Bohan Fang

A categorification of Morelli's theorem

We prove a theorem relating torus-equivariant coherent sheaves on toric varieties to polyhedrally-constructible sheaves on a vector space. At the level of K-theory, the theorem recovers Morelli’s description of the K-theory of a smooth projective toric variety (Morelli in Adv. Math. 100(2):154–182, 1993). Specifically, let X be a proper toric variety of dimension n and let

The Eynard–Orantin recursion and equivariant mirror symmetry for the projective line

M_{\mathbb{R}} = \mathrm{Lie}(T_{\mathbb{R}}^{\vee})\cong\mathbb {R}^{n}

T-duality and homological mirror symmetry for toric varieties

be the Lie algebra of the compact dual (real) torus

All Genus Open-Closed Mirror Symmetry for Affine Toric Calabi-Yau 3-Orbifolds

T_{\mathbb{R}}^{\vee}\cong U(1)^{n}

The Coherent–Constructible Correspondence for Toric Deligne–Mumford Stacks

. Then there is a corresponding conical Lagrangian Λ⊂T∗Mℝ and an equivalence of triangulated dg categories

On the Remodeling Conjecture for Toric Calabi-Yau 3-Orbifolds

\mathcal{P}\mathrm{erf}_{T}(X) \cong\mathit{Sh}_{cc}(M_{\mathbb{R}};\Lambda)

Open Gromov-Witten Invariants of Toric Calabi-Yau 3-Folds

, where

The Coherent-Constructible Correspondence and Homological Mirror Symmetry for Toric Varieties

\mathcal{P}\mathrm{erf}_{T}(X)

T-Duality and Equivariant Homological Mirror Symmetry for Toric Varieties

is the triangulated dg category of perfect complexes of torus-equivariant coherent sheaves on X and Shcc(Mℝ;Λ) is the triangulated dg category of complex of sheaves on Mℝ with compactly supported, constructible cohomology whose singular support lies in Λ. This equivalence is monoidal—it intertwines the tensor product of coherent sheaves on X with the convolution product of constructible sheaves on Mℝ.

A categorification of Morelli's theorem and homological mirror symmetry for toric varieties

We study the equivariantly perturbed mirror Landau-Ginzburg model of the projective line. We show that the Eynard-Orantin recursion on this model encodes all genus all descendants equivariant Gromov-Witten invariants of the projective line. The non-equivariant limit of this result is the Norbury-Scott conjecture, while by taking large radius limit we recover the Bouchard-Marino conjecture on simple Hurwitz numbers.