From the paper: The author gives the necessary and sufficient conditions for the existence of unitary local systems with prescribed local monodromies on \(\mathbb{P}^1-S\) where \(S\) is a finite set. This is used to give an algorithm to decide if a rigid local system on \(\mathbb{P}^1-S\) has finite global monodromy, thereby answering the following question of N. M. Katz [Rigid local systems, Ann. Math. Stud. 139 (1969; Zbl 0864.14013)]. Let \(L\) be a rigid local system on \(\mathbb{P}^1-\{p_1, \dots,p_s\}\) with finite monodromies at the punctures. When does the local system \(L\) have finite global monodromy (in terms of the Jordan canonical forms of the local monodromies around the punctures)? Clearly the local monodromies should be of finite order. But of course there are more conditions. The above problem is related to the following problem concerning \(\text{SU} (n)\): Let \(\overline A_1,\overline A_2,\dots,\overline A_s\) be conjugacy classes in \(\text{SU}(n)\). When can we lift to matrices \(A_i\) in \(\text{SU}(n)\) with the conjugacy class of \(A_i=\overline A_i\) and so that \(A_1 A_2\dots A_s=I\)?
We see that an affirmative answer for all \(\text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})\) conjugates of the local monodromies for the second problem is what is needed for the first problem. The \(\text{SU} (n)\) problem is related to quantum cohomology. I. Biswas had previously considered and solved this problem for \(\text{SU}(2)\).
Firstly, by the theorem of V. Mehta and C. Seshadri [Math. Ann. 248, 205–239 (1980; Zbl 0454.14006)] (modified to the \(\text{SU}(n)\) setting in the appendix to this paper) the existence problem for lifting is the same as the existence of a semistable parabolic vector bundle with prescribed local weights on \(\mathbb{P}^1\). Using the openness of semi-stability, this is reduced to checking if the trivial vector bundle with generic flags and the prescribed weights is semistable. This means that no subbundle should contradict semi-stability. The subbundles of a given rank and degree of the trivial vector bundle of rank \(n\) form an open subset of a Quot scheme and also of the moduli space of maps from \(\mathbb{P}^1\) to an appropriate Grassmann variety. Now the question of existence of a subbundle is translated into existence of a map from \(\mathbb{P}^1\) to a Grassmann variety, such that the prescribed points on \(\mathbb{P}^1\) go to appropriate generic Schubert cycles. The existence is (with a little bit more work) realized as the nonvanishing of certain Gromov-Witten numbers. The Harder-Narasimhan filtration is used to conclude that only inequalities corresponding to intersections which are numerically one need to be considered.
The above problem is related to one considered by A. A. Klyachko [Sel. Math., New Ser. 4, 419–445 (1998; Zbl 0915.14010)]. Using the Harder-Narasimhan filtration in that context the author concludes that only intersections which are numerically one need to be considered.