Abstract
We describe the geometry of generic heterotic backgrounds preserving minimal supersymmetry in four dimensions using the language of generalised geometry. They are characterised by an SU(3) × Spin(6 + n) structure within O(6, 6 + n) × ℝ+ generalised geometry. Supersymmetry of the background is encoded in the existence of an involutive subbundle of the generalised tangent bundle and the vanishing of a moment map for the action of diffeomorphisms and gauge symmetries. We give both the superpotential and the Kähler potential for a generic background, showing that the latter defines a natural Hitchin functional for heterotic geometries. Intriguingly, this formulation suggests new connections to geometric invariant theory and an extended notion of stability. Finally we show that the analysis of infinitesimal deformations of these geometric structures naturally reproduces the known cohomologies that count the massless moduli of supersymmetric heterotic backgrounds.
Highlights
Great amount of mathematics has been developed to use such constructions to find fourdimensional models with chiral fermions and Standard Model gauge groups [31,32,33,34,35,36,37,38]
Supersymmetry of the background further constrains these geometric structures, with preserved supersymmetry being equivalent to the existence of an integrable GNd ⊂ Hd structure [56, 60], where Hd is the double cover of Hd, and GNd depends on the dimension of the internal space d and the number of preserved supersymmetries N
We will see that integrability of the R+ × U(3) × Spin(6 + n) structure is given by an involutivity condition on the subbundle L−1. (Such an involutivity condition has previously appeared in a generalised geometric analysis of non-linear sigma models whose target spaces are “strong Kähler with torsion” [65].) Integrability of the full SU(3) × Spin(6 + n) structure, equivalent to supersymmetry for the background, requires an additional condition which takes the form of the vanishing of a moment map for the action of diffeomorphisms and gauge transformations on the space of SU(3)×Spin(6+n) structures
Summary
We begin with a review of the Hull-Strominger system [2, 67]. This is a set of equations describing the geometry of general N = 1 backgrounds of the heterotic string on a tendimensional manifold M that is a product of a six-dimensional manifold X with fourdimensional Minkowski space M = R3,1 × X, with trivial warp factor in the string frame. The condition D 2 = 0 is equivalent to the integrability of the conventional complex structure, the holomorphicity of the gauge bundle and the Bianchi identities for F , R and H.3. An (off-shell) configuration of the bosonic fields defines a generalised metric that reduces the structure group of E to SO(6) × SO(6) SU(4) × SU(4). As discussed in [48, appendix C], the existence of a nowhere-vanishing spinor that can parametrise N = 1 supersymmetry transformations in four dimensions requires a reduction of the structure group from that defined by the generalised metric, namely SU(4) × SU(4), to SU(3) × SU(4) ⊂ O(6, 6) × R+. J defines a generic reduction of the structure group of the generalised tangent bundle E to R+×U(3)×SU(4).. In analogy with a conventional complex structure, we can use J to decompose the generalised tangent space into eigenspaces. This expression guarantees that ψ is stabilised by the correct SU(3) × Spin(6 + n) group
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