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Actual All-Charge Sugawara Virasoro Law

Abstract

The actual quadratic modes satisfy all-integer Virasoro with central charge equal to lattice rank.

Assume precisely H Gc=Gc H=1 on the complex Gram matrix of the existing even lattice. The current commutator implies that the Virasoro defect commutes with all currents. Restricting to each charge sector, inverse-Gram differentiation and polynomial induction make the commutant scalar. This directly uses the frozen private eq_constant_of_partials_zero of ConditionalPolynomialRigidity. Weighted Euler kills the off-diagonal defect; diagonal evaluation retains and cancels the charged boundaries before computing the inverse-Gram trace and cubic oscillator sum.

Theorem 1.1 (All-integer Virasoro with central charge equal to rank).

Lean statement: D5/S3/VertexAlgebra/LatticeSugawaraVirasoro.sugawaraMode_virasoro

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/LatticeSugawaraVirasoro.sugawaraMode_virasoro (✓ std3). ∎

Citation. Bojko Bakalov and Victor G. Kac (2004). Twisted Modules over Lattice Vertex Algebras. DOI: 10.1142/9789812702562_0001. URL: https://arxiv.org/abs/math/0402315v1.

Commentary.

For every pair of integers m,n, the actual endomorphisms satisfy [L(m),L(n)]=(m-n)L(m+n) plus rank(D)(m^3-m)/12 times delta(m+n,0) id. The current law gives [L(m),h_i(q)]=-q h_i(m+q). The central defect commutes with all currents and is scalar independently on each charge sector. Weighted Euler kills the off-diagonal defect.

For a>0, evaluating L(a)L(-a) on single(beta,1) keeps both charged boundary terms. Subtracting 2a L(0) cancels their charge contribution. The remaining inverse-Gram trace is rank(D), and the oscillator sum is (a^3-a)/6. Skew symmetry supplies negative diagonal modes and the zero diagonal is zero. Consequently the scalar is the same on every integral charge sector.

The charge-sensitive lattice conformal construction is classical: Bakalov-Kac, arXiv math/0402315v1, section 4.1, equations (4.12)-(4.16). The finite commutator organization is adapted from Kytola at revision 5ff4245383b2cdd4eea7a0524bc1274c32041eb4. No positivity or fixed-charge Fock transfer is assumed.

References