Shape versus volume: Making large flat extra dimensions invisible

We show that the shape moduli associated with large extra dimensions can have a dramatic effect on the corresponding Kaluza-Klein spectra. Specifically, shape moduli can change the mass gap, induce level crossings, and even interpolate between theories with different numbers of compactified dimensions. We also show that in certain cases it is possible to maintain the ratio between the higher-dimensional and four-dimensional Planck scales while simultaneously increasing the Kaluza-Klein graviton mass gap by an arbitrarily large factor. These observations can therefore be used to alleviate many of the experimental bounds on theories with large extra spacetime dimensions.