arXiv subjects

Matthias Vancraeynest

Publications and source records attributed to Matthias Vancraeynest.

4 recordsLinked to original sources

arXiv2026

Weyl defects in skein theory and quantum cluster charts

We prove a modified invertibility property for the parabolic defects introduced in arXiv:2102.12283, arXiv:2505.14836. Namely, we show that the Borel defect cancels its dual, up to insertion of an invertible Weyl defect and up to restriction to an open-subcategory. The main result holds for an arbitrary reductive group $G$ -- for illustration we give extended computations for $G=\mathrm{SL}_3$. Along the way, we introduce a defect skein theory with defects in both codimension one and two, which is compatible with gluing of defect 3-manifolds and defect surfaces. Our results provide a potential defect skein theoretic construction of certain standard charts which appears frequently in quantum cluster varieties associated to surfaces.

math.QA↗

arXiv2025

The skein partition function of the mapping torus

We compute the dimensions of $\text{GL}_N$-skein modules of genus-one mapping tori $T^2\times_γS^1$, for an arbitrary diffeomorphism of $T^2$, and for generic quantum parameter. These are most cleanly expressed via a generating function over all $N$, which we dub the skein partition function, and for which we compute an explicit Euler product expansion.

math.QA↗

arXiv2025

Higher tensor categories and their extensions: notes from the Scottish Talbot On Algebra and Topology

These lecture notes are the product of a week-long learning workshop on the work of Johnson-Freyd and Reutter on the problem of the existence of minimal nondegenerate extensions of braided fusion categories (arXiv:2105.15167). They recount the mathematical arguments of the original paper from an expository angle, with background material covering the algebra and homotopy theory required to understand the statement and follow the proof. The notes are aimed at newcomers to the field of (braided) fusion 1- and 2-categories.

math.QA↗

arXiv2024

Classification of higher grade $\ell$ graphs for $\mathrm{U}(N)^2\times \mathrm{O}(D)$ multi-matrix models

The authors studied in [Ann. Inst. Henri Poincaré D 9, 367-433, (2022)], a complex multi-matrix model with $\mathrm{U}(N)^2 \times \mathrm{O}(D)$ symmetry, and whose double scaling limit where simultaneously the large-$N$ and large-$D$ limits were taken while keeping the ratio $N/\sqrt{D}=M$ finite and fixed. In this double scaling limit, the complete recursive characterization of the Feynman graphs of arbitrary genus for the leading order grade $\ell=0$ was achieved. In this current study, we classify the higher order graphs in $\ell$. More specifically, $\ell=1$ and $\ell=2$ with arbitrary genus, in addition to a specific class of two-particle-irreducible (2PI) graphs for higher $\ell \geqslant 3$ but with genus zero. Furthermore, we demonstrate that each 2PI graph with a single $\mathrm{O}(D)$-loop with an arbitrary $\ell$ corresponds to a reduced alternating knot diagram with $\ell$ crossings as listed in the Rolfsen knot table, or a resulting alternating knot diagram obtained after performing the Tait flyping moves. We generalize to 2PR by considering the connected sum and the Reidemeister move I.

math-ph↗