Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2607.11108

A TQFT perspective on satellite constructions and toral decompositions

Nikolaos Angelinos
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider the classical satellite and splice constructions of knot theory from the perspective of link-complement states in three-dimensional topological quantum field theory (TQFT). These states are prepared by a TQFT on a link-complement manifold, namely a manifold with multiple disjoint torus boundaries, obtained by removing a thickened link from a closed ambient three-manifold. In this setting, satellite and splice constructions are realized by gluing manifolds along torus boundaries, and the TQFT assigns to these building blocks multilinear operators acting on the corresponding torus Hilbert spaces. We give explicit prescriptions for these operators and show how they can be used to build link-complement states. This construction is closely related to the JSJ decomposition of link-complement manifolds. While splicing builds manifolds by gluing along tori, the JSJ decomposition cuts a manifold along essential tori into canonical pieces. From the TQFT point of view, this expresses a link-complement state as a network of elementary operators associated with the JSJ pieces. We illustrate this framework by studying the entanglement entropy of Seifert-fibered link complements, Hopf-chain complements, and complements of Whitehead-doubled Hopf chains, showing how their different JSJ structures lead to distinct entanglement patterns.

Explore related subjects

hep-thmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Nikolaos Angelinos. 2026-07-13. A TQFT perspective on satellite constructions and toral decompositions. https://arxiv.org/abs/2607.11108

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

KEEP EXPLORING

Related papers

arXiv2026

Gaillard-Zumino non-invertible symmetries

We uncover an infinite class of novel zero-form non-invertible symmetries in a broad family of four-dimensional models, studied years ago by Gaillard and Zumino (GZ), which includes several extended supergravities as particular subcases. The GZ models consist of abelian gauge fields coupled to a neutral sector, typically including a set of scalars, whose equations of motion are classically invariant under a continuous group $\mathscr{G}$ acting on the electric and magnetic field strengths via symplectic transformations. The standard lore holds that, at the quantum level, these symmetries are broken to an integral subgroup $\mathscr{G}_\mathbb{Z}$. We show that, in fact, a much larger subgroup $\mathscr{G}_\mathbb{Q}$ survives, albeit through non-invertible topological defects. We explicitly construct these defects and compute some of their fusion rules. As illustrative examples, we consider the axion-dilaton-Maxwell model and the bosonic sector of a class of $\mathcal{N}=2$ supergravities of the kind that appear in type II Calabi-Yau compactifications. Finally, we comment on how (part of) these non-invertible zero-form symmetries can be broken by gauging the $\mathscr{G}_\mathbb{Z}$ subgroup of invertible symmetries.

hep-th↗
arXiv2026

On the resolution of categorical symmetries in (Non-) Unitary Rational CFTs

We explore several aspects of categorical symmetry-resolved entanglement entropy (SREE) directly within two-dimensional rational conformal field theory (RCFT) (without invoking any SymTFT construction arXiv:2409.02806). We derive a general formula applicable whenever the action of the relevant topological defect lines on the annulus Hilbert space is known. This framework accommodates weakly and strongly symmetric boundaries, cloaking states, and fusion rings with multiplicities. We verify the formula in a range of diagonal unitary and non-unitary examples, including theories with generalized Haagerup-Izumi modular data. Furthermore, we extend the analysis to non-diagonal RCFTs. The $\frac{1}{2}E_6$ example demonstrates that closed-channel modular data and NIM-rep multiplicities alone do not suffice to determine the defect action on the complete open-channel Hilbert space.

hep-th↗
arXiv2026

Reflecting boundary conditions in critical loop models

In critical loop models, we call a boundary sticky if loops can attach to it, and reflecting otherwise. Using analytic bootstrap methods, we show that reflecting boundaries are characterised by one complex parameter, analogous to the boundary cosmological constant in Liouville theory. We determine disc 1-point functions, and write an explicit formula for disc 2-point functions as infinite combinations of conformal blocks. We also sketch the lattice interpretation of reflecting boundaries.

hep-th↗