Degenerations of K3, orientifolds and exotic branes

Abstract

A recently constructed limit of K3 has a long neck consisting of segments, each of which is a nilfold fibred over a line, that are joined together with Kaluza-Klein monopoles. The neck is capped at either end by a Tian-Yau space, which is non-compact, hyperkähler and asymptotic to a nilfold fibred over a line. We show that the type IIA string on this degeneration of K3 is dual to the type I ′ string, with the Kaluza-Klein monopoles dual to the D8-branes and the Tian-Yau spaces providing a geometric dual to the O8 orientifold planes. At strong coupling, each O8-plane can emit a D8-brane to give an O8 * plane, so that there can be up to 18 D8-branes in the type I ′ string. In the IIA dual, this phenomenon occurs at weak coupling and there can be up to 18 Kaluza-Klein monopoles in the dual geometry. We consider further duals in which the Kaluza-Klein monopoles are dualised to NS5-branes or exotic branes. A 3-torus with H-flux can be realised in string theory as an NS5-brane wrapped on T 3 , with the 3-torus fibred over a line. T-dualising gives a 4-dimensional hyperkähler manifold which is a nilfold fibred over a line, which can be viewed as a Kaluza-Klein monopole wrapped on T 2 . Further T-dualities then give non-geometric spaces fibred over a line and can be regarded as wrapped exotic branes. These are all domain wall configurations, dual to the D8-brane. Type I ′ string theory is the natural home for D8-branes, and we dualise this to find string theory homes for each of these branes. The Kaluza-Klein monopoles arise in the IIA string on the degenerate K3. T-duals of this give exotic branes on non-geometric spaces.

Special holonomy manifolds, domain walls, intersecting branes and T-folds

“…Here we will extend some of the analysis of [9] for the nilfold case to the solutions of [10]; we expect similar results will apply to the special holonomy spaces of [11,12,13,14]. The hyperkähler solution from the nilfold is dual to wrapped D8-branes or smeared NS5-branes, We will show here that each of the special holonomy spaces is dual to a system of intersecting branes wrapped on a torus that preserves precisely the same amount of supersymmetry.…”

“…Again this is governed by a piecewise linear function on the line, with the kinks associated with domain walls that are D8-branes wrapped on T 3 or NS5-branes smeared over a transverse T 3 . It is also T-dual to a T-fold fibred over a line [3,8,9]. In [9], the incorporation of these solutions into complete consistent string backgrounds was discussed.…”

“…It is also T-dual to a T-fold fibred over a line [3,8,9]. In [9], the incorporation of these solutions into complete consistent string backgrounds was discussed.…”

The doubled geometry of nilmanifold reductions

“…In [6], the solutions of [5] were generalised and, when combined with a Minkowski space factor, shown to be dual to intersecting brane solutions. In [4,6] it was also seen that further, dualities take these to T-folds [7,8] and to locally non-geometric spaces with R-flux.…”

“…It is not a solution of string theory, but it appears as the fibre in a 4-dimensional hyperkähler space which is a bundle over a line interval, and so can be incorporated into string theory in this way [1,2,3]. This case and its duals were explored in detail in [4]. This has an interesting generalisation to special holonomy spaces which arise as higher dimensional nilmanifolds fibred over a line [5], giving spaces of holonomy SU(3), SU(4), G 2 and Spin (7).…”