We give a construction that produces irreducible complex rigid local systems on $\mathbb {P}_{\mathbb {C}}^1-\{p_1,…,p_s\}$ 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 $\operatorname {SU}(n)$ (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 $\mathbb {P}^1-S$ 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\nonbreakingspace $n$. 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 $\mathbb {P}^1_{\mathbb {C}} -S$ 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 $\mathrm {SU}(n)$.

我们给出了一个构造，它通过量子舒伯特演算和奇异对偶在 $\mathbb {P}_{\mathbb {C}}^1-\{p_1,…,p_s\}$ 上产生不可约复刚性局部系统。这些局部系统是酉的，并且源于对控制特殊酉群 $\operatorname {SU}(n)$ 的乘法特征值问题的多面体中的顶点的研究（即，确定给定酉矩阵乘积的可能特征值）矩阵的特征值）。粗略地说，我们证明了这些多面体的最简单顶点的奇异对偶给出了所有可能的酉不可约刚性局部系统。因此，如果我们固定集合 $S=\{ p_1,…, p_s\}$ 并要求局部单调具有将 $n$ 划分为固定\非破坏空间 $n$ 的订单。回答 N. Katz 的一个问题，我们表明不存在秩大于 1 的不可约刚性局部系统，具有有限的全局单调，当 $n$ 是素数时，所有局部单调的阶数都除以 $n$。

我们还表明，$\mathbb {P}^1_{\mathbb {C}} -S$ 上所有具有有限局部单调的酉不可约刚性局部系统都是作为特殊线性群的保形块上的 Knizhnik-Zamalodchikov 方程的解而出现的. 在此过程中，概括作者和 J. Kiers 以前的工作，我们给出了一种归纳机制来确定 $\mathrm {SU}(n)$ 的乘法特征值问题中的所有顶点。