Holomorphic bundles for higher dimensional gauge theory

Abstract

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.

Instanton bundles on the flag variety F(0,1,2)

“…In order to check the (semi)-stability of rank 2 real instanton bundles, we are going to use the following version of the Hoppe's criterium (see [19]):…”

“…Finally, applying the horseshoe Lemma to (19), G has a linear O P 7 -resolution (1, 1)). On the other hand, from (1, 1)).…”

“…Tensoring again (19) by O A (−1, −1) we get a surjection E |A to O A ⊕ O {x,y} . Therefore, the only possibility is E |A ∼ = O 2 A .…”

Deformations of Nearly Kähler Instantons

“…Although instantons were first introduced in dimension four, the study of instantons on manifolds of dimension greater than four has a long history [1,6,7,13,14,16,17,[19][20][21]30,33,34,[36][37][38]40,41,43,[46][47][48][49][50][51][52][53][54][55][56][57][58][59] (not to mention the long and fruitful study of Hermitian-Yang-Mills connections). In favourable circumstances instantons solve the Yang-Mills equation; for this reason and others the study of instantons informs string theory, supergravity, and theoretical physics.…”

G2–instantons over twisted connected sums

“…This text addresses the existence problem of G 2 -instantons over twisted connected sums, as formulated by Walpuski and myself in [SEW15], and the production of the first examples to date of solutions obtained by a nontrivially transversal gluing process [MNSE17]. It is aimed at graduate students and researchers in nearby areas who might be interested in a condensed exposition of the main results spread over my articles [SE15,SEW15,JMPSE17,MNSE17] with Walpuski, Menet at al. and Menet-Nordström.…”