Instanton moduli space, stable envelopes and quantum difference equations

TL;DR

This paper establishes a connection between quantum difference equations of instanton moduli spaces and the Dubrovin connection, revealing an isomorphism between quantum algebras of Jordan type and quantum toroidal algebras.

Contribution

It proves the degeneration limit of quantum difference equations to the Dubrovin connection and identifies an algebraic isomorphism between quantum algebras.

Findings

Degeneration limit of quantum difference equations equals the Dubrovin connection.

Quantum algebra of Jordan type is isomorphic to quantum toroidal algebra.

Results bridge geometric and algebraic structures in instanton moduli spaces.

Abstract

In this article we prove the degeneration limit of the quantum difference equations of instanton moduli space for both algebraic one and the Okounkov-Smirnov geometric one is the Dubrovin connection for the instanton moduli space. As an application, we prove that the Maulik-Okounkov quantum algebra of Jordan type is isomorphic to the quantum toroidal algebra $U_{q_1,q_2}(\hat{\hat{\mathfrak{gl}}}_1)$ up to some centres.