Opuscula Math. 42, no. 3 (2022), 361-391
https://doi.org/10.7494/OpMath.2022.42.3.361
Opuscula Mathematica
Monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities
Abstract. We study the monodromy invariant Hermitian forms for second order Fuchsian differential equations with four singularities. The moduli space of our monodromy representations can be realized by certain affine cubic surface. In this paper we characterize the irreducible monodromies having the non-degenerate invariant Hermitian forms in terms of that cubic surface. The explicit forms of invariant Hermitian forms are also given. Our result may bring a new insight into the study of the Painlevé differential equations.
Keywords: Fuchsian differential equations, monodromy representation, monodromy invariant Hermitian form.
Mathematics Subject Classification: 34M35, 34M15.
- M. Acosta, Character varieties for real forms, Geometriae Dedicata 203 (2019), 257-277.
- S. Adachi, Y. Haraoka, Monodromy invariant Hermitian forms for non-rigid Fuchsian differential equations of order three, submitted.
- F. Beukers, G. Heckman, Monodromy for the hypergeometric function \({}_nF_{n-1}\), Invent. Math. 95 (1989), 325-354.
- P. Boalch, From Klein to Painlevé via Fourier, Laplace and Jimbo, Proc. London Math. Soc. (3) 90 (2005), 167-208.
- P. Calligaris, M. Mazzocco, Finite orbits of the pure braid group on the monodromy of the \(2\)-variable Garnier system, J. Integrable Syst. 3 (2018), no. 1, xyy005.
- V.S. Dotsenko, V.A. Fateev, Conformal algebra and multipoint correlation function in 2D statistical models, Nuclear Phys. B 240 (1984), 312-348.
- O. Gamayun, N. Iorgov, O. Lisovyy, Conformal field theory of Painlevé VI, J. High Energy Phys. 10 (2012), 038.
- Y. Haraoka, Finite monodromy of Pochhammer equation, Ann. Inst. Fourier. 44 (1994), 767-810.
- Y. Haraoka, Monodromy representations of systems of differential equations free from accessory parameters, SIAM J. Math. Anal. 25 (1994), 1595-1621.
- Y. Haraoka, S. Hamaguchi, Topological theory for Selberg type integral associated with rigid Fuchsian systems, Math. Ann. 353 (2012), 1239-1271.
- N. Iorgov, O. Lisovyy, J. Teschner, Isomonodromic tau-functions from Liouville conformal blocks, Comm. Math. Phys. 336 (2015), 671-694.
- K. Iwasaki, A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation, Proc. Japan Acad. 78, Ser. A (2002), 131-135.
- K. Iwasaki, An area-preserving action of the modular group on cubic surfaces and the Painlevé VI equation, Commun. Math. Phys. 242 (2003), 185-219.
- M. Jimbo, Monodromy problem and the boundary condition for some Painlevé equations, Publ. Res. Inst. Math. Sci. 18 (1982), 1137-1161.
- M. Kita, M. Yoshida, Intersection theory for twisted cycles, Math. Nachr. 166 (1994), 287-304.
- Shunya Adachi
- Kumamoto University, Graduate School of Science and Technology, 2-39-1 Kurokami, Chuo-ku, Kumamoto 860-8555, Japan
- Communicated by P.A. Cojuhari.
- Received: 2021-12-17.
- Accepted: 2022-01-23.
- Published online: 2022-04-29.
