Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.23644

A Topological Quantum Field Theory for Representation Varieties of Torus Links

Enrique Aycart-Maldonado·Alejandro Calleja
View original source ↗Read source document ↗Export metadata ↓

Abstract

We construct a Topological Quantum Field Theory (TQFT) that computes the virtual classes of $G$-representation varieties associated with the fundamental group of a torus link complement. Following the framework established by González-Prieto, Logares and Muñoz for $G$-representation varieties of surface groups, we introduce a suitable bordism whose iteration enables a recursive topological evaluation of these virtual classes. Via this TQFT, we explicitly compute the virtual class of these varieties for the groups $G=\mathrm{AGL}_1(\mathbb{C})$ and $G=\mathrm{SL}_2(\mathbb{C})$.

Explore related subjects

math.GTmath.AG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Enrique Aycart-Maldonado, Alejandro Calleja. 2026-10-16. A Topological Quantum Field Theory for Representation Varieties of Torus Links. https://arxiv.org/abs/2610.23644

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

KEEP EXPLORING

Related papers

arXiv2026

Small symplectic $4$-manifolds via contact gluing and some applications

We introduce a streamlined procedure for constructing small symplectic $4$-manifolds via contact gluing, based on a technique invented by David Gay around 2000. We give several applications of this procedure, which include results concerning embeddings of singular Lagrangian $RP^2$s, or embeddings of lens spaces as a hypersurface of contact type, in small rational surfaces such as $CP^2\#\overline{CP^2}$ and $S^2\times S^2$, as well as results on the uniqueness or classification of $Q$-homology ball symplectic fillings. Further work on the classification of singular Lagrangian $RP^2$s is suggested. Moreover, our investigation on the $S^1$-invariant contact structures suggests an interesting and fairly strong upper bound for the self-intersection of a rational unicuspidal curve with one Puiseux pair $(p,q)$ in any algebraic surface (the bound depends only on the values $p,q$), and for the symplectic version, we prove the existence of an ``optimal" symplectic rational unicuspidal curve in a rational $4$-manifold which realizes the upper bound for any given Puiseux pair $(p,q)$. Our results also suggest a revisit of the ``symplectic divisorial capping" problem first considered by Li and Mak. Further applications of the techniques developed in this paper hinge upon better understandings on the tightness and fillability criterions of $S^1$-invariant contact structures as well as their (small) symplectic fillings.

math.GT↗
arXiv2026

The Involution Rank of $\MCG(Σ_3)$ Is Four

For the closed orientable surface $Σ_3$, we prove \[ \irank\MCG(Σ_3)=4.\] Thus $\MCG(Σ_3)$ can be generated by four involutions, but not by three. This settles the genus-three case of Farb's problem on generation by involutions.

math.GT↗
arXiv2026

Regular Additive Scalar Invariants of ${\mathcal N}=1$ Hyperbolic Spin Supertriangles

We classify regular cyclic scalar invariants of non-degenerate hyperbolic ${\mathcal N}=1$ spin supertriangles which are additive under geodesic subdivision. After fixing the spin lift, the common-first gauge (introduced in earlier work by the second-named author and further developed here) gives a complete $3|4$ modulus of such supertriangles.. The space of additive scalar invariants is proved to have superdimension $3|1$: ordinary hyperbolic area in degree zero, two even generators in degree two, and one odd generator in degree three; the degree-one and degree-four sectors vanish. One degree-two generator is the fermionic first variation of body area, equivalently the nilpotent correction from ordinary body angle defect to super angle defect. The degree-three generator changes sign under reversal of the spin lift, while the other generators are fixed, so ordinary supertriangles carry only three even invariants.

math.GT↗