[Paper Review] BPZ equations for higher degenerate fields and nonperturbative Dyson-Schwinger equations

Saebyeok Jeong, Xinyu Zhang|arXiv (Cornell University)|Oct 19, 2017

Black Holes and Theoretical Physics1 references3 citations

TL;DR

This paper establishes a direct correspondence between the BPZ equations in 2D Liouville field theory involving higher degenerate fields and non-perturbative Dyson-Schwinger equations in 4D $υ=2$ superconformal quiver gauge theories. By deriving third-order differential equations from both sides—via null vector decoupling in Liouville theory and instanton constraints in gauge theory—it confirms the BPS/CFT correspondence at the non-perturbative level, identifying the instanton partition function with Liouville conformal blocks.

ABSTRACT

In the two-dimensional Liouville conformal field theory, correlation functions involving a degenerate field satisfy partial differential equations due to the decoupling of the null descendant field. On the other hand, the instanton partition function of a four-dimensional $\mathcal{N}=2$ supersymmetric theory in the $Ω$-background at a special point of the parameter space also satisfies a partial differential equation resulting from the constraints of the gauge field configurations. This partial differential equation can be proved using the nonperturbative Dyson-Schwinger equations. We show for the next-to-simplest case that the partial differential equations obtained from two different perspectives can be identified, thereby confirming an assertion of the BPS/CFT correspondence.

Motivation & Objective

• To establish a non-perturbative link between correlation functions in 2D Liouville field theory and partition functions in 4D $υ=2$ superconformal quiver gauge theories.
• To verify the BPS/CFT correspondence by identifying differential equations arising from degenerate fields in Liouville theory with those derived from non-perturbative Dyson-Schwinger equations in gauge theories.
• To analyze the case of the next-to-simplest degenerate field (level-3 null vector) and its associated third-order BPZ equation.
• To derive the instanton partition function’s differential equation using non-perturbative Dyson-Schwinger equations and match it with the Liouville side.
• To confirm the AGT correspondence beyond the Nekrasov-Shatashvili limit by identifying the same third-order differential structure on both sides.

Proposed method

• Utilizes the Alday-Gaiotto-Tachikawa (AGT) correspondence to relate Liouville correlation functions to instanton partition functions in $υ=2$ gauge theories.
• Applies the non-perturbative Dyson-Schwinger equations (NPDS) to derive a differential equation for the instanton partition function under gauge field configuration constraints.
• Derives the third-order BPZ equation for Liouville correlation functions involving a level-3 degenerate field via null descendant decoupling.
• Identifies the differential equations on both sides (Liouville and gauge theory) as equivalent, confirming structural consistency.
• Uses generalized hypergeometric functions and Pochhammer symbols to express solutions and verify differential equation structure.
• Extracts the $U(1)$ factor of the instanton partition function from NPDS equations, showing its dependence on $α$-parameters and $z$-variables.

Experimental results

Research questions

• RQ1 Can the third-order differential equation satisfied by Liouville correlation functions with a next-to-simplest degenerate field be matched with a differential equation derived from non-perturbative Dyson-Schwinger equations in gauge theory?
• RQ2 Does the non-perturbative Dyson-Schwinger approach yield a differential equation for the instanton partition function that matches the BPZ equation from Liouville theory?
• RQ3 How does the $U(1)$ factor of the instanton partition function emerge from the NPDS equations, and what is its structure in terms of $α$-parameters and $z$-variables?
• RQ4 To what extent does the AGT correspondence hold beyond the Nekrasov-Shatashvili limit, particularly at the level of differential equations?
• RQ5 Can the conformal block structure in Liouville theory be fully recovered from the instanton partition function via the NPDS framework?

Key findings

• The third-order differential equation derived from the Liouville correlation function with a level-3 degenerate field matches exactly with the differential equation obtained from the non-perturbative Dyson-Schwinger equations in the gauge theory side.
• The instanton partition function satisfies a differential equation that is structurally identical to the BPZ equation, confirming the BPS/CFT correspondence at the non-perturbative level.
• The $U(1)$ factor of the instanton partition function is derived as $Z^{\mathrm{instanton}}=\prod_{0\leq i<j\leq r}\left(1-\frac{z_{j}}{z_{i}}\right)^{-\frac{(\bar{a}_{i}-\bar{a}_{i+1})(\bar{a}_{j}-\bar{a}_{j+1}+\varepsilon)}{\varepsilon_{1}\varepsilon_{2}}}$, showing explicit dependence on $α$-differences and $z$-ratios.
• The residue calculation at $t = -z_j^{-1}$ yields a consistent expression for $\langle k_j - k_{j+1} \rangle$, linking the instanton partition function's logarithmic derivative to the $α$-parameters and $z$-variables.
• The generalized hypergeometric function ${}_3F_2$ arises as a solution to the derived differential equations, confirming the analytic structure of the conformal blocks.
• The matching of differential equations across both sides provides strong evidence for the validity of the AGT correspondence beyond the perturbative regime, particularly in the non-perturbative $υ=2$ gauge theory context.