Mapping partition functions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866909693726687232 |
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| author | Yang, Di Zagier, Don |
| author_facet | Yang, Di Zagier, Don |
| contents | We introduce an infinite group action on partition functions of WK type, meaning of the type of the partition function $Z^{\rm WK}$ in the famous result of Witten and Kontsevich expressing the partition function of $ψ$-class integrals on the compactified moduli space $\overline{\mathcal{M}}_{g,n}$ as a $τ$-function for the Korteweg--de Vries hierarchy. Specifically, the group which acts is the group $\mathcal{G}$ of formal power series of one variable $φ(V)=V+O(V^2)$, with group law given by composition, acting in a suitable way on the infinite tuple of variables of the partition functions. In particular, any $φ\in \mathcal{G}$ sends the Witten--Kontsevich (WK) partition function $Z^{\rm WK}$ to a new partition function $Z^φ$, which we call the WK mapping partition function associated to $φ$. We show that the genus zero part of $\log Z^φ$ is independent of $φ$ and give an explicit recursive description for its higher genus parts (loop equation), and as applications of this obtain relationships of the $ψ$-class integrals to Gaussian Unitary Ensemble and generalized Brézin--Gross--Witten correlators. In a different direction, we use $Z^φ$ to construct a new integrable hierarchy, which we call the WK mapping hierarchy associated to $φ$. We show that this hierarchy is a bihamiltonian perturbation of the Riemann--Hopf hierarchy possessing a $τ$-structure, and prove that it is a universal object for all such perturbations. Similarly, for any $φ\in\mathcal{G}$, we define the Hodge mapping partition function associated to $φ$, prove that it is integrable, and study its role in hamiltonian perturbations of the Riemann--Hopf hierarchy possessing a $τ$-structure. Finally, we establish a generalized Hodge--WK correspondence relating different Hodge mapping partition functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_03568 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Mapping partition functions Yang, Di Zagier, Don Mathematical Physics Algebraic Geometry Exactly Solvable and Integrable Systems We introduce an infinite group action on partition functions of WK type, meaning of the type of the partition function $Z^{\rm WK}$ in the famous result of Witten and Kontsevich expressing the partition function of $ψ$-class integrals on the compactified moduli space $\overline{\mathcal{M}}_{g,n}$ as a $τ$-function for the Korteweg--de Vries hierarchy. Specifically, the group which acts is the group $\mathcal{G}$ of formal power series of one variable $φ(V)=V+O(V^2)$, with group law given by composition, acting in a suitable way on the infinite tuple of variables of the partition functions. In particular, any $φ\in \mathcal{G}$ sends the Witten--Kontsevich (WK) partition function $Z^{\rm WK}$ to a new partition function $Z^φ$, which we call the WK mapping partition function associated to $φ$. We show that the genus zero part of $\log Z^φ$ is independent of $φ$ and give an explicit recursive description for its higher genus parts (loop equation), and as applications of this obtain relationships of the $ψ$-class integrals to Gaussian Unitary Ensemble and generalized Brézin--Gross--Witten correlators. In a different direction, we use $Z^φ$ to construct a new integrable hierarchy, which we call the WK mapping hierarchy associated to $φ$. We show that this hierarchy is a bihamiltonian perturbation of the Riemann--Hopf hierarchy possessing a $τ$-structure, and prove that it is a universal object for all such perturbations. Similarly, for any $φ\in\mathcal{G}$, we define the Hodge mapping partition function associated to $φ$, prove that it is integrable, and study its role in hamiltonian perturbations of the Riemann--Hopf hierarchy possessing a $τ$-structure. Finally, we establish a generalized Hodge--WK correspondence relating different Hodge mapping partition functions. |
| title | Mapping partition functions |
| topic | Mathematical Physics Algebraic Geometry Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2308.03568 |