One-dimensional super Calabi-Yau manifolds and their mirrors

Perturbative scattering amplitudes in Yang-Mills theory have many unexpected properties, such as holomorphy of the maximally helicity violating amplitudes. To interpret these results, we Fourier transform the scattering amplitudes from momentum space to twistor space, and argue that the transformed amplitudes are supported on certain holomorphic curves. This in turn is apparently a consequence of an equivalence between the perturbative expansion of ${\cal N}=4$ super Yang-Mills theory and the $D$-instanton expansion of a certain string theory, namely the topological $B$ model whose target space is the Calabi-Yau supermanifold $\Bbb{CP}^{3|4}$. Show more