Combinatorics and Topology of Kawai-Lewellen-Tye Relations

TL;DR

This work recasts open/closed string KLT relations as twisted period relations in twisted de Rham theory, showing that open amplitudes arise from pairings between twisted cycles and cocycles while closed amplitudes come from cocycle–cocycle pairings; the inverse KLT kernel $m_{α'}(β|γ)$ is realized as intersection numbers of twisted cycles, with each cycle modeled by an associahedron carrying monodromy data via higher-dimensional Pochhammer contours. It develops a complete combinatorial/topological framework: twisted cycles correspond to associahedra in the blown-up moduli space $\tilde{M}_{0,n}$, twisted cocycles form a logarithmic Parke–Taylor basis, and twisted period relations reproduce the KLT relations exactly. The paper provides explicit four- and five-point intersection calculations and proves a general recursion that expresses any intersection number in terms of lower-point data, thereby connecting the geometry of moduli spaces to the field-theory limit bi-adjoint amplitudes $m(β|γ)$. This framework unifies string-theoretic BCJ structures, Z- and J-integrals, and the KLT machinery, with potential extensions to higher genus and deeper links to motivic and combinatorial structures in scattering amplitudes.

Abstract

We revisit the relations between open and closed string scattering amplitudes discovered by Kawai, Lewellen, and Tye (KLT). We show that they emerge from the underlying algebro-topological identities known as the twisted period relations. In order to do so, we formulate tree-level string theory amplitudes in the language of twisted de Rham theory. There, open string amplitudes are understood as pairings between twisted cycles and cocycles. 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 [arXiv:1610.04230]. We show that they naturally arise as intersection numbers of twisted cycles. In this work we focus on the combinatorial and topological description of twisted cycles relevant for string theory amplitudes. In this setting, each twisted cycle is a polytope, known in combinatorics as the associahedron, together with an additional structure encoding monodromy properties of string integrals. In fact, this additional structure is given by higher-dimensional generalizations of the Pochhammer contour. An open string amplitude is then computed as an integral of a logarithmic form over an associahedron. We show that the inverse of the KLT kernel can be calculated from the knowledge of how pairs of associahedra intersect one another in the moduli space. In the field theory limit, contributions from these intersections localize to vertices of the associahedra, giving rise to the bi-adjoint scalar partial amplitudes.

Figures (2)

• Figure 4.1: Real slice of the moduli space of Riemann spheres with five punctures, $\mathcal{M}_{0,5}$, and its image under the blowup map $\pi^{-1}(\mathcal{M}_{0,5}) = \widetilde{\mathcal{M}}_{0,5}$.
• Figure 4.2: Combinatorial interpretation of the associahedra $K_2(\mathbb{I}_3)$, $K_3(\mathbb{I}_4)$, and $K_4(\mathbb{I}_5)$ in terms of rooted trees. Each face has an associated factorization channel.

Theorems & Definitions (18)

• Definition 3.1
• Definition 3.2
• Claim 3.1
• proof
• Claim 3.2
• proof
• Example 3.1
• Example 3.2
• Definition 4.1
• Lemma 4.1