arXiv subjects

Tianyue Liu

Publications and source records attributed to Tianyue Liu.

5 recordsLinked to original sources

arXiv2026

Turaev-Viro invariant from the modular double of $\mathrm {U}_{q}\mathfrak{sl}(2;\mathbb R)$

We define a family of Turaev-Viro type invariants of hyperbolic $3$-manifolds with totally geodesic boundary from the $6j$-symbols of the modular double of $\mathrm U_{q}\mathfrak{sl}(2;\mathbb R)$, and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the $1$-loop term the adjoint twisted Reidemeister torsion of the double of the manifolds.

math.GT↗

arXiv2026

$\mathrm {U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$ Turaev-Viro invariants for cusped $3$-manifolds

We define a family of Turaev-Viro type invariants for hyperbolic $3$-manifolds with cusp ends, extending the invariants introduced in \cite{LMSWY} for hyperbolic $3$-manifolds with totally geodesic boundary. These invariants are constructed from what we call the ideal $\mathrm{U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$-$6j$ symbols, which are variants of the $6j$-symbols associated with the positive representations of the modular double of $\mathrm{U}_q\mathfrak{sl}(2;\mathbb R)$. We also prove that these invariants decay exponentially, with the exponential decay rate determined by the hyperbolic volume of the manifold.

math.GT↗

arXiv2025

Asymptotics of $b$-$6j$ symbols and anti-de Sitter tetrahedra

In this paper, we study the asymptotics of the $6j$-symbols for the principal series of the modular double of $\mathrm U_q\mathfrak{sl}(2;\mathbb R)$, and of their analytic extension -- what we call the $b$-$6j$ symbols, relating them in various cases to the volume of truncated hyperideal tetrahedra in the hyperbolic and the anti-de Sitter geometry. To the best of our knowledge, this is the first time that the anti-de Sitter geometry appears in the asymptotics of quantum invariants. In addition, based on the connection to conformal field theory, we reveal a correspondence between the edge lengths and the dihedral angles of truncated hyperideal anti-de Sitter tetrahedra and the Fenchel-Nielsen coordinates of hyperbolic four-holed spheres. We also provide a concrete instance of $3D/2D$ holography, in the spirit of the AdS/CFT correspondence.

math-ph↗

arXiv2025

Toric Einstein 4-manifolds with nonnegative sectional curvature

We prove that $T^2$-invariant Einstein metrics with non-negative sectional curvature on a four-manifold are locally symmetric.

math.DG↗

arXiv2017

MaxDim of some simple groups

MaxDim, along with it relatives m and i, are group invariants analogous to the concept of dimension in vector spaces. In this paper we give the following result on the MaxDim of sporadic groups: 1. An inductive argument for the lower bounds of the MaxDim of those groups in the Suzuki tower, utilizing the fact that those groups act on strongly regular graphs. 2. A detailed computation of the MaxDim of Mathieu 24 group, utilizing the fact that it acts as automorphisms of Steiner system. 3. Estimations of the MaxDim for McLaughlin and Higmann-Sims Group.

math.GR↗