Gaiotto conjecture for $Rep_q(GL(N-1|N))$

TL;DR

This work proves a braided monoidal equivalence, for generic $q$, between the abelian category of $q$-monodromic perverse sheaves on the punctured determinant line bundle over the GL$(N-1)$ Grassmannian and the category ${\operatorname{Rep}}_q({GL}(N-1|N))$, by passing through the Sakellaridis–Wang zastava spaces of poles and the theory of factorizable sheaves. The main construction is a functor $F$ from the Gaiotto category to a category of factorizable sheaves ${\mathrm{FS}}$, built using a zastava-with-poles correspondence, whose exactness follows from semismallness and smoothness of the relevant maps. Rigidity and projectivity are established via a tautological family, a specialization to ${C=\mathbb{A}^1}$, and an Ext-vanishing argument that leverages linkage in the quantum supergroup ${U_q({\mathfrak{gl}}(N-1|N))}$. The derived equivalence then follows from the abelian braided tensor equivalence and the established FLE framework, generalizing geometric Satake to a quantum supergroup setting and linking representation theory with factorization geometry in a novel way. The results extend the geometric Langlands paradigm in the quantum realm and set the stage for broader applications to other quantum groups and supergroups.

Abstract

We prove D.Gaiotto's conjecture about geometric Satake equivalence for quantum supergroup $U_q({\mathfrak{gl}}(N-1|N))$ for generic $q$. The equivalence goes through the category of factorizable sheaves.

Theorems & Definitions (53)

• Remark 1.3.1
• Lemma 2.1.1: V. Serganova
• proof
• Definition 2.1.2
• Theorem 2.5.1
• Definition 3.4.1
• Lemma 3.4.2
• Proposition 3.5.1
• Lemma 3.5.2: J. Wang
• proof