Stringy K\"ahler moduli, mutation and monodromy

This paper gives the first description of derived monodromy on the stringy K\"ahler moduli space (SKMS) for a general irreducible flopping curve C in a 3-fold X with mild singularities. We do this by constructing two new infinite helices: the first consists of sheaves supported on C, and the second comprises vector bundles in a tubular neighbourhood. We prove that these helices determine the simples and projectives in iterated tilts of the category of perverse sheaves, and that all objects in the first helix induce a twist autoequivalence for X. We show that these new derived symmetries, along with established ones, induce the full monodromy on the SKMS. The helices have many further applications. We (1) prove representability of noncommutative deformations of all successive thickenings of a length l flopping curve, via tilting theory, (2) control the representing objects, characterise when they are not commutative, and their central quotients, and (3) give new and sharp theoretical lower bounds on Gopakumar-Vafa invariants for a curve of length l. When X is smooth and resolves an affine base, we furthermore (4) prove that the second helix classifies all tilting reflexive sheaves on X, and thus that (5) all noncommutative crepant resolutions arise from tilting bundles on X.