Morse-Bott inequalities, Topology Change and Cobordisms to Nothing

TL;DR

This paper uses Morse-Bott theory to analyze topology changes in spacetime decay processes like Bubbles of Nothing, providing bounds and insights into the structure of such cobordisms in string theory contexts.

Contribution

It introduces a topological framework based on Morse-Bott theory to study vacuum decay cobordisms and their topology changes, extending previous simple models.

Findings

Derived topological bounds on homology of decay mediating solutions.

Identified possible topology changes and defects in the compactification manifold.

Analyzed complex phenomena like bubble collisions and brane intersections.

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.