Positivity constraints for the gravitational path integral

For a quantum theory of gravity to have a well-defined Hilbert space, the inner product between different states of open and closed universes must be positive semi-definite. Positivity however is not manifest in the low-energy effective theory and in fact imposes nontrivial constraints on the theory. Working in the Gravitational Path Integral (GPI) approach, we derive the general set of positivity constraints on the closed and open universe Hilbert spaces.

In the case of AdS gravity, open universe positivity in principle follows from CFT unitarity, however the holographic description of closed universes remains unclear. Strikingly, we exhibit positivity of closed universes across many theories and prove that open positivity implies closed positivity, showing that the CFT 'knows' about the closed universe hilbert space.

We then analyze positivity constraints on gravitational theories coupled to axions. We present a method to compute off-shell axion wormholes in AdS and flat space which we use to show that positivity is violated if the axion shift symmetry is exact.
In low-energy EFTs where these wormholes are perturbatively stable, to restore positivity the wormhole must have a non-perturbative instability due to instantons that breaks the shift symmetry.
Positivity then leads to a proof of a sharp version of the Axion Weak Gravity Conjecture A-WGC, including precise numerical constants.
For the QCD axion this provides a bound on the axion decay constant which has phenomenological and experimental consequences for axion searches. In string theory, positivity gives a bound on the coupling between the axion and the dilaton in the low energy effective action.