Arrow Conventions

Sign Convention for Incoming vs. Outgoing

The Direction type assigns \(+1\) to Incoming and \(-1\) to Outgoing — a protocol-level requirement defined in the Type System. In the abstract index notation used throughout this section, incoming edges carry upper (contravariant) indices and outgoing edges carry lower (covariant) indices.

The signed sum of all magnetic quantum numbers must vanish for any non-zero basis element:

\[\sum_{\text{incoming}} m^i - \sum_{\text{outgoing}} m_j = 0\]

Incoming edges contribute \(+m\) and outgoing edges contribute \(-m\) to the conservation law.

Invariant Metric

The invariant metric \(g^{(j)}\) implements the canonical isomorphism between an irreducible representation and its dual. It must satisfy:

\[g^{(j)}_{m,\,m'} = (-1)^{j - m}\,\delta_{m,\,-m'}\]

This is a \((2j+1) \times (2j+1)\) real matrix. Its key properties are:

• Orthogonality: \(g^{(j)} (g^{(j)})^\top = I\)
• Square: \((g^{(j)})^2 = (-1)^{2j}\, I\)
• Transpose: \((g^{(j)})^\top = (-1)^{2j}\, g^{(j)}\) (symmetric for integer spins, antisymmetric for half-integer)
• Inverse: \((g^{(j)})^{-1} = (-1)^{2j}\, g^{(j)}\)

Arrow Reversal

To convert an incoming (upper) edge to an outgoing (lower) edge, lower the corresponding index with \(g_{ab}\):

\[\tilde{T}_{\ldots m'} = g^{(j)}_{m'm}\, T^{\ldots m}\]

To convert an outgoing (lower) edge to an incoming (upper) edge, raise the corresponding index with \(g^{ab}\):

\[\tilde{T}^{m'}_{\ldots} = g^{(j)m'm}\, T_{\ldots m} = T_{\ldots m} {[g^{(j)\top}]}^{mm;}\]

The two operations are exact inverses: \(g_{m'k}\, g^{km} = \delta_{m'}^{m}\).

Frobenius-Schur Indicator

The general definition of the Frobenius-Schur (FS) phase is given in Duality Interface — fs_phase. For SU(2) with spin \(j\) the formula specialises to:

\[\text{FS}(j) = (-1)^{2j}\]

It equals \(+1\) for integer spin (bosonic) representations and \(-1\) for half-integer spin (fermionic) representations. The FS indicator governs the phase accumulated when a contracted edge pair has its arrows flipped (bond inversion).

Bond Inversion

A bond is a contracted pair of edges between two tensors. The canonical pairing contracts an incoming (upper) edge of one tensor against an outgoing (lower) edge of the other:

\[\langle T, S \rangle = T^m\, S_m\]

When both edges of a bond share the same direction — for example, two leading edges (both incoming) are contracted together — a metric must be inserted to form a valid pairing. Contracting \(T\) and \(S\) along a bond with a metric insertion on \(S\):

\[\langle T,\, g S \rangle = T^m\, g_{mn}\, S^n\]

Bond inversion means moving the metric insertion from one side of the bond to the other. Using the transpose property \(g^\top = (-1)^{2j}\, g\):

\[\langle T,\, g S \rangle = \langle g^\top T,\, S \rangle = (-1)^{2j}\, \langle g T,\, S \rangle\]

Each time the metric is transferred across the bond, the contraction acquires a factor of \(\text{FS}(j) = (-1)^{2j}\). In practice, this means that computing an X-symbol with a bond whose arrows differ from the canonical configuration (e.g. both leading edges contracted, or both terminal edges contracted) produces a result that differs from the canonical-direction X-symbol by a factor of \(\text{FS}(j)\) per inverted bond.

Canonical Conjugate Pattern

The general contract for compute_conjugate is given in Duality Interface — compute_conjugate. For the SU(2) reference implementation, the canonical conjugate pattern is the direction-reversed counterpart of the canonical basis direction (Basis Conventions — Canonical Basis Direction): all leading edges outgoing and the terminal edge incoming.

This is the unique pattern consistent with the left-associative fusion tree when all arrows are reversed. An edge \(k\) of the input spec \(S\) is deviating if its direction differs from this canonical pattern. Each deviating edge contributes a factor of \(\text{FS}(r_k)\) to the accumulated conjugate phase:

\[\phi = \prod_{k\,\in\,\text{deviating edges}} \text{FS}(r_k) \in \{+1, -1\}\]

If \(S\) already matches the canonical conjugate pattern, then \(\phi = +1\).

Origin of the conjugate phase

The phase \(\phi\) is fixed by requiring that the full trace of \(\overline{T}\) against \(T\) yields the identity:

\[\langle \overline{T}_\mu, T_\nu \rangle = \delta_{\mu\nu}\]

For the canonical basis direction (leading edges incoming, terminal edge outgoing), the canonical conjugate \(\overline{T}^{\,\text{canon}}_\mu\) has all arrows reversed, and the trace pairs each upper index of \(T\) with the corresponding lower index of \(\overline{T}\) — no metric insertions are needed, so \(\phi = +1\).

When spec \(S\) deviates from the canonical basis direction on edge \(k\), the basis tensor \(T^S\) is obtained from the canonical tensor \(T^{\,\text{canon}}\) by applying \(g^{(r_k)}\) on edge \(k\):

\[T^S = g^{(r_k)}_k\, T^{\,\text{canon}}\]

For the trace \(\langle \overline{T}^S, T^S \rangle = \langle \overline{T}^S,\, g^{(r_k)}_k\, T^{\,\text{canon}} \rangle\) to equal the canonical trace, the extra metric factor on \(T^S\) must be transferred to \(\overline{T}^S\). By the bond inversion identity:

\[\langle \overline{T},\, g S \rangle = (-1)^{2j}\, \langle g\, \overline{T},\, S \rangle\]

moving \(g^{(r_k)}\) from \(T^S\) to \(\overline{T}^S\) costs a factor of \(\text{FS}(r_k) = (-1)^{2j_k}\). Each deviating edge contributes one such factor, giving the accumulated phase \(\phi = \prod_{k\,\in\,\text{deviating}} \text{FS}(r_k)\).