Spectral Sequence Computation of Higher Twisted $K$-Groups of $ SU(n)$
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| Format: | Preprint |
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2023
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| _version_ | 1866909983485984768 |
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| author | Evans, David E. Pennig, Ulrich |
| author_facet | Evans, David E. Pennig, Ulrich |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_00423 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Spectral Sequence Computation of Higher Twisted $K$-Groups of $ SU(n)$ Evans, David E. Pennig, Ulrich K-Theory and Homology Algebraic Topology Operator Algebras 19L50, 19L47, 46L80 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. |
| title | Spectral Sequence Computation of Higher Twisted $K$-Groups of $ SU(n)$ |
| topic | K-Theory and Homology Algebraic Topology Operator Algebras 19L50, 19L47, 46L80 |
| url | https://arxiv.org/abs/2307.00423 |