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The Actual Norm-Four Sugawara Complement

Abstract

The actual Sugawara and norm-four fields have the complete mixed OPE and a commuting Virasoro complement.

Let D be the production finite-rank even integral lattice data, G its complex Gram matrix, and H a complex matrix satisfying both HG=1 and GH=1. Let B(a,a)=4. On the full production Carrier(D), write vacuum for single(0,1), T for the charge-sensitive translation, Y for the actual state-field map, omega for the H-Sugawara state, and e=(1/16)single(0,chargePolynomial(a)^2)+(s/4)(single(a,1)+single(-a,1)). Normalized mode j is the Laurent coefficient at -j-1; Le_m=Y(e)(m+1) and Ltot_m=Y(omega)(m+1). No positivity, rootlessness or smaller carrier is assumed.

Theorem 1.1 (Both mixed OPE orientations and all integer mixed commutators).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourCosetVirasoro.mixed_products

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourCosetVirasoro.mixed_products (✓ std3). ∎

Source. Repository-derived.

Commentary.

For every complex s, both omega_j e and e_j omega equal T(e) at j=0, 2e at j=1, vacuum/4 at j=3, and zero at every other nonnegative integer j, including j=2 and every j at least four. Simultaneously, for all integers m,n, [Ltot_m,Le_n]=(m-n)Le_(m+n)+delta_(m+n,0)(m^3-m)I/24, as endomorphisms of the entire production carrier.

The actual current commutator gives [Ltot_m,a_(-1)]=a_(m-1). Applied twice to the vacuum, it gives 2 times the neutral quadratic state at m=0, B(a,a) times the vacuum at m=2, and zero at all other nonnegative m. The supplied charged-ground formulas treat both a and -a, while Ltot_(-1)=T. Thus the forward mixed OPE follows in the actual lattice carrier. The full integer Ward identity reduces to four terms; its integer binomial polynomial includes negative m+1. For the reverse orientation, omega=Ltot_(-2)vacuum and creativity give Le_m omega=(m+2)Le_(m-2)vacuum+delta_(m,2)vacuum/4 for m at least -1. The creation and translation coefficients and vacuum annihilation give all reverse branches and tail.

Theorem 1.2 (The actual complementary field commutes and has center rank minus one half).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourCosetVirasoro.coset_virasoro

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourCosetVirasoro.coset_virasoro (✓ std3). ∎

Source. Repository-derived.

Commentary.

If s squared equals one, set c=omega-e and Lc_m=Y(c)(m+1). The actual field is Y(c)=sugawaraField(D,H)-Y(e), and every integer mode is Lc_m=Ltot_m-Le_m. For all integers m,n, [Lc_m,Le_n]=0 and [Lc_m,Lc_n]=(m-n)Lc(m+n)+delta_(m+n,0)((rank(D)-1/2)/12)(m^3-m)I. Every operator acts on all of Carrier(D). State-field linearity binds the complement to its actual field before the two supplied Virasoro laws are combined with the proved mixed law.

At rank 24 this central charge is 24-1/2=47/2 by substitution. The norm-four formula is classical: Shimakura, arXiv:1202.6107, (1.3), attributes the Ising-vector assertion to Dong-Mason-Zhu, Theorem 6.3. The lattice fields follow Bakalov-Kac, math/0402315, section 4.1. Imported Carnahan and Kytola sources retain their licenses and attribution. The mixed calculation and concrete complement here establish commuting Virasoro actions, without asserting a whole commutant VOA, Baby Monster VOA, Ising simplicity, fusion rules or Monster completion.

References