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