Cobordism, singularities and the Ricci flow conjecture

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 $$ {\varOmega}_4^{SO} $$ Ω 4 SO 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 ℂℙn 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 ℂℙ2 is tackled by applying both the techniques developed in the previous sections and the classification of singularities in terms of ADE groups.

Geometric Flow of Bubbles

“…For what concerns fermionic matter, geometric flow equations might be instead derived after having expressed the metric in the tetrad formalism. The consequences of preserving the on-shell conditions on the matter content of the theory are furthermore left to be investigated, as well as any physical implication, within the context of the on-shell flow, of the flow singularities analysed in [159].…”

“…Strictly speaking, this does not rule out the fact that the actual microscopic dynamics general relativity emerges from in the real world might be different from the one described by superstring theory, nor it implies that every question concerning the nature, behaviour and non-perturbative features of space-time in such a framework has received a satisfying answer [146][147][148][149][150][151][152][153]. In contemporary research, the attempts at addressing some of the issues related to the fundamental properties of quantum gravity from the perspective of strings has produced numerous groundbreaking results, as those related to the cobordism conjecture [154][155][156][157][158][159][160][161][162]. With that being said, superstring theory, in the regimes over which we have control, can be safely assumed to be a well-behaved theory of quantum gravity.…”

“…The swampland distance conjecture has been widely explored and extended [224,[277][278][279][280][281][282][283][284], fruitfully applied in the context of cosmology [285][286][287][288] and related to either the restoration of global symmetries, which should be absent from any consistent theory of quantum gravity [262,263,[289][290][291][292][293][294][295][296], or the emergence of a critical string [297][298][299]. In a series of works [159,202,203,217,242,249,[300][301][302][303], some of which will constitute the backbone of part II, multiple generalisations of the conjecture to the space-time metric itself were put forward. Before considering the details of the results derived in [300] and [302], we will now provide a concise introduction to the motivations, guiding principles and typical tools employed when trying to construct a distance conjecture for space-time geometries.…”

“…A candidate statement was presented in [242], where the mathematics of geometric flows allowed to frame the AdS flat limit in a more encompassing context. Further refinements [202,203,217,303], generalisations [300,302] and applications [159,249,301] thereof were subsequently explored. Geometric flow equations were originally applied in the context of string theory and condensed matter physics by Daniel Friedan, while analysing an extension of Aleksandr Markovič Poljakov's discussion [304] of the O (N )-invariant nonlinear σ-model renormalisation in 2 + ϵ dimensions [305].…”

Dynamical Cobordism Conjecture: Solutions for End-of-the-World Branes

“…There has been a lot of progress in exploring the far reaching consequences of the Cobordism Conjecture recently. We would like to refer the reader to [7][8][9][10][11][12][13][14][15][16][17][18][19] for more insight on the recent developments regarding the Cobordism Conjecture.…”