Gelfand-Dickey Realizations of the supersymmetric classical W-algebras for $\mathfrak{gl}(n+1|n)$ and $\mathfrak{gl}(n|n)$
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| Format: | Preprint |
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2024
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| _version_ | 1866914891183423488 |
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| author | Carpentier, Sylvain Suh, UhiRinn |
| author_facet | Carpentier, Sylvain Suh, UhiRinn |
| contents | In this paper we realize the supersymmetric classical $W$-algebras $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ and $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ as differential algebras generated by the coefficients of a monic superdifferential operator $L$. In the case of $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ (resp. $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$) this operator is even (resp. odd). We show that the supersymmetric Poisson vertex algebra bracket on these supersymmetric W-algebras is the supersymmetric analogue of the quadratic Gelfand-Dickey bracket associated to the operator $L$. Finally, we construct integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A key observation is that to construct these hierarchies on the algebra $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ one needs to introduce a new concept of even supersymmetric Poisson vertex algebras. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_19717 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gelfand-Dickey Realizations of the supersymmetric classical W-algebras for $\mathfrak{gl}(n+1|n)$ and $\mathfrak{gl}(n|n)$ Carpentier, Sylvain Suh, UhiRinn Mathematical Physics 35Q53, 37K10, 37K30, 17B69, 17B80 In this paper we realize the supersymmetric classical $W$-algebras $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ and $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ as differential algebras generated by the coefficients of a monic superdifferential operator $L$. In the case of $\mathcal{W}(\overline{\mathfrak{gl}}(n|n))$ (resp. $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$) this operator is even (resp. odd). We show that the supersymmetric Poisson vertex algebra bracket on these supersymmetric W-algebras is the supersymmetric analogue of the quadratic Gelfand-Dickey bracket associated to the operator $L$. Finally, we construct integrable hierarchies of evolutionary Hamiltonian PDEs on both W-algebras. A key observation is that to construct these hierarchies on the algebra $\mathcal{W}(\overline{\mathfrak{gl}}(n+1|n))$ one needs to introduce a new concept of even supersymmetric Poisson vertex algebras. |
| title | Gelfand-Dickey Realizations of the supersymmetric classical W-algebras for $\mathfrak{gl}(n+1|n)$ and $\mathfrak{gl}(n|n)$ |
| topic | Mathematical Physics 35Q53, 37K10, 37K30, 17B69, 17B80 |
| url | https://arxiv.org/abs/2407.19717 |