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Spectral equivalence of nearby Lagrangians

Johan Asplund, Yash Deshmukh, Alex Pieloch

TL;DR

The paper builds a spectral wrapped Donaldson–Fukaya framework with coefficients in a commutative ring spectrum $R$, defining Lagrangian $R$-branes via lifts of Gauss maps and a corresponding $R$-oriented flow category whose Cohen–Jones–Segal realization yields wrapped Floer spectra $HW(-,-;R)$. It proves that any nearby Lagrangian $Lundle T^*Q$ equipped with an $R$-brane is isomorphic to the zero section $Q$ with an $R$-brane, and shows the cotangent fiber’s Floer spectrum aligns with the stable loop-space object $HW(F,F;R) imeq ig( S^ulletig) abla$; the Yoneda embedding further identifies morphism spaces, allowing a Whitehead-type bootstrap from discrete rings to general $R$. The work develops a robust, functorial setup of flow categories, bimodules, multimodules, and local systems, and provides a detailed CJS realization theory for these objects, enabling a spectrum-valued enhancement of wrap Floer theory and paving the way for further coefficient theories (e.g., $MO extstyleigl<kigr>$) in the sequel. Overall, it extends classical Floer–Hutchings–Abouzaid–Smith style results from $Z$ to spectral coefficients and clarifies how cotangent bundle geometry encodes stable homotopy information via loop spaces.

Abstract

Let $R$ be a commutative ring spectrum. We construct the wrapped Donaldson--Fukaya category with coefficients in $R$ of any stably polarized Liouville sector. We show that any two $R$-orientable and isomorphic objects admit $R$-orientations so that their $R$-fundamental classes coincide. Our main result is that any closed exact Lagrangian $R$-brane in the cotangent bundle of a closed manifold is isomorphic to an $R$-brane structure on the zero section in the wrapped Donaldson--Fukaya category, generalizing a well-known result over the integers. To achieve this, we prove that the Floer homotopy type of the cotangent fiber is given by the stable homotopy type of the based loop space of the zero section.

Spectral equivalence of nearby Lagrangians

TL;DR

The paper builds a spectral wrapped Donaldson–Fukaya framework with coefficients in a commutative ring spectrum , defining Lagrangian -branes via lifts of Gauss maps and a corresponding -oriented flow category whose Cohen–Jones–Segal realization yields wrapped Floer spectra . It proves that any nearby Lagrangian equipped with an -brane is isomorphic to the zero section with an -brane, and shows the cotangent fiber’s Floer spectrum aligns with the stable loop-space object ; the Yoneda embedding further identifies morphism spaces, allowing a Whitehead-type bootstrap from discrete rings to general . The work develops a robust, functorial setup of flow categories, bimodules, multimodules, and local systems, and provides a detailed CJS realization theory for these objects, enabling a spectrum-valued enhancement of wrap Floer theory and paving the way for further coefficient theories (e.g., ) in the sequel. Overall, it extends classical Floer–Hutchings–Abouzaid–Smith style results from to spectral coefficients and clarifies how cotangent bundle geometry encodes stable homotopy information via loop spaces.

Abstract

Let be a commutative ring spectrum. We construct the wrapped Donaldson--Fukaya category with coefficients in of any stably polarized Liouville sector. We show that any two -orientable and isomorphic objects admit -orientations so that their -fundamental classes coincide. Our main result is that any closed exact Lagrangian -brane in the cotangent bundle of a closed manifold is isomorphic to an -brane structure on the zero section in the wrapped Donaldson--Fukaya category, generalizing a well-known result over the integers. To achieve this, we prove that the Floer homotopy type of the cotangent fiber is given by the stable homotopy type of the based loop space of the zero section.

Paper Structure

This paper contains 72 sections, 114 theorems, 581 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Properties of the category
  4. Sketch of the proof of the main theorem
  5. Outline
  6. Background on flow categories
  7. $\langle k \rangle$-manifolds
  8. Collars and gluing
  9. Flow categories
  10. Flow bimodules
  11. Bordism of flow bimodules
  12. Flow multimodules
  13. Bordism of flow multimodules
  14. Orientations induced from bordisms
  15. Flow categories with local systems
  16. ...and 57 more sections

Key Result

Theorem 1.2

Suppose that $X$ is a stably polarized Liouville sector. There is a category $\mathcal{W}(X;R)$ that is enriched over the homotopy category of $R$-modules with objects given by Lagrangian $R$-branes and morphisms given by the homotopy classes of $R$-modules $HW(-,-;R)$, and the composition is induce

Theorems & Definitions (380)

  • Definition 1.1: \ref{['defn:Brane']}
  • Theorem 1.2: \ref{['dfn:wfuk_K']}
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5: \ref{['thm:modules_main']}
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9: \ref{['thm:inclusion_pushforward']}
  • Theorem 1.10: \ref{['thm:subcrit_trivial']}
  • ...and 370 more