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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2309.01537 |
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| _version_ | 1866914646586294272 |
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| author | Ahn, Changhyun Kim, Man Hea |
| author_facet | Ahn, Changhyun Kim, Man Hea |
| contents | 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 $h_1$ and $h_2$ appearing on the left hand sides in the (anti)commutators. By taking the linear combinations of these currents, the ${\cal N}=4$ supersymmetric linear $W_{\infty}[λ]$ algebra (and its ${\cal N}=4$ superspace description) for generic $λ$ is obtained explicitly. Moreover, we determine the ${\cal N}=2$ supersymmetric linear $W_{\infty}[λ]$ algebra for arbitrary $λ$. As a by product, the $λ$ deformed bosonic $W_{1+\infty}[λ] \times W_{1+\infty}[λ+\frac{1}{2}]$ 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 $(h_1+h_2-2)$. Each $w_{1+\infty}$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_01537 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The ${\cal N}=2,4$ Supersymmetric Linear $W_{\infty}[λ]$ Algebras for Generic $λ$ Parameter Ahn, Changhyun Kim, Man Hea High Energy Physics - 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 $h_1$ and $h_2$ appearing on the left hand sides in the (anti)commutators. By taking the linear combinations of these currents, the ${\cal N}=4$ supersymmetric linear $W_{\infty}[λ]$ algebra (and its ${\cal N}=4$ superspace description) for generic $λ$ is obtained explicitly. Moreover, we determine the ${\cal N}=2$ supersymmetric linear $W_{\infty}[λ]$ algebra for arbitrary $λ$. As a by product, the $λ$ deformed bosonic $W_{1+\infty}[λ] \times W_{1+\infty}[λ+\frac{1}{2}]$ 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 $(h_1+h_2-2)$. Each $w_{1+\infty}$ 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. |
| title | The ${\cal N}=2,4$ Supersymmetric Linear $W_{\infty}[λ]$ Algebras for Generic $λ$ Parameter |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2309.01537 |