# The Quantum DELL System

## TL;DR

This paper introduces quantum Hamiltonians for the double elliptic integrable system, providing its quantization, exploring eigenfunctions linked to gauge theory, and discovering new elliptic orthogonal polynomials.

## Contribution

It presents the first quantum Hamiltonians for the DELL system, connects eigenfunctions to 6d gauge theory partition functions, and introduces new elliptic orthogonal polynomials.

## Key findings

- Quantization of the classical DELL system achieved.
- Eigenfunctions correspond to instanton partition functions of 6d SU(N) gauge theory.
- Discovery of new symmetric orthogonal polynomials as elliptic generalizations of Macdonald polynomials.

## Abstract

We propose quantum Hamiltonians of the double elliptic many-body integrable system (DELL) and study its spectrum. These Hamiltonians are certain elliptic functions of coordinates and momenta. Our results provide quantization of the classical DELL system which was previously found in the string theory literature. The eigenfunctions for the N-body model are instanton partition functions of 6d SU(N) gauge theory with adjoint matter compactified on a torus with a codimension two defect. As a byproduct we discover new family of symmetric orthogonal polynomials which provide an elliptic generalization to Macdonald polynomials.

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Source: https://tomesphere.com/paper/1906.10354