Quantum States of Monodromy Groups

Abstract

Any irreducible finitely generated matrix group (with generators M1,…,Mp+1 satisfying the only relation M1…Mp+1 e I) is the monodromy group of some fuchsian linear system on Riemann's sphere. The eigenvalues of the matrices Mj define λk,j, the eigenvalues of the matrices-residues of the system only up to integers. There are always infinitely many possible choices of λk,j, a priori they must satisfy the only condition that their sum is 0. However, not always all a priori possible choices can be made. Some of them can be impossible due to the positions of the poles. Consider the a priori possible choices when the eigenvalues of only one matrix-residuum change (we presume that its pole is at 0). We show that infinitely many a priori possible choices are impossible if and only if the fuchsian system is obtained from another fuchsian system with a smaller number of poles and with a pole at 0 by the change of time t\mapsto t^k / (p_kt^k+p_{k-1}t^{k-1}+\ldots + p_0), p_i \in {\Bbb C}, p_0 \neq 0, k > 1. The result is applied to the Riemann–Hilbert problem.