Gromov-Hausdorff distance between filtered $A_{\infty}$ categories 1: Lagrangian Floer theory
Abstract
In this paper we introduce and study a distance, Gromov-Hausdorff distance, which measures how two filtered A $A_{\infty}$ categories are far away each other. In symplectic geometry the author associated a filtered$A_{\infty}$ category, Fukaya category, to a finite set of Lagrangian submanifolds. The Gromov-Hausdorff distance then gives a new invariant of a finite set of Lagrangian submanifolds. One can estimate it by the Hofer distance of Hamiltonian diffeomorphisms needed to send one Lagrangain submanifold to the other. A motivation to introduce Gromov-Hausdorff distance is to obtain a certain completion of Fukaya category. If we have a sequence of sets of Lagrangian submanifolds, which is a Cauchy sequence in the sense of Hofer metric, then the associated filtered A infinity categories also form a Cauchy sequence in Gromov-Hausdorff distance. In this paper we develop a theory to obtain an inductive limit of such a sequence of filtered $A_{\infty}$ categories. In other words, we give an affirmative answer to [Fu5] Conjecture 15.34.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2021
- DOI:
- 10.48550/arXiv.2106.06378
- arXiv:
- arXiv:2106.06378
- Bibcode:
- 2021arXiv210606378F
- Keywords:
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- Mathematics - Symplectic Geometry;
- Mathematics - Differential Geometry;
- Mathematics - Dynamical Systems;
- Mathematics - Quantum Algebra;
- 53D40;
- 53D37
- E-Print:
- 73 pages, 19 figures