A noncommutative Cartier formula for E1-ring spectra is proven and applied to show that p-curvature of the quantum connection computes quantum Steenrod operations for Calabi-Yau symplectic manifolds.
Integrality of mirror maps and arithmetic homological mirror symmetry for Greene--Plesser mirrors
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abstract
We prove the `integrality of Taylor coefficients of mirror maps' conjecture for Greene--Plesser mirror pairs as a natural byproduct of an arithmetic refinement of homological mirror symmetry. We also prove homological mirror symmetry for Greene--Plesser mirror pairs in all characteristics such that the B-side family has good reduction, generalizing work of the fifth author and Smith over the complex numbers. A key technical ingredient is a new versality argument which allows us to work throughout over a Novikov-type ring with integer coefficients.
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Noncommutative Cartier Formulae
A noncommutative Cartier formula for E1-ring spectra is proven and applied to show that p-curvature of the quantum connection computes quantum Steenrod operations for Calabi-Yau symplectic manifolds.