Mathematics – Algebraic Geometry
Scientific paper
2003-09-20
Internat. Math. Res. Notices, 2004, no. 1, 1-30
Mathematics
Algebraic Geometry
24pages, 4 figures, 3 eps files
Scientific paper
It is well known that the sixth Painlev\'e equation $\PVI$ admits a group of B\"acklund transformations which is isomorphic to the affine Weyl group of type $\mathrm{D}_4^{(1)}$. Although various aspects of this unexpectedly large symmetry have been discussed by many authors, there still remains a basic problem yet to be considered, that is, the problem of characterizing the B\"acklund transformations in terms of Riemann-Hilbert correspondence. In this direction, we show that the B\"acklund transformations are just the pull-back of very simple transformations on the moduli of monodromy representations by the Riemann-Hilbert correspondence. This result gives a natural and clear picture of the B\"acklund transformations. Key words: B\"acklund transformation, the sixth Painlev\'e equation, Riemann-Hilbert correspondence, isomonodromic deformation, affine Weyl group of type $\mathrm{D}_4^{(1)}$.
Inaba Michi-aki
Iwasaki Katsunori
Saito Masa-Hiko
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