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Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence

Yusuke Ohkubo, Hidetoshi Awata, Hiroki Fujino

TL;DR

The paper analyzes the $q \to 0$ crystallization of the deformed Virasoro algebra and the level-$N$ representation of the Ding-Iohara-Miki algebra, linking these limits to Hall-Littlewood functions and enabling explicit AGT-related results. By constructing a free-field realization with bosons and identifying PBW-type vectors with Hall-Littlewood functions, the authors show the disappearance of singular vectors and derive a crystallized Nekrasov partition function that equals the norm of the crystallized Whittaker vector. For $N=1$ and $N=2$, they demonstrate that the crystallized PBW-type vectors form bases, develop generalized Hall-Littlewood functions and integral forms, and obtain explicit 4-point correlation functions consistent with a crystallized 5D AGT picture. The work provides a tractable, zero-temperature–like instantiation of AGT where Nekrasov factors reduce to Hall-Littlewood–based expressions, offering a foundation for extending crystallization to higher $N$ and other scaling limits with potential physical and mathematical insights.

Abstract

In this paper, we consider the $q \rightarrow 0$ limit of the deformed Virasoro algebra and that of the level 1, 2 representation of Ding-Iohara-Miki algebra. Moreover, 5D AGT correspondence at this limit is discussed. This specialization corresponds to the limit from Macdonalds functions to Hall-Littlewood functions. Using the theory of Hall-Littlewood functions, some problems are solved. For example, the simplest case of 5D AGT conjectures is proven at this limit, and we obtain a formula for the 4-point correlation function of a certain operator.

Crystallization of deformed Virasoro algebra, Ding-Iohara-Miki algebra and 5D AGT correspondence

TL;DR

The paper analyzes the crystallization of the deformed Virasoro algebra and the level- representation of the Ding-Iohara-Miki algebra, linking these limits to Hall-Littlewood functions and enabling explicit AGT-related results. By constructing a free-field realization with bosons and identifying PBW-type vectors with Hall-Littlewood functions, the authors show the disappearance of singular vectors and derive a crystallized Nekrasov partition function that equals the norm of the crystallized Whittaker vector. For and , they demonstrate that the crystallized PBW-type vectors form bases, develop generalized Hall-Littlewood functions and integral forms, and obtain explicit 4-point correlation functions consistent with a crystallized 5D AGT picture. The work provides a tractable, zero-temperature–like instantiation of AGT where Nekrasov factors reduce to Hall-Littlewood–based expressions, offering a foundation for extending crystallization to higher and other scaling limits with potential physical and mathematical insights.

Abstract

In this paper, we consider the limit of the deformed Virasoro algebra and that of the level 1, 2 representation of Ding-Iohara-Miki algebra. Moreover, 5D AGT correspondence at this limit is discussed. This specialization corresponds to the limit from Macdonalds functions to Hall-Littlewood functions. Using the theory of Hall-Littlewood functions, some problems are solved. For example, the simplest case of 5D AGT conjectures is proven at this limit, and we obtain a formula for the 4-point correlation function of a certain operator.

Paper Structure

This paper contains 16 sections, 17 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Crystallization of the deformed Virasoro algebra and its Whittaker vector
  3. Review of the simplest 5D AGT correspondence
  4. Crystallization of the deformed Virasoro algebra and AGT correspondence.
  5. Crystallization of the representation of Ding-Iohara-Miki algebra
  6. Reargument of Ding-Iohara-Miki algebra and AGT correspondence
  7. Crystallization of $N=1$ case
  8. Crystallization of $N=2$ case
  9. Other types of limit
  10. Appendix
  11. Macdonald functions and Hall-Littlewood functions
  12. Partial orderings
  13. Other proofs of Lemma \ref{['lem:exlicit form of inv. shapovalov']}
  14. Explicit form of $\left\langle Q_{(s)}(-p_n), Q_{\lambda}(p_n;t) \right\rangle_{0,t}$
  15. Check of (\ref{['eq:N=1 conjecture']})
  16. ...and 1 more sections

Key Result

Proposition 2.5

The renormalization $\tilde{\Lambda}^2\mathbin{:=} \Lambda^2 (q/t)^{\frac{1}{2}}$ controls divergence at the $q \rightarrow 0$ limit ($\Lambda \rightarrow \infty$, $\tilde{\Lambda}$ : fixed):

Theorems & Definitions (44)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Proposition 3.1
  • Conjecture 3.2
  • Definition 3.3
  • Proposition 3.4
  • Definition 3.5
  • ...and 34 more