Toric Mirror Symmetry for Homotopy Theorists

TL;DR

This paper develops a spectral lift of toric homological mirror symmetry, constructing functors from torus-equivariant sheaves to constructible sheaves of spectra, and applies this to provide new insights into sheaves on projective spaces.

Contribution

It introduces functors that lift toric mirror symmetry to the spectral setting and establishes their monoidal and functorial properties, even over a base ring.

Findings

Spectral lift of toric mirror symmetry constructed.

New symmetric monoidal structures and functoriality results obtained.

Alternative proof of Beilinson's description of sheaves on projective spaces.

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.