We explore the geometrical structure of Higgs branches of quantum field theories with 8 supercharges in 3, 4, 5 and 6 dimensions. They are symplectic singularities, and as such admit a decomposition (or foliation) into so-called symplectic leaves, which are related to each other by transverse slices. We identify this foliation with the pattern of partial Higgs mechanism of the theory and, using brane systems and recently introduced notions of magnetic quivers and quiver subtraction, we formalise the rules to obtain the Hasse diagram which encodes the structure of the foliation. While the unbroken gauge symmetry and the number of flat directions are obtainable by classical field theory analysis for Lagrangian theories, our approach allows us to characterise the geometry of the Higgs branch by a Hasse diagram with symplectic leaves and transverse slices, thus refining the analysis and extending it to non-Lagrangian theories. Most of the Hasse diagrams we obtain extend beyond the cases of nilpotent orbit closures known in the mathematics literature. The geometric analysis developed in this paper is applied to Higgs branches of several Lagrangian gauge theories, Argyres-Douglas theories, five dimensional SQCD theories at the conformal fixed point, and six dimensional SCFTs.

This paper tests a conjecture on discrete non-Abelian gauging of 3d N = 4 supersymmetric quiver gauge theories. Given a parent quiver with a bouquet of n nodes of rank 1, invariant under a discrete S n global symmetry, one can construct a daughter quiver where the bouquet is substituted by a single adjoint n node. Based on the main conjecture in this paper, the daughter quiver corresponds to a theory where the S n discrete global symmetry is gauged and the new Coulomb branch is a non-Abelian orbifold of the parent Coulomb branch. We demonstrate and test the conjecture for three simply laced families of bouquet quivers and a non-simply laced bouquet quiver with C 2 factor in the global symmetry.

Three dimensional Coulomb branches have a prominent role in the study of moduli spaces of supersymmetric gauge theories with 8 supercharges in 3, 4, 5, and 6 dimensions. Inspired by simply laced 3d $$ \mathcal{N} $$ N = 4 supersymmetric quiver gauge theories, we consider Coulomb branches constructed from non-simply laced quivers with edge multiplicity k and no flavor nodes. In a computation of the Coulomb branch as the space of dressed monopole operators, a center-of-mass U(1) symmetry needs to be ungauged. Typically, for a simply laced theory, all choices of the ungauged U(1) (i.e. all choices of ungauging schemes ) are equivalent and the Coulomb branch is unique. In this note, we study various ungauging schemes and their effect on the resulting Coulomb branch variety. It is shown that, for a non-simply laced quiver, inequivalent ungauging schemes exist which correspond to inequivalent Coulomb branch varieties. Ungauging on any of the long nodes of a non-simply laced quiver yields the same Coulomb branch $$ \mathcal{C} $$ C . For choices of ungauging the U(1) on a short node of rank higher than 1, the GNO dual magnetic lattice deforms anisotropically such that it no longer corresponds to a Lie group, and therefore, the monopole formula yields a non-valid Coulomb branch. However, if the ungauging is performed on a short node of rank 1, the one-dimensional magnetic lattice is rescaled along its single direction i.e. isotropically and the corresponding Coulomb branch is an orbifold of the form $$ \mathcal{C} $$ C /ℤk . Ungauging schemes of 3d Coulomb branches provide a particularly interesting and intuitive description of a subset of actions on the nilpotent orbits studied by Kostant and Brylinski [1]. The ungauging scheme analysis is carried out for minimally unbalanced Cn, affine F4, affine G2, and twisted affine $$ {D}_4^{(3)} $$ D 4 3 quivers, respectively. The analysis is complemented with computations of the Highest Weight Generating functions.

The electron-vibrational problem of the general nonadiabatic molecular systems has been solved by means of the quasi-particle transformations. The SCF ab initio solution of the nonadiabatic fermion Hamiltonian yields stabilization of the electronic ground-state energy due to electron-phonon interaction and it also gives the corrections to the one-and two-particle terms. Two two-particle correction yields effective attractive electron-electron interaction, but in the form different from Frolich's effective electron-electron interaction term. In contrast to the standard electron-phonon Hamiltonian of solid-state physics that does not take into account the possible effects of nonadiabaticity of a system, the presented nonadiabatic theory yields also one-particle corrections. The presence of this term in the Hamiltonian might play a crucial role in the theory of superconductivity since the superconductors are nonadiabatic systems. Since the quasi-particle theory of vibrational energy calculations for nonadiabatic molecules is extremely extensive and will be published elsewhere [lo], we restrict ourselves only to the schematic way of derivation, with the focus being placed on the fermion part of the nonadiabatic e1.-vibr. Hamiltonian.

We develop a classification of minimally unbalanced 3d N = 4 quiver gauge theories. These gauge theories are important because the isometry group G of their Coulomb branch contains a single factor, which is either a classical or an exceptional Lie group. Concurrently, this provides a classification of hyperkähler cones with isometry group G which are obtainable by Coulomb branch constructions. HyperKähler cones such as Coulomb branches of 3d N = 4 quivers are indispensable tools for describing Higgs branches of different theories in various dimensions. In particular, they are used to describe Higgs branches of 5d N = 1 SQCD with gauge group SU(N c) and 6d N = (1, 0) SQCD with gauge group Sp(N c) at the respective UV fixed points.