与$\CP1$分支覆盖有关的矩阵Riemann-Hilbert问题
Matrix Riemann-Hilbert problems related to branched coverings of $\CP1$
摘要：
在这些注记中，我们解决了一类具有拟置换单值群的Riemann-Hilbert（逆单值）问题，这些问题对应于$\CP1$的非奇异分支覆盖。该解是根据基础黎曼曲面上的Szegö核给出的。特别是，我们的建设提供了一类新的解决方案的Schlesinger系统。给出了相应的τ函数的显式计算结果，并利用黎曼曲面的Jacobi流形上的θ因子描述了τ函数的零点因子（所谓的Malgrange因子）.讨论了τ函数与黎曼曲面上拉普拉斯算子行列式的关系。
英文摘要：
In these notes we solve a class of Riemann-Hilbert (inverse monodromy) problems with quasi-permutation monodromy groups which correspond to non-singular branched coverings of $\CP1$. The solution is given in terms of Szegö kernel on the underlying Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. We present some results on explicit calculation of the corresponding tau-function, and describe divisor of zeros of the tau-function (so-called Malgrange divisor) in terms of the theta-divisor on the Jacobi manifold of the Riemann surface. We discuss the relationship of the tau-function to determinant of Laplacian operator on the Riemann surface.
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