Exact partition functions of Higgsed 5d T N theories

We present a general prescription by which we can systematically compute exact partition functions of five-dimensional supersymmetric theories which arise in Higgs branches of the T _N theory. The theories may be realised by webs of 5-branes whose dual geometries are non-toric. We have checked our method by calculating the partition functions of the theories realised in various Higgs branches of the T ₃ theory. A particularly interesting example is the E ₈ theory which can be obtained by Higgsing the T ₆ theory. We explicitly compute the partition function of the E ₈ theory and find agreement with the field theory result as well as enhancement of the global symmetry to E ₈.