Inverse of the string theory KLT kernel

Abstract

Abstract:The field theory Kawai-Lewellen-Tye (KLT) kernel, which relates scattering amplitudes of gravitons and gluons, turns out to be the inverse of a matrix whose components are bi-adjoint scalar partial amplitudes. In this note we propose an analogous construction for the string theory KLT kernel. We present simple diagrammatic rules for the computation of the α -corrected bi-adjoint scalar amplitudes that are exact in α . We find compact expressions in terms of graphs, where the standard Feynman propagators 1/p 2 are replaced by either 1/ sin(πα p 2 /2) or 1/ tan(πα p 2 /2), as determined by a recursive procedure. We demonstrate how the same object can be used to conveniently expand open string partial amplitudes in a BCJ basis.

Combinatorics and topology of Kawai-Lewellen-Tye relations

“…We show how to calculate them on explicit examples and in general in Section 4. There, we also prove that combinatorial rules for finding intersection numbers are equivalent to the diagrammatic expansion found empirically in [1], establishing that the α -corrected bi-adjoint scalar amplitudes m α (β|γ) are given by intersection numbers of twisted cycles. Geometric and topological meaning of these objects can be easily pictured.…”

“…Similarly, closed string amplitudes are given as a pairing between two twisted cocycles. Finally, objects relating the two types of string amplitudes are the α -corrected bi-adjoint scalar amplitudes recently defined by the author [1]. We show that they naturally arise as intersection numbers of twisted cycles.…”

Biadjoint scalar tree amplitudes and intersecting dual associahedra

“…AHBHY construct (n − 1)!/2 associahedra in kinematic space K n by intersecting each of the cones C(α) with an appropriate hyperplane. 12 For each ordering α, with α(n) = n, define the hyperplane H(α) = s α(i),α(j) = constant | for all non-adjacent pairs i, j less than n .…”

“…We denote the inverse KLT kernel by m α (α|β), following Mizera. [12] In section 5.3 we present a new formula for the diagonal elements m α (α|α). This formula involves a discrete sum over a lattice which we introduce in section 5.1.…”

“…The kernel, KLT(α|β), can be inferred given knowledge of the string amplitudes. However, in [12], Mizera conjectured an algorithmic description of the inverse kernel, m α (α|β), that makes no reference to the string amplitudes (and their associated hypergeometric functions). In subsequent work, Mizera showed that his formulas for m α (α|β) can be regarded as (twisted) intersection pairings of the associahedra that tile M 0,n (R).…”