FQFT and cohomology
Gromov-Witten theory may be understood as providing a product on formal power series over certain abelian sheaf cohomology groups of Kähler manifolds . The quantum cohomology of is the resulting Frobenius algebra structure on these formal power series.
Together with Gromov-Witten theory quantum sheaf cohomlogy was discovered in and has its geometric roots as part of the data that describes certain 2-dimensional sigma-model quantum field theories with target space .
An introduction for readers familiar with basic concepts of Gromov-Witten theory is in
Slides of a talk for an audience of mathematical string theorists are
Quantum sheaf cohomology (pdf) Brandeis university (2010)
Quantum sheaf cohomology I (pdf)