Kaluza-Klein Reduction of Pure Gravity and its Implications for K3 Surface Compactifications
Kaluza demonstrated that a geometrical unification of Einsteinian gravity and Maxwell’s equations could occur in five (4+1) dimensions if the dependence on the fourth spatial coordinate is ignorable. Klein noted that the last assumption would be natural for a compact extra dimension (i.e., a circle, rather than a line) of very small size. Since this initial proposal dimensional reduction has been incorporated into string theory, where the compactification manifold of choice is a Calabi-Yau manifold. In this dissertation, we investigate reduction via the Kaluza-Klein mechanism by considering the general compactification from D to d (D > d) dimensions of pure gravity, wherein the internal metric moduli are promoted to moduli fields. An essential point is that D-dimensional equations of motion must be satisfied, even in the effective degrees of freedom (the moduli fields). If the d-dimensional equations of motion imply the D-dimensional equations the effective theory is consistent. As a first pass the truncation to massless modes is made, but with a special gauge choice, transverse/traceless gauge, imposed on the internal metric. Equivalently, compensating fields, which are intended to assure consistency, are included in the metric ansatz. It is concluded that the consistency of the compactification demands that all massless and massive Kaluza-Klein modes be included in the lower dimensional theory. Motivated by the importance and ubiquitousness of K3 compactifications, a review of K3 geometry is presented. The E8 ⊕ E8 ⊕ U31,1 and Sp(32)/Z2 ⊕ U31,1 decompositions of the (co)homology lattice ofthe K3 are exhibited explicitly in terms of a natural orbifold basis, which augments the abstract derivations available in the literature. A novel feature is introduced – anapproximate, but explicit, metric on K3, which exactly generates a K3 metric in thelimit of small fiber and large base.
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