Incidence Geometry in a Weyl Chamber I: $G{L}_n$

Abstract

We study the central hyperplane arrangement whose hyperplanes are the vanishing loci of the weights of the first and the second fundamental representations of ${\mathfrak{gl}}_n$ restricted to the dual fundamental Weyl chamber. We obtain generating functions that count flats and faces of a given dimension. This counting is interpreted in physics as the enumeration of the phases of the Coulomb and mixed Coulomb-Higgs branches of a five dimensional gauge theory with 8 supercharges in presence of hypermultiplets transforming in the fundamental and antisymmetric representation of a U(n) gauge group as described by the Intriligator-Morrison-Seiberg superpotential.

25 pages, 6 figures

References in corpus (4)

• F-Theory Compactifications with Multiple U(1)-Factors: Constructing Elliptic Fibrations with Rational Sections
• A New Model for Elliptic Fibrations with a Rank One Mordell-Weil Group: I. Singular Fibers and Semi-Stable Degenerations
• Tate Trees for Elliptic Fibrations with Rank one Mordell-Weil group
• M-theory on Elliptic Calabi-Yau Threefolds and 6d Anomalies

Cited by in corpus (10)

• Flopping and Slicing: SO(4) and Spin(4)-models
• Mordell-Weil Torsion, Anomalies, and Phase Transitions
• Singular Geometry and Higgs Bundles in String Theory
• The Geometry of SO(3), SO(5), and SO(6) models
• The Geometry of G${}_2$, Spin(7), and Spin(8)-models
• The Geometry of F${}_4$-Models
• Characteristic numbers of elliptic fibrations with non-trivial Mordell-Weil groups
• 48 Crepant Paths to $\text{SU}(2)\times \text{SU}(3)$
• The suspended pinch point and SU($2$)$\times$U($1$) gauge theories
• Flops and Fibral Geometry of E${}_7$-models