Looking for structure in the cobordism conjecture

Abstract

The cobordism conjecture of the swampland program states that the bordism group of quantum gravity must be trivial. We investigate this statement in several directions, on both the mathematical and physical side. We consider the Whitehead tower construction as a possible organising principle for the topological structures entering the formulation of the conjecture. We discuss why and how to include geometric structures in bordism groups, such as higher U(1)-bundles with connection. The inclusion of magnetic defects is also addressed in some detail. We further elaborate on how the conjecture could predict Kaluza--Klein monopoles, and we study the gravity decoupling limit in the cobordism conjecture, with a few observations on NSNS string backgrounds. We end with comments in relation to T-duality, as well as the finiteness conjecture.

Cobordism, singularities and the Ricci flow conjecture

“…The reader is strongly suggested to refer to [20][21][22][23][24][25] for comprehensive discussions of the techniques that are usually employed when studying Ricci flow of differentiable manifolds. On the other hand, the recent Cobordism conjecture [26][27][28][29][30][31][32][33][34], stating that viable quantum gravity backgrounds should have trivial cobordism class and thus be cobordant to one another, can arguably be regarded as a refinement of the No global symmetries conjecture. It must be noted that the Ricci flow conjecture typically deals with locally defined geometric flow equations and properties, while the Cobordism conjecture comes as a genuine global statement.…”

“…The precise and rigorous way in which this is done is covered in [36], where tangential structures are introduced and treated in detail. A more condensed but readable account of this is contained in the introduction of [31].…”

“…The problem here is that cobordisms are, in a sense, "only sensitive to the topology of a manifold", and not even completely at that. By this we mean that the bordism class of a manifold can be affected by such things as characteristic numbers (the Pontrjagin and Stiefel-Whitney numbers), tangential structures, such as in [31], etc. However, the main appeal of bordisms is that they may relate two manifolds whose topologies differ completely.…”

“…If we decide to leave the SO-structure, we could apply the same logic to more complicated structures, such as Ω Spin d , Ω String d , etc. As shown in [31], these particular ones mentioned can be obtained by further constraining the vanishing of certain charactersitic numbers. For example, Ω Spin d is obtained from Ω SO d by requiring that w 2 = 0, as expected.…”

“…Then, we will progressively add more structure, defining some of the concepts when necessary. Some other introductions are given in [26,31], as well as [35,Chapters 4,17]. For a more technical and in-depth treatment of bordisms the reader is redirected to the classical reference, [36].…”