Mixed network calculus

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 N = 2 and N = 4 quiver gauge theories and demonstrate how 3d S-duality appears in our setup.

Automated summary

The paper demonstrates the combination of modified topological vertices with conventional topological vertices, showing a description of interacting D5-NS5-D3 brane systems. Additionally, new types of intertwining operators in the Ding-Iohara-Miki algebra are introduced, providing an algebraic description of the Hanany-Witten brane creation effect. The efficient computation of brane factors in 3d N = 2 and N = 4 quiver gauge theories is discussed, along with the appearance of 3d S-duality in the setup.