Building blocks of closed and open string amplitudes

TL;DR

This paper reviews the relations between open and closed string amplitudes at tree-level and genus one, highlighting the holomorphic factorisation, single-valuedness, and new proofs of amplitude properties.

Contribution

It provides a comprehensive review of the connections between open and closed string amplitudes, including new insights into their single-valued nature at genus one.

Findings

KLT relations derive from holomorphic factorisation.

Tree-level closed string amplitudes have single-valued coefficients.

A new proof of single-valuedness for genus-one 2-point amplitudes.

Abstract

In this text we review various relations between building blocks of closed and open string amplitudes at tree-level and genus one. We explain that KLT relations between tree-level closed and open string amplitudes follow from the holomorphic factorisation of conformal correlation functions on conformal blocks. We give a simple hands-on evaluation of the $\alpha'$-expansion of tree-level closed string amplitudes displaying the special single-valued nature of the coefficients. We show that the same techniques can be used also at genus-one, where we give a new proof of the single-valued nature of the coefficients of 2-point closed string amplitudes. We conclude by giving an overview of some open problems.