Parabolic Degrees and Lyapunov Exponents for Hypergeometric Local Systems

Abstract

Consider the flat bundle on ${\mathbb{P}}^1-\{0,1,\infty \}$ corresponding to solutions of the hypergeometric differential equation

$${\prod }_{i=1}^n(\mathrm{D}-{\alpha }_i)-z{\prod }_{j=1}^n(\mathrm{D}-{\beta }_j)=0,\text{where\hspace{0.5em}}\mathrm{D}=z\frac{\mathrm{d}}{\mathrm{d}z}$$

For α_i and β_j real numbers, this bundle is known to underlie a complex polarized variation of Hodge structure. Setting the complete hyperbolic metric on ${\mathbb{P}}^1-\{0,1,\infty \},$ we associate n Lyapunov exponents to this bundle. We study the dependence of these exponents on parameters ${\alpha }_i,{\beta }_j$ through algebraic computations and numerical simulations, and point out new equality cases of the exponents with parabolic degrees of these bundles.

Keywords:

• hypergeometric functions
• Higgs bundle
• variation of Hodge structure
• parabolic degree
• Lyapunov exponent

MATHEMATICS SUBJECT CLASSIFICATION:

• 34D08; 37D50; 14D07; 33C20; 22E40

Acknowledgments

These experiments were suggested by Maxim Kontsevich, I am very grateful to him for suggesting this problem, sharing his initial experiments, and for his involvement. I thank dearly Jeremy Daniel for his curiosity to the subject and his answer to my myriad of questions as well as Bertrand Deroin; Anton Zorich for his flawless support and attention, Martin Müller and Roman Fedorov for taking time to explain their understanding of the parabolic degrees and Hodge invariants at MPIM in Bonn. I am also very thankful to Carlos Simpson for his kind answers and encouragements, and to Yuri Manin for pointing out a possible link to cosmology.