Webs of Quantum Algebra Representations in 5d $${\mathcal {N}}=1$$ Super Yang–Mills

Abstract

Instanton partition functions of 5d \({\mathcal {N}}=1\) Super Yang–Mills reduced on \(S^1\) are engineered in type IIB string theory from webs of (p, q)-branes. Branes intersections are associated to the (refined) topological vertex, while the web diagram provides gluing rules. These partition functions are covariant under the action of a quantum toroidal algebra, the Ding–Iohara–Miki algebra. In fact, a web of representations can be associated to the brane web diagram, where (p, q)-branes correspond to representations of levels (q, p), and topological vertices to intertwiners. Using this correspondence, the \({\mathcal {T}}\)-operator of a new type of quantum integrable systems can be constructed. Its vacuum expectation value reproduces the Nekrasov instanton partition function, while further insertion of algebra elements provides the qq-characters.