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Higgs bundles for M-theory on \(G_2\)-manifolds. (English) Zbl 1414.81183

Summary: M-theory compactified on \(G_2\)-holonomy manifolds results in 4d \( \mathcal{N}=1\) supersymmetric gauge theories coupled to gravity. In this paper we focus on the gauge sector of such compactifications by studying the Higgs bundle obtained from a partially twisted 7d super Yang-Mills theory on a supersymmetric three-cycle \(M_3\). We derive the BPS equations and find the massless spectrum for both abelian and non-abelian gauge groups in 4d. The mathematical tool that allows us to determine the spectrum is Morse theory, and more generally Morse-Bott theory. The latter generalization allows us to make contact with twisted connected sum (TCS) \(G_2\)-manifolds, which form the largest class of examples of compact \(G_2\)-manifolds. M-theory on TCS \(G_2\)-manifolds is known to result in a non-chiral 4d spectrum. We determine the Higgs bundle for this class of \(G_2\)-manifolds and provide a prescription for how to engineer singular transitions to models that have chiral matter in 4d.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
83E30 String and superstring theories in gravitational theory
53Z05 Applications of differential geometry to physics
81T60 Supersymmetric field theories in quantum mechanics
83D05 Relativistic gravitational theories other than Einstein’s, including asymmetric field theories
70S15 Yang-Mills and other gauge theories in mechanics of particles and systems
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