| Summary: | <p>Gravity amplitudes, at tree level, can be computed from gauge theory amplitudes using the (field theory) KLT relation. Underlying the KLT relation are the `Kleiss-Kuijf' and 'fundamental BCJ' identities satisfied by gauge theory partial amplitudes. These identities, and the KLT relation, are proved in an elementary way using properties of Lie polynomials, and the properties of a Lie bracket we call the 'S bracket', which has appeared in examples in superstring calculations. The S bracket is also closely related to string-like formulas which express partial amplitudes as sums of residues of rational functions on <em>M</em><sub>0,n</sub>. In this context, identities amoung rational functions on <em>M</em><sub>0,n</sub> allow us to prove a formula for non-linear sigma model partial amplitudes.</p>
<p>In the last decade, organising the computation of tree and loop amplitudes into the volumes of polytopes has led to new formulas for partial amplitudes in special cases. More generically, the ABHY associahedron is a polytopal realisation of the associahedron that is directly related to the tree level partial amplitudes of all gauge theories. It is known that the ABHY associahedron can be derived by studying A<sub>n</sub> quiver representations. Generalised ABHY polytopes are defined here, associated to Artinian module categories. The faces of an ABHY polytope are themselves isomorphic to ABHY polytopes of lower dimension. When this generalised definition is applied to categories associated to surfaces, the ABHY polytopes relate to the 'loop level' and 'multi-trace' partial amplitudes of the conventional Feynman perturbation series for biadjoint scalar theory. It is suggested that understanding these partial amplitudes and their ABHY polytopes will be important for stating and proving a correct generalisation of the (field theory) KLT relation to the full amplitudes of gauge theory and gravity.</p>
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