Quantum Hypermultiplet Moduli Spaces in 𝒩=2 String Vacua: A Review

The hypermultiplet moduli space M H M_H in type II string theories compactified on a Calabi-Yau threefold X X is largely constrained by supersymmetry (which demands quaternion-Kählerity), S-duality (which requires an isometric action of S L ( 2 , Z ) SL(2,Z) ) and regularity. Mathematically, M H M_H ought to encode all generalized Donaldson-Thomas invariants on X X consistently with wall-crossing, modularity and homological mirror symmetry. We review recent progress towards computing the exact metric on M H M_H , or rather the exact complex contact structure on its twistor space.