Generalising G2 geometry: involutivity, moment maps and moduli

Abstract

We analyse the geometry of generic Minkowski $$ \mathcal{N} $$ N = 1, D = 4 flux compactifications in string theory, the default backgrounds for string model building. In M-theory they are the natural string theoretic extensions of G2 holonomy manifolds. In type II theories, they extend the notion of Calabi-Yau geometry and include the class of flux backgrounds based on generalised complex structures first considered by Graña et al. (GMPT). Using E7(7) × ℝ+ generalised geometry we show that these compactifications are characterised by an SU(7) ⊂ E7(7) structure defining an involutive subbundle of the generalised tangent space, and with a vanishing moment map, corresponding to the action of the diffeomorphism and gauge symmetries of the theory. The Kähler potential on the space of structures defines a natural extension of Hitchin’s G2 functional. Using this framework we are able to count, for the first time, the massless scalar moduli of GMPT solutions in terms of generalised geometry cohomology groups. It also provides an intriguing new perspective on the existence of G2 manifolds, suggesting possible connections to Geometrical Invariant Theory and stability.

Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory

“…Its moduli space is quaternionic-Kähler and parameterises the hypermultiplet degrees of freedom in the D = 5 theory. The analogous objects were first introduced in E 7(7) × R + geometry in the context of D = 4 backgrounds of string and M-theory in [24], and analogous structures were later found in the O(6, 6 + n) geometry relevant to heterotic strings [25]. This work can be viewed as an extension of these ideas to D = 5 Minkowski backgrounds.…”

“…In this paper we defined and classified a new object in E 6(6) ×R + generalised geometry which we called an exceptional complex structure and used it to analyse generic supersymmetric D = 5 Minkowski backgrounds of M-theory. These are the analogue of SL(3, C) structures in conventional geometry, or SU (3,3) structures in Hitchin's generalised geometry, and they extend the definition of exceptional complex structures in [24,25] to D = 5 backgrounds. We saw that ECSs fell into three families labelled type and class.…”

“…The SU(2) action on the triplet J α means that, as in the hyperkähler case, there is an S 2 CP 1 SU(2)/U(1) of such choices. In fixing one, we find a structure analogous to the ECS that was used to describe four-dimensional N = 1 Mtheory and type II geometries in [24], and the corresponding heterotic backgrounds in [25] backgrounds. As in those cases, we show that integrability is naturally defined in terms of an involutive subbundle of the generalised tangent bundle.…”

“…First, we define a notion of type similar to that of generalised complex structures defined in [2], and also defined in exceptional geometry in [24]. We then go further and show that ECS are also characterised by a second important property that we call class.…”

“…This particular choice of complement is convenient as one can use the Killing spinor equations 24 to show [Σ, Σ] ⊆ Σ. We can therefore decompose the exterior derivative as…”