Towards a categorification of scattering amplitudes

TL;DR

This paper develops a categorification framework for scalar field theory scattering amplitudes, connecting them to advanced algebraic structures like cluster categories and introducing pseudo-periodic categories.

Contribution

It introduces a novel categorification approach for scattering amplitudes using cluster categories and proposes a generalization to pseudo-periodic categories for polynomial potentials.

Findings

Amplitudes for cubic theories expressed via projectives of hearts of t-structures.

Extension of categorification to theories with higher polynomial potentials.

Proposal of pseudo-periodic categories as a new mathematical structure.

Abstract

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.