Orbifolds and Kaluza-Klein-Monopoles in Heterotic E₈ x E₈ String Theory Preserving Eight Supercharges

Abstract

We give a detailed account on heterotic E₈ x E₈ orbifold models with N=(1,0), D=6 supersymmetry using the operator approach. It is shown that there is a weak and a strong form of the level matching condition. The weak form, which is the only form known in previous literature, allows for classification of orbifold models whereas the strong form is needed to fix transformation phases of twisted states under the orbifold twist. It is shown that only the weak form of the level matching conditions translates into a local matching of fractional parts of gravitational and gauge instanton numbers from a ten-dimensional viewpoint. This, in turn, is used to show that all orbifold models considered in this work can be classified by flat E₈ x E₈ bundles on orbifolds with the fixed points taken out, under the only constraint that the fractional parts of gravitational and gauge instanton numbers match. This directly carries over to M-theory on S^1/Z₂.<br /> We construct multiple Kaluza-Klein-monopole solutions in Wilson line backgrounds and verify the result by computing the correct quantum numbers and comparing to the charge spectrum of toroidal compactifications of the heterotic string. We propose that the moduli space of a single SU(2) non-abelian instanton on a Kaluza-Klein-Monopole background is given by the t'Hooft ansatz as in flat space and explicitly show that instantons can become pointlike at orbifold singularities even though their scale parameter remains finite. The case of N non-abelian instantons on N Kaluza-Klein-monopoles is analyzed in the limit where all instantons and Kaluza-Klein-monopoles are located within a region of scale much smaller than the size of the Kaluza-Klein direction. Based on index calculations for manifolds with boundary, we argue that this situation is locally identical to heterotic string theory on Z_N orbifold singularities for general N in the standard embedding. This implies, that heterotic E₈ x E₈ orbifolds with standard embedding allow for small instantons singularities in their moduli spaces.<br /> Applying these results to orbifold models, we give new evidence that higgsing of the models leads to smooth K3 compactifications of heterotic E₈ x E₈ string theory. Based on index calculations as well as details of the Higgs mechanism in N=(0,1), D=6 supersymmetry, we argue that massless modes ascribed to the supergravity multiplet in ten dimensions, such as the geometric moduli of K3, have to appear in twisted sectors as massless modes charged with respect to gauge groups of the orbifold models. Turning to M-theory on S^1/Z₂, we give further evidence that these modes are eleven-dimensional bulk modes which have to live in the interior of the interval. Especially, these modes cannot be localized on the ends of the eleven-dimensional interval as suggested in the previous literature.