Integrable systems and quantum cohomology

Abstract

Abstract In this final chapter we look at the map W : CrGr(s+1) in the case of quantum cohomology and, more generally, abstract quantum cohomology. That is, we apply the fundamental construction of Chapter 8 to the situation of Chapters 5 and 6. It should not be surprising that this yields benefits, as the map W is a geometrical manifestation of the quantum cohomology D-module, and we have seen already that W is of great importance in other integrable systems such as the KdV equation and the harmonic map equation. In fact these expectations will be exceeded handsomely: in the case of the quantum cohomology of a Calabi–Yau manifold, W can be interpreted as a variation of Hodge structure (VHS) on a ‘mirror dual’ Calabi–Yau manifold.