Topological field theories (TFTs) are crucial for understanding the fundamental properties of systems with a mass gap and the global symmetries of quantum field theories. Cummings and Heckman, from the University of Pennsylvania and the Kavli Institute for Theoretical Physics, demonstrate that TFTs, despite appearing decoupled from gravity, exhibit a surprising sensitivity to Newton’s constant through complex saddle‑point configurations. Their research reveals that this dependence extends to local metric fluctuations, suggesting TFTs may not exist as truly independent entities within the broader landscape of physical theories—a situation often described as residing in the “Swampland.” This finding, together with earlier work on global symmetries, shows that topological operators in boundary systems possessing a gravitational dual are inherently non‑topological in the bulk, fundamentally altering our understanding of their behaviour.

The study establishes that TFTs are intrinsically linked to local metric fluctuations, effectively placing them within the Swampland. The breakthrough stems from an analysis of how TFT sectors behave in asymptotically AdS spacetimes, revealing a non‑perturbative sensitivity to Newton’s constant through a summation over topologically distinct spacetime configurations. By meticulously tracking the fate of decoupling in the boundary dual of these gravitational systems, the authors uncover a surprising connection between topology and gravity.

A “topological equivalence principle” emerges: TFT fields, while seemingly independent of the local metric, are constrained to propagate on the same topological manifold as gravitational degrees of freedom. The authors examine the full path integral, approximating it as a weighted sum over different spacetimes, and demonstrate that even exact correlators of the TFT depend non‑perturbatively on Newton’s constant. This dependence is inherent to the structure of the path integral itself, not merely a consequence of higher‑order corrections. The work highlights that the fixed‑manifold factorization of gravitational and TFT fields, crucial for decoupling, holds exactly only for topological field theories.

Building on earlier findings concerning the absence of global symmetries in theories exhibiting subregion‑subregion duality, the authors further solidify the conclusion that topological operators in boundary systems with a gravity dual are invariably non‑topological in the bulk. Any topological operator in the boundary theory corresponds to a dynamical brane insertion in the bulk, inevitably coupling to local metric fluctuations. The authors rigorously demonstrate that a factorization of the boundary‑theory Hilbert space into a conventional CFT component and a separate “edge” component is untenable, as the topological symmetry operators necessitate a coupling to the bulk geometry. These results have significant implications for our understanding of quantum gravity and the nature of topological structures within it.

By establishing that TFTs cannot remain entirely decoupled from gravity, the study challenges conventional assumptions about the behaviour of these theories in gravitational settings. The topological equivalence principle provides a novel lens through which to examine the relationship between boundary and bulk theories in the context of quantum gravity.

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Gravitational Path Integrals and Topological Field Theories

The research presents a rigorous investigation into the compatibility of TFTs with gravitational systems, specifically challenging the notion that TFTs can remain entirely decoupled from gravity. The authors examine the gravitational path integral, represented as a sum over spacetime backgrounds, and its interaction with a decoupled TFT sector. The full path integral is written as

[

Z_{\text{full}}=\int [\mathcal{D}\Phi_{\text{grav}}][\mathcal{D}\Phi_{\text{TFT}}],

e^{-S_{\text{grav}}[\Phi_{\text{grav}}]-S_{\text{TFT}}[\Phi_{\text{TFT}}]},

]

and is approximated as a weighted sum over manifolds

[

Z_{\text{full}}\simeq \sum_{M} w_{M}, Z_{\text{TFT}}[M],

]

with the weights (w_{M}) determined by details of the gravitational sector.

From this formulation the authors demonstrate that even if a TFT is insensitive to local metric data, its fields must propagate on the same topological manifold as the gravitational degrees of freedom—a statement they term the topological equivalence principle. The principle is shown to be an exact feature of the path integral, not merely a one‑loop approximation.

Focusing on asymptotically AdS spacetimes, the authors anticipate a large‑(N) CFT dual description. A decoupled TFT sector would imply a factorization of the boundary‑theory Hilbert space

[

\mathcal{H}{\text{boundary}} = \mathcal{H}{\text{CFT}} \otimes \mathcal{H}_{\text{edge}},

]

where (\mathcal{H}_{\text{edge}}) would capture potential edge modes. The study shows that topological symmetry operators in the dual CFT, when linking interacting degrees of freedom, correspond to dynamical branes in the bulk that couple to local metric fluctuations. This imposes strong constraints on the existence of truly decoupled bulk TFT sectors.

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TFTs and Gravity Linked by Newton’s Constant

The authors demonstrate a fundamental connection between TFTs and gravity: seemingly decoupled TFT sectors are non‑perturbatively sensitive to Newton’s constant. This sensitivity arises from a summation over topologically distinct saddle‑point configurations in asymptotically AdS spacetimes, challenging the idea of independent topological structures in quantum gravity.

The full path integral can be expressed as

[

Z_{\text{full}} = \sum_{M} w_{M}, Z_{\text{TFT}}[M],

]

showing an explicit dependence on Newton’s constant through the weights (w_{M}). The factorization of gravitational and TFT fields holds exactly, meaning the TFT partition function remains unaffected by local metric fluctuations only at a superficial level; the underlying dependence is exact.

The authors introduce a topological equivalence principle, analogous to the standard equivalence principle, stating that all fields couple to the same local metric data. Consequently, if a boundary system possesses a dual gravity description, its topological operators are always non‑topological in the bulk.

By considering bulk gravitational systems on asymptotically AdS spacetimes and their large‑(N) CFT duals, the authors find that the Hilbert space factorization

[

\mathcal{H}{\text{boundary}} = \mathcal{H}{\text{CFT}} \otimes \mathcal{H}_{\text{edge}}

]

fails because topological symmetry operators in the CFT correspond to dynamical branes with non‑zero tension, directly coupling to metric fluctuations. This insight shows that TFTs are not truly “in the Swampland” as independent entities; rather, they are inextricably linked to gravity through their dependence on the underlying spacetime geometry.

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Topological Field Theories and Gravitational Coupling

Through analysis of asymptotically AdS spacetimes, the authors establish that seemingly decoupled TFT sectors exhibit a non‑perturbative sensitivity to Newton’s constant, indicating an interaction with local metric fluctuations. This finding implies that traditional TFTs do not exist within the landscape of consistent gravitational theories—a situation described as the Swampland.

The work extends earlier observations regarding the absence of global symmetries in certain dual field theories, reinforcing the idea that topological operators in boundary systems always correspond to non‑topological objects in the bulk gravitational description. Even when a topological symmetry appears isolated, it inevitably couples to the gravitational sector via dynamical branes. The authors suggest future research should explore specific models and further refine the understanding of the interplay between topology and gravity.