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The Euclidean path integral is well approximated by instantons. If instantonsare dynamical, then instantons are necessarily complexified. These fuzzyinstantons can have various physical applications. In slow-roll inflationmodels, fuzzy instantons can explain the probability distribution of theinitial conditions of the universe. Even though the potential shape does notsatisfy the slow-roll conditions following the swampland criteria, still thefuzzy instantons can explain the origin of the universe. If we extend theEuclidean path integral beyond the no-boundary proposal, we may study fuzzyEuclidean wormholes that have various physical applications in cosmology andblack hole physics. We summarize them and discuss possible future researchtopics.

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