Abstract
We describe how the geometry of the Higgs branch of 5d superconformal field theories is transformed under movement along the extended Coulomb branch. Working directly with the (unitary) magnetic quiver, we demonstrate a correspondence between Fayet-Iliopoulos deformations in 3d and 5d mass deformations. When the Higgs branch has multiple cones, characterised by a collection of magnetic quivers, the mirror map is not globally well-defined, however we are able to utilize the correspondence to establish a local version of mirror symmetry. We give several detailed examples of deformations, including decouplings and weak-coupling limits, in (Dn, Dn) conformal matter theories, TN theory and its parent PN, for which we find new Lagrangian descriptions given by quiver gauge theories with fundamental and anti-symmetric matter.
Highlights
Higgs branch is realized as its deformation space
We describe how the geometry of the Higgs branch of 5d superconformal field theories is transformed under movement along the extended Coulomb branch
The most literal way to answer this question is to work directly with the magnetic quiver, and understand, for instance, how to decouple matter or take a weak coupling limit explicitly in the magnetic quiver. We show that this is achieved by Fayet-Iliopoulos (FI) deformations, and argue that, for any extended Coulomb branch deformation of the 5d SCFT, implemented as partial resolutions of the GTP [31, 41, 58], there is an FI deformation that enacts the corresponding change in the magnetic quiver, extrapolating between the Higgs branches emanating from the two points in the extended Coulomb branch
Summary
We would like to understand FI deformations in a unitary quiver. Let us start by reviewing how FI deformations affect the equations of motion at a given gauge node in the quiver. Considering each node in the larger quiver in turn, and combining this with the identity Tr(P f Pf ) = Tr(Pf P f ), we conclude that for any quiver with unitary gauge groups and bifundamental hypermultiplets the FI parameters should obey the relation. The vev for the bifundamentals is obtained from (2.5) by picking the tensor product with the k ×k identity matrix Ik. All the groups along the subquiver are broken as U(ni) → U(ni − k) and we need to add a U(k) node associated with the surviving gauge symmetry. The U(1) node in blue is introduced to rebalance in the second subtraction, and, as we have explained before, is not connected to the blue U(2) node Overall, this FI deformation describes the transition from the E8 to the E7 theory. We clearly see here that generically there are multiple choices of FI deformation which implement a given mass deformation
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