2-Representations of sl2 from Quasi-Maps

TL;DR

This paper introduces a new approach to 2-representations of sl2 using coherent sheaves on zastava spaces, connecting geometric constructions with algebraic representations and extending to simple 2-representations via matrix factorizations.

Contribution

It constructs 2-Verma modules for sl2 using coherent sheaves on zastava spaces and realizes simple 2-representations through superpotentials and matrix factorizations.

Findings

Coherent sheaves on zastavas form 2-Verma modules for sl2.

Superpotentials enable realization of simple 2-representations.

Zastavas are smooth affine spaces in the sl2 case.

Abstract

We describe a new type of $2$-representations, using coherent sheaves. Feigin, Finkelberg, Kuznetsov, Mirkovi\'c and Braverman have provided a construction of Verma modules for complex semi-simple Lie algebras using based quasi-map spaces from $\mathbb{P}^1$ to flag varieties (zastavas). We consider here the case of $\mathfrak{sl}_2$, where the zastavas are smooth, and are mere affine spaces. We show that coherent sheaves on zastavas provide a $2$-Verma module for $\mathfrak{sl}_2$ in the sense of Naisse-Vaz. Adding a superpotential and considering matrix factorizations, we obtain a realization of simple $2$-representations of $\mathfrak{sl}_2$.