The mirror Lagrangian cobordism for the Euler exact sequence

TL;DR

This paper explores the mirror symmetry correspondence for the Euler sequence on projective space, describing the associated Lagrangian cobordism and the mirror Lagrangian of the cotangent sheaf within the Fukaya category.

Contribution

It provides a detailed description of the Lagrangian cobordism corresponding to the Euler sequence via mirror symmetry, specifically for projective space.

Findings

Mirror Lagrangian of the cotangent sheaf identified

Lagrangian cobordism corresponding to the Euler sequence described

Connections established within the Fukaya category framework

Abstract

For $X = \mathbb{P}^n$ the Euler sequence is given by $$ 0 \rightarrow \Omega^1_{\mathbb{P}^n} \rightarrow \mathcal{O}_{\mathbb{P}^n}^{n+1}(-1) \rightarrow \mathcal{O}_{\mathbb{P}^n} \rightarrow 0 $$ We describe the Lagrangian cobordism corresponding to this sequence via mirror symmetry, in the sense of Biran-Cornea. In particular, we describe the mirror Lagrangian of the cotangent sheaf $\Omega^1_{\mathbb{P}^n} \in \mathcal{D}^b(\mathbb{P}^n)$ in the mirror Fukaya category $Fuk(U_{\Delta})$.