# Combinatorics and Topology of Kawai-Lewellen-Tye Relations

## TL;DR

This paper explores the topological and combinatorial structures underlying Kawai-Lewellen-Tye relations in string theory, using twisted de Rham theory and associahedra to unify open and closed string amplitudes.

## Contribution

It introduces a novel algebro-topological framework for string amplitudes, connecting twisted cycle intersections with KLT relations and bi-adjoint scalar amplitudes.

## Key findings

- KLT relations emerge from twisted period identities.
- Open string amplitudes are integrals over associahedra.
- Intersection numbers of associahedra relate to the inverse KLT kernel.

## 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 $\alpha'$-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.