Compactifications of the heterotic string are a viable route to phenomenologically realistic vacua and interesting new mathematics. While supergravity aspects of heterotic compactifications are largely well-understood their worldsheet description remains largely unexplored. We review recent work in developing linear sigma model techniques aimed at elucidating the underlying worldsheet description.

Type IIB toroidal orientifolds are among the earliest examples of flux vacua. By applying T-duality, we construct the first examples of massive IIA flux vacua with Minkowski space-times, along with new examples of type IIA flux vacua. The backgrounds are surprisingly simple with no four-form flux at all. They serve as illustrations of the ingredients needed to build type IIA and massive IIA solutions with scale separation. To check that these backgrounds are actually solutions, we formulate the complete set of type II supergravity equations of motion in a very useful form that treats the R-R fields democratically.

We construct the first known example of a near horizon supergravity solution for a pair of NS5-branes, intersecting on R-1.3 and localised in all directions except a single transverse circle. We do this by establishing an explicit map between the conifold metric and the near horizon geometry of two intersecting NS5-branes. clarifying and correcting a number of open issues in the literature en route. Our technique is general in nature and may be applied to a whole class of 1/4-BPS five-brane webs and their geometric duals. These 1/4-BPS solutions may have an interesting holographic interpretation in terms of little string theory.

The effective field theory of heterotic vacua that realise ℝ^3,1 preserving N = 1 supersymmetry is studied. The vacua in question admit large radius limits taking the form ℝ^3,1 × X, with X a smooth threefold with vanishing first Chern class and a stable holomorphic gauge bundle E. In a previous paper we calculated the kinetic terms for moduli, deducing the moduli metric and Kähler potential. In this paper, we compute the remaining couplings in the effective field theory, correct to first order in α` . In particular, we compute the contribution of the matter sector to the Kähler potential and derive the Yukawa couplings and other quadratic fermionic couplings. From this we write down a Kähler potential K and superpotential W.

Using mirror symmetry, we resolve an old puzzle in the linear sigma model description of the spacetime Higgs mechanism in a heterotic string compactification with (2,2) worldsheet supersymmetry. The resolution has a nice spacetime interpretation via the normalization of physical fields and suggests that with a little care deformations of the linear sigma model can describe heterotic Higgs branches.

We consider a family of perturbative heterotic string backgrounds. These are complex threefolds X with c₁ = 0, each with a gauge field solving the Hermitian Yang-Mill's equations and compatible B and H fields that satisfy the anomaly cancellation conditions. Our perspective is to consider a geometry in which these backgrounds are fibred over a parameter space. If the manifold X has coordinates x, and parameters are denoted by y, then it is natural to consider coordinate transformations x → x (with tilde) (x, y) and y → y (with tilde)(y). Similarly, gauge transformations of the gauge field and B field also depend on both x and y. In the process of defining deformations of the background fields that are suitably covariant under these transformations, it turns out to be natural to extend the gauge field A to a gauge field A on the extended (x, y)-space. Similarly, the B, H, and other fields are also extended. The total space of the fibration of the heterotic structures is the Universal Geometry of the title. The extension of gauge fields has been studied in relation to Donaldson theory and monopole moduli spaces. String vacua furnish a richer application of these ideas. One advantage of this point of view is that previously disparate results are unified into a simple tensor formulation. In a previous paper, by three of the present authors, the metric on the moduli space of heterotic theories was derived, correct through O(α`), and it was shown how this was related to a simple Kähler potential. With the present formalism, we are able to rederive the results of this previously long and involved calculation, in less than a page.

We study a system of D-branes localized near an intersection of Neveu-Schwarz fivebranes, that is known to exhibit a landscape of supersymmetric and (metastable) supersymmetry breaking vacua. We show that early universe cosmology drives it to a particular long-lived supersymmetry breaking ground state.

We prove that N = 2 theories that arise by taking n free hypermultiplets and gauging a subgroup of Sp(n), the non-R global symmetry of the free theory, have a remaining global symmetry, which is a direct sum of unitary, symplectic, and special orthogonal factors. This implies that theories that have SU(N) but not U(N) global symmetries, such as Gaiotto's T_N theories, are not likely to arise as IR fixed points of RG flows from weakly coupled N = 2 gauge theories.

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.