Cobordism, Singularities and the Ricci Flow Conjecture

TL;DR

This paper explores a novel approach to connect Ricci flow singularities with cobordism theory, aiming to reconcile two conjectures in quantum gravity by using surgery techniques and defect insertions.

Contribution

It introduces a method to trivialize cobordism classes during Ricci flow via surgery and defects, linking geometric flow singularities with cobordism classifications.

Findings

Detailed analysis of $ ext{Ω}^{SO}_4$ cobordism class.

Connection established between blowing up points and connected sums with $ ext{CP}^n$.

Application of ADE classification to Ricci flow singularities.

Abstract

In the following work, an attempt to conciliate the Ricci flow conjecture and the Cobordism conjecture, stated as refinements of the Swampland distance conjecture and of the No global symmetries conjecture respectively, is presented. This is done by starting from a suitable manifold with trivial cobordism class, applying surgery techniques to Ricci flow singularities and trivialising the cobordism class of one of the resulting connected components via the introduction of appropriate defects. The specific example of $\Omega^{SO}_4$ is studied in detail. A connection between the process of blowing up a point of a manifold and that of taking the connected sum of such with $\mathbb{CP}^n$ is explored. Hence, the problem of studying the Ricci flow of a $K3$ whose cobordism class is trivialised by the addition of $16$ copies of $\mathbb{CP}^2$ is tackled by applying both the techniques developed in the previous sections and the classification of singularities in terms of ADE groups.