplectic Verlinde algebras.

Contribution

It introduces new string topology operations in twisted K-theory and computes Verlinde algebra completions for symplectic groups, connecting algebraic and topological frameworks.

Findings

Computed twisted K-theory of loop spaces of quaternionic projective spaces.

Related Verlinde algebra completions to twisted K-theory of pro-spectra.

Identified the algebraic structures with topological constructions in string topology.

Abstract

We construct string topology operations in twisted K-theory. We study the examples given by symplectic Grassmannians, computing the twisted K-theory of the loop spaces of quaternionic projective spaces in detail. Via the work of Freed-Hopkins-Teleman, these computations are related to completions of the Verlinde algebras of Sp(n). We compute these completions, and other relevant information about the Verlinde algebras. We also identify the completions with the twisted K-theory of the Gruher-Salvatore pro-spectra. Further comments on the field theoretic nature of these constructions are made.