Bootstrapping BPS spectra of 5d/6d field theories

We propose a systematic approach to computing the BPS spectrum of any 5d/6d supersymmetric quantum field theory in Coulomb phases, which admits either gauge theory descriptions or geometric descriptions, based on the Nakajima-Yoshioka's blowup equations. We provide a significant generalization of the blowup equation approach in terms of both properly quantized magnetic fluxes on the blowup C-2 and the effective prepotential for 5d/6d field theories on the Omega background which is uniquely determined by the Chern-Simons couplings on their Coulomb branches. We employ our method to compute BPS spectra of all rank-1 and rank-2 5d Kaluza-Klein (KK) theories descending from 6d N = (1, 0) superconformal field theories (SCFTs) compactified on a circle with/without twist. We also discuss various 5d SCFTs and KK theories of higher ranks, which include a few exotic cases such as new rank-1 and rank-2 5d SCFTs engineered with frozen singularity as well as the 5d SU(3)(8) gauge theory currently having neither a brane web nor a smooth shrinkable geometric description. The results serve as non-trivial checks for a large class of non-trivial dualities among 5d theories and also as independent evidences for the existence of certain exotic theories.

Automated summary

The proposed method offers a systematic approach to computing the BPS spectrum of 5d/6d supersymmetric quantum field theories in Coulomb phases, utilizing blowup equations and the Omega background. It has been successfully applied to calculate the BPS spectra of various rank-1 and rank-2 5d Kaluza-Klein (KK) theories, with discussions on exotic cases such as new 5d SCFTs and the 5d SU(3)(8) gauge theory. These results serve as important checks for dualities among 5d theories and provide independent evidence for the existence of certain exotic theories.