Quantum difference equations and Maulik-Okounkov quantum affine algebras of affine type $A$

TL;DR

This paper establishes an isomorphism between two quantum algebra structures of affine type A using monodromy representations, linking algebraic and geometric quantum difference equations.

Contribution

It proves the isomorphism of positive halves of quantum toroidal and Maulik-Okounkov quantum affine algebras of affine type A.

Findings

Degeneration limits of algebraic and geometric quantum difference equations coincide.

Monodromy representation effectively captures the algebraic structure.

Main tool is the comparison of quantum difference equations in different limits.

Abstract

In this paper we prove the isomorphism of the positive half of the quantum toroidal algebra and the positive half of the Maulik-Okounkov quantum affine algebra of affine type $A$ via the monodromy representation for the Dubrovin connection. The main tool is on the proof of the fact that the degeneration limit of the algebraic quantum difference equation is the same as that of the Okounkov-Smirnov geometric quantum difference equation.