Topological vertex for Higgsed 5d TN theories

We analyse the computation of the partition function of 5d T-N theories in Higgs branches using the topological vertex. The theories are realised by a web of (p, q) 5-branes whose dual description may be given by an M-theory compactification on a certain local non-toric Calabi-Yau threefold. We explicitly show how it is possible to directly apply the topological vertex to the non-toric geometry. Using this novel technique, which considerably simplifies the computation by the existing method, we are able to compute the partition function of the higher rank E-6, E-7 and E-8 theories. Moreover we show how in some specific cases similar results can be extended to the computation of the partition function of 5d T-N theories in the Higgs branch using the refined topological vertex. These cases require a modification of the refined topological vertex.