Holonomic Quantum Fields. II — The Riemann–Hilbert Problem —

Abstract

In 1857 Riemann [4] (>i<) posed the question whether there exists, for given <?!, • • •, a n eP^ and M l9 • • •, M n <E:GL (m, C) , a differential equation (2.0. 1) which has a solution matrix Y(x) having precisely the monodromic property (2.0. 2).He imposed a further condition that Y(x) be at most regularly singular at the branch points a y (v = l, •••,?*, oo); namely that its singularities there be of the form (2. 0. 4) y(x) =0,(.r)• Cr-O~L> (v = l, ••-,«, oo) ( ** } where 0 y (x) is an invertible meromorphic matrix at x = a v , and L y is a constant matrix such that e ZniL » = M v .Since in 1900 Hilbert [5] included the above problem to his mathematical problems, it has often been called the Riemann-Hilbert problem.Considerable efforts has been made by a number of people [6], [7], [8], [10], [11], [12], [13], [14], toward the solution of the Riemann-Hilbert problem.Among them we note the names of (1) J. Plemelj [10], who presented an existence proof of a solution on P^ for arbitrary m and 11, (2) G. D. Birkhoff [11], who solved independently the original problem and its various generalizations previously proposed by himself, and (3) H. Rohrl [14], who extended the result of Plemelj to an arbitrary Riemann surface.A solution Y(x) to the Riemann-Hilbert problem (and hence the coefficient A (x) of the differential equation (2.0. 1)) depends on the initially specified branch points a v and the monodromy matrices M v , sometimes referred to as the Riemann data.L. Schlesinger [9] discussed this point as a deformation theory of differential equation (2.0. 1).Assuming that A (x) has the form Y] -, he asked for the condition v=i xa y for (2.0. 1) to have constant monodromy under the variation of the position of branch points, and obtained his celebrated equations (see § 2. 3, Proposition 2. 3. 12)(2. 0. 5)The methods so far employed to solve the Riemann-Hilbert problem (t) Riemann himself treated an w-th order equation for one unknown function.(**) p or y -oo we replace x-a y by \/x.••• \dxi'»dxm (resp.!••• \dui-"dum) means I dxv \ dxm (resp.I dui-I du m )