Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Quantum Corner VOA and the Super Macdonald Polynomials

Panupong Cheewaphutthisakun, Jun'ichi Shiraishi, Keng Wiboonton

TL;DR

This work establishes a precise link between the quantum corner VOA $q\widetilde{Y}_{L,0,N}[\Psi]$ and Sergeev-Veselov super Macdonald polynomials, showing that correlation functions of the currents correspond to the super Macdonald polynomials under a specific map. The authors build on Miura transformations and horizontal Fock tensor products of the quantum toroidal $\mathfrak{gl}_1$ algebra to realize the currents and prove a main Theorem via a combinatorial lemma expressed as sums over reverse SSYBTs. The strategy reduces the proof to a detailed lemma about partitions and reverse tableaux, treated case-by-case: $(N,0)$, $(N,M)$, and $(0,M)$, with extensive inductive and cancellation arguments. The results extend the known quantum $W_N$–Macdonald correspondence to the corner-VOA setting and its super-analogue, with potential implications for five-dimensional gauge theories and related geometric representation theory. The paper also sketches several avenues for future exploration, including broader color patterns, higher-order currents, and deeper structural connections between VOAs and partially symmetric polynomials.

Abstract

In this paper, we establish a relation between the quantum corner VOA $q\widetilde{Y}_{L,0,N}[Ψ]$, which can be regarded as a generalization of quantum $W_N$ algebra, and Sergeev-Veselov super Macdonald polynomials. We demonstrate precisely that, under a specific map, the correlation functions of the currents of $q\widetilde{Y}_{L,0,N}[Ψ]$, coincide with the Sergeev-Veselov super Macdonald polynomials.

Quantum Corner VOA and the Super Macdonald Polynomials

TL;DR

This work establishes a precise link between the quantum corner VOA and Sergeev-Veselov super Macdonald polynomials, showing that correlation functions of the currents correspond to the super Macdonald polynomials under a specific map. The authors build on Miura transformations and horizontal Fock tensor products of the quantum toroidal algebra to realize the currents and prove a main Theorem via a combinatorial lemma expressed as sums over reverse SSYBTs. The strategy reduces the proof to a detailed lemma about partitions and reverse tableaux, treated case-by-case: , , and , with extensive inductive and cancellation arguments. The results extend the known quantum –Macdonald correspondence to the corner-VOA setting and its super-analogue, with potential implications for five-dimensional gauge theories and related geometric representation theory. The paper also sketches several avenues for future exploration, including broader color patterns, higher-order currents, and deeper structural connections between VOAs and partially symmetric polynomials.

Abstract

In this paper, we establish a relation between the quantum corner VOA , which can be regarded as a generalization of quantum algebra, and Sergeev-Veselov super Macdonald polynomials. We demonstrate precisely that, under a specific map, the correlation functions of the currents of , coincide with the Sergeev-Veselov super Macdonald polynomials.

Paper Structure

This paper contains 24 sections, 47 theorems, 282 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Super Macdonald Polynomials
  3. Partitions, (reverse) semi-standard Young tableaus, (reverse) semi-standard Young bitableaus
  4. The algebra of symmetric polynomials
  5. Macdonald polynomials
  6. Super Macdonald polynomials
  7. Quantum Corner VOA
  8. Quantum toroidal algebra of type $\mathfrak{gl}_1$
  9. Quantum corner VOA
  10. Quantum Corner VOA/Super Macdonald correspondence
  11. Proof of lemma \ref{['lemm44-1106-29jan']} for the case $(N,0)$ where $N \in \mathbb{Z}^{>0}$
  12. Basis step $N = 1$
  13. Inductive step
  14. Proof of lemma \ref{['lemm44-1106-29jan']} for the case $(N,M)$ where $N, M \in \mathbb{Z}^{>0}$
  15. Basis step $M = 1$
  16. ...and 9 more sections

Key Result

Proposition 2.20

Let $T \in \operatorname{RSSYBT}(N,M;\lambda)$, and let $T_1$ be the subdiagram of $T$ consisting of the boxes with super numbers. Then, the shape of $T_1$ is a Young diagram of a partition.

Figures (6)

  • Figure 1: The positions highlighted in yellow represent the allowed positions for the boxes containing the number $1$.
  • Figure 2: $T(i_1,\dots,i_k ; \lambda) \in \operatorname{RSSYBT}(N,1;\lambda)$ with $c$ boxes labeled by $N+1$.
  • Figure 3: $T(i^\prime_1,\dots,i^\prime_k ; \lambda)$ obtained by replacing all $\textcolor{blue}{N+1}$ in $T(i_1,\dots,i_k ; \lambda)$ with $N+1,N+2,\dots,N+c$.
  • Figure 4: The blue boxes represent the possible positions of the boxes with super number $N+1$.
  • Figure 5: The reverse SSYBT $T_1(i_1,\dots,i_k)$ contains boxes assigned the super number $N+1$ within the row intervals $\gamma_1 = 2, \gamma_2 =3$, and $\gamma_3 = 4$. The number of boxes in each interval is $c_1 = 2, c_2 = 1,$ and $c_3 = 1$, respectively.
  • ...and 1 more figures

Theorems & Definitions (134)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Example 2.7
  • Definition 2.9: Arm length and leg length
  • Definition 2.11
  • Definition 2.12
  • ...and 124 more