Rigid local systems and the multiplicative eigenvalue problem

• 成果类型: Article
• 署名作者: Belkale, Prakash
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2022.195.3.3
• 发表日期: 2022
• 页码: 911-995
• 关键词: level-rank duality; conformal blocks; honeycomb model; strange duality; vector-bundles; PRODUCTS; MODULI; COHOMOLOGY; monodromy; unitarity

摘要

We give a construction that produces irreducible complex rigid local systems on P-C(1) - {p(1), ..., P-s} via quantum Schubert calculus and strange duality. These local systems are unitary and arise from a study of vertices in the polytopes controlling the multiplicative eigenvalue problem for the special unitary groups SU(n) (i.e., determination of the possible eigenvalues of a product of unitary matrices given the eigenvalues of the matrices). Roughly speaking, we show that the strange duals of the simplest vertices of these polytopes give all possible unitary irreducible rigid local systems. As a consequence we obtain that the ranks of unitary irreducible rigid local systems, including those with finite global monodromy, on P-1 - S are bounded above if we fix the cardinality of the set S = {p(1), ..., p(s)} and require that the local monodromies have orders that divide n for a fixed n. Answering a question of N. Katz, we show that there are no irreducible rigid local systems of rank greater than one, with finite global monodromy, all of whose local monodromies have orders dividing n, when n is a prime number. We also show that all unitary irreducible rigid local systems on P-C(1) - S with finite local monodromies arise as solutions to the Knizhnik-Zamalodchikov equations on conformal blocks for the special linear group. Along the way, generalizing previous works of the author and J. Kiers, we give an inductive mechanism for determining all vertices in the multiplicative eigenvalue problem for SU(n).