Mixed network calculus

TL;DR

This paper develops a unified formalism combining Higgsed and refined topological vertices to describe complex brane systems, providing algebraic tools and insights into dualities and brane interactions in gauge theories.

Contribution

It introduces a novel extended formalism for topological vertices, including new intertwining operators, and offers an algebraic description of brane creation effects and dualities.

Findings

Unified formalism for brane systems using topological vertices

New intertwining operators for Ding-Iohara-Miki algebra

Algebraic description of Hanany-Witten brane creation effect

Abstract

We show how to combine higgsed topological vertices introduced in our previous work 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.