The symplectic Verlinde algebras and string K-theory

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.

💡 Research Summary

The paper develops a systematic bridge between string topology and twisted K‑theory, and then uses this bridge to compute concrete examples involving symplectic Grassmannians, quaternionic projective spaces, and the Verlinde algebras of the compact symplectic groups Sp(n).

The first part of the work revisits the Chas–Sullivan string‑topology operations, which give a graded‑commutative product on the homology of the free loop space LX of a closed manifold X. The authors show how to lift these operations from ordinary homology to (complex) K‑theory by incorporating a twisting class τ∈H³(X,ℤ). For a given τ they define the twisted K‑theory groups K⁎_τ(X) via the spectrum of τ‑twisted vector bundles (or, equivalently, via a bundle of compact operators). The key technical step is to construct a transfer map for the evaluation fibration ev: LX→X and a parallel‑transport map along loops that respects the τ‑twist. By composing these maps they obtain a product \