Abstract
We study the physics of globally consistent four-dimensional $\mathcal{N}=1$ supersymmetric M-theory compactifications on $G_2$ manifolds constructed via twisted connected sum; there are now perhaps fifty million examples of these manifolds. We study a rich example that exhibits $U(1)^3$ gauge symmetry and a spectrum of massive charged particles that includes a trifundamental. Applying recent mathematical results to this example, we compute membrane instanton corrections to the superpotential and spacetime topology change in a compact model; the latter include both the (non-isolated) $G_2$ flop and conifold transitions. The conifold transition spontaneously breaks the gauge symmetry to $U(1)^2$, and associated field theoretic computations of particle charges make correct predictions for the topology of the deformed $G_2$ manifold. We discuss physical aspects of the abelian $G_2$ landscape broadly, including aspects of Higgs and Coulomb branches, membrane instanton corrections, and some general aspects of topology change.
Highlights
Singular limit of the compactification manifold so that there is no large-volume approximation, and (3) relatively little is known about the relevant seven-manifolds compared to, for example, Calabi-Yau threefolds
We study the physics of globally consistent four-dimensional N = 1 supersymmetric M-theory compactifications on G2 manifolds constructed via twisted connected sum; there are perhaps fifty million examples of these manifolds
For the G2 manifold X that we study, we will show that M-theory on X yields an N = 1 supersymmetric four-dimensional supergravity theory at low energies with U(1)3 gauge symmetry and a spectrum of massive charged particles including trifundamentals
Summary
We review Kovalev’s construction [1] for obtaining compact G2 manifolds from twisted connection sums. This is a compact seven-manifold which admits a closed G2 structure that is determined by the G2 structures on M±; they are not a priori torsion-free This leads to: Theorem 1 [Kovalev’s Theorem] Let (V±, ω±, Ω±) be two ACyl Calabi-Yau three-folds with asymptotic ends of the form R+ × S1 × S± for a pair of hyperKahler K3 surfaces S±, and suppose that there exists a diffeomorphism r : S+ → S− preserving the Ricci-flat metrics and satisfying (2.11). The fundamental group and the Betti numbers were computed for early examples of G2 manifolds, but thanks to [5], it is possible to compute the full integral cohomology for many twisted connected sums, including the torsional components of H3(X, Z) and H4(X, Z), as well as the first Pontryagin class p1 If it weren’t for a general observation that we will discuss, the explicit knowledge of p1 in examples would play a critical role [22] in determining the quantization of M-theory flux in those examples. It gives the first construction technique for compact rigid associative submanifolds in compact G2 manifolds, and it is possible to compute the form of membrane instanton corrections to the superpotential in examples; see the following
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