by A Varchenko (Univ. North Carolina)

This book recounts the connections between multidimensional hypergeometric functions and representation theory. In 1984, physicists Knizhnik and Zamolodchikov discovered a fundamental differential equation describing correlation functions in conformal field theory. The equation is defined in terms of a Lie algebra. Kohno and Drinfeld found that the monodromy of the differential equation is described in terms of the quantum group associated with the Lie algebra. It turns out that this phenomenon is the tip of the iceberg. The Knizhnik–Zamolodchikov differential equation is solved in multidimensional hypergeometric functions, and the hypergeometric functions yield the connection between the representation theories of Lie algebras and quantum groups. The topics presented in this book are not adequately covered in periodicals.

• Introduction
• Construction of Complexes Calculating Homology of the Complement of a Configuration
• Construction of Homology Complexes for Discriminantal Configuration
• Algebraic Interpretation of Chain Complexes of a Discriminantal Configuration
• Quasiisomorphism of Two-Sided Hochschild Complexes to Suitable One-Sided Hochschild Complexes
• Bundle Properties of a Discriminantal Configuration
• R-Matrix for the Two-Sided Hochschild Complexes
• Monodromy
• R-Matrix Operator as the Canonical Element, Quantum Doubles
• Hypergeometric Integrals
• Kac–Moody Lie Algebras Without Serre's Relations and Their Doubles
• Hypergeometric Integrals of a Discriminantal Configuration
• Resonances at Infinity
• Degenerations of Discriminantal Configurations
• Remarks on Homology Groups of a Configuration with Coefficients in Local Systems More General than Complex One-Dimensional