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Global Fukaya category II: applications

Yasha Savelyev

Abstract

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.

Global Fukaya category II: applications

Abstract

To paraphrase, part I constructs a bundle of 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 -theory of the integers and the rationals. We find new curvature constraint phenomena for smooth and singular -connections on principal -bundles over , where is or . Even for the classical group 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 -theory of the integers and the rationals, defined in ~\cite{cite_SavelyevAlgKtheory} following Toën, admits a injection in degree . This gives a path from Floer theory to number theory.

Paper Structure

This paper contains 28 sections, 15 theorems, 145 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Method of Cartesian and Kan fibrations
  3. An application to curvature bounds of singular and smooth connections
  4. A non-metric measure of curvature
  5. Curvature bounds
  6. Abstract resolutions of singular connections
  7. First quantum obstruction and smooth invariants
  8. First quantum obstruction as a manifold invariant
  9. Acknowledgements
  10. Preliminaries
  11. Outline
  12. From a Cartesian fibration to a Kan fibration
  13. Qualitative description of the perturbation data
  14. Extending $\mathcal{D} _{pt}$ to higher dimensional simplices
  15. The main lemma and immediate consequences
  16. ...and 13 more sections

Key Result

Theorem 1.1

For $M=S ^{2}$ the functors $F _{\mathbb{Z}}, F _{\mathbb{Q} }$ are not null-concordant. Furthermore, $F _{P,\mathbb{Q} }$ is not concordant to $F _{P', \mathbb{Q} }$ if $P \not\simeq P'$ are Hamiltonian $S ^{2}$ bundles over $S ^{4}$.

Figures (3)

  • Figure 1:
  • Figure 2: The labels $L _{0}$ indicate that the Lagrangian subbundle is constant with corresponding fiber $L _{0}$. The curve $c _{r}$ bounds $R _{-}$.
  • Figure 3: Over the boundary components with black labels $L _{i}$ the Lagrangian subbundle $\mathcal{L} _{r} ^{\mathfrak {n}}$ is constant with corresponding fiber $L _{i}$. Over the $i$'th red boundary component the Lagrangian subbundle corresponds to the path of Lagrangians $p _{i}$. Likewise over the right boundary component of the blue region, the Lagrangian subbundle corresponds to the path of Lagrangians $p _{0} ^{-1}$. In the red striped regions we have removed the curvature of the connection, in the blue striped region we have added it.

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3: Of Theorem \ref{['thm:lowerboundsingular']} and of Theorem \ref{['prop:alternative']}
  • Remark 1.4
  • Definition 1.5
  • Definition 1.6
  • Theorem 1.7
  • Example 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 35 more