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  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Sugawara modes act on affine currents with weight one

Statement

Let V be a restricted affine-current module of any complex level k≠−h∨ as in Restricted affine-current module. For the modes of Sugawara modes and their normal ordering, every x∈g and m,n∈Z satisfy [Lm,Jx(n)]=−nJx(m+n). In particular this holds on all positive-energy unitary modules in the previously considered nonnegative integral level branch.

Facts & Assumptions

Given: The restricted affine-current module and its noncritical level.

[F1]

Sugawara current commutators on every restricted module proves the current commutator with precisely these hypotheses and normal ordering.

[F2]

Sugawara modes and their normal ordering defines locally finite modes whenever k+h∨≠0.

Proof

1.1givenF1F2

The assumed noncritical level makes the denominator in [F2] nonzero, and restrictedness makes all its mode operators defined on every vector. These are exactly the modes used by [F1].

2.1F1step 1.1∎

Apply [F1] to each x,m,n. It yields the asserted identity as an equality of endomorphisms of V. Since no extra hypotheses were used, the positive-energy unitary subclass is included whenever its level is noncritical.

Depends on

Used by

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Sources