Mixed network calculus

Abstract

Abstract We show how to combine higgsed topological vertices introduced in [7] with conventional refined topological vertices. We demonstrate that the extended formalism describes very general interacting D5-NS5-D3 brane systems. In particular, we introduce new types of intertwining operators of Ding-Iohara-Miki algebra between different types of Fock representations corresponding to the crossings of NS5 and D5 branes. As a byproduct we obtain an algebraic description of the Hanany-Witten brane creation effect, give an efficient recipe to compute the brane factors in 3d$$ \mathcal{N} $$ N = 2 and $$ \mathcal{N} $$ N = 4 quiver gauge theories and demonstrate how 3d S-duality appears in our setup.