Abstract
Resolutions of certain toroidal orbifolds, like T6/ℤ2× ℤ2, are far from unique, due to triangulation dependence of their resolved local singularities. This leads to an explosion of the number of topologically distinct smooth geometries associated to a single orbifold. By introducing a parameterisation to keep track of the triangulations used at all resolved singularities simultaneously, (self-)intersection numbers and integrated Chern classes can be determined for any triangulation configuration. Using this method the consistency conditions of line bundle models and the resulting chiral spectra can be worked out for any choice of triangulation. Moreover, by superimposing the Bianchi identities for all triangulation options much simpler though stronger conditions are uncovered. When these are satisfied, flop-transitions between all different triangulations are admissible. Various methods are exemplified by a number of concrete models on resolutions of the T6/ℤ2× ℤ2 orbifold.
Highlights
String theory provides a perturbatively consistent approach to quantum gravity
An important advantage of string theory is that its consistency requirements mandate the existence of the gauge and matter structures that form the bedrock of the Standard Model of particle physics
Its internal consistency predicts the existence of a specific number of extra quantum fields propagating on a two dimensional string worldsheet, which in some guise can be interpreted as extra spacetime dimensions beyond those observed in the physical world
Summary
String theory provides a perturbatively consistent approach to quantum gravity. An important advantage of string theory is that its consistency requirements mandate the existence of the gauge and matter structures that form the bedrock of the Standard Model of particle physics. While the symmetry structure of the Z2 × Z2 orbifold can be used to reduce this number by some factor, it still leaves a huge number (of the order of 1033) genuinely distinct choices This is not a minor complication, as many physical properties of the resulting effective field theories are sensitively dependent on the triangulation chosen. The only way to overcome this complication was by side stepping it: one makes some choice for the triangulation of all these resolved singularities and analyses the resulting physics in that particular case This led to some insights in the structure of the theory in a somewhat larger part of the moduli space, but it seemed hopeless to extract any meaningful generic information about the properties of resolved T 6/Z2 × Z2 orbifolds. In the end 3 Ri and 48 Er provide via the Poincaré duality a basis of the real cohomology group, i.e. of the (1, 1)-forms, on the resolved manifold
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