Building Momentum Kernel from Shapovalov Form

Abstract: These notes are an extended version of the talks given by the authors at the XIV International Workshop on Lie Theory and Its Applications in Physics, Sofia, Bulgaria, 20-26 June 2021. The concise version published in the proceedings of the workshop contains additional discussions for the $q$-deformed scenario: \noindent\href{https://link.springer.com/chapter/10.1007/978-981-19-4751-3_23}{https://link.springer.com/chapter/10.1007/978-981-19-4751-3\_23}. In these notes we identify KLT kernel with the Shapovalov form on Verma module with its highest/lowest weight given by the reference momentum and rest of the momenta as roots. We then take a step forward and show how the Feynman diagrams emerge naturally as the Shapovalov duals of the Verma module basis vectors. We show such algebraic construct offers a compact expression for the BCJ numerators. Explicit examples are shown for the nonlinear sigma model and the HEFT pre-numerators.