6j symbols for U_q(sl_2) and non-Euclidean tetrahedra

TL;DR

The paper links the semiclassical asymptotics of quantum 6j symbols for $U_q(\mathfrak{sl}_2)$ at roots of unity or positive real $q$ to the geometry of non-Euclidean tetrahedra and WZW conformal blocks, extending the classical $q=1$ results of Wigner, Ponzano–Regge, and Roberts. It develops a robust recursion-based approach (Schulten–Gordon) to show that both the quantum 6j symbols and their conjectured asymptotics satisfy the same second-order difference equation, enabling precise Euclidean-limit matching and extension to degenerate and hyperbolic cases. The authors derive a comprehensive set of asymptotic formulas across multiple degeneracy regimes (non-degenerate, degenerate faces, classically forbidden, hyperbolic), including explicit expressions that involve tetrahedral dihedral angles, volumes, and Gram determinants, with connections to the moduli space of flat SU(2) connections and Verlinde theory. These results illuminate the semiclassical structure underlying quantum invariants and conformal blocks, and they raise further questions about uniform formulations and extensions to broader quantum groups and geometric settings.

Abstract

We relate the semiclassical asymptotics of the 6j symbols for the representation theory of the quantized enveloping algebra U_q(sl_2) at q a primitive root of unity, or q positive real, to the geometry of non-Euclidean tetrahedra. The formulas are motivated by the geometry of conformal blocks in the Wess-Zumino-Witten model; they generalize formulas in the case q = 1 of Wigner, Ponzano and Regge, and Schulten and Gordon, proved by J. Roberts.

Figures (6)

• Figure 1: The spherical tetrahedon $\tau$
• Figure 2: Seven types of tetrahedra and the corresponding Lagrangians
• Figure 3: Quantum $6j$ symbols versus the asymptotic formulas
• Figure 4: The pentagon identity
• Figure 5: Two degenerations of $(\mathbb{P}^1;p_{1},p_{2},p_{3},p_{4})$ in $\overline{\mathcal{C}}_{0,4}$.

Theorems & Definitions (30)

• Theorem 1.0.1
• Proposition 2.4.1
• proof
• Lemma 2.4.2
• proof
• Corollary 2.4.3
• proof
• Proposition 4.1.1
• Theorem 4.1.2
• Remark 4.1.3