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Toric Mirror Symmetry for Homotopy Theorists

Qingyuan Bai, Yuxuan Hu

TL;DR

This work extends toric mirror symmetry to spectral algebraic geometry by constructing a symmetric monoidal, fully faithful functor $\kappa$ from QCoh on toric quotient stacks to sheaves of spectra on real vector spaces, with image characterized by the FLTZ conic skeleton $\Lambda_\Sigma$ and the polyhedral stratification $\mathcal{S}_\Sigma$. It develops a robust combinatorial-constructible framework, including a monoidal equivalence $\Phi_M: \mathrm{Fun}(M, \mathrm{Sp}) \simeq \mathrm{QCoh}(B\mathbb{T})$ and a global functor $\kappa$ that respects torus actions, the six-functor formalism, and Day convolution. The paper proves that $\kappa$ is fully faithful for smooth projective fans and identifies its essential image as $\mathcal{S}\mathrm{hv}_{\Lambda_\Sigma}(M_\mathbb{R}; \mathrm{Sp})$, enabling a microlocal description of the image and compact generators via twisted polytopes $D_x$. A de-equivariantization yields a non-equivariant equivalence, which, together with exodromy techniques, recovers Beilinson-type descriptions for projective spaces and supports a broader toric-fibration and log-perfectoid mirror framework in the spectral setting.

Abstract

We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror symmetry theorem of Fang-Liu-Treumann-Zaslow (arXiv:1007.0053). Along the way, we obtain symmetric monoidal structures and functoriality results concerning those functors, which are new even over a field $k$. We also explain how the `non-equivariant' version of the theorem would follow from this functoriality via the de-equivariantization technique. As a concrete application, we obtain an alternative proof of Beilinson's linear algebraic description of quasi-coherent sheaves on projective spaces with spectral coefficients.

Toric Mirror Symmetry for Homotopy Theorists

TL;DR

This work extends toric mirror symmetry to spectral algebraic geometry by constructing a symmetric monoidal, fully faithful functor from QCoh on toric quotient stacks to sheaves of spectra on real vector spaces, with image characterized by the FLTZ conic skeleton and the polyhedral stratification . It develops a robust combinatorial-constructible framework, including a monoidal equivalence and a global functor that respects torus actions, the six-functor formalism, and Day convolution. The paper proves that is fully faithful for smooth projective fans and identifies its essential image as , enabling a microlocal description of the image and compact generators via twisted polytopes . A de-equivariantization yields a non-equivariant equivalence, which, together with exodromy techniques, recovers Beilinson-type descriptions for projective spaces and supports a broader toric-fibration and log-perfectoid mirror framework in the spectral setting.

Abstract

We construct functors sending torus-equivariant quasi-coherent sheaves on toric schemes over the sphere spectrum to constructible sheaves of spectra on real vector spaces. This provides a spectral lift of the toric homolgoical mirror symmetry theorem of Fang-Liu-Treumann-Zaslow (arXiv:1007.0053). Along the way, we obtain symmetric monoidal structures and functoriality results concerning those functors, which are new even over a field . We also explain how the `non-equivariant' version of the theorem would follow from this functoriality via the de-equivariantization technique. As a concrete application, we obtain an alternative proof of Beilinson's linear algebraic description of quasi-coherent sheaves on projective spaces with spectral coefficients.
Paper Structure (28 sections, 54 theorems, 326 equations, 1 figure)

This paper contains 28 sections, 54 theorems, 326 equations, 1 figure.

Table of Contents

  1. Introduction
  2. What is done in this note?
  3. Inspirations and technicalities
  4. Acknowledgments
  5. Conventions
  6. Combinatorial model
  7. Toric geometry
  8. Recollections on toric geometry
  9. Quasi-coherent sheaves
  10. Combinatorial v.s. quasi-coherent
  11. Constructible sheaves
  12. Convolution product for sheaves on real vector spaces
  13. Digression: multiplicative structures on Betti homology
  14. Combinatorial v.s. constructible
  15. Polyhedral stratification
  16. ...and 13 more sections

Key Result

Theorem A

Let $N$ be a lattice and $\Sigma$ be a smooth projective fan in $N_\mathbb{R}\mathrel{\mathop:}= N\otimes \mathbb{R}$ (see notation:fan). Let $M$ and $M_\mathbb{R}$ be the dual lattice and vector space. There exists a fully-faithful, symmetric monoidal functor where $X_\Sigma$ is the flat toric scheme associated to $\Sigma$ and $\mathbb{T}=\mathop{\mathrm{Sp\acute{e}t}}\nolimits(\mathbb{S}[M])$ i

Figures (1)

  • Figure 1: An illustration of a sheaf in $\mathcal{S}\mathrm{hv}_{\overline\Lambda_{\mathbb{P}^2}}(\mathbb{R}^2/\mathbb{Z}^2)$, drawn in a fundamental domain of $\mathbb{R}^2/\mathbb{Z}^2$. The short directional strokes—drawn along the edges and diagonal, fanning out at the corners—schematically represent $\overline\Lambda_{\mathbb{P}^2}$ in each cotangent fiber. Three distinguished stalks and ways that they are allowed to exit are drawn.

Theorems & Definitions (180)

  • Theorem A
  • proof
  • Remark 1.1.1
  • Theorem B
  • proof
  • Theorem C: \ref{['nonequivariant version of the correspondence']}
  • Theorem D: \ref{['beilinson']}
  • Remark 1.2.1: Proof ideas from the literature
  • Remark 1.2.2: Dropping assumptions on smoothness and projectivity
  • Remark 1.2.3: Necessity of higher algebra
  • ...and 170 more