Combinatorics and topology of KawaiLewellenTye relations
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 algebrotopological identities known as the twisted period relations. In order to do so, we formulate treelevel 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 biadjoint scalar amplitudes recently defined by the author [1]. 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 higherdimensional 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 biadjoint scalar partial amplitudes.
 Publication:

Journal of High Energy Physics
 Pub Date:
 August 2017
 DOI:
 10.1007/JHEP08(2017)097
 arXiv:
 arXiv:1706.08527
 Bibcode:
 2017JHEP...08..097M
 Keywords:

 Scattering Amplitudes;
 Bosonic Strings;
 Field Theories in Higher Dimensions;
 Superstrings and Heterotic Strings;
 High Energy Physics  Theory;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Combinatorics
 EPrint:
 51 pages