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We construct novel web diagrams with a trivalent or quadrivalent gluing for various 6d/5d theories from certain Higgsings of 6d conformal matter theories on a circle. The theories realized on the web diagrams include 5d Kaluza-Klein theories from circle compactifications of the 6d $G_2$ gauge theory with 4 flavors, the 6d $F_4$ gauge theory with 3 flavors, the 6d $E_6$ gauge theory with 4 flavors and the 6d $E_7$ gauge theory with 3 flavors. The Higgsings also give rise to 5d Kaluza-Klein theories from twisted compactifications of 6d theories including the 5d pure SU(3) gauge theory with the Chern-Simons level 9 and the 5d pure SU(4) gauge theory with the Chern-Simons level 8. We also compute the Nekrasov partition functions of the theories by applying the topological vertex formalism to the newly obtained web diagrams.
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We propose a systematic approach to computing the BPS spectra of any 5d/6d supersymmetric quantum field theory in Coulomb phases, which admits either gauge theory descriptions or geometric descriptions, based on the Nakajima-Yoshiokas blowup equations. We provide a significant generalization of the blowup equation approach in terms of both properly quantized magnetic fluxes on the blowup $hat{mathbb{C}}^2$ 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 $mathcal{N}=(1,0)$ 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.
We generalize Nakajima-Yoshiokas blowup formula to calculate the partition functions counting the spectrum of bound states to half-BPS Wilson loop operators in 5d (and 6d) supersymmetric field theories. The partition function in the presence of a Wilson loop operator on the $Omega$-background is factorized when put on the blowup $hat{mathbb{C}}^2$ into two Wilson loop partition functions under the localization. This structure provides a set of blowup equations for Wilson loop operators. We explain how to formulate the blowup equations and solve them to compute the partition functions of Wilson loop operators. We test this idea by explicitly calculating the Wilson loop partition functions in various 5d/6d field theories and comparing them against known results and expected dualities.
We classify 5d N=1 gauge theories carrying a simple gauge group that can arise by mass-deforming 5d SCFTs and 6d SCFTs (compactified on a circle, possibly with a twist). For theories having a 6d UV completion, we determine the tensor branch data of the 6d SCFT and capture the twist in terms of the tensor branch data. We also determine the dualities between these 5d gauge theories, thus determining the sets of gauge theories having a common UV completion.
We propose Type IIB 5-brane configurations that engineer the 6d $mathcal{N}=(1,0)$ SCFTs with $SO(N)$ gauge symmetry coupled to a single tensor multiplet on a circle, by considering RG flows on Higgs branches of $D$-type conformal matter theories. We test the brane systems against known Calabi-Yau threefolds for the 6d SCFTs on a circle. In addition we study a new RG flow involving Higgs vevs of scalar operators with Kaluza-Klein momentum along the circle. The new RG flow results in the 5-brane webs for the 6d SCFTs of $D_N$ gauge symmetry compactified on a circle with $Z_2$ outer-automorphism twist.
Gauge theories in four dimensions can exhibit interesting low energy phenomena, such as infrared enhancements of global symmetry. We explore a class of 4d N=1 gauge theories arising from a construction that is motivated by duality walls in 5d gauge theories. Their quiver descriptions bear a resemblance to 4d theories obtained by compactifying 6d N=(1,0) superconformal field theories on a torus with fluxes, but with lower number of flavours and different number of gauge singlets and superpotentials. One of the main features of these theories is that they exhibit a flavour symmetry enhancement, and with supersymmetry enhancement for certain models, in the infrared. Properties of the superconformal fixed points of such theories are investigated in detail.
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