K-Contact and Sasakian Structures

Monopoles on Sasakian Three-folds

“…Indeed, if X [2] −→ Σ is the fiber product of X with itself, we have a Z-fold cover X [2] of X [2] , given as pairs of points x , y in the fiber over X plus a homotopy class of paths from x to y along the fiber. (The inverse image of a point in Σ would thus be S 1 × R.) Given a bundle E on X, there is a natural bundle Hom E on X [2] and by lifting on X [2] , given over (x , y) by Hom(E x , E y ). This has natural maps Hom(E x , E y ) ⊗ Hom(E y , E z ) −→ Hom(E x , E z ), and this is one of the essential properties of a bundle gerbe, defined by Murray in the rank one case.…”

“…This has natural maps Hom(E x , E y ) ⊗ Hom(E y , E z ) −→ Hom(E x , E z ), and this is one of the essential properties of a bundle gerbe, defined by Murray in the rank one case. Our remark is that in our case we have a natural section of Hom E , given by our integrating ∇ c over X [2] along the fibers, and this section respects the multiplication, so that s(x, y) × s(y, z) = s(x, z).…”

“…If π : S −→ Σ is the spectral curve, taking the fiber product Y = S × Σ X ⊂ P 1 × X. There is a well defined line bundle L, and so another line bundle Hom L along Y [2] , again equipped with a natural section when one lifts to the Z-cover Y [2] .…”

“…In parallel, work of Nahm [21], and then Hitchin [10], tied this complex interpretation to the Nahm transform, giving a very effective dictionary which allowed the classification of monopoles in 1983 by Donaldson [7]. The work, originally done for the gauge group SU (2), was extended to classical gauge groups by Murray and Hurtubise in a series of papers [19,12,11], and then by Jarvis to arbitrary reductive groups [14].…”

“…Let X be a compact quasiregular Sasakian three-fold, with metric g. These are manifolds with a contact structure and a metric, compatible in the sense that there is a unit (Reeb) vector field ξ orthogonal to the contact planes, which acts on the manifold as a Killing field. (A useful reference is the book of Boyer and Galicki [2].) The orbits of ξ are compact; under this hypothesis the manifold has a circle action by isometries, and the quotient…”