Hamiltonian elements in algebraic K-theory
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911637137522688 |
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| author | Savelyev, Yasha |
| author_facet | Savelyev, Yasha |
| contents | A Hamiltonian bundle $M \hookrightarrow P \to X$ (with monotone compact fibers) induces via Floer theory a type of ``bundle of $A _{\infty}$ categories'' over $X$, with fiber given by the Fukaya category of $M$. Morita theory of $A _{\infty} $ categories, the above picture for $X=S ^{m}$, and geometric representation theory yield the following: if $G$ is a compact Lie group and $R$ is a commutative ring then there is a natural group homomorphism $π_{m} (BG) \to K ^{Cat}_{m}(R) $, where $K ^{Cat} _{m} (R)$ are a type of categorified algebraic $K$-theory groups of $R$, analogous to Toën's secondary $K$-theory. We also construct underlying maps of this type to classical algebraic $K$-theory of $R$. This framework gives a geometry-powered proof that $K ^{Cat} _{2} (\mathbb{Z} )$ is infinitely generated (with the details to appear in a future work). This is in contrast to Quillen's finite generation result for standard algebraic $K$-theory of $\mathbb{Z} $. Taking the Langlands dual of $G$, we explore a conjectural relationship between the images of the corresponding homomorphisms above. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21003 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hamiltonian elements in algebraic K-theory Savelyev, Yasha Algebraic Topology Symplectic Geometry A Hamiltonian bundle $M \hookrightarrow P \to X$ (with monotone compact fibers) induces via Floer theory a type of ``bundle of $A _{\infty}$ categories'' over $X$, with fiber given by the Fukaya category of $M$. Morita theory of $A _{\infty} $ categories, the above picture for $X=S ^{m}$, and geometric representation theory yield the following: if $G$ is a compact Lie group and $R$ is a commutative ring then there is a natural group homomorphism $π_{m} (BG) \to K ^{Cat}_{m}(R) $, where $K ^{Cat} _{m} (R)$ are a type of categorified algebraic $K$-theory groups of $R$, analogous to Toën's secondary $K$-theory. We also construct underlying maps of this type to classical algebraic $K$-theory of $R$. This framework gives a geometry-powered proof that $K ^{Cat} _{2} (\mathbb{Z} )$ is infinitely generated (with the details to appear in a future work). This is in contrast to Quillen's finite generation result for standard algebraic $K$-theory of $\mathbb{Z} $. Taking the Langlands dual of $G$, we explore a conjectural relationship between the images of the corresponding homomorphisms above. |
| title | Hamiltonian elements in algebraic K-theory |
| topic | Algebraic Topology Symplectic Geometry |
| url | https://arxiv.org/abs/2407.21003 |