Building Momentum Kernel from Shapovalov Form

TL;DR

The paper develops an algebraic framework for BCJ duality by identifying the KLT momentum kernel with the Shapovalov form on Verma modules and constructing a dual basis whose Gram matrix is the inverse of the momentum kernel. It then expresses amplitudes and numerators in a universal, theory-agnostic form using a fixed reference vector $U_{ ext{1,...,n-1}}$ and basis vectors $V( extsum k_i, extsigma)$, yielding compact formulas like $N( extGamma)=\frac{1}{2s_{123\cdots n-1}}\langle U,V(\textGamma)\rangle$ and $A(1\textsigma n)=\langle U,V^*(\textsum k_i,\textsigma)\rangle$. The nonlinear sigma model serves as a primary example where numerators are shown to be momentum-kernel expressions and where a structure-constant formulation is developed; the framework is then extended to HEFT pre-numerators and, further, to gauge theory via an extended root system inspired by string theory to incorporate polarization data. Overall, the work unifies BCJ numerators and amplitudes within a cohesive algebraic setting, with clear routes to higher-point constructions and potential loop generalizations.

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.

Figures (2)

• Figure 1: Cubic trees at 5-point.
• Figure 2: The $t$-channel graph at $4$-point.