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Kac determinant and singular vector of the level N representation of Ding-Iohara-Miki algebra

Yusuke Ohkubo

TL;DR

The paper derives a closed-form Kac determinant for the level-$N$ representation of the Ding-Iohara-Miki (DIM) algebra, revealing how the PBW-type vectors form a basis for generic parameter choices. It achieves this by decomposing the level-$N$ representation into a deformed $W_N$ sector and a $U(1)$ Heisenberg sector via a linear boson transformation and employing screening currents to construct singular vectors, which in turn determine the determinant’s vanishing lines. A key result is that singular vectors of the DIM algebra coincide with generalized Macdonald functions, the q-deformed AFLT basis, with an explicit identification $|\chi_{\vec{r},\vec{s}}\rangle \propto |P_{\Theta_{\vec{r},\vec{s}}}\rangle$. The work connects DIM representation theory to AGT/Nekrasov structures by providing a complete basis and clear links between generalized and ordinary Macdonald functions, broadening the resonance with 5D gauge theories and $qW$-algebras.

Abstract

In this paper, we obtain the formula for the Kac determinant of the algebra arising from the level $N$ representation of the Ding-Iohara-Miki algebra. It is also discovered that its singular vectors correspond to generalized Macdonald functions (the q-deformed version of the AFLT basis).

Kac determinant and singular vector of the level N representation of Ding-Iohara-Miki algebra

TL;DR

The paper derives a closed-form Kac determinant for the level- representation of the Ding-Iohara-Miki (DIM) algebra, revealing how the PBW-type vectors form a basis for generic parameter choices. It achieves this by decomposing the level- representation into a deformed sector and a Heisenberg sector via a linear boson transformation and employing screening currents to construct singular vectors, which in turn determine the determinant’s vanishing lines. A key result is that singular vectors of the DIM algebra coincide with generalized Macdonald functions, the q-deformed AFLT basis, with an explicit identification . The work connects DIM representation theory to AGT/Nekrasov structures by providing a complete basis and clear links between generalized and ordinary Macdonald functions, broadening the resonance with 5D gauge theories and -algebras.

Abstract

In this paper, we obtain the formula for the Kac determinant of the algebra arising from the level representation of the Ding-Iohara-Miki algebra. It is also discovered that its singular vectors correspond to generalized Macdonald functions (the q-deformed version of the AFLT basis).

Paper Structure

This paper contains 6 sections, 7 theorems, 109 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Kac determinant of level $N$ representation of DIM algebra
  3. Proof of Theorem \ref{['thm:KacDet']}
  4. Singular vectors and generalized Macdonald functions
  5. Macdonald functions
  6. Definition of DIM algebra and level $N$ representation

Key Result

Theorem 2.6

Let $\mathrm{det}_n \mathbin{:=} \det \left( \Braket{X_{\vec{\lambda}}|X_{\vec{\mu}}} \right)_{\vec{\lambda}, \vec{\mu} \vdash n}$. Then where $b_{\lambda}(q) \mathbin{:=} \prod_{i\geq 1} \prod_{k=1}^{m_i} (1-q^k)$, $b'_{\lambda}(q) \mathbin{:=} \prod_{i\geq 1} \prod_{k=1}^{m_i} (-1+q^k)$. $P^{(N)}(n)$ denotes the number of $N$-tuples of Young diagrams of size $n$, i.e., $\# \{ \vec{\lambda}=(\la

Figures (3)

  • Figure 1: Young diagram with $N-1$ edges
  • Figure 2: Young diagram corresponding to singular vector in the case that $r_{k+1} \geq r_k$ for all $k$.
  • Figure 3: The example of $\Theta_{\vec{r},\vec{s}}$ when $N=6$ and $r_1>r_4>r_3>r_5>r_2$.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Definition 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Lemma 3.1
  • Definition 4.1
  • Theorem 4.3
  • ...and 5 more