[Paper Review] Cobordism Classes and the Swampland

Jacob McNamara, Cumrun Vafa|arXiv (Cornell University)|Sep 23, 2019

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TL;DR

The authors argue that nontrivial cobordism groups in quantum gravity place theories in the Swampland and that vanishing cobordism implies the existence of defects (including D-branes and new non-supersymmetric ones) to trivialize potential cobordism classes.

ABSTRACT

We argue that any proposed quantum theory of gravity with non-trivial cobordism classes in the space of configurations belongs to the Swampland. The argument is based on the assumption that there are no global symmetries in a consistent theory of quantum gravity. The triviality of the cobordism classes requires the existence of certain stringy defects that trivialize the potential cobordism classes. We provide evidence for this conjecture by identifying those defects demanded by this argument that could preserve supersymmetry, and predict the existence of new non-supersymmetric defects in string theory.

Motivation & Objective

• Motivate a topological, cobordism-based criterion for consistency in quantum gravity and string theory.
• Define and interpret cobordism groups of quantum gravity as Ω_k^QG and relate them to potential global symmetries.
• Argue that Ω_k^QG = 0 is necessary to avoid global symmetries and to ensure the existence of symmetry-breaking or gauging defects.
• Show how known string theory objects (D-branes, O-planes, etc.) and proposed new defects implement the trivialization of cobordism classes.

Proposed method

• Define Ω_k^QG as the abelian group of k-dimensional quantum-gravity backgrounds modulo dynamically allowed topology-changing processes.
• Use the black hole information/evaporation argument to motivate why global cobordism charges should be absent in quantum gravity.
• Introduce the concept of apparent cobordism groups ṼΩ_k^QG to discuss how approximations may require additional defects or gauge fields to achieve Ω_k^QG = 0.
• Apply cobordism reasoning to p-form gauge fields and D-branes, spin cobordism, and various string theory contexts (K3, S^1_p, orientifolds, F-theory, heterotic).
• Argue that completeness of the spectrum for p-form gauge fields follows from Ω_k^QG = 0 and that defects corresponding to magnetic monopoles realize this condition.
• Discuss how different string theories kill or gauge cobordism classes via known and proposed defects.

Experimental results

Research questions

• RQ1 Does a nontrivial Ω_k^QG imply the existence of a corresponding global (d−k−1)-form symmetry and potential inconsistencies in quantum gravity?
• RQ2 Can known string theory objects (D-branes, O-planes, pin+ structures, F-theory) kill or gauge cobordism classes, thereby driving Ω_k^QG to zero?
• RQ3 What new non-supersymmetric defects are required to trivialize cobordism classes that are not killed by known ingredients?
• RQ4 How do cobordism invariants (e.g., signature, p_1) constrain allowable compactifications like K3, S^1_p, orientifolds, and heterotic backgrounds?
• RQ5 How do apparent cobordism groups guide refinements of string theory backgrounds when completing the spectrum or including defects?

Key findings

• Nontrivial cobordism groups Ω_k^QG signal potential global symmetries and inconsistencies in quantum gravity.
• Vanishing Ω_k^QG requires the existence of defects (including D-branes and other stringy objects) that trivialize cobordism classes.
• In various examples, known string theory ingredients (D-branes, O-planes, nonorientable manifolds, F-theory, spin/cobordism refinements) can kill or gauge cobordism classes.
• The analysis predicts new non-supersymmetric defects needed to maintain consistency when cobordism classes cannot be killed by known ingredients.
• Complete elimination of cobordism obstructions may require moving beyond geometric backgrounds to include more general backgrounds and non-perturbative effects.