Resultants and Gravity Amplitudes

Two very different formulations of the tree-level S-matrix of N=8 Einstein supergravity in terms of rational maps are known to exist. In both formulations, the computation of a scattering amplitude of n particles in the k R-charge sector involves an integral over the moduli space of certain holomorphic maps of degree d=k-1. In this paper we show that both formulations can be simplified when written in a manifestly parity invariant form as integrals over holomorphic maps of bi-degree (d,n-d-2). In one formulation the full integrand becomes directly the product of the resultants of each of the two maps defining the one of bi-degree (d,n-d-2). In the second formulation, a very different structure appears. The integrand contains the determinant of a (n-3)x(n-3) matrix and a 'Jacobian'. We prove that the determinant is a polynomial in the coefficients of the maps and contains the two resultants as factors.Comment: 21 page

Computation of Contour Integrals on M0,n{\cal M}_{0,n}

Contour integrals of rational functions over ${\cal M}_{0,n}$, the moduli space of $n$-punctured spheres, have recently appeared at the core of the tree-level S-matrix of massless particles in arbitrary dimensions. The contour is determined by the critical points of a certain Morse function on ${\cal M}_{0,n}$. The integrand is a general rational function of the puncture locations with poles of arbitrary order as two punctures coincide. In this note we provide an algorithm for the analytic computation of any such integral. The algorithm uses three ingredients: an operation we call general KLT, Petersen's theorem applied to the existence of a 2-factor in any 4-regular graph and Hamiltonian decompositions of certain 4-regular graphs. The procedure is iterative and reduces the computation of a general integral to that of simple building blocks. These are integrals which compute double-color-ordered partial amplitudes in a bi-adjoint cubic scalar theory.Comment: 36+11 p