R-matrices from Feynman Diagrams in 5d Chern-Simons Theory and Twisted M-theory

In this work we study the analogues of R-matrices that arise in 5d non-commutative topological-holomorphic Chern-Simons theory, which is known to describe twisted M-theory. We first study the intersections of line and surface operators in 5d Chern-Simons theory, which correspond to M2- and M5-branes, respectively. A Feynman diagram computation of the correlation function of this configuration furnishes an expression reminiscent of an R-matrix derivable from 4d Chern-Simons theory. We explain how this object is related to a Miura operator that is known to realize (matrix-extended) $W_{\infty}$-algebras. For 5d Chern-Simons theory with nonabelian gauge group, we then perform a Feynman diagram computation of coproducts for deformed double current algebras and matrix-extended $W_{\infty}$-algebras from fusions of M2-branes, M5-branes, and M2-M5 intersections.