A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory

Abstract

The string corrections of tree-level open-string amplitudes can be described by Selberg integrals satisfying a Knizhnik–Zamolodchikov (KZ) equation. This allows for a recursion of the α′-expansion of tree-level string corrections in the number of external states using the Drinfeld associator. While the feasibility of this recursion is well-known, we provide a mathematical description in terms of twisted de Rham theory and intersection numbers of twisted forms. In particular, this leads to purely combinatorial expressions for the matrix representation of the Lie algebra generators appearing in the KZ equation in terms of directed graphs. This, in turn, admits efficient algorithms for symbolic and numerical computations using adjacency matrices of directed graphs and is a crucial step towards analogous recursions and algorithms at higher genera.

Two dialects for KZB equations: generating one-loop open-string integrals

“…As an example, let us state the first couple of orders of the series expansion of a typical integral Z tree n : The analogous expressions for arbitrary orders in the α -expansion of n-point disk integrals can for instance be generated from the Drinfeld associator [12,45] or Berends-Giele recursions [15]. 3 The Berends-Giele method in ref.…”

“…, b n ) whose decomposition in the (n−3)! bases expanded in [12,45] can be generated from the techniques in ref. [52].…”

“…For the discussion to follow, it is useful to define a graphical representation for the corresponding integrands, extending the graphical notation of ref. [45] to genus one: we are going to represent each occurring label as a vertex and each integration kernel as a directed edge 1…”

“…In turn, a differential equation with respect to a variable which disappears during the evaluation of the iterated integral, is not very useful. The solution to this problem has been spelt out in both mathematics [13,14] and physics [12,45] literature: one can introduce an additional auxiliary insertion point z 0 and establish a differential equation with respect to z 0 for a basis vector of augmented integrals Z tree 0,n . For the integrals Z τ n at genus one, a similar augmentation can be introduced leading to augmented one-loop integrals Z τ 0,n whose constituents will be reviewed in subsection 3.3 and whose differential equations in section 4 are a central result of this work.…”

Vector Space of Feynman Integrals and Multivariate Intersection Numbers

“…Moreover, it was observed that using recursion relations for computing intersection numbers can be further refined by relating them to dlog forms at each step of the recursive algorithm [20]. Other recent intersection-theory approaches include [21][22][23].…”