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Quantum Maslov classes

Yasha Savelyev

TL;DR

The paper introduces Quantum Maslov classes as a higher-dimensional analogue of the Seidel/Hu-Lalonde-Seidel framework, aimed at linking monotone Lagrangian branes, Hamiltonian fibrations, and Fukaya-category structures to Hofer geometry. By developing taut Hamiltonian structures and their moduli spaces of holomorphic sections, it constructs a map $\Psi$ from path-homotopy groups of Lagrangian paths to Floer homology, and extends this to an $A_\infty$-functor from the exact path category to the Fukaya category. The main geometric payoff is a rigidity phenomenon for the Hofer 2-systole of Lagrangian equators in $S^2$, contrasting with the known flexibility of Hofer girth, and providing a framework for further applications to global Fukaya categories. The results are achieved through explicit energy-area identities, gluing arguments for Hamiltonian structures, and a concrete computation using energy-minimizing perturbation data, culminating in nontrivial quantum Maslov classes that persist under taut concordances and yield nonzero Floer-theoretic invariants. This work sets up a conjectural $A_\infty$ functor to the Fukaya category and suggests deep semi-classical relations between Maslov-type invariants and classical characteristic classes.

Abstract

We give a construction of ``quantum Maslov characteristic classes'', generalizing to higher dimensional cycles the Hu-Lalonde-Seidel morphism. We also state a conjecture extending this to an $A _{\infty}$ functor from the exact path category of the space of monotone Lagrangian branes to the Fukaya category. Quantum Maslov classes are used here for the study of Hofer geometry of Lagrangian equators in $S ^{2}$, giving a rigidity phenomenon for the Hofer metric 2-systole, which stands in contrast to the flexibility phenomenon of the closely related Hofer metric girth studied by Rauch ~\cite{cite_Itamar}, in the same context of Lagrangian equators of $S ^{2}$. More applications appear in ~\cite{cite_SavelyevGlobalFukayacategoryII}.

Quantum Maslov classes

TL;DR

The paper introduces Quantum Maslov classes as a higher-dimensional analogue of the Seidel/Hu-Lalonde-Seidel framework, aimed at linking monotone Lagrangian branes, Hamiltonian fibrations, and Fukaya-category structures to Hofer geometry. By developing taut Hamiltonian structures and their moduli spaces of holomorphic sections, it constructs a map from path-homotopy groups of Lagrangian paths to Floer homology, and extends this to an -functor from the exact path category to the Fukaya category. The main geometric payoff is a rigidity phenomenon for the Hofer 2-systole of Lagrangian equators in , contrasting with the known flexibility of Hofer girth, and providing a framework for further applications to global Fukaya categories. The results are achieved through explicit energy-area identities, gluing arguments for Hamiltonian structures, and a concrete computation using energy-minimizing perturbation data, culminating in nontrivial quantum Maslov classes that persist under taut concordances and yield nonzero Floer-theoretic invariants. This work sets up a conjectural functor to the Fukaya category and suggests deep semi-classical relations between Maslov-type invariants and classical characteristic classes.

Abstract

We give a construction of ``quantum Maslov characteristic classes'', generalizing to higher dimensional cycles the Hu-Lalonde-Seidel morphism. We also state a conjecture extending this to an functor from the exact path category of the space of monotone Lagrangian branes to the Fukaya category. Quantum Maslov classes are used here for the study of Hofer geometry of Lagrangian equators in , giving a rigidity phenomenon for the Hofer metric 2-systole, which stands in contrast to the flexibility phenomenon of the closely related Hofer metric girth studied by Rauch ~\cite{cite_Itamar}, in the same context of Lagrangian equators of . More applications appear in ~\cite{cite_SavelyevGlobalFukayacategoryII}.

Paper Structure

This paper contains 27 sections, 22 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Taut conditions and Hofer geometry
  3. Lagrangian equators in $S ^{2}$
  4. Hamiltonian fibrations and taut structures
  5. Coupling forms
  6. Hamiltonian structures on fibrations
  7. Relative section classes of Hamiltonian structures
  8. Families of Hamiltonian structures.
  9. Moduli spaces of sections of Hamiltonian structures
  10. Family version
  11. Area of fibrations
  12. Gluing Hamiltonian structures
  13. Gluing Hamiltonian structures with estimates
  14. Quantum Maslov classes
  15. Definition of the quantum Maslov classes
  16. ...and 12 more sections

Key Result

Theorem 1.2

For all $d$, there is a natural and non-trivial additive group homomorphism:

Figures (1)

  • Figure 1:

Theorems & Definitions (55)

  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 45 more