Towards a supersymmetric w 1 + ∞ symmetry in the celestial conformal field theory

The four different kinds of currents are given by the multiple (β, γ) and (b, c) ghost systems with a multiple product of derivatives. We determine their complete algebra where the structure constants depend on the deformation parameter λ appearing in the conformal weights of above fields nontrivially and depend on the generic spins h1 and h2 appearing on the left hand sides in the (anti)commutators. By taking the linear combinations of these currents, the $$ \mathcal{N} $$ N = 4 supersymmetric linear W∞[λ] algebra (and its $$ \mathcal{N} $$ N = 4 superspace description) for generic λ is obtained explicitly. Moreover, we determine the $$ \mathcal{N} $$ N = 2 supersymmetric linear W∞[λ] algebra for arbitrary λ. As a by product, the λ deformed bosonic W1+∞[λ] × W1+∞$$ \left[\lambda +\frac{1}{2}\right] $$ λ + 1 2 subalgebra (a generalization of Pope, Romans and Shen’s work in 1990) is obtained. The first factor is realized by (b, c) fermionic fields while the second factor is realized by (β, γ) bosonic fields. The degrees of the polynomials in λ for the structure constants are given by (h1 + h2 – 2). Each w1+∞ algebra from the celestial holography is reproduced by taking the vanishing limit of other deformation prameter q at λ = 0 with the contractions of the currents.

The $$ \mathcal{N} $$ = 2, 4 supersymmetric linear W∞[λ] algebras for generic λ parameter

Ahn,

Kim

2024

Supersymmetry and the celestial Jacobi identity

Ball,

Spradlin,

Srikant

et al. 2024

The $$ \mathcal{N} $$ = 2 supersymmetric w1+∞ symmetry in the two-dimensional SYK models

Ahn

2022

We identify the rank (qsyk + 1) of the interaction of the two-dimensional $$ \mathcal{N} $$ N = (2, 2) SYK model with the deformation parameter λ in the Bergshoeff, de Wit and Vasiliev (in 1991)’s linear W∞[λ] algebra via $$ \lambda =\frac{1}{2\left({q}_{\mathrm{syk}}+1\right)} $$ λ = 1 2 q syk + 1 by using a matrix generalization. At the vanishing λ (or the infinity limit of qsyk), the $$ \mathcal{N} $$ N = 2 supersymmetric linear $$ {W}_{\infty}^{N,N} $$ W ∞ N , N [λ = 0] algebra contains the matrix version of known $$ \mathcal{N} $$ N = 2 W∞ algebra, as a subalgebra, by realizing that the N-chiral multiplets and the N-Fermi multiplets in the above SYK models play the role of the same number of βγ and bc ghost systems in the linear $$ {W}_{\infty}^{N,N} $$ W ∞ N , N [λ = 0] algebra. For the nonzero λ, we determine the complete $$ \mathcal{N} $$ N = 2 supersymmetric linear $$ {W}_{\infty}^{N,N} $$ W ∞ N , N [λ] algebra where the structure constants are given by the linear combinations of two different generalized hypergeometric functions having the λ dependence. The weight-1,$$ \frac{1}{2} $$ 1 2 currents occur in the right hand sides of this algebra and their structure constants have the λ factors. We also describe the λ = $$ \frac{1}{4} $$ 1 4 (or qsyk = 1) case in the truncated subalgebras by calculating the vanishing structure constants.