Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.23752

Spectral networks, finite webs, and the $T_3$ cubic differential

Omar Kidwai·Guillaume Tahar
View original source ↗Read source document ↗Export metadata ↓

Abstract

We describe spectral networks for cubic differentials on $\mathbf{P}^1$ with exactly three poles all of order two, following observations of Hollands-Neitzke. To do so, we give a precise definition and establish some basic properties of a class of finite webs (corresponding to BPS states in physics) in the cubic differential setting, including their non-vanishing central charge. Turning to the example, we prove that at rational angles the network is totally degenerate; the number of walls and Stokes regions is explicitly computed and depends on an arithmetic criterion. At all other angles, the spectral network is dense in the surface and contains no finite webs, as expected. Physically, this example corresponds to a fundamental building block for a large class of four-dimensional $\mathcal{N}=2$ QFTs with gauge group $\mathrm{SU}(3)$.

Explore related subjects

math.AGhep-thmath-phmath.DGmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Omar Kidwai, Guillaume Tahar. 2026-10-16. Spectral networks, finite webs, and the $T_3$ cubic differential. https://arxiv.org/abs/2610.23752

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Some Cubic and Quartic Inequalities of Four Variables

Let $\mathcal{H} \subset \mathcal{H}_{n,d} := \mathbb{R}[x_1$,$\ldots$, $x_n]_d$ be a vector space, and $A$ be a compact semialgebraic subset of $\mathbb{P}_{\mathbb{R}}^{n-1}$. We shall study some PSD cones $\mathcal{P} = \mathcal{P}(A$, $\mathcal{H}) := \big\{f \in \mathcal{H}$ $\big|$ $f(a) \geq 0$ ($\forall a \in A$)$\big\}$. Our interests are (1) to determine the extremal elements of $\mathcal{P}$, (2) to determine discriminants of $\mathcal{P}$, (3) to describe $\mathcal{P}$ as a union of basic semialgebraic subsets, and (4) to find a nice test set when $\dim \mathcal{H}$ is low. In this article, we present (1), (2), (3) and (4) for $\mathcal{P}(\mathbb{R}^4$, $\mathcal{H}_{4,4}^{s0})$ and $\mathcal{P}(\mathbb{R}_+^4$, $\mathcal{H}_{4,4}^{s0})$, where $\mathcal{H}_{n,d}^{s0} := \big\{f \in \mathcal{H}_{n,d}$ $\big|$ $f$ is symmetric and $f(1,\ldots,1)=0 \big\}$. We also provide (1) -- (4) for $\mathcal{P}(\mathbb{R}_+^4$, $\mathcal{H}_{4,3}^{c0})$, where $\mathcal{H}_{n,d}^{c0} := \big\{f \in \mathcal{H}_{n,d}$ $\big|$ $f$ is cyclic and $f(1,\ldots,1)=0 \big\}$.

math.AG↗
arXiv2026

Categorical wall-crossing for higher-framed DT/PT quivers

We study a framed three-loop quiver with $a+r$ incoming arrows and $a$ outgoing arrows. For a generic window parameter, we construct a semiorthogonal decomposition of the bounded derived category of the positive GIT quotient into products of unframed window categories and categories of the negative quotient. The arrow imbalance $r$ determines an interval of admissible slopes of length $r$. The proof separates a lower-convex-hull decomposition of dominant weights, Koszul induction, full faithfulness and semiorthogonality of Hall functors, and the two window equivalences. We extend the decomposition to trace potentials satisfying block additivity. Applications include a precisely specified open chart of the framed conifold quiver, an algebraically defined extended ADHM critical model, and the critical presentation of the Quot scheme of points on $\mathbb{A}^3$. We do not identify the extended ADHM model with moduli of higher-rank pairs having a generically rank-one reduced-curve sheaf. A formal MacMahon identity is proved separately, and the localized categorical rank formula is deduced using the localized strict-slope product formula of Pădurariu--Toda.

math.AG↗
arXiv2026

A dimension bound towards the Casas-Alvero Conjecture

The Casas-Alvero conjecture predicts that every univariate polynomial over an algebraically closed field of characteristic zero sharing a common factor with each of its Hasse-Schmidt derivatives is a power of a linear polynomial. The conjecture for polynomials of a fixed degree $n$ is equivalent to the projective variety of such polynomials being one-dimensional. In this paper, we produce dimension bounds for this variety (in degree $n$) over any algebraically closed field of characteristic zero using an arithmetic invariant $B(n)$ of $n$. Consequently for infintely many degrees $n$, we show that the associated arithmetic Casas-Alvero scheme has finitely many rational points over any characteristic zero field. Along the way, we prove several rigidity results towards the conjecture and study arithmetic properties of $B(n)$.

math.AG↗