G 2 -manifolds and associative submanifolds via semi-Fano 3 -folds

Abstract

Abstract. We provide a significant extension of the twisted connected sum construction of G2-manifolds, ie Riemannian 7-manifolds with holonomy group G2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Our extension allows us to prove many new results about compact G2-manifolds and leads to new perspectives for future research in the area. Some of the main contributions of the paper are:(i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a subclass of weak Fano 3-folds) can be used as components in the twisted connected sum construction. (iii) We construct many new topological types of compact G2-manifolds by applying the twisted connected sum to asymptotically Calabi-Yau 3-folds of semi-Fano type studied in [18]. (iv) We obtain much more precise topological information about twisted connected sum G2-manifolds; one application is the determination for the first time of the diffeomorphism type of many compact G2-manifolds. (v) We describe "geometric transitions" between G2-metrics on different 7-manifolds mimicking "flopping" behaviour among semi-Fano 3-folds and "conifold transitions" between Fano and semi-Fano 3-folds. (vi) We construct many G2-manifolds that contain rigid compact associative 3-folds. (vii) We prove that many smooth 2-connected 7-manifolds can be realised as twisted connected sums in numerous ways; by varying the building blocks matched we can vary the number of rigid associative 3-folds constructed therein. The latter result leads to speculation that the moduli space of G2-metrics on a given 7-manifold may consist of many different connected components, and opens up many further questions for future study. For instance, the higher-dimensional gauge theory invariants proposed by Donaldson may provide ways to detect G2-metrics on a given 7-manifold that are not deformation equivalent.

Compact, singular G2-holonomy manifolds and M/heterotic/F-theory duality

“…, where 5) where B 2 is the same base as in the G 2 building blocks (1.3). The axio-dilaton corresponds to the compex structure of the elliptic curve E. Furthermore, putting all this together, implies that X F is also elliptically fibered by E as follows…”

“…Recently progress in the construction of compact G 2 -manifolds has been made within mathematics [5,6], based on earlier results by Kovalev [7]. These so-called twisted connected sum (TCS) G 2 -manifolds have a K3-fiberation over a three-manifold.…”

$\mathrm{G}_2$–instantons over twisted connected sums: An example

“…If E ± | Σ ± is spherical, then c 1 (E ± ) must be non-zero; hence, Theorem 1.3 cannot be applied to any of the G 2 -manifolds mass-produced in [CHNP12] via perpendicular matching. This means that the list of currently known G 2 -manifolds to which Theorem 1.3 could potentially be applied is relatively short.…”

“…Let us briefly recall the twisted connected sum construction (mostly) following Corti-Haskins-Nordström-Pacini [CHNP12]. Definition 1.1.…”

Lightest Visible-Sector Supersymmetric Particle is Likely to be Unstable

“…We define a hidden sector as containing states in the low-energy effective theory that are uncharged at tree-level under the SM gauge symmetries, but can be charged under their own symmetry group G H . Compactified string/M -theory solutions will generically have hidden sectors, containing, at a minimum, the gauge fields and gauginos associated the various group factors contained in G H , when the UV solution is supersymmetric [2]- [10]. Such sectors have already been used for model-building purposes, in particular for the breaking of supersymmetry (SUSY), which takes place in a hidden sector and is then mediated to the visible sector.…”