arXiv 2301.040682023-01math-phmath.AGmath.MP

从扩展环方程导出的 Blobbed 拓扑递归

Blobbed topological recursion from extended loop equations

Alexander Hock, Raimar Wulkenhaar

备注：: 49 pages. v2: discussion of the loop insertion operator and of symmetry of the ω^{(g)}_n added; references updated and expanded; minor corrections
发表信息：: Journal of Geometry and Physics Volume 212, June 2025, 105457
DOI：: 10.1016/j.geomphys.2025.105457

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AI总结：

针对具有特定测度的埃尔米特矩阵模型，本文提出一种通过满足扩展环方程的六个亚纯函数系统的新方法，推导出全局线性与二次环方程及递归核表示，从而证明并推广了亚纯微分族的 blobbed 拓扑递归猜想。

摘要：

我们考虑测度为 $dμ_{E,λ}(M)=\frac{1}{Z} \exp(-\frac{λN}{4} \mathrm{tr}(M^4)) dμ_{E,0}(M)$ 的 $N\times N$ 埃尔米特矩阵模型，其中 $dμ_{E,0}$ 是协方差为 $\langle M_{kl}M_{mn}\rangle=\frac{δ_{kn}δ_{lm}}{N(E_k+E_l)}$ 的高斯测度，给定 $E_1,...,E_N>0$。此前已知该设定会产生两个由 $y(z)=-x(-z)$ 强关联的黎曼球分歧覆盖 $x,y$，以及一族被猜想服从 Borot 和 Shadrin 提出的 blobbed 拓扑递归的亚纯微分 $ω^{(g)}_{n}$。我们通过一个满足扩展环方程的六个亚纯函数系统，为该问题开发了一种新方法。其中两个函数在 $x$ 的原像处对称，可由其一致性关系确定。在 $\infty$ 处展开给出了 $ω^{(g)}_{n}$ 的全局线性与二次环方程。这些全局方程不仅在 $x$ 的分歧点附近提供 $ω^{(g)}_{n}$，还在位于相对角线 $z_i+z_j=0$ 和 $z_i=0$ 处的所有其他极点附近提供 $ω^{(g)}_{n}$。我们推导出了至少对 $g\leq 1$ 有效的递归核表示。

英文摘要：

We consider the $N\times N$ Hermitian matrix model with measure $dμ_{E,λ}(M)=\frac{1}{Z} \exp(-\frac{λN}{4} \mathrm{tr}(M^4)) dμ_{E,0}(M)$, where $dμ_{E,0}$ is the Gaussian measure with covariance $\langle M_{kl}M_{mn}\rangle=\frac{δ_{kn}δ_{lm}}{N(E_k+E_l)}$ for given $E_1,...,E_N>0$. It was previously understood that this setting gives rise to two ramified coverings $x,y$ of the Riemann sphere strongly tied by $y(z)=-x(-z)$ and a family $ω^{(g)}_{n}$ of meromorphic differentials conjectured to obey blobbed topological recursion due to Borot and Shadrin. We develop a new approach to this problem via a system of six meromorphic functions which satisfy extended loop equations. Two of these functions are symmetric in the preimages of $x$ and can be determined from their consistency relations. An expansion at $\infty$ gives global linear and quadratic loop equations for the $ω^{(g)}_{n}$. These global equations provide the $ω^{(g)}_{n}$ not only in the vicinity of the ramification points of $x$ but also in the vicinity of all other poles located at opposite diagonals $z_i+z_j=0$ and at $z_i=0$. We deduce a recursion kernel representation valid at least for $g\leq 1$.

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