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.

A concrete model for a 7-dimensional gauge theory under special holonomy is proposed, within the paradigm of Donaldson and Thomas [D-T], over the asymptotically cylindrical G2−manifolds provided by Kovalev's solution to a noncompact version of the Calabi conjecture [Kov1, Kov2]. One obtains a solution to the G2−instanton equation from the associated Hermitian Yang-Mills problem, to which the methods of Simpson et al. are applied, subject to a crucial asymptotic stability assumption over the 'boundary at infinity'. 4 Conclusion 37 4.

We present an explicit numerical method to obtain the Cartan-Khaneja-Glaser decomposition of a general element G ∈ SU (2 N ) in terms of its 'Cartan' and 'non-Cartan' components. This effectively factors G in terms of group elements that belong in SU (2 n ) with n < N , a procedure that an be iterated down to n = 2. We show that every step reduces to solving the zeros of a matrix polynomial, obtained by truncation of the Baker-Campbell-Hausdorff formula, numerically. All computational tasks involved are straightforward and the overall truncation errors are well under control.

ABSTRACT. Motivated by gauge theory under special holonomy, we present techniques to produce holomorphic bundles over certain noncompact 3−folds, called building blocks, satisfying a stability condition 'at infinity'. Such bundles are known to parametrise solutions of the Yang-Mills equation over the G2−manifolds obtained from asymptotically cylindrical Calabi-Yau 3−folds studied by Kovalev, Haskins et al. and Corti et al..The most important tool is a generalisation of Hoppe's stability criterion to holomorphic bundles over smooth projective varieties X with Pic X ≃ Z l , a result which may be of independent interest.Finally, we apply monads to produce a prototypical model of the curvature blow-up phenomenon along a sequence of asymptotically stable bundles degenerating into a torsion-free sheaf.