[Paper Review] Conformal blocks and Painlevé functions

Hajime Nagoya|arXiv (Cornell University)|Nov 28, 2016

Advanced Algebra and Geometry17 references9 citations

TL;DR

This paper proposes a conjectural combinatorial expansion formula for the three-point irregular conformal block with one irregular singular point of rank one and two regular singular points, expressed in terms of pairs of skew Young diagrams. The formula involves tau functions of Painlevé equations, with coefficients derived from products over hook-lengths and partition-dependent functions, extending the AGT correspondence framework to irregular conformal blocks via a novel combinatorial structure.

ABSTRACT

This paper is based on my presentation at RIMS workshop on "Theory of Integrable Systems and Its Applications in Various Fields" held in Kyoto on 19--21, August 2015. The aim of the present paper is to give a short account of recent studies on relations between conformal blocks in the two-dimensional conformal field theory and Painlevé functions. In addition, we present a conjecture on a combinatorial expansion formula of the three-point irregular conformal block at an irregular singular point, with two regular singular points and one irregular singular point. Our conjectural expansion formula is written in terms of pairs of skew Young diagrams, while the four-point regular conformal block, by AGT correspondence, is written in terms of pairs of Young diagrams.

Motivation & Objective

• To establish a combinatorial expansion formula for three-point irregular conformal blocks with one irregular and two regular singular points.
• To extend the AGT correspondence framework from regular four-point blocks (in terms of Young diagrams) to irregular three-point blocks.
• To propose a conjectural series expansion of the Painlevé tau function in terms of skew Young diagram pairs, generalizing known Fourier-type expansions.
• To identify the algebraic structure of coefficients in the expansion, conjecturing them to be non-negative integers with specific symmetries.
• To provide a systematic method for computing coefficients via partition functions and hook-length products.

Proposed method

• Proposes a series expansion of the irregular conformal block at an irregular singular point in terms of pairs of skew Young diagrams (λ, μ) with nested partitions ν ⊂ λ, η ⊂ μ.
• Introduces auxiliary functions: Uλ/ν = ∏(i,j)∈λ/ν (2(β−θ)+i−j), Vμ/η = ∏(i,j)∈μ/η (−2β+i−j), and Sλ,μ = ∏(i,j)∈λ [(β+i−j)²−θt²]/hλ(i,j)² × ∏(i,j)∈μ [(θ−β+i−j)²−θ₀²]/hμ(i,j)².
• Derives the expansion as t^{-2θt²−2β(θ−β)} e^{β/t} ∑λ,μ t^{|λ|+|μ|} ∑ν⊂λ,η⊂μ (−1)^{|ν|} cλ,μ^{ν,η} Uλ/ν Vμ/η Sλ,μ.
• Uses known results from degeneration limits of Nekrasov partition functions and integral representations of irregular vectors in Verma modules as foundational inputs.
• Employs the AGT correspondence as a guiding principle, extending the known Fourier expansion of PVI tau functions to irregular conformal blocks.
• Validates the structure via low-order coefficient calculations (t⁻², t⁻³, t⁻⁴), showing consistency with conjectured coefficient forms and symmetries.

Experimental results

Research questions

• RQ1 Can a combinatorial expansion formula be constructed for the three-point irregular conformal block with one rank-one irregular singular point and two regular singular points?
• RQ2 How do the coefficients in the expansion relate to Young diagram combinatorics, particularly skew diagrams and their symmetries?
• RQ3 Do the coefficients cλ,μ^{ν,η} remain non-negative integers across all terms, and what is their algebraic structure?
• RQ4 Is there a generalization of the AGT correspondence to irregular conformal blocks via skew Young diagram pairs?
• RQ5 Can the structure of the expansion be derived from degeneration limits of Nekrasov partition functions or integral representations of irregular vectors?

Key findings

• The conjectural expansion formula for the 3-point irregular conformal block is expressed as a sum over pairs of skew Young diagrams with coefficients cλ,μ^{ν,η} ∈ ℤ≥₀.
• The coefficient cλ,μ^{∅,∅} = 1, cλ,μ^{(1),(1)} = 2|λ||μ|, and cλ,μ^{(2),(2)} = qλ qμ, where qλ is a partition-dependent function involving hook-lengths and row/column sums.
• The coefficient cλ,μ^{(2),(1,1)} = 3 qλ qμ′, indicating a symmetry under transpose and index exchange.
• The expansion exhibits full symmetry: cλ,μ^{ν,η} = cμ,λ^{η,ν} = cλ′,μ′^{ν′,η′}, confirming invariance under partition duality and index swapping.
• The structure of the t⁻³ and t⁻⁴ coefficients is fully reproduced by the conjectured formula, with terms involving Uλ/ν, Vμ/η, and Sλ,μ, confirming consistency at low orders.
• The formula generalizes the known Fourier-type expansion of PVI tau functions to irregular blocks, replacing Young diagrams with skew Young diagram pairs.