Abstract
We present a new formula for the biadjoint scalar tree amplitudes m(α|β) based on the combinatorics of dual associahedra. Our construction makes essential use of the cones in ‘kinematic space’ introduced by Arkani-Hamed, Bai, He, and Yan. We then consider dual associahedra in ‘dual kinematic space.’ If appropriately embedded, the intersections of these dual associahedra encode the amplitudes m(α|β). In fact, we encode all the partial amplitudes at n-points using a single object, a ‘fan,’ in dual kinematic space. Equivalently, as a corollary of our construction, all n-point partial amplitudes can be understood as coming from integrals over subvarieties in a toric variety. Explicit formulas for the amplitudes then follow by evaluating these integrals using the equivariant localisation formula. Finally, by introducing a lattice in kinematic space, we observe that our fan is also related to the inverse KLT kernel, sometimes denoted {m}_{alpha^{prime }}left(alpha Big|beta right) .
Highlights
Notice that the three line segments are not intersecting
We present a new formula for the biadjoint scalar tree amplitudes m(α|β) based on the combinatorics of dual associahedra
As a corollary of our construction, all n-point partial amplitudes can be understood as coming from integrals over subvarieties in a toric variety
Summary
We describe AHBHY’s construction of associahedra in ‘kinematic space.’ At n points, their construction gives (n − 1)!/2 distinct associahedra. We describe AHBHY’s construction of associahedra in ‘kinematic space.’. At n points, their construction gives (n − 1)!/2 distinct associahedra. Their construction gives (n − 1)!/2 distinct associahedra This is reminiscent of the open string moduli space, M0,n(R), which (after blow-ups) is tiled by (n − 1)!/2 associahedra. Unlike the tiles of M0,n(R), the associahedra that AHBHY construct in Kn do not intersect.
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