The Topological Equivalence Principle: On Decoupling TFTs from Gravity

Abstract: Topological field theories (TFTs) play an important role in characterizing the deep infrared (IR) of many quantum systems with a mass gap, as well as the global symmetries of quantum field theories (QFTs) decoupled from gravity. In gravitational asymptotically AdS spacetimes, TFT sectors which are putatively decoupled from local metric data are nevertheless non-perturbatively sensitive to Newton's constant via a sum over topologically distinct saddle point configurations. Tracking the fate of this non-decoupling in the boundary dual, we argue that in spite of appearances, this dependence on Newton's constant extends to local metric fluctuations. Said differently, TFTs are in the Swampland. In tandem with earlier results on the absence of global symmetries in theories with subregion-subregion duality, this also establishes that topological operators of boundary systems with a gravity dual are always non-topological in the bulk.

• The paper demonstrates that TFT sectors cannot decouple from gravity, as gravitational saddle points enforce a non-perturbative topological coupling.
• It employs holographic duality and explicit examples like Chern-Simons and BF theories to expose inconsistencies in decoupled TFT proposals.
• The findings support the Swampland paradigm by proving that all bulk topological operators inevitably interact with gravitational modes.

The Topological Equivalence Principle and the Non-Decoupling of Topological Field Theories from Gravity

Introduction and Motivation

Topological Field Theories (TFTs) serve as essential tools in the infrared characterization of quantum field theories (QFTs), particularly in systems with mass gaps and in the analysis of generalized global symmetries. Their central feature is insensitivity to local metric data, granting a robust way to describe topological phases and defects in weakly coupled QFTs. Recent advances—motivated in part by the AdS/CFT correspondence—raise a critical question: can TFT sectors be strictly decoupled from quantum gravity? In other words, is it possible for a topological sector to remain completely independent of Newton’s constant and the dynamical content of the gravitational sector when considering full quantum gravity, particularly in spacetimes with non-trivial topology?

This paper thoroughly demonstrates that, contrary to any apparent decoupling, TFT sectors are inherently and non-perturbatively coupled to gravity. As a result, all boundary topological operators in a holographic CFT with a gravitational dual correspond to bulk operators that are necessarily non-topological: they always couple—directly or indirectly—to local metric fluctuations. This "topological equivalence principle" generalizes the usual equivalence principle: all fields, including those of a TFT, are subject to the same global topological and geometric data as the gravitational sector, invalidating true decoupling.

Decoupling Proposals and the Factorization Fallacy

A hypothetical decoupled TFT sector would suggest, in AdS/CFT, the following:

• The bulk path integral factorizes as $$Z_{\text{bulk}} = Z_{\text{grav}} \cdot Z_{\text{TFT}}$$, and
• The dual boundary Hilbert space factorizes as $$\mathcal{H}_{\text{CFT}} \otimes \mathcal{H}_{\text{edge}}$$.

Within this framework, topological operators $\mathcal{U}$ of the boundary TFT are assumed to act solely on $\mathcal{H}_{\text{edge}}$ and to commute with the full CFT operator algebra, reflecting their purported topological and localized nature. However, the gravitational path integral embodies a sum over inequivalent spacetime topologies—an unavoidable source of global coupling between all sectors, including the TFT.

Bulk Gravitational Saddles and Non-Perturbative Coupling

The essential technical observation is that the full bulk partition function is dominated by saddle points of differing topology, especially in asymptotically AdS spacetimes. For example, with a boundary geometry $S^1 \times S^{D-1}$, the bulk admits both thermal AdS and AdS-Schwarzschild black hole saddles, with their dominance shifting across the Hawking-Page transition. The non-trivial cycles (e.g., $S^1$ or $S^{D-1}$) present in these saddles play a direct role in the evaluation of topological correlators.

In such cases:

• TFT observables (e.g., expectation values of defect operators wrapping non-contractible cycles) receive different contributions depending on the dominant bulk topology.
• The transition between saddle points, controlled by gravitational parameters such as the AdS radius $L_{\text{AdS}}$ and the Planck length $L_{\text{Pl}}$, establishes a non-perturbative dependence of TFT correlators on Newton’s constant.

