Proving AGT relations in the large-c limit

TL;DR

This work analyzes the Virasoro 4-point conformal block in the large central charge limit $c\to\infty$, where the block becomes a hypergeometric function and the AGT relation reduces to a sum over chiral Nekrasov functions. By taking $\epsilon_1\to 0$ (or equivalently $c\to\infty$) with appropriate adjustments to the external dimensions, the authors show that only chiral Nekrasov functions with $(Y,Y')=([1^n],\emptyset)$ contribute, and the conformal block equals a $_2F_1$ series with explicit coefficients that match the Nekrasov sum. They derive a general selection rule and compute the first few orders in the $x$-expansion, demonstrating exact agreement with the hypergeometric block. The results extend the existing AGT proofs from special external states to generic external states in this limit and discuss implications for instanton-geometry regularization and potential connections to RG and Seiberg-Witten theory.

Abstract

In the limit of large central charge $c$ the 4-point Virasoro conformal block becomes a hypergeometric function. It is represented by a sum of chiral Nekrasov functions, which can also be explicitly evaluated. In this way the known proof of the AGT relation is extended from special to generic set of external states, but in the special limit of c=\infty.