α'expansion of open superstring amplitudesOn this website we are providing the explicit form of the α'expansion of superstring disk amplitudes involving N massless states from the gauge multiplet. According to [1106.2645] and [1106.2646], the (N−3)! independent colorordered open string tree amplitudes A_{open} are obtained from an (N − 3)! × (N − 3)!matrix F acting on the (N−3)!component vector of independent YangMills subamplitudes
A_{YM}\; =\; A_{YM}(1, \sigma(2), \ldots \sigma(N2), N1, N)\; ,\hskip2cm \sigma\in S_{N3}
A_{\rm open}\; =\; F \; \cdot \; A_{YM}\; .
The inverse string tension α' enters F through the dimensionless Mandelstam invariants
s_{ij}:=\alpha' (k_i+k_j)^2,\; \; \; s_i:=s_{i,i+1}, \; \; \; {\rm and} \\[5mm]
t_i:=\alpha' (k_i+k_{i+1}+k_{i+2})^2\equiv s_{i,i+1,i+2}, \; \; \;
{\rm where} \; \; \; i \simeq i+N \; \; \; {\rm and}\; \; \; k_i^2=0,
i.e. its Taylor expansion in s_{ij} encodes the lowenergy behaviour of the string amplitude. The pattern of multiple zeta values (MZVs)
\zeta_{n_1,\ldots,n_r} \; =\; \sum_{0 < k_1 < \ldots < k_r}
\; \; \prod_{l=1}^r k_l^{n_l} \; , \; n_l \in {\bf N}^+ \; , \; n_r \geq 2
M_{2k+1}:=\left. F\;\right_{\zeta_{2k+1}} \; \; \; {\rm and}\; \; \;
P_{2k}:=\left. F\;\right_{(\zeta_2)^k}
are sufficient to reconstruct the α'dependence along with any other MZV product, e.g.:
A suitable Hopfalgebra inspired MZV representation paving the way for an extrapolation to all weights is explained in [1205.1516], together with the techniques to also reduce closedstring tree amplitudes to matrices M_{w} and vectors A_{YM}. The polynomial structure of the matrices M_{w} and P_{w} was systematically addressed in [1304.7267] and [1304.7304]. The purpose of this website is to make the explicit results of these references publicly available:
At higher weights, we keep the data size under control by suppressing redundant information: starting at weight w=9 for N=5 and weight w=6 for N=7, the links lead to the ﬁrst row of the matrices M_{w} and P_{w} associated with the canonical color ordering 1, 2, 3, . . . , N . As spelled out in [1304.7267], the remaining entries can be obtained by suitable relabelling. According to [1304.7304], the complete α'expansion of the Npoint tree amplitude can be obtained from the Drinfeld associator Φ(e_{0}, e_{1}) recursively. Herein, e_{0} and e_{1} are (N − 2)! × (N − 2)!matrices linear in s_{ij} acting on worldsheet functions of the (N − 1)point amplitude. For multiplicities 5 through 9, the matrices can be downloaded here:

