Weak $E_2$-Morita equivalences via quantization of the 1-shifted cotangent bundle

TL;DR

This paper studies the quantization of the 1-shifted cotangent bundle in positive characteristic, showing a weak Morita equivalence between the quantized algebra and the structure sheaf on the Frobenius twist.

Contribution

It establishes a weak $E_2$-Morita equivalence for the quantization of the 1-shifted cotangent bundle over a perfect field, linking module categories via explicit equivalences.

Findings

The quantization is an $E_2$-algebra over the Frobenius twist.

The module categories over the quantization and the structure sheaf are equivalent.

Explicit description of the Morita equivalence in terms of coherent modules.

Abstract

In this paper, we investigate the structure of the convergent quantization of the 1-shifted cotangent bundle $S$ of a smooth scheme $X$ over a perfect field of positive characteristic. The quantization is an $E_2$-algebra over the Frobenius twist $S'$ of the 1-shifted cotangent bundle which restricted to the zero section $X'\rightarrow S'$ is weakly $E_2$-Morita equivalent to the structure sheaf $\mathscr O_{X'}$ of the Frobenius twist $X'$ of $X$. Explicitly, we show that the $(\infty,2)$-category of coherent (left-)modules over $\mathscr O_{X'}$ is equivalent to the full subcategory of the $(\infty,2)$-category of coherent (left-)modules over the quantization restricted to the zero section generated by $\mathscr O_X$ are equivalent.