A simple construction of Grassmannian polylogarithms

We give a simple explicit construction of the Grassmannian n-logarithm, which is a multivalued analytic function on the quotient of the Grassmannian of generic n-dimensional subspaces in 2n-dimensional coordinate complex vector space by the action of the 2n-dimensional coordinate torus. We study Tate iterated integrals, which are homotopy invariant integrals of 1-forms dlog(rational functions). We introduce the Hopf algebra of integrable symbols related to an algebraic variety, which controls the Tate iterated integrals We give a simple explicit formula for the Tate iterated integrals related to the Grassmannian polylogarithms.

💡 Research Summary

The paper presents a new, elementary construction of the Grassmannian n‑logarithm, a multivalued analytic function defined on the quotient of the Grassmannian of generic n‑dimensional subspaces in a 2n‑dimensional complex coordinate space by the action of the 2n‑dimensional coordinate torus. The authors achieve this by introducing two central concepts: Tate iterated integrals and the Hopf algebra of integrable symbols associated with an algebraic variety.

First, a Tate iterated integral is defined as a homotopy‑invariant iterated integral of 1‑forms of the type d log (f), where f is a rational function. For a matrix A∈Mat_{n×2n}(ℂ) representing a point of the Grassmannian G(n,2n), the Plücker coordinates Δ_I (determinants of n×n minors) give rise to rational functions r_{I,J}=Δ_I/Δ_J. The 1‑forms ω_{I,J}=d log r_{I,J} are then used to build iterated integrals
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