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

TL;DR

This work develops a comprehensive framework for equivariant higher twists of $K$-theory on $SU(n)$ induced by exponential functors, extending classical twist theory via graded Fell bundles and stable $C^*$-algebras. The authors establish a spectral-sequence approach that reduces the computation to the maximal torus and affine Weyl-group cohomology, then prove a rational collapse yielding a precise description: $K^G_{ ext{dim}(G)}(C^* ilde{ ext{E}}) ens{}Q \uparrow ext{ isomorphic to } R_F(G) ens{}Q / J_{F,Q}$ and $K^G_{ ext{dim}(G)+1}(C^* ilde{ ext{E}}) ens{}Q=0$, with $J_{F,Q}$ generated by derivatives of a potential related to the exponential functor. The paper unifies classical twists (via the determinant/power of the basic gerbe) with higher twists and reveals a derivative- or potential-based presentation of the higher fusion ideal, mirroring structures from loop-group representations and Grassmannian cohomology. In addition to recovering the Verlinde ring in the classical case, the construction furnishes a noncommutative determinant analogue for the exterior-algebra functor, bridging operator-algebraic twists with geometric and representation-theoretic data. Overall, the results offer a rigorous rational structure for higher equivariant twists and illuminate deeper ties to conformal field theory and Grassmannian geometry.

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.

Figures (2)

• Figure 1: The group $G_I$ is associated to the barycentre of $\Delta_I$. The picture shows these groups for $SU(3)$ for all $I$.
• Figure 2: The $\Lambda$-CW-complex structure of $\mathfrak{t}$ for $n=2$ and the boundary of the $2$-cell $c_{12}$.

Theorems & Definitions (28)

• Definition 2.1
• Example 2.2
• Remark 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof