Topological strings and 5d T N partition functions

Abstract

We evaluate the Nekrasov partition function of 5d gauge theories engineered by webs of 5-branes, using the refined topological vertex on the dual Calabi-Yau three-folds. The theories include certain non-Lagrangian theories such as the T N theory. The refined topological vertex computation generically contains contributions from decoupled M2-branes which are not charged under the 5d gauge symmetry engineered. We argue that, after eliminating them, the refined topological string partition function agrees with the 5d Nekrasov partition function. We explicitly check this for the T 3 theory as well as Sp(1) gauge theories with N f = 2, 3, 4 flavors. In particular, our method leads to a new expression of the Sp(1) Nekrasov partition functions without any contour integrals. We also develop prescriptions to calculate the partition functions of theories obtained by Higgsing the T N theory. We compute the partition function of the E 7 theory via this prescription, and find the E 7 global symmetry enhancement. We finally discuss a potential application of the refined topological vertex to non-toric web diagrams. © 2014 The Author(s).