Discrete θ and the 5d superconformal index

Abstract

5d Yang-Mills theory with an Sp(N ) gauge group admits a discrete analog of the θ parameter. We describe the origin of this parameter in N = 1 theories from Type I' string theory, and study its effect on the 5d superconformal fixed point theories with an Sp(1) = SU(2) gauge group by computing the superconformal index. Our result confirms the lack of global symmetry enhancement in the so-calledẼ 1 theory.

Duality and enhancement of symmetry in 5d gauge theories

“…In the pure case there are 3 different SU(2) webs not related by an SL(2, Z) transformation corresponding to different U(2) CS levels [11,15]. These are also given from the general web of figure 18.…”

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.…”

“…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].…”