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Fredenhagen–Jörß local fields glue under refinement

Statement

Let a conformal net and Fredenhagen–Jörß field family satisfy the definitions given earlier on this page. Fix a finite cover of $S^{1}$ by intervals $I_{1}, \dots, I_{N}$ and a smooth partition of unity ${ϕ_{i}}_{i = 1}^{N}$ subordinate to that cover. For a quasi-primary $a$ and $f \in C^{\infty} (S^{1})$ , define on $H^{\infty}$ $Y_{ϕ} (a, f) b = \sum_{i = 1}^{N} Y_{I_{i}} (a, ϕ_{i} f) b .$ If, for each $i$ , the partition is refined by finitely many smooth functions $ψ_{i r}$ supported in intervals $J_{i r} \subset I_{i}$ with $\sum_{r} ψ_{i r} = 1$ on $supp ϕ_{i}$ , then replacing each summand by $\sum_{r} Y_{J_{i r}} (a, ψ_{i r} ϕ_{i} f) b$ leaves $Y_{ϕ} (a, f) b$ unchanged for every $b \in H^{\infty}$ . Thus the global smearing is well-defined on the stated domain and independent of such local refinements.

Statement references

Möbius and conformal nets on the circle

Fredenhagen–Jörß fields and their energy bounds

Smooth partitions of unity subordinate to an open cover

Facts & Assumptions

Given: A finite interval cover of $S^{1}$ , a subordinate smooth partition ${ϕ_{i}}_{i = 1}^{N}$ , refinement functions ${ψ_{i r}}$ as stated, a quasi-primary vector $a$ , a smooth test function $f$ , and $b \in H^{\infty}$ .

[F1]

The local FJ fields are linear in test functions on $H^{\infty}$ and compatible under interval inclusion (Fredenhagen–Jörß fields and their energy bounds).

[F2]

A subordinate smooth partition has $supp ϕ_{i} \subseteq I_{i}$ and $\sum_{i} ϕ_{i} = 1$ (Smooth partitions of unity subordinate to an open cover).

[A1]

For each $i$ , $supp ψ_{i r} \subseteq J_{i r} \subset I_{i}$ and $\sum_{r} ψ_{i r} = 1$ on $supp ϕ_{i}$ . No existence of a refinement is used in the conditional statement.

Proof

1.1givenF1F2

Since $supp ϕ_{i} \subset I_{i}$ , each test function $ϕ_{i} f$ is supported in $I_{i}$ . By [F1], every term $Y_{I_{i}} (a, ϕ_{i} f) b$ is defined on $H^{\infty}$ ; the sum is finite, so it defines a vector in $H$ for every $b \in H^{\infty}$ .

1.2givenF1A1

For each $i, r$ , the support inclusion in [A1] gives $supp (ψ_{i r} ϕ_{i} f) \subset J_{i r} \subset I_{i}$ . By the interval compatibility in [F1], $Y_{J_{i r}} (a, ψ_{i r} ϕ_{i} f) b = Y_{I_{i}} (a, ψ_{i r} ϕ_{i} f) b$ .

2.1givenF1A1step 1.2algebra∎

Sum the equalities in 1.2 over $r$ . Linearity in [F1] and $\sum_{r} ψ_{i r} = 1$ on $supp ϕ_{i}$ give $\sum_{r} Y_{J_{i r}} (a, ψ_{i r} ϕ_{i} f) b = Y_{I_{i}} ⁣ (a, \sum_{r} ψ_{i r} ϕ_{i} f) b = Y_{I_{i}} (a, ϕ_{i} f) b .$ Summing this identity over $i$ proves the stated refinement invariance on $H^{\infty}$ .

Depends on

Dependency tree · 2 levels

Sources

Sebastiano Carpi, Yasuyuki Kawahigashi, Roberto Longo, and Mihály Weiner, From Vertex Operator Algebras to Conformal Nets and Back, arXiv:1503.01260v4