We provide a new perspective on the cosmological constant by exploring the background-independent Wheeler-DeWitt quantization of general relativity. The Chern-Simons-Kodama state of quantum gravity, a generalization of the Hartle-Hawking and Vilenkin states, has a striking structural similarity to the topological field theory of the quantum Hall effect. As a result, we study the gravitational topological $\theta$ sectors in analogy to Yang-Mills theory. We find that the cosmological constant $\mathrm{\Lambda }$ is intimately linked to the $\theta$ parameter by $\theta =12{\pi }^2/(\mathrm{\Lambda }{\ell }_{\mathrm{Pl}}^2)\text{mod}2\pi$ due to the fact that Chern-Simons-Kodama state must live in a particular $\theta$ sector. This result is shown in the canonical, nonperturbative formalism. Furthermore, we explain how the physics of the Hamiltonian constraint is analogous to the quantum Hall effect, with the cosmological constant playing the role of a quantum gravitational Hall resistivity. These relations suggest that $\mathrm{\Lambda }$ is topologically protected against perturbative graviton loop corrections, analogous to the robustness of quantized Hall conductance against disorder in a metal.

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• Quantum Hall effect
• Quantum cosmology
• Quantum gravity
• Symmetries
• Topology