Fundamental solutions of the KZ equation on moduli spaces and the Riemann-Hilbert problem

2015 Fiscal Year Final Research Report

• Project/Area Number: 25400054
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Waseda University
• Principal Investigator: UENO KIMIO 早稲田大学, 理工学術院, 教授 (70160190)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: KZ方程式 / 多重対数関数 / 多重ゼータ値 / リーマン・ヒルベルト問題 / モノドロミー保存変形

Outline of Final Research Achievements

The inversion formulas of polylogarithms can be regarded as a recursive Riemann-Hilbert (RH, for short) problem of additive type. Solving this problem, we can chracterize the polylogarithms. This problem generalizes to the case for multiple polylogarithms (MPL, for short). From the inversion formulas of MPL, one can also obtain a recursive RH problem of additive type, and the unique solvability can be shown. Furthemore, one can show that this problem is equivalent to the multiplicative Riemann-Hilbert problem of the fundamental solution for the KZ equation of one variable. The article summarizing these results is now submited to a mathematical journal. Recently, we investigate the connection to monodromy preserving deformation, and addressed at the annual meetings of Japan Mathematical Society.

Free Research Field

自然科学