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Register a new userFive-brane webs, Higgs branches and unitary/orthosymplectic magnetic quivers
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We study the Higgs branch of 5d superconformal theories engineered from brane webs with orientifold five-planes. We propose a generalization of the rules to derive magnetic quivers from brane webs pioneered in arXiv:2004.04082, by analyzing theories that can be described with a brane web with and without O5 planes. Our proposed magnetic quivers include novel features, such as hypermultiplets transforming in the fundamental-fundamental representation of two gauge nodes, antisymmetric matter, and $mathbb{Z}_2$ gauge nodes. We test our results by computing the Coulomb and Higgs branch Hilbert series of the magnetic quivers obtained from the two distinct constructions and find agreement in all cases.
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It is widely considered that the classical Higgs branch of 4d $mathcal{N}=2$ SQCD is a well understood object. However there is no satisfactory understanding of its structure. There are two complications: (1) the Higgs branch chiral ring contains nilpotent elements, as can easily be checked in the case of $mathrm{SU}(N)$ with 1 flavour. (2) the Higgs branch as a geometric space can in general be decomposed into two cones with nontrivial intersection, the baryonic and mesonic branches. To study the second point in detail we use the recently developed tool of magnetic quivers for five-brane webs, using the fact that the classical Higgs branch for theories with 8 supercharges does not change through dimensional reduction. We compare this approach with the computation of the hyper-Kahler quotient using Hilbert series techniques, finding perfect agreement if nilpotent operators are eliminated by the computation of a so called radical. We study the nature of the nilpotent operators and give conjectures for the Hilbert series of the full Higgs branch, giving new insights into the vacuum structure of 4d $mathcal{N}=2$ SQCD. In addition we demonstrate the power of the magnetic quiver technique, as it allows us to identify the decomposition into cones, and provides us with the global symmetries of the theory, as a simple alternative to the techniques that were used to date.
Magnetic quivers have led to significant progress in the understanding of gauge theories with 8 supercharges at UV fixed points. For a given low-energy gauge theory realised via a Type II brane construction, there exist magnetic quivers for the Higgs branches at finite and infinite gauge coupling. Comparing these moduli spaces allows to study the non-perturbative effects when transitioning to the fixed point. For 5d $mathcal{N}=1$ SQCD, 5-brane webs have been an important tool for deriving magnetic quivers. In this work, the emphasis is placed on 5-brane webs with orientifold 5-planes which give rise to 5d theories with orthogonal or symplectic gauge groups. For this set-up, the magnetic quiver prescription is derived and contrasted against a unitary magnetic quiver description extracted from an O$7^-$ construction. Further validation is achieved by a derivation of the associated Hasse diagrams. An important class of families considered are the orthogonal exceptional $E_n$ families ($-infty < n leq 8$), realised as infinite coupling Higgs branches of $mathrm{Sp}(k)$ gauge theories with fundamental matter. In particular, the moduli spaces are realised by a novel type of magnetic quivers, called unitary-orthosymplectic quivers.
We discuss the effective Chern-Simons levels for 3d $mathcal{N}=2$ gauge theories and their relations to the relative angles between NS5-brane and NS5-brane. We find that turning on real masses for chiral multiplets leads to various equivalent brane webs that are related by flipping the sign of mass parameters. This flip can be interpreted as 3d mirror symmetry for abelian theories. Each of these webs has a corresponding mathematical quiver structure. We check the equivalence of vortex partition functions for these brane webs by implementing topological vertex method. In addition, we compute the vortex partition functions of nonabelian theories with gauge group $U(N)$ and find the associated quiver structures and brane webs. We find that on Higgs branch nonabelian brane webs are broken to abelian brane webs with gauge group $U(1)^{otimes N}$. We also discuss the Ooguri-Vafa invariants for nonabelian theories and the movement of flavor D5-branes that leads to equivalent brane webs.
For any gauge theory, there may be a subgroup of the gauge group which acts trivially on the matter content. While many physical observables are not sensitive to this fact, the identification of the precise gauge group becomes crucial when the magnetic spectrum of the theory is considered. This question is addressed in the context of Coulomb branches for $3$d $mathcal{N}=4$ quiver gauge theories, which are moduli spaces of dressed monopole operators. Since monopole operators are characterized by their magnetic charge, the identification of the gauge group is imperative for the determination of the magnetic lattice. It is well-known that the gauge group of unframed unitary quivers is the product of all unitary nodes in the quiver modded out by the diagonal $mathrm{U}(1)$ acting trivially on the matter representation. This reasoning generalises to the notion that a choice of gauge group associated to a quiver is given by the product of the individual nodes quotiented by any subgroup that acts trivially on the matter content. For unframed (unitary-) orthosymplectic quivers composed of $mathrm{SO}(textrm{even})$, $mathrm{USp}$, and possibly $mathrm{U}$ gauge nodes, the maximal subgroup acting trivially is a diagonal $mathbb{Z}_2$. For unframed unitary quivers with a single $mathrm{SU}(N)$ node it is $mathbb{Z}_N$. We use this notion to compute the Coulomb branch Hilbert series of many unitary-orthosymplectic quivers. Examples include nilpotent orbit closures of the exceptional E-type algebras and magnetic quivers that arise from brane physics. This includes Higgs branches of theories with 8 supercharges in dimensions $4$, $5$, and $6$. A crucial ingredient in the calculation of exact refined Hilbert series is the alternative construction of unframed magnetic quivers from resolved Slodowy slices, whose Hilbert series can be derived from Hall-Littlewood polynomials.
Superconformal five dimensional theories have a rich structure of phases and brane webs play a crucial role in studying their properties. This paper is devoted to the study of a three parameter family of SQCD theories, given by the number of colors $N_c$ for an $SU(N_c)$ gauge theory, number of fundamental flavors $N_f$, and the Chern Simons level $k$. The study of their infinite coupling Higgs branch is a long standing problem and reveals a rich pattern of moduli spaces, depending on the 3 values in a critical way. For a generic choice of the parameters we find a surprising number of 3 different components, with intersections that are closures of height 2 nilpotent orbits of the flavor symmetry. This is in contrast to previous studies where except for one case ($N_c=2, N_f=2$), the parameters were restricted to the cases of Higgs branches that have only one component. The new feature is achieved thanks to a concept in tropical geometry which is called stable intersection and allows for a computation of the Higgs branch to almost all the cases which were previously unknown for this three parameter family apart form certain small number of exceptional theories with low rank gauge group. A crucial feature in the construction of the Higgs branch is the notion of dressed monopole operators.
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