MODULI SPACES OF G₂MANIFOLDS

Abstract

This paper is a review of current developments in the study of moduli spaces of G 2 manifolds. G 2 manifolds are 7-dimensional manifolds with the exceptional holonomy group G 2 . Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi-Yau manifolds in 7 dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M -theory compactifications. Despite the physical motivation, many of the results are of purely mathematical interest. Here we cover the basics of G 2 manifolds, local deformation theory of G 2 structures and the local geometry of the moduli spaces of G 2 structures.

Infinitesimal moduli of G2 holonomy manifolds with instanton bundles

“…For a generic vector bundle E, we must specify a metric G xy ∈ Λ 0 (Sym(E * ⊗ E * )), in order to define the inner product 11) JHEP11 (2016)016 for {α x , β y } ∈ Λ * (Y, E). As in the case of endomorphism bundles, we may choose a trivial metric δ xy , but other choices may be more natural.…”

“…Deformations of G 2 holonomy manifolds, and their associated moduli space, have been thoroughly studied, both by mathematicians and theoretical physicists [4][5][6][7][8][9][10] (see [11] for a recent review). It has been shown, by Joyce [4,5], that, for compact spaces, the third Betti number sets the dimension of the infinitesimal moduli space.…”

Compact G2 holonomy spaces from SU(3) structures

“…The Poincaré dual Π of this homology class will therefore be a torsion element of H 3 (X, Z) and we take it to be of order N , that is [N Π] = 0. 6 Moreover to obtain the tadpole we need to consider what happens to the term f ·F 2 . However given the way we constructed it, f · Υ is always an exact form, and therefore it disappears when taking (2.24) in cohomology.…”

Marginal deformations of heterotic G2 sigma models

“…For more details, see e.g. [41,42]. Preservation of spacetime N = 1 supersymmetry for the ansatz (1.1) implies the existence of a globally defined real spinor η on Y .…”

“…With e i substituted by dx i in this expression, where x i are coordinates on R 7 , this defines a G 2 -structure ϕ 0 on R 7 , regarded as the imaginary octonions (see e.g. [42] for further details). Thus, simply stated, the positivity condition locally identifies (Y, ϕ) with (R 7 , ϕ 0 ).…”