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

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. Charlie Cummings and Jonathan J. 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 this dependence extends to local metric fluctuations, suggesting TFTs may not exist as truly independent entities within the broader landscape of physical theories , a concept known as residing within the ‘Swampland’. This finding, alongside previous work on global symmetries, establishes 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 research establishes that TFTs, despite appearing decoupled from gravity, are intrinsically linked to local metric fluctuations, effectively placing them within the “Swampland”, theoretical constructs incompatible with a consistent quantum gravitational theory. This 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. The team achieved this by meticulously tracking the fate of decoupling in the boundary dual of these gravitational systems, uncovering a surprising connection between topology and gravity.

The study unveils a “topological equivalence principle” wherein TFT fields, while seemingly independent of the local metric, are constrained to propagate on the same topological manifold as gravitational degrees of freedom. Researchers approached this problem by examining the full path integral, approximating it as a weighted sum over different spacetimes, and demonstrating that even exact correlators of the TFT depend non-perturbatively on Newton’s constant. This analysis moves beyond perturbative approximations, asserting that the observed 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.

This research builds upon earlier findings concerning the absence of global symmetries in theories exhibiting subregion-subregion duality, further solidifying the conclusion that topological operators in boundary systems with a gravity dual are invariably non-topological in the bulk. Experiments show that any topological operator in the boundary theory corresponds to a dynamical brane insertion in the bulk, inevitably coupling to local metric fluctuations. The team rigorously demonstrates 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. The implications of this work extend to 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 behavior of these theories in gravitational settings. This breakthrough opens avenues for exploring the interplay between topology, gravity, and quantum field theory, potentially leading to new insights into the fundamental laws governing the universe and the constraints on consistent theoretical frameworks. The topological equivalence principle established here provides a novel lens through which to examine the relationship between boundary and bulk theories in the context of quantum gravity.

Gravitational Path Integrals and Topological Field Theories

The research detailed a rigorous investigation into the compatibility of topological field theories (TFTs) with gravitational systems, specifically challenging the notion that TFTs can remain entirely decoupled from gravity. Scientists approached this problem by examining the gravitational path integral, represented as a sum over spacetime backgrounds, and its interaction with a decoupled TFT sector. The study mathematically formulated the full path integral as Zfull = ∫[DΦgrav][DΦTFT] exp−Sgrav[Φgrav]−STFT[ΦTFT], where Sgrav represents the gravitational action and STFT denotes the action for the TFT. Researchers then approximated this integral as a weighted sum over different spacetimes, expressed as Zfull ≃ ΣM wMZTFT[M], with weights wM dependent on details within the gravitational sector.

This approach enabled the team to demonstrate that even if a TFT is insensitive to local metric data, its fields are constrained to propagate on the same topological manifold as gravitational degrees of freedom, a principle termed the “topological equivalence principle”. The work posits this relationship is not merely a one-loop approximation but an exact characteristic of the path integral, holding true due to the unique properties of TFTs. To further refine this understanding, the study focused on asymptotically AdS spacetimes, anticipating a large N conformal field theory (CFT) dual description. The team hypothesized that a decoupled TFT sector would imply a factorization of the boundary theory Hilbert space into HCFT ⊗Hedge, where HCFT represents the large N degrees of freedom and Hedge characterizes potential edge modes. Crucially, the research established that topological symmetry operators in the dual CFT, when linking interacting degrees of freedom, correspond to dynamical branes in the bulk that demonstrably couple to local metric fluctuations. This finding imposes significant constraints on the existence of truly decoupled bulk TFT sectors, suggesting that topological operators in boundary systems with a gravity dual are inherently non-topological in the bulk.

TFTs and Gravity Linked by Newton’s Constant

Scientists have demonstrated a fundamental connection between topological field theories (TFTs) and gravity, revealing that seemingly decoupled TFT sectors are, in fact, non-perturbatively sensitive to Newton’s constant. The research establishes this sensitivity through a summation over topologically distinct saddle point configurations in asymptotically AdS spacetimes, challenging the notion of truly independent topological structures in quantum gravity. Experiments revealed that even though TFTs are initially formulated independent of local metric data, their fields are constrained to propagate on the same topological manifold as gravitational degrees of freedom. The team measured this interdependence by examining the gravitational path integral, approximating it as a weighted sum over different spacetimes.

This analysis shows that the full path integral, denoted as Zfull, can be expressed as a sum over manifolds (M) of weighted TFT partition functions, ZTFT[M], demonstrating a non-trivial dependence on Newton’s constant. Measurements confirm that the factorization of the gravitational and TFT fields holds exactly, meaning the TFT partition function remains unaffected by local metric fluctuations at a fundamental level. This finding is not a perturbative approximation but rather an exact result of the path integral calculation. Further work established the “topological equivalence principle,” analogous to the standard equivalence principle, which posits that all fields couple to the same local metric data.

The study highlights that if a boundary system possesses a dual gravity description, its topological operators are always non-topological in the bulk. Scientists achieved this by considering bulk gravitational systems on asymptotically AdS spacetimes and their large N CFT dual descriptions, where the Hilbert space factors into a CFT component and an ‘edge’ component characterizing bulk TFT degrees of freedom. Results demonstrate that topological symmetry operators within the dual CFT, which interact with other degrees of freedom, can be interpreted as dynamical branes in the bulk with non-zero tension, directly coupling to local metric fluctuations. This breakthrough delivers a crucial insight: TFTs are not truly ‘in the Swampland’, a theoretical region of inconsistency, but are inextricably linked to gravity through their dependence on the underlying spacetime geometry. The research opens avenues for exploring the interplay between topology, gravity, and quantum field theory, potentially reshaping our understanding of fundamental symmetries and the nature of spacetime itself.

Topological Field Theories and Gravitational Coupling

This work demonstrates a fundamental connection between topological field theories and gravity, challenging the notion that these theories can be entirely decoupled from gravitational dynamics. Through analysis of asymptotically AdS spacetimes, the authors establish that seemingly decoupled topological field theory sectors exhibit a non-perturbative sensitivity to Newton’s constant, indicating an interaction with local metric fluctuations. This finding implies that topological field theories, as traditionally understood, do not exist within the landscape of consistent gravitational theories, a concept known as the Swampland. Furthermore, the research 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. The authors rigorously show that even in scenarios where a topological symmetry appears isolated, it inevitably couples to the gravitational sector via dynamical branes. While acknowledging the complexity of fully characterizing all possible topological configurations, the authors suggest future research should focus on exploring the implications of these findings for specific models and further refining the understanding of the interplay between topology and gravity.