Shift operators from the simplex representation in momentum-space CFT

A bstract We derive parametric integral representations for the general n -point function of scalar operators in momentum-space conformal field theory. Recently, this was shown to be expressible as a generalised Feynman with topology an ( ? 1)-simplex, featuring arbitrary cross ratios. Here, we show all graph polynomials can expressed terms first and second minors Laplacian matrix simplex. Computing effective resistance between nodes corresponding electrical network, inverse parametrisation is found determinant Cayley-Menger matrix. These parametrisations reveal new families weight-shifting operators, determinants, that connect functions spacetime dimensions differing by two. Moreover, action previously known preserving dimension manifest. Finally, enable validity Ward identities established directly, without recourse recursion number points.