group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Gromov-Witten theory may be understood as providing a product on formal power series over certain abelian sheaf cohomology groups of Kähler manifolds $X$. The quantum cohomology of $X$ 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 $X$.
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