Exact partition functions of Higgsed 5d T N theories

Abstract

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 .

Topological vertex for Higgsed 5d T N theories

“…After the removal of some extra factors independent of the Coulomb branch moduli it was possible to compute the partition function of the T N theory [21,22]. The enhancement of the flavour symmetries in its superconformal index 3 was also checked in [14,21,22], and a crucial point in obtaining enhancement of the global symmetry of five-dimensional superconformal field theories was the correct removal of the extra factors independent of the Coulomb branch moduli [12,14,21,22,[24][25][26][27][28]. 4 On the other hand the web diagrams corresponding to Higgsed 5d T N theories are not dual to toric geometries and therefore simple application of the refined topological vertex does not work.…”

“…4 On the other hand the web diagrams corresponding to Higgsed 5d T N theories are not dual to toric geometries and therefore simple application of the refined topological vertex does not work. The formalism to compute their partition function was developed in [21,28] 5 by making use of the technique in [33][34][35][36][37][38]. The prescription to compute the partition function of Higgsed 5d T N theories is a three step process.…”

“…However it is still possible to compute the partition function of the IR theory realised by a Higgsed diagram by first computing the partition function of a UV theory which embeds the IR theory and then applying a suitable tuning condition to the parameters of the UV partition function. After eliminating some singlet hypermultiplets in the Higgs vacuum we obtain the partition function of the IR theory [21,28]. For example the Higgsing in figure 2 requires a tuning…”

“…Let us compare the result with the one computed by the original method using the tuned UV diagram of the left figure of figure 8. In this example it is also possible to directly use the result of the topological string partition function for a Higgsed T 3 computed in [28] and perform an additional Higgsing in the diagram. The result is simple and can be written in terms of infinite products…”

“…Another fundamental piece in the correct computation of the partition function in the Higgs branch of 5d theories is the correct identification of the contributions of singlet hypermultiplets to the partition function for these contributions need to be subtracted as well from the partition function in order to get the correct result. The rule that was used in [21,28] to correctly identify these contributions is that these hypermultiplets will come from M2-branes wrapping curves connecting pairs of legs in the diagram that become external after entering the Higgs branch. Remarkably these contributions are totally absent if the partition function is computed using the usual topological vertex rule for Higgsed diagrams.…”