Essays on the theory of elliptic hypergeometric functions

Abstract

We give a brief review of the main results of the theory of elliptic hypergeometric functions -- a new class of special functions of mathematical physics. We prove the most general univariate exact integration formula generalizing Euler's beta integral, which is called the elliptic beta integral. An elliptic analogue of the Gauss hypergeometric function is constructed together with the elliptic hypergeometric equation for it. Biorthogonality relations for this function and its particular subcases are described. We list known elliptic beta integrals on root systems and consider symmetry transformations for the corresponding elliptic hypergeometric functions of the higher order.

62 pages

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• Comment on star-star relations in statistical mechanics and elliptic gamma-function identities
• Baxter operators with deformed symmetry
• A new pentagon identity for the tetrahedron index
• Heterotic Surface Defects and Dualities from 2d/4d Indices
• Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model
• Elliptic Hypergeometric Laurent Biorthogonal Polynomials with a Dense Point Spectrum on the Unit Circle
• Elliptic beta integrals and solvable models of statistical mechanics
• Integrable (2+1)-dimensional systems of hydrodynamic type
• The Endless Beta Integrals
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• Surface defects as transfer matrices
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• A noncommutative weight-dependent generalization of the binomial theorem
• Charging the Superconformal Index
• New elliptic solutions of the Yang-Baxter equation
• Supersymmetric Casimir Energy and $\mathrm{SL}(3,\mathbb{Z})$ Transformations
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• Elliptic Ding-Iohara Algebra and the Free Field Realization of the Elliptic Macdonald Operator
• From Principal Series to Finite-Dimensional Solutions of the Yang-Baxter Equation
• Mathematical structures behind supersymmetric dualities
• From rarefied elliptic beta integral to parafermionic star-triangle relation
• A duality web of linear quivers
• Universal volume of groups and anomaly of Vogel's symmetry
• Limits of elliptic hypergeometric biorthogonal functions
• Intertwining operators for Sklyanin algebra and elliptic hypergeometric series
• Self-similar potentials in quantum mechanics and coherent states
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• An Elliptic Garnier System from Interpolation
• A parafermionic hypergeometric function and supersymmetric 6j-symbols
• Padé interpolation for elliptic Painlevé equation
• Elliptic Well-Poised Bailey Transforms and Lemmas on Root Systems
• Integral pentagon relations for 3d superconformal indices
• Modified elliptic gamma functions and 6d superconformal indices
• $C_2$ generalization of the van Diejen model from the minimal $(D_5,D_5)$ conformal matter
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• 4d $\mathcal{N}=1$ quiver gauge theories and the ${\mathrm{A}}_{\mathrm{n}}$ Bailey lemma
• S-move matrix in the NS sector of $N=1$ super Liouville field theory
• Rank $Q$ E-String on Spheres with Flux
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• Rational hypergeometric identities
• Electric potentials and field lines for uniformly-charged tube and cylinder expressed by Appell's hypergeometric function and integration of $Z(u|m)\mathrm{sc}(u|m)$
• Complex rational Ruijsenaars model. The two-particle case
• Complex and rational hypergeometric functions on root systems
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• A study of N =1 SCFT derived from N =2 SCFT: index and chiral ring
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