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Trace pairing and a complete resolution of a split projector

Statement

Let $C$ be a complex modular ribbon fusion category with its spherical pivotal trace, let $Y$ be an object, and let $P : Y \to Y$ satisfy $P^{2} = P$ . For each simple label $i$ , put $E_{i} = {e : Y \to U_{i} ∣ e P = e}, F_{i} = {f : U_{i} \to Y ∣ P f = f} .$ The bilinear pairing $E_{i} \times F_{i} \to C$ , $(e, f) \mapsto {tr}_{j} (e f)$ , is nondegenerate. For arbitrary bases $(e_{i, a})$ and $(f_{i, b})$ of these two spaces, its matrix $C_{a b}^{(i)} = {tr}_{j} (e_{i, a} f_{i, b})$ is invertible and $P = \sum_{i, a, b} d_{i} (C^{(i)}^{- 1})_{b a} f_{i, b} e_{i, a}, d_{i} = {tr}_{j} ({id}_{U_{i}}) .$ Zero-dimensional blocks contribute an empty sum; the inverse of their empty matrix denotes the unique automorphism of the zero vector space. This identity does not divide by ${tr}_{j} (P)$ or by $d_{Y}$ , either of which may vanish.

Statement references

Modular ribbon fusion category and Hopf-link data

Simple ribbon fusion dimensions are nonzero

Facts & Assumptions

Given: The object, idempotent, trace, fixed Hom spaces, and bases in the Statement.

[F1]

Modular ribbon fusion category and Hopf-link data gives finite semisimplicity, $End (U_{i}) = C {id}_{U_{i}}$ , and the pivotal trace. An idempotent on a finite direct sum of simple objects splits by the ordinary image/kernel decomposition of its finite multiplicity matrices.

[F2]

Simple ribbon fusion dimensions are nonzero gives $d_{i} \neq 0$ . Linearity of the pivotal trace gives ${tr}_{j} (t {id}_{U_{i}}) = t d_{i}$ for $t \in C$ .

Proof

1.1givenF1algebra

Split $P$ as $Y \overset{r}{\to} Z \overset{h}{\to} Y$ with $r h = {id}_{Z}$ and $h r = P$ , using the image/kernel decompositions in [F1]. The maps $e \mapsto e h$ and $f \mapsto r f$ identify $E_{i}$ with $Hom (Z, U_{i})$ and $F_{i}$ with $Hom (U_{i}, Z)$ . Their inverses are $a \mapsto a r$ and $b \mapsto h b$ : for example $(e h) r = e P = e$ and $r (h b) = b$ . Under these maps $e f = (e h) (r f)$ , so the pairing is unchanged.

2.1F1F2step 1.1algebra

Write $Z = ⨁_{i, t} U_{i}$ with inclusions $h_{i, t} : U_{i} \to Z$ and projections $r_{i, t} : Z \to U_{i}$ . They satisfy $r_{i, t} h_{k, s} = δ_{i k} δ_{t s} {id}_{U_{i}}$ when $i = k$ , and zero when $i \neq k$ , and $\sum_{i, t} h_{i, t} r_{i, t} = {id}_{Z}$ . Thus the pairing between $Hom (Z, U_{i})$ and $Hom (U_{i}, Z)$ has matrix $d_{i} I$ in these projection/inclusion bases. By [F2] it is nondegenerate, and this property persists under every invertible change of bases and the isomorphisms of step 1.1. Consequently every $C^{(i)}$ in the Statement is square and invertible.

3.1givenF1F2step 2.1algebra

Define $R$ to be the displayed sum in the Statement. It satisfies $P R = R = R P$ , because every $e_{i, a}$ and $f_{i, b}$ is fixed on its stated side. For $f_{k, c}$ in a chosen basis, Schur's lemma gives $e_{i, a} f_{k, c} = 0$ if $i \neq k$ , and $e_{k, a} f_{k, c} = (C_{a c}^{(k)} / d_{k}) {id}_{U_{k}}$ if $i = k$ . Therefore $R f_{k, c} = \sum_{a, b} d_{k} (C^{(k)}^{- 1})_{b a} f_{k, b} \frac{C_{a c}^{(k)}}{d_{k}} = f_{k, c} .$ This calculation divides only by the nonzero simple dimension $d_{k}$ .

4.1step 1.1step 2.1step 3.1algebra∎

The maps $h h_{i, t} : U_{i} \to Y$ from steps 1.1 and 2.1 belong to $F_{i}$ and span its simple inclusions, so step 3.1 gives $R h h_{i, t} = h h_{i, t}$ for all $i, t$ . Summing against $r_{i, t}$ gives $R h = h$ . Since $R = R P = R h r$ , it follows that $R = h r = P$ . If $Z = 0$ , all these sums are empty and $P = 0$ , so the same conclusion holds. This proves both nondegeneracy and the resolution, without a nonzero total-dimension assumption.

Remarks

This lemma supplies the finite algebraic resolution after a boundary projector and its field-space identifications have been constructed. It does not identify a disk RT correlator with this trace pairing and does not construct the FRS connecting ribbon graph or the marked-torus projector.

Depends on

Dependency tree · 2 levels

Sources

Fjelstad, Fuchs, Runkel and Schweigert, TFT Construction of RCFT Correlators V, arXiv:hep-th/0503194v2