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Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces

Benjamin Gammage, Maxim Jeffs

TL;DR

The paper establishes a functorial form of homological mirror symmetry for very affine hypersurfaces, showing that natural symplectic operations correspond to algebraic functors under HMS. It develops sectorial Liouville glueing for the A-side and matches it with derived gluing of toric B-sides, using Knörrer periodicity to relate two mirrors of the complement ($Z$ and $\tilde{Z}$) and to account for non-geometric equivalences. The main results prove Auroux's conjectures (modulo grading adjustments) by constructing and identifying restriction, lifting, and monodromy-related functors between wrapped Fukaya categories and coherent/singularity/matrix-factorization categories, with precise graded enhancements. The work introduces a robust framework for presenting Liouville manifolds as gluings of Liouville sectors and combines this with a graded enhancement of Orlov-type Knörrer equivalences to realize full HMS compatibilities.

Abstract

We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has in fact two natural mirrors, one of which is a derived scheme. These two mirrors are related via a non-geometric equivalence mediated by Knörrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors.

Homological mirror symmetry for functors between Fukaya categories of very affine hypersurfaces

TL;DR

The paper establishes a functorial form of homological mirror symmetry for very affine hypersurfaces, showing that natural symplectic operations correspond to algebraic functors under HMS. It develops sectorial Liouville glueing for the A-side and matches it with derived gluing of toric B-sides, using Knörrer periodicity to relate two mirrors of the complement ( and ) and to account for non-geometric equivalences. The main results prove Auroux's conjectures (modulo grading adjustments) by constructing and identifying restriction, lifting, and monodromy-related functors between wrapped Fukaya categories and coherent/singularity/matrix-factorization categories, with precise graded enhancements. The work introduces a robust framework for presenting Liouville manifolds as gluings of Liouville sectors and combines this with a graded enhancement of Orlov-type Knörrer equivalences to realize full HMS compatibilities.

Abstract

We prove that homological mirror symmetry for very affine hypersurfaces respects certain natural symplectic operations (as functors between partially wrapped Fukaya categories), verifying conjectures of Auroux. These conjectures concern compatibility between mirror symmetry for a very affine hypersurface and its complement, itself also a very affine hypersurface. We find that the complement of a very affine hypersurface has in fact two natural mirrors, one of which is a derived scheme. These two mirrors are related via a non-geometric equivalence mediated by Knörrer periodicity; Auroux's conjectures require some modification to take this into account. Our proof also introduces new techniques for presenting Liouville manifolds as gluings of Liouville sectors.
Paper Structure (24 sections, 51 theorems, 49 equations, 10 figures)

This paper contains 24 sections, 51 theorems, 49 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Statement of Results
  4. Notation and Terminology
  5. $A$-side Definitions
  6. $B$-side Definitions
  7. An Example
  8. Summary of Notation
  9. Summary of Methods
  10. Gradings
  11. Matrix factorization categories
  12. Bigradings
  13. Gluing Diagrams
  14. Liouville sectors for Laurent polynomials
  15. Compatibility of Liouville structures
  16. ...and 9 more sections

Key Result

THEOREM 1

Under the mirror symmetry equivalences of Kuwagaki and BenVivek as in Theorem thm:alphas, composed with the Knörrer periodicity equivalences of Theorem prop:orlov and Lemma lemma:small_orlov: and furthermore there is an exact triangle (Corollary cor:triangle):

Figures (10)

  • Figure 1: From proof of Proposition \ref{['prop:sector_gluing']}: $U$ (in green) is a product neighborhood of the fiber $F_{R_1}$ (black dot)
  • Figure 2: From proof of Proposition \ref{['prop:A2']}: $X_{A_2}$ is in green; $X_{A_1}$ is in red.
  • Figure 3: A sketch of the deformed Liouville vector field from the proof of Proposition \ref{['prop:A2']}: the hypersurface $\mathrm{Re}(z) = -\rho/2$ is in red. The two black points are zeroes of the Liouville vector field.
  • Figure 4: From proof of Proposition \ref{['prop:A2']}: the hypersurfaces $F_{R_1 \mathrm{e}^{i \delta}}$ and $F_{R_1}$ and the sectorial hypersurface $G$.
  • Figure 5: From proof of Proposition \ref{['prop:A2']}: the image $G_1$ of the sectorial hypersurface $G$ after applying the isotopy that exchanges $F_{R_1 \mathrm{e}^{i \delta}}$ and $F_{R_1}$
  • ...and 5 more figures

Theorems & Definitions (114)

  • CONJECTURE 1.2.1: Speculations
  • THEOREM
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Remark 5
  • DEFINITION 2.1.1
  • DEFINITION 2.1.2
  • DEFINITION 2.1.3
  • ...and 104 more