Global Fukaya category II: applications
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908379130101760 |
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| author | Savelyev, Yasha |
| author_facet | Savelyev, Yasha |
| contents | To paraphrase, part I constructs a bundle of $A _{\infty}$ categories given the input of a Hamiltonian fibration over a smooth manifold. Here we show that this bundle is generally non-trivial by a sample computation. One principal application is differential geometric, and the other is about algebraic $K$-theory of the integers and the rationals. We find new curvature constraint phenomena for smooth and singular $\mathcal{G}$-connections on principal $\mathcal{G}$-bundles over $S ^{4}$, where $\mathcal{G}$ is $\operatorname {PU} (2)$ or $\operatorname {Ham} (S ^{2} )$. Even for the classical group $\operatorname {PU} (2)$ these phenomena are inaccessible to known techniques like the Yang-Mills theory. The above mentioned computation is the geometric component used to show that the categorified algebraic $K$-theory of the integers and the rationals, defined in ~\cite{cite_SavelyevAlgKtheory} following Toën, admits a $\mathbb{Z} $ injection in degree $4$. This gives a path from Floer theory to number theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_19362 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Global Fukaya category II: applications Savelyev, Yasha Symplectic Geometry To paraphrase, part I constructs a bundle of $A _{\infty}$ categories given the input of a Hamiltonian fibration over a smooth manifold. Here we show that this bundle is generally non-trivial by a sample computation. One principal application is differential geometric, and the other is about algebraic $K$-theory of the integers and the rationals. We find new curvature constraint phenomena for smooth and singular $\mathcal{G}$-connections on principal $\mathcal{G}$-bundles over $S ^{4}$, where $\mathcal{G}$ is $\operatorname {PU} (2)$ or $\operatorname {Ham} (S ^{2} )$. Even for the classical group $\operatorname {PU} (2)$ these phenomena are inaccessible to known techniques like the Yang-Mills theory. The above mentioned computation is the geometric component used to show that the categorified algebraic $K$-theory of the integers and the rationals, defined in ~\cite{cite_SavelyevAlgKtheory} following Toën, admits a $\mathbb{Z} $ injection in degree $4$. This gives a path from Floer theory to number theory. |
| title | Global Fukaya category II: applications |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/2505.19362 |