All Tree Amplitudes of 6D $(2,0)$ Supergravity: Interacting Tensor Multiplets and the $K3$ Moduli Space

TL;DR

The paper solves the problem of constructing the tree-level S-matrix for 6D $\,(2,0)$ supergravity with 21 abelian tensor multiplets, the low-energy limit of Type IIB on $\mathrm{K3}$. It introduces a localized, twistor-like rational-map formula that, via SUSY reduction from a $(2,2)$ theory, extends to multi-flavor tensor multiplets using the ${\rm Pf}\,\mathcal{X}_{n_2}$ structure; soft limits of the amplitudes reveal the moduli-space coset ${SO(5,21)}/{SO(5)\times SO(21)}$ and yield new double-soft theorems mixing flavor and R-symmetry. Dimensional reduction to 4D yields a novel tree-level S-matrix for $\mathcal{N}=4$ Einstein-Maxwell theory, consistent with known 4D results and tying the 6D construction to lower-dimensional physics. The work therefore provides a concrete S-matrix realization of the K3 moduli space, confirms duality-consistent reductions, and opens paths toward incorporating $\alpha'$ corrections in a unified framework.

Abstract

We present a twistor-like formula for the complete tree-level S matrix of 6D $(2,0)$ supergravity coupled to $21$ abelian tensor multiplets. This is the low-energy effective theory that corresponds to Type IIB superstring theory compactified on a $\mathrm{K}3$ surface. The formula is expressed as an integral over the moduli space of certain rational maps of the punctured Riemann sphere. By studying soft limits of the formula, we are able to explore the local moduli space of this theory, ${SO(5,21)\over SO(5)\times SO(21)}$. Finally, by dimensional reduction, we also obtain a new formula for the tree-level S matrix of 4D $\mathcal{N}=4$ Einstein-Maxwell theory.