5-brane webs, symmetry enhancement, and duality in 5d supersymmetric gauge theory

Abstract

We present a number of investigations of 5d N = 1 supersymmetric gauge theories that make use of 5-brane web constructions and the 5d superconformal index. These include an observation of enhanced global symmetry in the 5d fixed point theory corresponding to SU(N ) gauge theory with Chern-Simons level ±N , enhanced global symmetries in quiver theories, and dualities between quiver theories and non-quiver theories. Instanton contributions play a crucial role throughout.

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.…”

“…be computed by the powerful technique of the refined topological vertex [19,20] after eliminating what we call decoupled factor which is a contribution associated to strings between parallel external legs [21,22,25]. Technical details of the refined topological vertex as well as the decoupled factor are summarised in appendix A.…”

“…The claim of [21,22,24,25] is that the correct partition function of the 5d field theory can be computed from the topological string partition function by appropriately cancelling these contributions, namely the 5d Nekrasov partition function has the following form…”

“…The refined topological string partition itself does not reproduce the Nekrasov partition function of the five-dimensional theory realised by the M-theory compactification on the toric Calabi-Yau threefold X, but we need to eliminate some factors which come from BPS states with no gauge charge [21,22,24,25]. In the dual brane picture, those BPS states come from strings between parallel external legs.…”

“…The non-full spin content appears since the moduli space of the P 1 inside O(−2) ⊕ O fibered over P 1 is non-compact. Therefore, as for removing the decoupled factor, we may also say that we remove the contribution of components which do not fill the full spin content of the vector multiplet or recover the invariance under the exchange between q and t [22,24,25].…”