The Topological Equivalence Principle: On Decoupling TFTs from Gravity

TL;DR

This work argues that topological field theories cannot remain decoupled from gravity in holographic spacetimes. By analyzing Euclidean saddles and Lorentzian thermofield double constructions, it shows that bulk gravity funneled through varying saddles and Hawking-Page transitions induces nonperturbative mixing with TFT sectors, implying bulk TFTs couple to local metric fluctuations. The results enforce the absence of bulk global symmetries and place TFTs squarely in the Swampland, motivating further exploration of SymTFTs and cobordism ideas within quantum gravity. Overall, the paper provides a nonperturbative consistency check that decoupled TFT sectors cannot exist in AdS/CFT, sharpening constraints on how topological data can manifest in holographic theories.

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.

Figures (8)

• Figure 1: Gluing manifolds computes the TFT quantum expectation value $\bra{\Psi'} \widetilde{\mathcal{U}} \ket{\Psi}$.
• Figure 2: A graph of $\langle \mathcal{U} \rangle_{\beta}$ as a function of $\beta$, as computed in the bulk dual with fixed boundary conditions. The AdS-Schwarzschild phase is in blue, and the thermal AdS phase is in red. Generically, $\langle \mathcal{U} \rangle_\beta$ is constant in each phase, and the phase transition occurs at the inverse Hawking-Page temperature $\beta_{\mathrm{HP}}$. On the other hand, in the CFT dual, this correlator is controlled by an edge mode system which does not depend on $N$, i.e., the correlation function is expected to be the same on both backgrounds. This signals a contradiction.
• Figure 3: Depiction of Euclidean thermal AdS. Topologically, this is an $S^1$ fibered over $\mathrm{Cone}(S^{D-1})$. A zero-form symmetry operator $\mathcal{U}$ wrapping the boundary $S^{D-1}$ can be pushed into the bulk, with dual $\widetilde{\mathcal{U}}$. The bulk cycle supporting $\widetilde{\mathcal{U}}$ is contractible, so the thermal correlation function of $\mathcal{U}$ evaluates to the quantum dimension of $\widetilde{\mathcal{U}}$.
• Figure 5: Left: Penrose diagram of thermal AdS ($\beta > \beta_{\mathrm{HP}}$). Right: Penrose diagram of AdS-Schwarzschild ($\beta < \beta_{\mathrm{HP}}$). For both spacetimes, a Cauchy slice drawn in blue. The TFD is prepared by Euclidean evolution, and continuation along this Cauchy slice.
• Figure 6: A Cauchy slice $\Sigma$ of thermal AdS, along with a manifold-with-boundary preparing boundary conditions $J$ and $\overline{J}$, respectively. We have suppressed the timelike direction.