This feature precludes any true decoupling: even if local fluctuations of the metric do not interact with the TFT sector at the Lagrangian level, the full quantum path integral’s sum over topologically distinct manifolds necessarily induces coupling.

Contradiction in Holographic Duality

The purported decoupling would require that boundary computations—factorized between CFT and TFT sectors and independent of bulk gravitational data—match the full bulk predictions for all values of Newton’s constant. The explicit calculation shows that this cannot be true: changing the gravitational parameters (and hence the dominant saddle) alters the value of boundary TFT observables when computed via the dual bulk description.

This mismatch arises because, in the boundary calculation, the TFT sector is completely insensitive to $N$ (the rank parameter controlling $L_{\text{AdS}}/L_{\text{Pl}}$), while in the bulk description the expectation value of TFT operators varies discontinuously at phase transitions triggered by $N$-dependent topological change (e.g., across the Hawking-Page transition). Thus, any assumption of strict decoupling leads to a contradiction, ruling out all but the trivial TFT.

Consequences for Bulk Symmetries and the Swampland

A notable corollary is the final elimination of global symmetries, including generalized $p$-form symmetries, in all holographic quantum gravity theories. All candidate topological operators in the bulk—ostensibly encoding global symmetries—are dynamically realized as branes or defects with tension, always coupling to the gravitational sector in a manner dictated by the bulk topology [Heckman et al., (Heckman et al., 2024)].

The absence of decoupled TFT sectors aligns with the Swampland paradigm: truly decoupled topological sectors are not realized in quantum gravity. This result is compatible with and provides nontrivial support for the Swampland Cobordism conjecture [McNamara & Vafa, (McNamara et al., 2019)], which posits the triviality of the bordism group in theories of quantum gravity.

Lorentzian Picture and Factorization Failure

A Lorentzian analysis, especially in the context of the thermofield double (TFD) state and black hole geometries, confirms and extends these conclusions. The paper establishes that, even with apparently decoupled edge mode sectors, the Hilbert space structure in the TFD (and in particular, its embedding across Hawking-Page transitions) forces the triviality of all defect sectors. Any would-be nontrivial TFT sector would generate a form of the factorization problem: the inability to express the total Hilbert space as a pure tensor product across the two boundaries due to wormhole-induced correlations, reflecting the global correlation structure imposed by quantum gravity [Torres & Yu, (Torres et al., 7 Oct 2025)].

Explicit Examples and Surgery Arguments

The appendix extends the argument to paradigmatic classes of TFTs, including:

• Chern-Simons theory in 3D, where modular $S$-matrix constraints imply that only the trivial theory is consistent when the gravitational sector is dynamical,
• BF theories in arbitrary dimensions, where linking arguments similarly enforce triviality of defect sectors.

These findings further underscore the universality of the paper's main claim across standard classes of TFTs.

Implications and Future Directions

The conclusion that TFTs cannot be decoupled from gravity has broad implications for theories relying on robust infrared topological sectors or attempting to classify global symmetries in gravitational contexts. It suggests that all physical topological terms must ultimately couple to the dynamical content of quantum gravity, and no nontrivial global superselection sectors remain once gravity is included.

Looking forward, these results challenge the construction of non-Lagrangian topological sectors in gravitational theories and suggest that further studies should focus on the nontrivial transformations and anomalies associated with topology change, possible generalizations to non-AdS holographic dualities, and the rigorous classification of all consistent symmetry/topological couplings in quantum gravity.

Conclusion

The paper rigorously establishes that, in quantum gravity with holographic duality, the topological equivalence principle precludes the existence of TFT sectors that are decoupled from gravity. Any attempt to realize a truly topological sector independent of gravitational data runs into contradiction either via bulk/topology-changing saddle dominance or through the boundary dual's Hilbert space structure. All bulk topological operators are thereby rendered non-topological, establishing that no global symmetries—of any form—survive in quantum gravity and that all candidate TFTs reside within the Swampland (2601.09781).