Abstract
We analyse the geometry of generic Minkowski mathcal{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.
Highlights
Supersymmetric string backgrounds play a central role in our understanding of string phenomenology and the AdS/CFT correspondence
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
In type II theories, truly N = 1 backgrounds necessarily have non-zero flux, and are generically not of Calabi-Yau type. This raises several natural questions: how does one extract the properties of the low-energy theory from the geometry of this much larger class? Do they have an “nice” geometrical description in analogy to that of special holonomy spaces? What tools do we have to find the number of massless moduli or construct examples? Does incorporating them in a larger class shed any light on the nature of G2 manifolds?
Summary
Supersymmetric string backgrounds play a central role in our understanding of string phenomenology and the AdS/CFT correspondence. The second supersymmetry condition is the vanishing of a moment map, defined for the action of generalised diffeomorphisms (that is, conventional diffeomorphisms plus form-field gauge transformations) and in the G2 case imposes the condition d φ = 0 This reformulation puts the analysis of generic supersymmetric backgrounds, and G2 structures in particular, in the same setting as many classic problems in differential geometry: we have a complex condition (the involutivity of the subbundle) together with an infinite-dimensional moment map. We will discuss how the extra differential conditions that promote these structures to integrable SL(3, C) and generalised CalabiYau structures come from a moment map for the action of diffeomorphisms and, in the latter case, gauge symmetries These two examples will provide the model for how we analyse generic four-dimensional N = 1 flux backgrounds
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