Proving Topological Identities

Another advantage of QRecoupling.jl is its ability to verify axioms of 3D topological quantum field theories (TQFTs).

In this tutorial, we will prove the two most important topological identities in the Turaev-Viro/Ponzano-Regge state sum models: The Orthogonality Relation (Bubble Move) and the Biedenharn-Elliott Pentagon Identity (Pachner 2-3 Move).

1. Exact Orthogonality (The Bubble Move)

In a triangulated 3-manifold, integrating out an internal edge corresponds to the "Bubble Move". Mathematically, this enforces the orthogonality of the $6j$-symbols.

We can prove this exactly (without floating-point errors) using the exact=true condition.

using QRecoupling

function test_orthogonality(j1, j2, j3, j4, j5, j6;k=k)
    # determine valid bounds for x
    x_min = max(abs(j1 - j2), abs(j3 - j4))
    x_max = min(j1 + j2, j3 + j4)
    
    LHS = 0
    
    for x in x_min:x_max
        # sum over admissible spins 
        if (j1 + j2 + x) <= k && (j3 + j4 + x) <= k
            dim_x = qdim(x, k=k, exact=true)
            sym1  = q6j(j1, j2, x, j3, j4, j5, k=k, exact=true)
            sym2  = q6j(j1, j2, x, j3, j4, j6, k=k, exact=true)
            
            LHS += dim_x * sym1 * sym2
        end
    end
    
    # evaluate exact RHS
    RHS = 0
    if j5 == j6 && abs(j1 - j2) <= j5 <= (j1 + j2) && (j1 + j2 + j5) <= k && abs(j3 - j4) <= j5 <= (j3 + j4) && (j3 + j4 + j5) <= k
        RHS = 1/qdim(j5, k=k, exact=true)
    end
    
    diff = LHS-RHS

    println("Orthogonality Check for SU(2)_$k")
    println("--------------------------------")
    println("LHS (Sum): ", LHS)
    println("RHS (Exact): ", RHS)
    println("LHS - RHS = ", diff)
    println(iszero(diff)  ? "Identity Holds!" : "Identity Failed!")
end


julia> test_orthogonality(1,1,1,1,1,1, k=5)
Orthogonality Check for SU(2)_5
--------------------------------
LHS (Sum): Exact Algebraic Result in ℚ(ζ₁₄):
  Value: (-ζ^4 + ζ^3)
RHS (Exact): -ζ^4 + ζ^3
LHS - RHS = Exact Algebraic Result in ℚ(ζ₁₄):
  Value: 0
Identity Holds!

julia> test_orthogonality(0.5, 0.5, 0.5, 0.5, 1.0, 1.0, k=10)
Orthogonality Check for SU(2)_10
--------------------------------
LHS (Sum): Exact Algebraic Result in ℚ(ζ₂₄):
  Value: (-1//2*ζ^6 + ζ^2 - 1//2)
RHS (Exact): -1//2*ζ^6 + ζ^2 - 1//2
LHS - RHS = Exact Algebraic Result in ℚ(ζ₂₄):
  Value: 0
Identity Holds!


julia> test_orthogonality(12,15,17,18,13,14, k=60)
Orthogonality Check for SU(2)_60
--------------------------------
LHS (Sum): Exact Algebraic Result in ℚ(ζ₁₂₄):
  Value: 0
RHS (Exact): 0
LHS - RHS = Exact Algebraic Result in ℚ(ζ₁₂₄):
  Value: 0
Identity Holds!

By leveraging QRecoupling.jl, the radical prefactors of the $6j$-symbols perfectly annihilate each other during multiplication, keeping the entire computation division-free and strictly inside the cyclotomic field $\mathbb{Q}(\zeta)$.

2. The Biedenharn-Elliott Pentagon Identity

The Pentagon Identity guarantees that topological invariants are independent of the chosen triangulation. It relates the product of two $6j$-symbols to a sum over the product of three $6j$-symbols.

Because this involves a large summation and many multiplications, it is the perfect candidate for our high-speed :numeric engine.

using QRecoupling

#TODO: test  ∑ {6j}{6j}{6j} = {6j}{6j}