B\"acklund Transformations of the Sixth Painlev\'e Equation in Terms of Riemann-Hilbert Correspondence

It is well known that the sixth Painlev'e equation $\text{PVI}$ 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 ${\mathrm{D}}_4^{(1)}$.