Instantons and Special Geometry

Abstract

We survey and discuss constructions of instantons on noncompact complete manifolds of special holonomy from the viewpoint of evolution equations and give several explicit examples.To Simon Salamon on the occasion of his 60th birthday * ϕ ∧ Ω = 0.(1.5) Remark 1.2. If a G 2 -manifold M is a product R 3 × X (or T 3 × X) where X is a hyperKähler 4-manifold with hyperKähler triple σ then, if (x 1 , x 2 , x 3 ) are local coordinates on R 3 (or T 3 ), we have the product G 2 -structure

Instantons on flat space: Explicit constructions

“…subject to the initial constraint F A pr 0 q ^σpr 0 q 2 " 0, (2.12) see, e.g., [15]. Here we are using the decomposition (2.9) and ˚r denotes the Hodge star on the 6-sphere of radius r.…”

“…For our connection ansatz (1.1) we could again substitute β " Ω in (2.15) to obtain the Spinp7q-instanton condition. However, we may also rephrase the Spinp7q-instanton condition for the family Aprq of connections over S 7 in (1.1) as 16) for details see [15]. Here we used the decomposition (2.14) and ˚r is the Hodge star on the 7-sphere of radius r.…”

“…3 pdx 14 `dx 27 ´2dx 58 q, β 9 " 1 ? 6 p´dx 14 ´dx 27 `3dx 36 ´dx 58 q, β 10 " dx 15 `dx 48 , β 11 " 1 ? 3 p´dx 15 `2dx 26 `dx 48 q, β 12 " 1 ?…”

Deformation Theory of Asymptotically Conical Spin(7)-Instantons

“…Donaldson-Thomas [17] and Carrion [13] have discussed Spin(7)-instantons more generally, and around the same time, in 1998, Lewis also discussed Spin (7)-instantons in his PhD thesis [38]. In recent years, Spin(7)-instantons have been studied by Sa Earp [46], Tanaka [50], Walpuski [51], Lotay-Madsen [40] and many others.…”