On the geometry of almost complex 6-manifolds

Abstract

Abstract. This article discusses some basic geometry of almost complex 6-manifolds. A 2-parameter family of intrinsic first-order functionals on almost complex structures on 6-manifolds is introduced and their Euler-Lagrange equations are computed.A natural generalization of holomorphic bundles over complex manifolds to the almost complex case is introduced. The general almost complex manifold will not admit any nontrivial bundles of this type, but there is a class of nonintegrable almost complex manifolds for which there are such nontrivial bundles. For example, the G 2 -invariant almost complex structure on the 6-sphere admits such nontrivial bundles. This class of almost complex manifolds in dimension 6 will be referred to as quasi-integrable and a corresponding condition for unitary structures is considered.Some of the properties of quasi-integrable structures (both almost complex and unitary) are developed and some examples are given.However, it turns out that quasi-integrability is not an involutive condition, so the full generality of these structures in Cartan's sense is not well-understood. The failure of this involutivity is discussed and some constructions are made to show, at least partially, how general these structures can be.

Langlands Duality and G2 Spectral Curves

“…For three-forms in six dimensions there is a similar story -for a 6-dimensional complex vector space W , there is an open orbit in Λ 3 W * under the action of GL(6) whose stabilizer is SL(3) × SL(3) × Z 2 (see [19], [20], [24], [8], [6]). If x 1 , x 2 , x 3 , y 1 , y 2 , y 3 is a basis with dual basis ξ 1 , ξ 2 , ξ 3 , η 1 , η 2 , η 3 , a normal form is…”

“…Let z be a local coordinate on Σ, so that f = g(z)dz 2 and q = r(z)dz 6 . The tautological section x of p * K on K is then just wdz, and the spectral curve has equation…”

DWSB in heterotic flux compactifications

“…As written in (5.9), the pseudo-HYM condition can be split into two parts, which can be seen as F-flatness and D-flatness conditions, along the lines of [41]. The first, F 0,2 = 0, demands the bundle to be pseudo-holomorphic [33] and can be seen as an Fflatness condition, while the second one can be seen as a D-flatness condition. Now, suppose that we can solve the F-flatness condition F 0,2 = 0.…”

“…Recall that the almost-complex structure of M is not integrable. Following [33], we can say that (4.25) requires the bundle to be pseudo-HYM and in particular that the condition F 0,2 = 0 requires the bundle to be pseudo-holomorphic. Clearly, one cannot use the standard theory of bundles on Kähler spaces.…”

An intrinsic volume functional on almost complex 6-manifolds and nearly Kähler geometry

“…There is a significant overlap with our research, though the presentation and terminology are different. The property (1-3) (which is equivalent to the existence of Hermitian connection with totally antisymmetric curvature) is called "Nijenhuis tensor of real type" in [Bryant 2006]. The main focus of that preprint is the so-called "quasiintegrable almost complex manifold": manifolds with Nijenhuis tensor of real type, which is at every point of M either nondegenerate (of constant signature) or zero.…”