Smooth Moduli Spaces of Associative Submanifolds

Abstract

Let M 7 be a smooth manifold equipped with a G 2 -structure φ, and Y 3 be a closed compact φ-associative submanifold. In [17], R. McLean proved that the moduli space M Y,φ of the φ-associative deformations of Y has vanishing virtual dimension. In this paper, we perturb φ into a G 2 -structure ψ in order to ensure the smoothness of M Y,ψ near Y . If Y is allowed to have a boundary moving in a fixed coassociative submanifold X, it was proved in [7] that the moduli space M Y,X of the associative deformations of Y with boundary in X has finite virtual dimension. We show here that a generic perturbation of the boundary condition X into X ′ gives the smoothness of M Y,X ′ . In another direction, we use Bochner's technique to prove a vanishing theorem that forces M Y or M Y,X to be smooth near Y . For every case, some explicit families of examples will be given. MSC 2000: 53C38 (35J55, 53C21, 58J32).

Deformations of homogeneous associative submanifolds in nearly parallel $G_2$-manifolds

“…Proof. The ellipticity of D is shown in [7]. For any ψ, ψ ′ ∈ C ∞ (M, ν), we compute by Definition 2.7 and Lemma A.1…”

“…For any Riemannian manifold (Y, g), consider its Riemannian cone (C(Y ), g) = (R >0 × Y, dr 2 + r 2 g). A Riemannian 7-manifold (Y, g) is called a nearly parallel G 2 -manifold if the holonomy group of g is contained in Spin (7). The existence of a nearly parallel G 2 -structure is equivalent to that of a spin structure with a real Killing spinor ( [2]), which is also used in supergravity and superstring theory in physics.…”

“…where vol g0 is a volume form of g 0 , i(·) is the interior product, and v i ∈ T (R 7 ). Similarly, Spin (7) fixes the standard metric h 0 = 7 i=0 (dx i ) 2 and the orientation on R 8 . We have the following identities:…”

Second-Order deformations of associative submanifolds in nearly parallel G2-manifolds

“…Lemma B.5. C Irreducible decomposition of spin (7) In this section, we give an irreducible decomposition of the Lie algebra spin (7) of Spin (7) under an SU(2)-action. First, we study the su(4) ⊂ spin (7) case.…”

“…where vol g0 is a volume form of g 0 and v i ∈ T (R 7 ). Similarly, Spin (7) fixes the standard metric h 0 = 7 i=0 (dx i ) 2 and the orientation on R 8 . We have the following identities:…”