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Restricted affine-current fields are mutually local

Statement

Let $g$ be a finite-dimensional complex simple Lie algebra with normalized invariant form $B$ , let $k \in C$ , and let $V$ be a restricted affine-current module of level $k$ as in Restricted affine-current module for the untwisted central extension of Untwisted affine central extension. Write $J_{x} (n)$ for the current mode of $x \in g$ at $n \in Z$ , so $[J_{x} (m), J_{y} (n)] = J_{[x, y]} (m + n) + m k B (x, y) δ_{m + n, 0} I_{V} .$ For independent formal variables $z, w$ , define $J^{x} (z) : = \sum_{n \in Z} J_{x} (n) z^{- n - 1}, δ (z, w) : = \sum_{m \in Z} z^{- m - 1} w^{m} .$ Restrictedness makes $J^{x} (z)$ a field on $V$ , meaning that for each $v \in V$ , $J^{x} (z) v$ has only finitely many negative powers of $z$ . Coefficientwise on $V$ the commutator is $[J^{x} (z), J^{y} (w)] = J^{[x, y]} (w) δ (z, w) + k B (x, y) \partial_{w} δ (z, w) I_{V} .$ In particular $(z - w)^{2} [J^{x} (z), J^{y} (w)] = 0$ , so the currents are pairwise local of order at most two. Define the two formal expansions by $ι_{z, w} (z - w)^{- 1} : = \sum_{n \geq 0} z^{- n - 1} w^{n}, ι_{w, z} (z - w)^{- 1} : = - \sum_{n \geq 0} w^{- n - 1} z^{n},$ and define the corresponding expansions of $(z - w)^{- 2}$ by differentiating in $w$ . The notation $A (z) B (w) \sim C (w) / (z - w)^{2} + D (w) / (z - w)$ records these singular terms with those two expansions; their first-minus-second difference is $C (w) \partial_{w} δ (z, w) + D (w) δ (z, w)$ . In this convention the singular OPE is $J^{x} (z) J^{y} (w) \sim \frac{k B (x, y) I_{V}}{(z - w)^{2}} + \frac{J^{[x, y]} (w)}{z - w} .$

Statement references

Restricted affine-current module

Untwisted affine central extension

Facts & Assumptions

Given: The finite-dimensional complex simple Lie algebra, its normalized invariant form, the complex level $k$ , and the restricted current module $V$ .

[F1]

Restricted affine-current module: for every $v \in V$ and $x \in g$ , all sufficiently large positive modes $J_{x} (n)$ annihilate $v$ .

[F2]

Untwisted affine central extension: the affine bracket is $[J_{x} (m), J_{y} (n)] = J_{[x, y]} (m + n) + m k B (x, y) δ_{m + n, 0} I_{V}$ .

Proof

1.1F1givenalgebra

For fixed $v$ , [F1] gives an integer $N$ such that $J_{x} (n) v = 0$ for $n \geq N$ . Hence only finitely many terms of $J^{x} (z) v$ have negative powers of $z$ , so $J^{x} (z)$ is a field on $V$ .

2.1F2step 1.1algebra

Expand the commutator coefficientwise and apply [F2]. In the noncentral sum, reindex by $r = m + n$ ; then $\sum_{m, n} J_{[x, y]} (m + n) z^{- m - 1} w^{- n - 1} = J^{[x, y]} (w) \sum_{m} z^{- m - 1} w^{m} = J^{[x, y]} (w) δ (z, w) .$ The central sum is $k B (x, y) \sum_{m} m z^{- m - 1} w^{m - 1} = k B (x, y) \partial_{w} δ (z, w) .$ Each coefficient in the two-variable distribution is a single mode commutator, so these reindexings are coefficient identities.

3.1step 2.1algebra

Index shifting in the defining series gives $(z - w) δ (z, w) = 0$ . Differentiating with respect to $w$ gives $(z - w) \partial_{w} δ (z, w) = δ (z, w)$ , and multiplying once more by $z - w$ yields $(z - w)^{2} \partial_{w} δ (z, w) = 0$ . Substitution in step 2.1 proves $(z - w)^{2} [J^{x} (z), J^{y} (w)] = 0$ .

4.1step 2.1algebra∎

The two expansions of $(z - w)^{- 1}$ differ by $δ (z, w)$ , and differentiating them in $w$ shows that the two expansions of $(z - w)^{- 2}$ differ by $\partial_{w} δ (z, w)$ . Thus the commutator in step 2.1 is exactly the difference of the two expansions of the displayed singular OPE, with coefficients $J^{[x, y]} (w)$ and $k B (x, y) I_{V}$ .

Restricted affine-current fields are mutually local

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources