Swampland Cobordism Conjecture

• The Swampland Cobordism Conjecture is a topological condition in quantum gravity that requires all cobordism classes to vanish, thereby forbidding global symmetries.
• It links cobordism invariants to physical charges and mandates the existence of string theory defects—both supersymmetric and non-supersymmetric—to ensure consistency.
• The conjecture imposes strict selection rules in effective field theory, guiding model building by preventing nontrivial compactification backgrounds incompatible with quantum gravity.

The Swampland Cobordism Conjecture is a principle in quantum gravity and string theory stating that all compact quantum gravity backgrounds—parametrized as elements of certain cobordism groups—must be trivial in a consistent ultraviolet (UV) completion. This topological requirement directly forbids global symmetries and mandates the existence of specific physical defects, which include both supersymmetric and predicted non-supersymmetric objects responsible for trivializing the corresponding cobordism invariants. The conjecture serves as a sharp criterion for distinguishing the “Landscape” (consistent low-energy effective theories with quantum gravity UV completions) from the “Swampland” (inconsistent or incomplete theories).

1. Cobordism Classes and Their Physical Interpretation

A cobordism class is a topological invariant describing equivalence relations among compact, k-dimensional manifolds (or compactification spaces) used as background configurations in quantum gravity. Two manifolds, $M$ and $N$, of the same dimension are cobordant if there exists a compact $(k+1)$-dimensional manifold $W$ with boundary $\partial W = M \,\sqcup\, N$. The set of such equivalence classes forms the abelian group $\Omega_k^{PG}$, with group operation given by disjoint union and inverse given by orientation reversal.

In quantum gravity, global cobordism invariants in $\Omega_k^{PG}$ correspond to conserved $(d-k-1)$-form charges, with $d$ the spacetime dimension. These can be understood as labeling “defects” or nonlocal charges associated with nontrivial topology—e.g., cycles wrapped by branes or flux backgrounds.

If $\Omega_k^{PG} \neq 0$, then there must exist global charges (topological “superselection sectors”), which directly contradicts the widely held belief—motivated by black hole arguments—that quantum gravity admits no exact global symmetries.

Table 1: Cobordism Classes and Physical Interpretation

Cobordism Group	Example Generator	Physical Interpretation
$\Omega_1^{\rm Spin}$	$S^1_p$	2-form symmetry, $d-3$-brane charge
$\Omega_4^{\rm Spin}$	K3	5-brane charge, heterotic fivebrane coupling
$\Omega_k^{PG}$	general	$(d-k-1)$-form global symmetry

2. Link to Global Symmetries and Quantum Gravity Consistency

A foundational assumption is that consistent quantum gravity theories prohibit any unbroken exact global symmetry. Black hole complementarity and information erasure arguments imply that any such charge can be lost in black hole evaporation, leading to inconsistencies such as the proliferation of stable, Planck-scale remnants labeled by global charges.

If a cobordism group $\Omega_k^{PG}$ is nontrivial, it supports a global $(d-k-1)$-form symmetry. This is made precise through explicit maps such as:

$$Q: \Omega_{p+1}^{SO,U(1)_p} \longrightarrow \mathbb{Z}, \qquad Q(M, A_p) = \int_M F_{p+1}$$

where $A_p$ is a $p$-form gauge field, $F_{p+1} = dA_p$, and the charge is invariant under cobordism due to $dF_{p+1} = 0$. The existence of such a conserved charge signals a forbidden global symmetry.

Therefore, the full quantum gravity theory must have all relevant cobordism groups trivialized, i.e., $\Omega_k^{PG} = 0$ for all $k$.

3. Dynamical Trivialization by Stringy Defects

The resolution, enforced by the Cobordism Conjecture, is that string theory and M-theory must contain a complete spectrum of dynamical defects—which include D-branes, orientifolds, and other stringy objects—that can trivialize all cobordism classes. These defects act as physical “walls” or domain walls connecting different compactification backgrounds within the same cobordism class.

Examples:

• Type IIA: The O8-plane/Hořava–Witten wall trivializes the decompactification point and the $S^1_p$ generator, acting as an end-of-the-world boundary for certain backgrounds.
• Type IIB: Under T-duality, the O8-plane maps to two O7-planes, killing the analogous cobordism class in the spin$^c$ group.
• Heterotic theory: The K3 surface carries 24 units of fivebrane charge via $S \supset \tfrac{1}{2}\int B \wedge p_1(R)$, which matches the cobordism invariant from its signature.

Whenever no known supersymmetric defect is available to trivialize a given class, the conjecture predicts the necessary existence of a non-supersymmetric defect. For instance, constructing a domain wall between two supersymmetric string vacua in different dimensions cannot preserve any supersymmetry; thus, its existence is required purely by consistency of the swampland principle.

Table 2: Examples of Cobordism Trivialization by Defects

Theory	Generator	Defect Type	Breaks SUSY?
Type IIA	$S^1_p$ / pt$^+$	O8/HW wall	Sometimes
Type IIB	$S^1_p$	2 $\times$ O7-plane	Sometimes
Heterotic	K3	Fivebrane charge	No
General	—	(Unknown)	Yes (if SUSY-preserving defect not found)

4. Mathematical Formulation and Topological Invariants

The cobordism group, in the context of quantum gravity, is the quotient:

$\Omega_k^{PG} = \frac{\{\text{compact $k$-dimensional backgrounds}\}}{\text{topology-changing processes}}$

where the “topology-changing” processes are realized as dynamical finite-energy transitions, often mediated by the presence of defects (domain walls, branes, orientifolds).

Key mathematical tools involve:

• The Hirzebruch signature theorem for 4-manifolds, relating the signature $\sigma(M)$ to the first Pontrjagin class via

$\sigma(M) = \frac{1}{3}\int_M p_1(R)$

with $$p_1(R) = \frac{1}{8\pi^2}\operatorname{tr}(R\wedge R)$$.

• In heterotic string theory, the Green–Schwarz modified Bianchi identity

$dH = \frac{p_1(R) - p_1(F)}{2}$

imposes a “string structure” condition ($p_1/2=0$) forcing triviality of the corresponding cobordism group in relevant dimensions for consistency.

5. Predictions and Evidence in String Theory

A central piece of evidence arises from explicit calculations of spin, spin$^c$, and string cobordism groups in the low-energy limit (where gauge fields and singularities can be ignored). These calculations are nontrivial and show that if the full spectrum of string theory defects is not included, cobordism groups are generally nonzero—implying forbidden global symmetries. The presence of additional stringy defects restores consistency.

The conjecture predicts new phenomena:

• Existence of yet-unknown non-supersymmetric defects, necessary to trivialize cobordism classes uncovered by these calculations.
• The domain walls associated with these defects could, in principle, be probed through global charge measurements at horizons (e.g., black hole considerations) or via flux quantization.

Black hole physics provides indirect evidence: The absence of stable remnants and the completeness of the charge spectrum naturally follows if and only if all cobordism charges can be absorbed (trivialized) by dynamical processes.

6. Operational Role in Effective Field Theory and Implications

From an effective field theory (EFT) viewpoint, the conjecture enforces powerful selection rules:

• Any effective theory admitting a nontrivial compactification background with a conserved $(d-k-1)$-form charge that cannot be dynamically trivialized by known (stringy) defects is inconsistent with quantum gravity and is part of the Swampland.
• This imposes tight restrictions on model building in string theory, constraints on allowed gauge and symmetry structures, and has implications for the spectrum of admissible branes, domain walls, and topological sectors.

It also provides a sharp distinction with conjectures such as the Trans-Planckian Censorship Conjecture (TCC): whereas the Swampland Cobordism Conjecture imposes a topological requirement on the structure of the theory (the group $\Omega_k^{PG}$), the TCC constrains the types of cosmological histories or solutions allowed in a theory.

7. Summary and Impact

The Swampland Cobordism Conjecture asserts that the only consistent quantum gravity theories are those in which all cobordism classes vanish ($\Omega_k^{PG} = 0$). This requires the existence of a complete set of stringy defects—sometimes supersymmetric, but in certain cases necessarily non-supersymmetric—that dynamically trivialize all possible global topological charges. The conjecture is underpinned by both explicit topological calculations and physical arguments relating to the fate of black hole remnants and the absence of global symmetries.

As a criterion for UV completeness, it provides a topological selection rule for quantum gravity vacua, predicts new objects within string theory, and offers a framework for understanding possible topology-changing processes in the quantum gravity landscape, thereby deeply influencing the structure of effective field theories compatible with quantum gravity.