2d-4d Connection between q-Virasoro/W Block at Root of Unity Limit and Instanton Partition Function on ALE Space

TL;DR

This paper establishes a connection between q-Virasoro/W blocks at roots of unity and instanton partition functions on ALE spaces, providing a unified framework that simplifies previous case-by-case conjectures.

Contribution

It introduces a limiting procedure linking 2d q-deformed conformal blocks with 4d instanton partition functions on ALE spaces, revealing algebraic structures like super-Virasoro and parafermions.

Findings

Generated super-Virasoro algebra in the root of unity limit.

Derived the projection of 5d instanton partition functions onto ALE spaces.

Identified parafermion-like currents at roots of unity.

Abstract

We propose and demonstrate a limiting procedure in which, starting from the q-lifted version (or K-theoretic five dimensional version) of the (W)AGT conjecture to be assumed in this paper, the Virasoro/W block is generated in the r-th root of unity limit in q in the 2d side, while the same limit automatically generates the projection of the five dimensional instanton partition function onto that on the ALE space R^4/Z_r. This circumvents case-by-case conjectures to be made in a wealth of examples found so far. In the 2d side, we successfully generate the super-Virasoro algebra and the proper screening charge in the q -> -1, t -> -1 limit, from the defining relation of the q-Virasoro algebra and the q-deformed Heisenberg algebra. The central charge obtained coincides with that of the minimal series carrying odd integers of the N=1 superconformal algebra. In the r-th root of unity limit in q in the 2d side, we give some evidence of the appearance of the parafermion-like currents. Exploiting the q-analysis literatures, q-deformed su(n) block is readily generated both at generic q, t and the r-th root of unity limit. In the 4d side, we derive the proper normalization function for general (n, r) that accomplishes the automatic projection through the limit.