Abstract
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 Nc for an SU Nc gauge theory, number of fundamental flavors Nf, 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 (Nc = 2, Nf = 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.
Highlights
Between seven branes, providing the Higgs branch moduli, one modulus per each such sub web
The combinatorial data is not an exclusive feature of 3d, but rather of any dimension that includes 3! It is a feature of a co-dimension 3 object. This is further supported by the brane picture of type [5] in which a Higgs branch modulus is characterized by a Dp brane in between two D(p + 2) branes where the boundary conditions remove a vector multiplet, but leave a compact scalar which, together with one of the 3 real scalars, exponentiate to form monopole operators in the construction of the moduli space
The conclusion of this paper is that tropical geometry holds the key to solve a problem in supersymmetric quantum field theory that had not been probed with the tools that were developed in the past
Summary
On the right of the same figure the gauge coupling is taken to infinity and all the masses as well as the VEV of the adjoint scalar field is set to zero (i.e. at the origin of the Coulomb branch) Before taking this limit, there is a single web that can move along the 7-branes, and the Higgs branch is trivial, remembering to factor out overall position moduli. One can think of 1 D3 brane which is stretched between the sub webs per each intersection point On this particular example, we have found a way to read the 3d quivers that describe (via equations (2.1), (2.2) and (2.3)) the H∞ of the 5d theory. The final answer on reading the 3d quiver for the whole 3 parameter family of 5d SQCD theories, an answer which is more general and applies for any five brane web
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