Heterotic vacua of string theory are realised, at large radius, by a compact threefold with vanishing first Chern class together with a choice of stable holomorphic vector bundle. These form a wide class of potentially realistic four-dimensional vacua of string theory. Despite all their phenomenological promise, there is little understanding of the metric on the moduli space of these. What is sought is the analogue of special geometry for these vacua. The metric on the moduli space is important in phenomenology as it normalises D-terms and Yukawa couplings. It is also of interest in mathematics, since it generalises the metric, first found by Kobayashi, on the space of gauge field connections, to a more general context. Here we construct this metric, correct to first order in α , in two ways: first by postulating a metric that is invariant under background gauge transformations of the gauge field, and also by dimensionally reducing heterotic supergravity. These methods agree and the resulting metric is Kähler, as is required by supersymmetry. Checking the metric is Kähler is intricate and the anomaly cancellation equation for the H field plays an essential role. The Kähler potential nevertheless takes a remarkably simple form: it is the Kähler potential of special geometry with the Kähler form replaced by the α -corrected hermitian form.arXiv:1605.05256v4 [hep-th]

We study half-twisted linear sigma models relevant to (0,2) compactifications of the heterotic string. Focusing on theories with a (2,2) locus, we examine the linear model parameter space and the dependence of genus zero half-twisted correlators on these parameters. We show that in a class of theories the correlators and parameters separate into A and B types, present techniques to compute the dependence, and apply these to some examples. These results should bear on the mathematics of (0,2) mirror symmetry and the physics of the moduli space and Yukawa couplings in heterotic compactifications.

Using F-theory/heterotic duality, we describe a framework for analyzing nongeometric T 2 -fibered heterotic compactifications to six-and four-dimensions. Our results suggest that among T 2 -fibered heterotic string vacua, the non-geometric compactifications are just as typical as the geometric ones. We also construct fourdimensional solutions which have novel type IIB and M-theory dual descriptions.These duals are non-geometric with three-and four-form fluxes not of (2, 1) or (2, 2) Hodge type, respectively, and yet preserve at least N = 1 supersymmetry.

We consider compactifications of M-theory and type IIA string theory to four dimensions. For Minkowski space-time, a supergravity no-go theorem forbids flux supported in the internal space. We show how to evade this no-go theorem by exhibiting new sources of brane charge: in string theory, the basic physical phenomenon is the generation of new brane charges from D-branes in transverse fluxes. In M-theory, there is a new source of M5-brane charge from novel higher derivative couplings that involve fluxes as well as curvatures. We present some explicit orientifold examples with both N = 1 and N = 2 space-time supersymmetry. Finally, we explain the status of massive type IIA flux compactifications.