Parameterize the higher-point determinant factor in invariant cross ratios

Determine a parametrization of the factor \(\det^{\prime}(P)\) for \(n>4\) in terms of invariant cross ratios \(\vec{\sigma}\), subject to the condition \(\sum_{i=1}^{n-1}\lambda_i=0\), in order to analyze the Okuda–Penedones limit of higher-point \(\mathcal{N}=4\) SYM correlators.

Background

For higher-point correlators, the Okuda–Penedones limit relates the boundary correlator to a flat-space amplitude through an integral transform involving the nonzero eigenvalues of the matrix $P_{ij}$. The resulting expression contains the factor $\det^{\prime}(P)=\prod_{i=1}^{n-1}\lambda_i$, where the eigenvalues must be expressed in terms of the invariant cross ratios that remain fixed in the limit.

The paper notes that the eigenvalues satisfy $\sum_{i=1}^{n-1}\lambda_i=0$, so a suitable cross-ratio parametrization is needed for generic configurations. Such a parametrization would make the higher-point transform and the corresponding SYM correlators more explicit, but the paper does not provide one for $n>4$.