Morse-Bott inequalities, Topology Change and Cobordisms to Nothing

Abstract

The Cobordism Conjecture predicts spacetime-ending configurations, such as Bubbles of Nothing (BoN), being commonplace. These correspond to vacuum decays in which the compactification manifold ${\mathcal{C}}_n$ shrinks to a point, with the instability expanding at the speed of light and leaving nothing (not even spacetime) behind. Most constructions of BoN or cobordisms to nothing found in the literature feature simple instances of ${\mathcal{C}}_n$ or singular cobordisms, which cannot be approached from the effective field theory. Assuming the solution mediating such decay to nothing is homeomorphic to a smooth description, we are able to go a step further, and obtain topological bounds on its homology for generic ${\mathcal{C}}_n$. Through the use of Morse-Bott theory we then translate this into information on the number and types of topology changes the compact manifold experiences as we move towards the tip of the bordism, as well as the location of possible cobordism defects. We illustrate our results with different detailed examples coming from String Theory. Furthermore, with this approach, we are able to study more complicated arrangements such as BoN collisions or intersection of End of the World branes.

43 pages + appendices, 12 figures, 24 footnotes. v2 updated to published version

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• Hitchhiker's Guide to the Swampland: The Cosmologist's Handbook to the string-theoretical Swampland Programme