G₂–instantons over twisted connected sums

Abstract

We review a method to construct G2-instantons over compact G2-manifolds arising as the twisted connected sum of a matching pair of Calabi-Yau 3-folds with cylindrical end, based on the series of articles [SE15, SEW15, JMPSE17, MNSE17] by the author and others. The construction is based on gluing G2-instantons obtained from holomorphic bundles over such building blocks, subject to natural compatibility and transversality conditions. Explicit examples are obtained from matching pairs of semi-Fano 3-folds by an algorithmic procedure based on the Hartshorne-Serre correspondence.

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

“…Building on this, Sá Earp and the author [SEW13] developed a method for constructing G 2 -instantons over twisted connected sums provided a pair of such bundles and a liftr : E + | Σ + → E − | Σ − of the hyperkähler rotation r, which pulls back A ∞,− to A ∞,+ (and assuming certain transversality conditions). The following is a very special case of the main result of [SEW13]. Theorem 1.3.…”

“…Finally, it should be pointed out that there is work in progress by Sá Earp et al to find more examples to which the more general main result of [SEW13] can be applied to construct G 2 -instantons.…”

“…In [SEW13] Sá Earp and the author developed a method for constructing G 2 -instantons over twisted connected sums. So far, however, we were unable to find a single example of the input required for this construction.…”

Holomorphic bundles for higher dimensional gauge theory

“…blow-ups of so-called weak Fanos), which shall be addressed in the future. For a more detailed exposition of building blocks, we suggest the Introduction of [SEW15] and references therein.…”

Generalised Chern-Simons Theory and G2-Instantons over Associative Fibrations

“…Indeed Theorem 2 yields nontrivial, albeit nongeneric, examples of G 2 -instanton moduli, whenever a complex vector bundle E → T 4 admits SD connections. The interested reader will find other examples in works of Walpuski, Clarke and the author [2,11,12,13,17]. In the high-energy physics community, solutions to a very similar problem in the context of G 2 -structures with torsion have been found eg.…”