# Symmetries of the Simply-Laced Quantum Connections and Quantisation of Quiver Varieties

## TL;DR

This paper explores the symmetries of simply-laced quantum connections, extending classical isomonodromy symmetries through quantum Hamiltonian reduction of quiver representation varieties.

## Contribution

It introduces a new group of symmetries for quantum connections, generalising quantum/Howe duality and classical symmetries via quantisation of quiver varieties.

## Key findings

- Identifies symmetries as a quantisation of classical isomonodromy symmetries.
- Constructs quantum Hamiltonian reduction for simply-laced quivers.
- Links quantum symmetries to classical dualities and quiver varieties.

## Abstract

We will exhibit a group of symmetries of the simply-laced quantum connections, generalising the quantum/Howe duality relating KZ and the Casimir connection. These symmetries arise as a quantisation of the classical symmetries of the simply-laced isomonodromy systems, which in turn generalise the Harnad duality. The quantisation of the classical symmetries involves constructing the quantum Hamiltonian reduction of the representation variety of any simply-laced quiver, both in filtered and in deformation quantisation.