Miura operators as R-matrices from M-brane intersections

TL;DR

This paper links Miura operators to R-matrices within quantum algebras, realized through M-brane intersections in M-theory, and demonstrates their role in constructing qq-characters and understanding gauge theories.

Contribution

It proposes that Miura operators are R-matrices of quantum algebras and realizes them via M-brane intersections, connecting string theory, algebra, and gauge theory.

Findings

Miura operators are identified as R-matrices of quantum toroidal algebras.

M-brane intersections realize these R-matrices physically.

qq-characters are constructed from the Miura transformation.

Abstract

We propose that Miura operators are R-matrices of certain infinite-dimensional quantum algebras. We test our proposal by realizing Miura operators of $q$-deformed $W$- and $Y$-algebras in terms of R-matrices of the quantum toroidal algebra of $\mathfrak{gl}(1)$. Physically, the representations of this toroidal algebra arise from the algebra of local operators on M2-branes and M5-branes, in M-theory subject to an $\Omega$-background. We associate an R-matrix to each M2-M5 brane crossing, by studying its description as a gauge-invariant intersection of a topological line defect and a holomorphic surface defect in 5-dimensional non-commutative Chern-Simons theory. The Miura transformation is engineered using multiple M2-M5 intersections, relying crucially on the properties of the underlying R-matrices. We thereby identify each R-matrix with a Miura operator. In a dual Type IIB frame, the components of the Miura transformation are shown to coincide with the half-index of a 3d supersymmetric gauge theory on a Hanany-Witten system of D3-NS5 branes. As a further application, we demonstrate that $qq$-characters can be algebraically constructed from the Miura transformation.