  Given a quiver $Q$, a formal potential is called analytic if its coefficients
are bounded by the terms of a geometric series. As shown by Toda, the
potentials appearing in the deformation theory of complexes of coherent sheaves
on complex projective Calabi-Yau threefolds are analytic. Our paper consists of
two parts. In the first part, we establish the foundations of the differential
calculus of quivers with analytic potentials and prove two fundamental results:
the inverse function theorem and Moser's trick. As an application, we prove
finite determinacy of analytic potentials with finite-dimensional Jacobi
algebra, answering a question of Ben Davison. We also prove a Mather-Yau type
theorem for analytic potentials with finite-dimensional Jacobi algebra,
extending previous work by the first author with Zhou. In the second part, we
study Donaldson-Thomas theory of quivers with analytic potentials. First, we
construct a canonical perverse sheaf of vanishing cycles on the moduli stack of
finite dimensional modules over the Jacobi algebra of an arbitrary iterated
mutation via a separation lemma for analytic potentials. Finally, we provide a
transformation formula for DT invariants (weighted by the Behrend function)
under iterated mutations as a counterpart of Nagao's result on topological DT
invariants. Our result leads to a perverse analogue of the $F$-series in
cluster algebras. It also yields a Behrend--weighted Caldero--Chapoton formula
for what we call anti-cluster algebras.
