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

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.

arXiv comment44 pages; v3. two appendices added: Relation with quantum toroidal gl(p) algebra & Examples of qq-characters

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• A Note on Quiver Quantum Toroidal Algebra
• Quantum toroidal algebras and solvable structures in gauge/string theory
• 5d/6d Wilson loops from blowups
• More on topological vertex formalism for 5-brane webs with O5-plane
• 5d AGT correspondence of supergroup gauge theories from quantum toroidal ${\mathfrak{gl}}_1$
• Gauge origami and quiver W-algebras
• di-Langlands correspondence and extended observables
• Parallel surface defects, Hecke operators, and quantum Hitchin system
• New dimer integrable systems and defects in five dimensional gauge theory
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials
• Intertwining operator and integrable hierarchies from topological strings