Heterotic non-Kähler geometries via polystable bundles on Calabi–Yau threefolds

Abstract

In arXiv:1008.1018 it is shown that a given stable vector bundle V on a Calabi-Yau threefold X which satisfies c 2 (X) = c 2 (V ) can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explicit examples of polystable bundles on elliptically fibered Calabi-Yau threefolds where it applies. The polystable bundle is given by a spectral cover bundle, for the visible sector, and a suitably chosen bundle, for the hidden sector. This provides a new class of heterotic flux compactifications via non-Kähler deformation of Calabi-Yau geometries with polystable bundles. As an application, we obtain examples of non-Kähler deformations of some three generation GUT models. 1 andreas@math.hu-berlin.de 2 mariogf@qgm.au.dk, Supported by QGM (Centre for Quantum Geometry of Moduli Spaces) funded by the Danish National Research Foundation.

New Examples of Stable Bundles on Calabi–Yau Three-Folds

“…Condition (10) is the integrability condition for the existence of a solution of the heterotic anomaly equation. For the case [W ] = 0 it has been shown that X and E can be deformed to a solution of the anomaly equation even already on the level of differential forms [16], [17] (generalizing results of [18], [19]). Thus it is of interest to see if a given stable vector bundle satisfies (10) and so provides a solution to the basic consistency constraint imposed by heterotic string theory.…”

Supersymmetric String Vacua with Torsion and Geometric Flows

“…Strominger [36], Dasgupta, Rajesh, and Sethi [7], Becker, Becker, Fu, Tseng, and Yau [3], Carlevaro and Israel [5], Andreas and Garcia-Fernandez [2], and others).…”

“…Second, the expression Tr(Rm ∧ Rm), which appears in the equation (2) and is fundamental to the Green-Schwarz anomaly cancellation in string theory, does not seem to have been studied before as a curvature condition in complex differential geometry. What sets it apart from much studied conditions such as constant scalar or constant Ricci curvature is that it is quadratic in the curvature tensor.…”

“…Of course a crucial and difficult step will be to prove that this condition is preserved along the flow. Hereρ is defined such that i∂∂ f ∧ ρ =ρ jk f¯k jω 2 2! for any function f .…”

Higher dimensional generalizations of twistor spaces

“…In the compactification of heterotic string theory to four dimensional Minkowski spacetime [12,61,52], the internal six-manifolds can become non-Kähler in the presence of fluxes. Various models of constructing heterotic manifolds and their vector-bundles have been put forward, see for example [52,9,17,18,10,5,36,6,13,4]. They play an important role in searching for realistic string theory vacua with four dimensional Minkowski spacetime.…”