Abstract
We present a general procedure to construct 6-dimensional manifolds with SU(3)-structure from SU(2)-structure 5-manifolds. We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-manifolds. Both half-flat and nearly Kahler 6-manifolds are prominent in flux compactifications of string theory. Subsequently, we investigate instanton equations for connections on vector bundles over these half-flat manifolds. A suitable ansatz for gauge fields on these 6-manifolds reduces the instanton equation to a set of matrix equations. We finally present some of its solutions and discuss the instanton configurations obtained this way.
Highlights
String vacua with p-form fields along the extra dimensions (“flux compactifications”) have been intensively studied in recent years
We thereby obtain half-flat cylinders and sine-cones over 5-manifolds with Sasaki-Einstein SU(2)-structure. They are nearly Kahler in the special case of sine-cones over Sasaki-Einstein 5-manifolds. Both half-flat and nearly Kahler 6-manifolds are prominent in flux compactifications of string theory
On Calabi-Yau manifolds the introduction of fluxes partially resolves the vacuum degeneracy problem by giving masses to problematic moduli, but they lead to non-integrable SU(3)-structures on the internal compact 6-manifolds
Summary
We begin by introducing several geometric structures that will become important in the constructions of this paper. As in [54], an almost contact metric manifold is an odddimensional Riemannian manifold (M 2m+1, g) such that there exists a reduction of the structure group SO(2m + 1) of the bundle of orthonormal frames on T M to U(m). Let M 5 be 5-manifold with an SU(2)-structure, i.e. the frame bundle of M 5 can be reduced to an SU(2) principal subbundle It has been proven in [57] that an SU(2)-structure is determined by a quadruplet (η, ω1, ω2, ω3) of differential forms, wherein η ∈ Ω1(M 5) and ωα ∈ Ω2(M 5) for α = 1, 2, 3. As shown in [57], SU(2)-structures in 5 dimensions always induce a nowhere-vanishing spinor on M 5 This will be generalized Killing if and only if the SU(2)-structure is hypo, and Killing if and only if the SU(2)-structure is Sasaki-Einstein.
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