Shadows of the Planck Scale: The Changing Face of Compactification Geometry

By studying the effects of the shape moduli associated with toroidal compactifications, we demonstrate that Planck-sized extra dimensions can cast significant ``shadows'' over low-energy physics. These shadows can greatly distort our perceptions of the compactification geometry associated with large extra dimensions, and place a fundamental limit on our ability to probe the geometry of compactification simply by measuring Kaluza-Klein states. We also discuss the interpretation of compactification radii and hierarchies in the context of geometries with non-trivial shape moduli. One of the main results of this paper is that compactification geometry is effectively renormalized as a function of energy scale, with ``renormalization group equations'' describing the ``flow'' of geometric parameters such as compactification radii and shape angles as functions of energy.