TQFT & Quantum Gravity

QRecoupling.jl is purpose-built to provide the exact, high-performance topological kernels required for 3D Quantum Gravity (spin foam models), string-Net condensates, and state-sum invariants like the Turaev-Viro model.

This page demonstrates how to compute physical observables, manipulate tensor networks, and seamlessly transition between quantum and classical spacetime limits.

The Quantum Recoupling Symbols

The core topological vertex in $SU(2)_k$ recoupling theory is the quantum $6j$-symbol (or Racah-Wigner symbol). It represents the probability amplitude of a tetrahedral quantum geometry. Other useful tensors are quantum $3j$ symbols.

To evaluate a $6j$-symbol, simply provide the six half-integer spins $j_i$ and the topological level $k$.

using QRecoupling

# Evaluate at a root of unity (Turaev-Viro regime)
val = q6j(1, 1, 1, 1, 1, 1, k=10)
println(val) # 0.1547005383792515

# Evaluate the EXACT cyclotomic field representation
exact_val = q6j(1, 1, 1, 1, 1, 1, k=10, exact=true)
println(exact_val) # -2//3*ζ^6 + 4//3*ζ^2 - 1

Complex Deformations

If you are studying analytic continuation, or generic quantum groups, you can evaluate the geometry at any continuous complex deformation parameter $q$

complex_val = q6j(1, 1, 1, 1, 1, 1, q=exp(0.5im))

The Classical Limit (Ponzano-Regge)

As the level $k \to \infty$, the quantum deformation parameter $q \to 1$. This limit reduces the Turaev-Viro spherical spacetime to a flat Ponzano-Regge geometry.

Because standard cyclotomic evaluation fails at $q=1$ (due to $0/0$ division singularities), QRecoupling.jl mathematically detects q=1 and routes the calculation through a safe, exact geometric reduction path

# Fast floating-point classical limit
classical_float = q6j(1, 1, 1, 1, 1, 1, q=1)

# Exact zero-loss rational classical limit
classical_rational = q6j(1, 1, 1, 1, 1, 1, q=1, exact=true)
1//6

Topological Tensors

When constructing massive tensor networks, you often need composite symbols that include specific phase conventions or quantum dimension regularizations.

The API provides direct access to these networks

k = 5

# Wigner 3j symbol (coupling of angular momenta)
val_3j = q3j(1, 1, 1, 0, 0, 0, k=k)

# F-Symbol (fusion tree crossing matrix)
f_val = fsymbol(1, 1, 1, 1, 1, 1, k=k)

# G-Symbol (tetrahedrally symmetric invariant)
g_val = gsymbol(1, 1, 1, 1, 1, 1, k=k)

High-Performance Tensor Contractions

To contract large networks (like an $S^3$ manifold or a knot complement), performance is critical. You can bypass the internal algebraic graph construction and use the eager=true flag to push floating-point data straight into the Log-Sum-Exp numerical solver.

# Bypasses DCR graph allocation for maximum speed inside tight loops
fast_val = q6j(10, 10, 10, 10, 10, 10, k=50, eager=true)

Note: eager=true is only available when an integer level k is provided.