MQCD, ('Barely') G_2 Manifolds and (Orientifold of) a Compact Calabi-Yau

TL;DR

This paper explores nonperturbative superpotentials from M2-branes in G_2-manifolds and their relation to compact Calabi-Yau manifolds, connecting various string theories and M-theory through complex geometric and physical analyses.

Contribution

It establishes new links between G_2-manifolds, Calabi-Yau spaces, and M-theory, including evaluations of superpotentials and conjectures on involutions affecting period integrals.

Findings

Comparison of M2-brane superpotentials with heterotic instantons

Evaluation of RP^2-instanton superpotential in Calabi-Yau compactification

Summation of series for supersymmetric 3-cycle embeddings in G_2-manifolds

Abstract

We begin with a discussion on two apparently disconnected topics - one related to nonperturbative superpotential generated from wrapping an M2-brane around a supersymmetric three cycle embedded in a G_2-manifold evaluated by the path-integral inside a path-integral approach of [1], and the other centered around the compact Calabi-Yau CY_3(3,243) expressed as a blow-up of a degree-24 Fermat hypersurface in WCP^4[1,1,2,8,12]. For the former, we compare the results with the ones of Witten on heterotic world-sheet instantons [2]. The subtopics covered in the latter include an N=1 triality between Heterotic, M- and F-theories, evaluation of RP^2-instanton superpotential, Picard-Fuchs equation for the mirror Landau-Ginsburg model corresponding to CY_3(3,243), D=11 supergravity corresponding to M-theory compactified on a `barely' G_2 manifold involving CY_3(3,243) and a conjecture related to the action of antiholomorphic involution on period integrals. We then show an indirect connection between the two topics by showing a connection between each one of the two and Witten's MQCD [3]. As an aside, we show that in the limit of vanishing "\zeta", a complex constant that appears in the Riemann surfaces relevant to definining the boundary conditions for the domain wall in MQCD, the infinite series of [4] used to represent a suitable embedding of a supersymmetric 3-cycle in a G_2-mannifold, can be summed.