Weighted Laplacians, cocycles and recursion relations

Abstract

Hodge's formula represents the gravitational MHV amplitude as the determinant of a minor of a certain matrix. When expanded, this determinant becomes a sum over weighted trees, which is the form of the MHV formula first obtained by Bern, Dixon, Perelstein, Rozowsky and rediscovered by Nguyen, Spradlin, Volovich and Wen. The gravity MHV amplitude satisfies the Britto, Cachazo, Feng and Witten recursion relation. The main building block of the MHV amplitude, the so-called half-soft function, satisfies a different, Berends-Giele-type recursion relation. We show that all these facts are illustrations to a more general story. We consider a weighted Laplacian for a complete graph of n vertices. The matrix tree theorem states that its diagonal minor determinants are all equal and given by a sum over spanning trees. We show that, for any choice of a cocycle on the graph, the minor determinants satisfy a Berends-Giele as well as Britto-Cachazo-Feng-Witten type recursion relation. Our proofs are purely combinatorial.Comment: 12 pages, some figures embedded in the tex

Self-dual gravity

“…[40]. It can also be seen that the current is given by an expansion of a certain (reduced) determinant, see [40]. Precisely the same arguments as in the case of self-dual YM theory show that all tree-level amplitudes with n > 3 vanish.…”

“…A general expression can be found in e.g. [40]. It can also be seen that the current is given by an expansion of a certain (reduced) determinant, see [40].…”

“…The first of these follows because the curvature of the self-dual part of the Levi-Civita connection is self-dual as a 2-form. The second follows because the equation (40) says that there is only the scalar curvature, but no self-dual part of Weyl. More details on the above material, together with proofs of many of the facts mentioned, can be found in e.g.…”

Pure connection formalism for gravity: recursion relations

“…An application of the matrix tree theorem also allows to represent the current as a certain matrix determinant. As is explained in [11], and will be reviewed below, there is also a BCFW-type recursion relation that can be written for such quantities.…”

“…Its solution readily presents itself by working out the few first amplitudes. We don't need to give a proof of the general formula here, because it is already spelled out in [11] in greater generality. In Section 4 we derive and analyse the recursion relation for the all but one same helicity on-shell legs current.…”