[Paper Review] Polytopes and Skeleta

David Treumann, Eric Zaslow|arXiv (Cornell University)|Sep 20, 2011

Algebraic structures and combinatorial models13 references4 citations

TL;DR

This paper constructs a Legendrian space $\Lambda^\infty$ from a reflexive polytope pair, equipping it with a sheaf of dg categories via Kashiwara-Schapira theory. The authors conjecture this sheaf models the Fukaya category of the large volume limit $Z^\vee_\infty$, and provide computational evidence via homology calculations matching Betti numbers from Danilov-Khovanskiû2011. The work supports homological mirror symmetry at the large complex structure limit for toric hypersurfaces.

ABSTRACT

To each simplicial reflexive polytope in Z^(n+1), we attach an n-dimensional space, Lambda^\infty. It is the Legendrian boundary of a conic Lagrangian considered in work of the authors, Fang and Liu, and because of this it carries a sheaf of dg categories called the "Kashiwara-Schapira sheaf." We discuss some conjectures and results about the role that Lambda^\infty and the Kashiwara-Schapira sheaf should play in homological mirror symmetry.

Motivation & Objective

• To define a topological space $\Lambda^\infty$ that could serve as a Lagrangian skeleton for the large volume limit $Z^\vee_\infty$ of a mirror toric hypersurface.
• To equip $\Lambda^\infty$ with a sheaf of dg categories using Kashiwara-Schapira theory, conjecturing it models the Fukaya category of $Z^\vee_\infty$.
• To provide computational evidence that the homology of $\Lambda^\infty$ matches the Betti numbers of $Z^\vee_\infty$, as computed by Danilov and Khovanskiïb2011.
• To support Kontsevich's homological mirror symmetry conjecture at the large complex structure and large volume limits for toric hypersurfaces.
• To establish a framework where the global sections of the sheaf on $\Lambda^\infty$ recover $\mathrm{Perf}(Z^\infty)$, linking geometry and category theory.

Proposed method

• Construct $\Lambda^\infty$ as a Legendrian submanifold in the contact boundary of the cotangent bundle of a compact torus, derived from the dual polytope $\triangle^\vee$.
• Use the Kashiwara-Schapira sheaf machinery to assign a sheaf of dg categories to $\Lambda^\infty$, leveraging its singular Lagrangian structure.
• Define a chain complex $C_*$ modeling $\Lambda^\infty$'s homology, with generators from vertices and edges of $\triangle^\vee$, and differentials encoded by combinatorics of face inclusions.
• Compute the homology of $C_*$ using matrix representations of maps between $G_F$-groups associated to faces, reducing to Smith normal form via Sage.
• Automate the computation for all 194 facet-simplicial three-dimensional reflexive polytopes using a Sage worksheet, with matrices encoding face inclusion maps.
• Compare the resulting Betti numbers with those from Danilov-Khovanskiïb2011 to verify agreement.

Experimental results

Research questions

• RQ1 Is $\Lambda^\infty$ homotopy equivalent to the affine hypersurface $Z^\vee_\infty$ at the large volume limit?
• RQ2 Does the Kashiwara-Schapira sheaf on $\Lambda^\infty$ model the Fukaya category of compact Lagrangian branes on $Z^\vee_\infty$?
• RQ3 Does the Leray spectral sequence for $\Lambda^\infty \to \partial\triangle^\vee$ degenerate at $E^2$ over $\mathbb{Q}$, as conjectured?
• RQ4 Can the homology of $\Lambda^\infty$ be computed algorithmically and shown to match the Betti numbers of $Z^\vee_\infty$?
• RQ5 Is the category of global sections of the sheaf on $\Lambda^\infty$ equivalent to $\mathrm{Perf}(Z^\infty)$, as expected in mirror symmetry?

Key findings

• The homology of $\Lambda^\infty$ computed via the chain complex $C_*$ yields $H_0(\Lambda^\infty) \cong \mathbb{Q}$ and $H_1(\Lambda^\infty) \cong \mathbb{Q}^3 \oplus \mathbb{Q}^7$, matching the Betti numbers of $Z^\vee_\infty$.
• For the standard simplex $\triangle^\vee = \mathrm{conv}\{v_1,v_2,v_3\}$, the differential matrix $\partial: \mathbb{Q}^9 \to \mathbb{Q}^3$ has rank 2, leading to $\dim H_0 = 1$ and $\dim H_1 = 10$, consistent with $Z^\vee_\infty$ being an elliptic curve minus nine points.
• The computed Betti numbers for all 194 facet-simplicial three-dimensional reflexive polytopes agree with those from Danilov and Khovanskiïb2011.
• The $E^2$ page of the Leray spectral sequence for $\Lambda^\infty \to \partial\triangle^\vee$ is conjectured to degenerate over $\mathbb{Q}$, supporting the homotopy equivalence of $\Lambda^\infty$ and $Z^\vee_\infty$.
• The algorithmic implementation in Sage correctly computes the ranks and kernels of maps between $G_F$-groups, enabling automated verification of homology over $\mathbb{Z}$.
• The sheaf of dg categories on $\Lambda^\infty$ is conjectured to be equivalent to the Fukaya category on $Z^\vee_\infty$, and its global sections are expected to recover $\mathrm{Perf}(Z^\infty)$.