Superstring Amplitudes as a Mellin Transform of Supergravity

TL;DR

The paper establishes a precise relationship between tree-level open-string N-gluon MHV amplitudes and N-graviton MHV amplitudes in supergravity by mapping disk-boundary vertex positions to helicity spinors and expressing both as tree-graph minors of a generalized Laplacian. By promoting vertex data to a larger Mellin-coordinate space of cross-ratios $u_{i,j}$, the authors show that string amplitudes are (multiple) Mellin transforms of gravity amplitudes from Mellin position space to the dual kinematic space, with inverse transforms reproducing gravity from string data. A unified framework introduces a fundamental function ${rak M}_N(z,z',s)$ and a Pascal-triangle structure of nonlinear constraints that connect world-sheet integrals to Mellin-space representations, revealing a direct and computable link between string form factors and gravity amplitudes. The work suggests a deeper string–supergravity correspondence, offering a potential dual, delta-function–based description of string amplitudes in Mellin space that could enable new recursion relations and more direct computations of scattering processes.

Abstract

At the tree level, the maximally helicity violating amplitudes of N gauge bosons in open superstring theory and of N gravitons in supergravity are known to have simple representations in terms of tree graphs. For superstrings, the graphs encode integral representations of certain generalized Gaussian hypergeometric functions of kinematic invariants while for supergravity, they represent specific kinematic expressions constructed from spinor-helicity variables. We establish a superstring/supergravity correspondence for this class of amplitudes, by constructing a mapping between the positions of gauge boson vertices at the disk boundary and the helicity spinors associated to gravitons. After replacing vertex positions by a larger set of N(N-3)/2 coordinates, the superstring amplitudes become (multiple) Mellin transforms of supergravity amplitudes, from the projective space into the dual Mellin space of N(N-3)/2 kinematic invariants. Similarly, inverse Mellin transforms transmute open superstrings into supergravity. We elaborate on the properties of multiple Mellin and inverse Mellin transforms in the framework of superstring/supergravity correspondence.

Figures (5)

• Figure 2: Cayley graphs for $n=3$.
• Figure 3: Cayley graphs for $n=4$.
• Figure 4: Partional fractioning converts a vertex $j$ of $deg(j)=3$ to a vertex of $deg(j)=2$.
• Figure 5: Hamilton graphs corresponding to a minimal basis subject to partial fractionning.
• Figure :