The Hodge theory of Knizhnik-Zamolodchikov equations and Rigid Local Systems

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The Knizhnik-Zamolodchikov differential equations link many areas of mathematics and physics, including representation theory (the study of symmetries), algebraic geometry (study of solutions of polynomial equations), and conformal field theory (from mathematical physics). The goal of this project is to develop Hodge theoretic techniques, which are complex analytic in nature, in the study of these differential equations. This work will produce new structures relating representation theory, Hodge theory, the theory of motives, and mathematical physics, which will further develop the relations between these fields and their computational aspects. The project will provide research training opportunities for graduate students.In more detail, three interrelated research projects will be undertaken. The first, is to develop a motivic factorization formula for invariants and conformal block local systems (which are local systems on the moduli of smooth pointed curves associated to a simple Lie algebra and representations) in genus zero. The second is to determine the structure of the Hodge-Galois fusion rings and representation rings that arise as a consequence of motivic factorization. Finally, the third is to compute examples, particularly for cases when the conformal blocks local system is rigid where the global monodromy is finite, using Hodge theoretic criteria.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.