Towards a categorification of scattering amplitudes

Categorification of scattering amplitudes for planar Feynman diagrams in scalar field theories with a polynomial potential is reported. Amplitudes for cubic theories are directly written down in terms of projectives of hearts of intermediate $t$-structures restricted to the cluster category of quiver representations, without recourse to geometry. It is shown that for theories with $\phi^{m+2}$ potentials those corresponding to $m$-cluster categories are to be used. The case of generic polynomial potentials is treated and our results suggest the existence of a generalization of higher cluster categories which we call pseudo-periodic categories. An algorithm to obtain the projectives of hearts of intermediate $t$-structures for these types is presented.