# New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds

## TL;DR

This paper introduces a new class of quantum toroidal algebras derived from 5D supersymmetric gauge theories on orbifolds, providing algebraic tools to compute instanton partition functions and related invariants.

## Contribution

It constructs a deformation of quantum toroidal algebras for orbifold gauge theories and develops their representations and vertex operators, linking algebraic structures to physical partition functions.

## Key findings

- Defined a new quantum toroidal algebra for orbifold theories
- Developed vertical and horizontal representations of the algebra
- Reconstructed Nekrasov partition functions and $qq$-characters

## Abstract

Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D $\mathcal{N}=1$ supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold $S^1\times\mathbb{C}^2/\mathbb{Z}_p$ where the action of $\mathbb{Z}_p$ is determined by two integer parameters $(\nu_1,\nu_2)$. The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito's vertex representations of the quantum toroidal algebra of $\mathfrak{gl}(p)$. We construct the vertex operator intertwining between these two types of representations. This object is identified with a $(\nu_1,\nu_2)$-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the $qq$-characters of the quiver gauge theories.