Topological Equivalence Principle Demonstrates Gravity’s Non-Perturbative Sensitivity Via Sums over Configurations

Key Takeaways

• TFTs couple non‑perturbatively to Newton’s constant
• Summation over manifolds links topology to gravity
• Factorization of boundary Hilbert space fails
• Topological operators become dynamical branes in bulk
• Findings reshape Swampland criteria for topological sectors

Summary

Cummings and Heckman introduce a topological equivalence principle showing that topological field theories (TFTs), previously thought to be independent of gravity, actually exhibit a non‑perturbative sensitivity to Newton’s constant via a sum over distinct spacetime configurations. By analyzing asymptotically AdS backgrounds, they demonstrate that TFT correlators depend on metric fluctuations, invalidating the notion of a fully decoupled TFT sector. This links topological operators in boundary theories to dynamical branes in the bulk, challenging the factorization of the boundary Hilbert space and placing TFTs in the Swampland context.

Pulse Analysis

Topological field theories have long been prized as solvable sectors that capture global features of quantum systems without reference to local geometry. In high‑energy physics they serve as probes of mass‑gap dynamics, anomaly inflow, and the classification of symmetry‑protected phases. The prevailing view held that such theories could be grafted onto a gravitational background and remain completely insulated from metric fluctuations, offering a clean laboratory for studying pure topology within a quantum‑gravity framework. This assumption underpinned many Swampland arguments that treat TFTs as isolated building blocks.

Cummings and Heckman overturn that picture by formulating a “topological equivalence principle” that forces every TFT field to propagate on the same topological manifold as the graviton sector. By evaluating the full path integral as a weighted sum over distinct spacetime configurations, they expose a non‑perturbative dependence of the TFT partition function on Newton’s constant, encoded in the weights of each manifold. The dependence survives even when the TFT action itself lacks metric terms, proving that the decoupling is only apparent and that topology and gravity are inseparably linked.

The consequences ripple through holography and Swampland research. The expected factorization of the boundary Hilbert space into a CFT and an independent “edge” sector collapses, because topological symmetry operators map to dynamical branes that couple to bulk metric fluctuations. This forces a revision of criteria that label pure TFTs as safe islands in the landscape; they now occupy a constrained region where gravity cannot be ignored. Future work will likely target explicit AdS/CFT models, explore higher‑form analogues, and refine Swampland conjectures to incorporate the topological equivalence principle.