Higgs bundles for N=1 gauge theories

Thesis

We consider two constructions of 4d N = 1 gauge theories in M-theory. The first is purely geometric and involves compactifying M-theory on singular manifolds of G2-holonomy. The geometries studied are ALE-fibered over a compact base M3. This fibration admits a description in terms of a Higgs bundle with three-dimensional base M3 which arises as the BPS equations of M-theory reduced adiabatically along the ALE fibers. The gauge theory sector of these compactifications originates from M2-branes wrapped on vanishing cycles of the ALE fibration with interactions set by Euclidean M2-branes wrapped on associative submanifolds traced out by the vanishing cycles. The first part of this thesis develops the physics of these models and introduces a colored supersymmetric quantum mechanics organizing and quantifying non-perturbative effects due to M2-brane instantons. In this framework we study the local models of twisted connected sum G2-manifolds and describe their possible chiral deformations.

The second construction considered in this thesis utilizes M5-branes wrapped on Rie- mann surfaces embedded in local Calabi-Yau threefolds. We focus on the subset of such configurations derived from 4d N = 2 theories of class S breaking to N = 1. Their BPS equations construct Higgs bundles differing from the standard class S Higgs bundle by an additional Higgs field. The associated N = 1 curve collects the spectral data of both Higgs fields and different curves describe distinct vacua of the same gauge theory. The second part of this thesis studies the construction of such Higgs bundles and derives the confinement properties of each vacuum from the associated N = 1 curve. This allows for the study of confinement in non-Lagrangian N = 1 theories which is illustrated by constructing an infinite class of non-Lagrangian N = 1 theories that contain confining vacua.

Authors: Hübner, M

• DOI: 10.5287/ora-eyy1nnyyn
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: G2-manifold; M-theory; Higgs Bundle; class S; supersymmetric quantum mechanics
• Subjects: Mathematics; Mathematical physics