AFLT-type Selberg integrals

Abstract

In their 2011 paper on the AGT conjecture, Alba, Fateev, Litvinov and Tarnopolsky (AFLT) obtained a closed-form evaluation for a Selberg integral over the product of two Jack polynomials, thereby unifying the well-known Kadell and Hua--Kadell integrals. In this paper we use a variety of symmetric functions and symmetric function techniques to prove generalisations of the AFLT integral. These include (i) an ${\mathrm{A}}_n$ analogue of the AFLT integral, containing two Jack polynomials in the integrand; (ii) a generalisation of (i) for $\gamma =1$ (the Schur or GUE case), containing a product of $n+1$ Schur functions; (iii) an elliptic generalisation of the AFLT integral in which the role of the Jack polynomials is played by a pair of elliptic interpolation functions; (iv) an AFLT integral for Macdonald polynomials.

arXiv comment53 pages; v2 contains minor corrections and changes of notation

References in corpus (28)

• Liouville Correlation Functions from Four-dimensional Gauge Theories
• Liouville theory revisited
• Liouville Field Theory -- A decade after the revolution
• On combinatorial expansion of the conformal blocks arising from AGT conjecture
• An Algorithm for the Microscopic Evaluation of the Coefficients of the Seiberg-Witten Prepotential
• Advanced Determinant Calculus
• Proving AGT conjecture as HS duality: extension to five dimensions
• Instantons on Quivers and Orientifolds
• Integrable structure, W-symmetry and AGT relation
• Whittaker vectors of the Virasoro algebra in terms of Jack symmetric polynomial
• Hankel hyperdeterminants and Selberg integrals
• 2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space
• Selberg Integral and SU(N) AGT Conjecture
• Selberg Type Integrals Associated with $s{l}_3$
• Correspondence between conformal field theory and Calogero-Sutherland model
• A Selberg integral for the Lie algebra A_n
• The sl_3 Selberg integral
• Cubic curves from instanton counting
• Elliptic Selberg integrals
• (Shifted) Macdonald Polynomials: q-Integral Representation and Combinatorial Formula
• Trinion Conformal Blocks from Topological strings
• Elliptic hypergeometric functions associated with root systems
• Bisymmetric functions, Macdonald polynomials and sl_3 basic hypergeometric series
• Transformation formulae for multivariable basic hypergeometric series
• Evaluation of the $B{C}_n$ elliptic Selberg integral via the fundamental invariants
• Limits of multivariate elliptic hypergeometric biorthogonal functions
• Determinantal elliptic Selberg integrals
• A New Scheme for NMSSM in Gauge Mediation

Cited by in corpus (6)

• A slow review of the AGT correspondence
• An Elliptic Hypergeometric Function Approach to Branching Rules
• $q$-Difference Systems for the Jackson Integral of Symmetric Selberg Type
• AGT basis in SCFT for c=3/2 and Uglov Polynomials
• Proof of 5D $A_n$ AGT conjecture at $\beta =1$
• Proof of $A_n$ AGT conjecture at $\beta =1$