Fuzzy instantons in landscape and swampland: review of the Hartle-Hawking wave function and several applications

Abstract

The Euclidean path integral is well approximated by instantons. If instantons are dynamical, then instantons are necessarily complexified. These fuzzy instantons can have various physical applications. In slow-roll inflation models, fuzzy instantons can explain the probability distribution of the initial conditions of the universe. Even though the potential shape does not satisfy the slow-roll conditions following the swampland criteria, still the fuzzy instantons can explain the origin of the universe. If we extend the Euclidean path integral beyond the no-boundary proposal, we may study fuzzy Euclidean wormholes that have various physical applications in cosmology and black hole physics. We summarize them and discuss possible future research topics.