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The level-zero trivial sl2 module

Example

The one-dimensional trivial representation of the loop central extension, with central level $k = 0$ , is irreducible, unitary, and positive-energy in degree zero. Every Sugawara mode is zero on it, and its Virasoro central charge is $0$ .

Facts & Assumptions

Given: The one-dimensional complex Hilbert space $H = C$ , with every loop-current mode and the central generator acting by zero, and the degree-zero grading.

[F1]

Positive-energy unitary module for an affine current algebra: a positive-energy unitary module has nonnegative energy grading, skew-adjoint compact modes, and a scalar central level.

[F2]

Sugawara modes and their normal ordering: each Sugawara mode is a scalar multiple of a sum of normally ordered products of two current modes.

[F3]

Sugawara modes satisfy Virasoro at every noncritical complex level: the central charge is $k \dim (g) / (k + h^{\lor})$ in this module regime.

[F4]

Dual Coxeter number for the normalized invariant form: $h^{\lor} > 0$ for the normalized compact simple algebra.

Verification

1.1givenF1algebra

The zero action respects the affine bracket because both the current bracket and its central term act as zero; $H_{0} = H$ is one-dimensional and every positive energy space is zero. The inner product is positive and the adjoint relation $0^{*} = - 0$ holds, so the stated representation meets [F1] and is irreducible.

2.1F2F3F4algebra∎

Every summand in [F2] is a product of zero operators, hence $L_{m} = 0$ for all $m$ . In [F3], $k = 0$ and $h^{\lor} > 0$ by [F4], so the denominator is nonzero and $c = 0$ . Thus the Virasoro relation is the zero identity on this example.

Depends on

Used by

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Sources

Antony Wassermann, Kac–Moody and Virasoro Algebras, Chapter III §6 (standard reference, not scraped)