The Endless Beta Integrals

Abstract

We consider a special degeneration limit ${\omega }_1\to -{\omega }_2$ (or $b\to \mathrm{i}$ in the context of $2d$ Liouville quantum field theory) for the most general univariate hyperbolic beta integral. This limit is also applied to the most general hyperbolic analogue of the Euler-Gauss hypergeometric function and its $W(E_7)$ group of symmetry transformations. Resulting functions are identified as hypergeometric functions over the field of complex numbers related to the $\mathrm{SL}(2,\mathbb{C})$ group. A new similar nontrivial hypergeometric degeneration of the Faddeev modular quantum dilogarithm (or hyperbolic gamma function) is discovered in the limit ${\omega }_1\to {\omega }_2$ (or $b\to 1$).

arXiv commentequations (62), (69) and (72) corrected

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• A parafermionic hypergeometric function and supersymmetric 6j-symbols
• Introduction to the theory of elliptic hypergeometric integrals
• Elliptic hypergeometric function and $6j$-symbols for the SL(2,$\mathbb{C}$) group
• On the Dotsenko-Fateev complex twin of the Selberg integral and its extensions
• Complex and rational hypergeometric functions on root systems
• Rational hypergeometric identities
• BC-type open $SL(2,\mathbb{C})$ spin chain
• From hyperbolic to complex Euler integrals