In mirror symmetry conjecture, we add what is called "B-filed" $B \in H^2(X,\mathbb{R}/\mathbb{Z})$ in the K"hlar moduli space so that the K"hlar moduli space has enough freedom comparable to the complex moduli space of a mirror manifold. In string theory, one may twist the Lagrangian by this auxiliary 2-form $B$.

The usual K"hlar moduli space parametrizes the volume of 2-cycles. Naively $B$ parametrizes the imaginary volume of the cycles, but is this really a useful concept? What is the role of B-field in mathematics?