Spectral Sequence Computation of Higher Twisted $K$-Groups of $ SU(n)$

TL;DR

This paper computes the rationalized higher twisted K-groups of SU(n) using spectral sequences, revealing their structure as quotients of the representation ring and connecting to conformal field theory and non-commutative geometry.

Contribution

It introduces a spectral sequence approach to compute higher twisted K-theory of SU(n), identifying generators of the higher fusion ideal and relating to exponential functors and non-commutative bundles.

Findings

Rational higher twists are non-trivial only in degree dim(G).

The higher fusion ideal is generated by derivatives of a potential.

The determinant bundle over LSU(n) has a non-commutative analogue.

Abstract

Motivated by the Freed-Hopkins-Teleman theorem we study graded equivariant higher twists of $K$-theory for the groups $G = SU(n)$ induced by exponential functors. We compute the rationalisation of these groups for all $n$ and all non-trivial functors. Classical twists use the determinant functor and yield equivariant bundles of compact operators that are classified by Dixmier-Douady theory. Their equivariant $K$-theory reproduces the Verlinde ring of conformal field theory. Higher twists give equivariant bundles of stable UHF algebras, which can be classified using stable homotopy theory. Rationally, only the $K$-theory in degree $\dim(G)$ is again non-trivial. The non-vanishing group is a quotient of a localisation of the representation ring $R(G) \otimes \mathbb{Q}$ by a higher fusion ideal $J_{F,\mathbb{Q}}$. We give generators for this ideal and prove that these can be obtained as derivatives of a potential. For the exterior algebra functor, which is exponential, we show that the determinant bundle over $LSU(n)$ has a non-commutative counterpart where the fibre is the unitary group of the UHF algebra.