arXiv subjects

Natalia Amburg

Publications and source records attributed to Natalia Amburg.

4 recordsLinked to original sources

arXiv2024

The three-point Gaudin model and branched coverings of the Riemann sphere

We study the three-point quantum $\mathfrak{sl_2}$-Gaudin model. In this case the compactification of the parameter space is $\overline{M_{0,4}(\mathbb{C})}$, which is the Riemann sphere. We analyze sphere coverings by the joint spectrum of the Gaudin Hamiltonians treating them as algebraic curves. We write equations of these curves as determinants of tridiagonal matrices and deduce some consequences regarding the geometric structure of the Gaudin coverings.

math-ph↗

arXiv2024

Belyi Function Decompositions for The Icosahedron of Genus 4

The icosahedron $I_4$ of genus 4 is a dessin d'enfant embedded in Bring's curve $\mathcal{B}$. The dessin $I_4$ is related in some sense to a regular icosahedron $I_0$ embedded in the complex Riemann sphere. In particular, decompositions of Belyi functions $β_{I_0}: \mathbb{CP}^1 \rightarrow \mathbb{CP}^1$ and $β_{I_4}: \mathcal{B} \rightarrow \mathbb{CP}^1$ for $I_0$ and $I_4$ have the same lattice. The diagram of $β_{I_0}$ decompositions is already known. In the present paper we find $β_{I_4}$ decompositions. Note that $β_{I_0}$ decomposes into rational functions on $\mathbb{C}P^1$, while in case of $β_{I_4}$ we deal with maps between different algebraic curves.

math.AG↗

arXiv2023

Belyi pair for the orientation cover of $ \overline{{\mathcal M}_{0,5}^{\mathbb R}}$

Let $\overline{{\mathcal M}_{0,5}^{\mathbb R}}$ be the Deligne-Mumford compactification of the moduli space of genus 0 real algebraic curves with 5 marked points. By ${\mathcal L}(\overline{{\mathcal M}_{0,5}^{\mathbb R}})$ we denote its orientation cover. The cell decomposition of ${\mathcal L}(\overline{{\mathcal M}_{0,5}^{\mathbb R}})$ is a dessin d'enfant of the genus 4. In this paper we compute the Belyi pair for this dessin. In particular, it appears that the corresponding curve is a celebrated Bring curve.

math.AG↗

arXiv2022

On Products of Random Matrices

We introduce a family of models, which we name matrix models associated with children's drawings -- the so-called dessin d'enfant. Dessins d'enfant are graphs of a special kind drawn on a closed connected orientable surface (in the sky). The vertices of such a graph are small disks that we call stars. We attach random matrices to the edges of the graph and get multimatrix models. Additionally, to the stars we attach source matrices. They play the role of free parameters or model coupling constants. The answers for our integrals are expressed through quantities that we call the "spectrum of stars." The answers may also include some combinatorial numbers, such as Hurwitz numbers or characters from group representation theory.

math-ph↗