Deformation of superintegrability in the Miwa-deformed Gaussian matrix model
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866909301107326976 |
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| author | Mironov, A. Morozov, A. Popolitov, A. Shakirov, Sh. |
| author_facet | Mironov, A. Morozov, A. Popolitov, A. Shakirov, Sh. |
| contents | We consider an arbitrary deformation of the Gaussian matrix model parameterized by Miwa variables $z_a$. One can look at it as a mixture of the Gaussian and logarithmic (Selberg) potentials, which are both superintegrable. The mixture is not, still one can find an explicit expression for an arbitrary Schur average as a linear transform of a {\it finite degree} polynomial made from the values of skew Schur functions at the Gaussian locus $p_k=δ_{k,2}$. This linear operation includes multiplication with an exponential $ e^{z_a^2/2}$ and a kind of Borel transform of the resulting product, which we call multiple and enhanced. The existence of such remarkable formulas appears intimately related to the theory of auxiliary $K$-polynomials, which appeared in {\it bilinear} superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in the very detail the generating function of correlators $<(\Tr X)^k>$ in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian $β$-ensemble. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_09670 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Deformation of superintegrability in the Miwa-deformed Gaussian matrix model Mironov, A. Morozov, A. Popolitov, A. Shakirov, Sh. High Energy Physics - Theory Mathematical Physics We consider an arbitrary deformation of the Gaussian matrix model parameterized by Miwa variables $z_a$. One can look at it as a mixture of the Gaussian and logarithmic (Selberg) potentials, which are both superintegrable. The mixture is not, still one can find an explicit expression for an arbitrary Schur average as a linear transform of a {\it finite degree} polynomial made from the values of skew Schur functions at the Gaussian locus $p_k=δ_{k,2}$. This linear operation includes multiplication with an exponential $ e^{z_a^2/2}$ and a kind of Borel transform of the resulting product, which we call multiple and enhanced. The existence of such remarkable formulas appears intimately related to the theory of auxiliary $K$-polynomials, which appeared in {\it bilinear} superintegrable correlators at the Gaussian point (strict superintegrability). We also consider in the very detail the generating function of correlators $<(\Tr X)^k>$ in this model, and discuss its integrable determinant representation. At last, we describe deformation of all results to the Gaussian $β$-ensemble. |
| title | Deformation of superintegrability in the Miwa-deformed Gaussian matrix model |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2403.09670 |