My planar contractions currently lead to space mismatches.

[Gertian]

In my first commit I managed to make the relevant diagrams planar trough the introduction of braiding symbols.

The resulting code however gives a spacemismatch on the contraction :

function northwest_corner(E4, C1, E1, peps_above, peps_below=peps_above)
    @planar opt=true corner[-1 -2 -3; -4 -5 -6] := 
        E4[-1 1 2; 3] *
        C1[3; 4] *
        E1[4 5 6; -4] *
        peps_above[21; 11 10 18 17] *
        conj(peps_below[21; 13 14 15 16])*
        τ[5 7;6 8]*τ[-5 9;-6 10]*τ[8 12;9 11]*τ[7 14;12 13]*τ[15 2;16 19]*τ[19 1;17 20]*τ[20 -2;18 -3]
end

The stack leads to this final function :

function planarcontract!(C::AbstractTensorMap{S,N₁,N₂},
                         A::BraidingTensor{S}   , (oindA, cindA)::Index2Tuple{2,2}, 
                         B::AbstractTensorMap{S}, (cindB, oindB)::Index2Tuple{2,N₃},
                         (p1, p2)::Index2Tuple{N₁,N₂},
                         α::Number, β::Number,
                         backend::Backend...) where {S,N₁,N₂,N₃}

which spits out :
ERROR: SpaceMismatch("(ℂ^1 ⊗ ℂ^1) ← ((ℂ^1)' ⊗ ℂ^1) ≠ permute(((ℂ^1)' ⊗ ℂ^1) ← (ℂ^1 ⊗ (ℂ^1)')[(2, 4), (1, 3)] * (ℂ^1 ⊗ ℂ^1 ⊗ (ℂ^1)') ← ℂ^1[(2, 3), (1, 4)], ((1, 2), (3, 4))")
Stacktrace:

I suspect that the origin of this bug is a mislabeling of my tau indices but I dont know which is the appropriate labeling.
@Jutho ?

[Gertian]

Update : The following version seems to work just fine I'll find out why tomorrow :)

    function northwest_corner_planar(E4, C1, E1, peps_above, peps_below=peps_above)
        @planar opt=true corner[L1 S s4; L4 E3 e3] :=
            E4[L1 W w3; L2] *
            C1[L2; L3] *
            E1[L3 N3 n2; L4] *
            peps_above[P1; N1 E1 S W] *
            conj(peps_below[P3;n1 e1 s1 w1]) *
            τ[e1 n1; n2 e2] * τ[w2 s2; s1 w1] * τ[P2 s3; s2 P3] * τ[P1 N1; N2 P2] *
            τ[e2 E2; E3 e3] * τ[E1 s4; s3 E2] * τ[w3 N2; N3 w2]
    end
    ```

[Gertian]

Update : it was indeed the convention of \tau indices.
The contractions now work but in all likeliness there are still conventions that I don't understand !

[Jutho]

Hi Gertian, what is the confusion about the tau indices? It's as always:
outgoing1 outgoing2 incoming1 incoming2

where the tau tensor maps incoming1 to outgoing2 and vice versa (whether it is braided above or below does not matter for fermions/symmetric braiding).

[Gertian]

Lets say we have two lines a, b crossing. Naively I thought that all definitions such as
tau[a1 b1; b2 a2] --> a1, b1 cross into b2 a2
and
tau[a1 b2;b1 a2] --> a1, b2 cross into b1 a2
would both lead to a planar diagram but this is not true.

As I understand these different choices easily arise when rotating crossings and naively taking the same rule for indices as is depicted below where the purple line shows my naive separation between domain and codomain.

I suspect that there is some rule related to drawing the crossings that I'm not aware of that solves this ambiguity ?

[Jutho]

The correct way to think of the tau is that

[Gertian]

Ok that makes sense.
So in a contraction this would be tau[a1 a2; b1 b2]

Could you however reinterpret this as a crossing from a2,b2 to b1 a1 and hence use tau[a2 b2;a1 b1] ?