On Positive Celestial Geometry: ABHY in the Sky

TL;DR

The paper develops a geometric, positive-geometry framework for celestial amplitudes by embedding them in a dual kinematic space at null infinity and realizing tree-level $\phi^3$ amplitudes as canonical forms on an ABHY-like celestial associahedron. It provides explicit Mellin-transform results in $D\geq n-1$, extends the construction to scalar-scaffolded YM/GR theories, and reveals a deep connection to string world-sheet via scattering equations, linking celestial data to worldsheet saddle points. The work shows that delta-function support arises intrinsically from the geometry and posits a broad, top-down path toward celestial holography, with potential ties to CCFT, BMS symmetries, and celestial OPEs. Overall, it unifies Mellin integrals, positive geometry, and holographic boundary data into a single geometric picture with wide-ranging implications for flat-space holography and string theory.

Abstract

Celestial amplitudes are multiple Mellin transforms w.r.t. conformal dimensions. For arbitrary multiplicity $n$ of massless states in sufficiently high space--time dimension $D$ we perform all Mellin integrations and find an associahedron description in celestial space. The latter expresses celestial tree--level $φ^3$ amplitudes as the canonical forms associated with this positive geometry. This yields a geometric interpretation of celestial amplitudes in terms of the underlying boundary geometry. In particular, distributional support on the celestial sphere is not imposed but arises geometrically. Our universal treatment of Mellin integrals in $D$ dimensions also provides a unified description of celestial amplitudes arising from different bulk theories, including (scalar-scaffolded) gluons and gravitons.

Figures (2)

• Figure 1: Celestial associahedra for $n=4,5$ and $n=6$.
• Figure 2: From left to right: Five-point deformed geometry for $(c_{1,3},c_{1,4},c_{2,4})=(4,2,1), (1,3,5), (2,3,1)$, respectively.