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The Norm-Four Peirce Annihilator on the Whole Fixed Space

Abstract

The actual norm-four Virasoro zero mode obeys the Peirce quartic on the complete theta-fixed energy-two space.

Let D be an even integral lattice whose real-cast Gram matrix is positive definite, and suppose every integral charge has norm different from two. Let a have norm four and let the complex scalar s satisfy s squared equal to one. Write A for the actual zero-charge state with polynomial chargePolynomial(a) squared and E(b) for single(b,1)+single(-b,1). The production vector is A/16+s E(a)/4, and L is its actual Y coefficient at integer mode one, namely NormFourVirasoroVector.mode(D,a,s,0).

Theorem 1.1 (The full actual fixed energy-two space is annihilated).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourFixedPeirce.actual_fixed_energy_two_peirce

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

Source. Repository-derived.

Acknowledgement. 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.

Acknowledgement. Igor B. Frenkel, James Lepowsky, and Arne Meurman (1988). Vertex Operator Algebras and the Monster. DOI: 10.1016/S0079-8169(08)X6136-7.

Commentary.

For every vector in the literal intersection of fixedSpace(D) with the eigenvalue-two space of the Sugawara zero mode formed using the inverse complex Gram matrix, the ambient endomorphism product L(L-1/16)(L-1/2)(L-2) sends that vector to zero. The identities are in the production finite-support lattice carrier and use exact complex rational scalars. There is no premise of invariance, diagonalization, an assumed multiplication table or an Ising representation.

For a neutral quadratic Q(i,j)=single(0,X(i,0)X(j,0)), let g_i be the Gram contraction with a and R_i=sum_k a_k Q(k,i). The actual contractions give LQ(i,j)=g_i R_j/8+g_j R_i/8+s g_i g_j E(a)/4 and LR_i=R_i/2+g_i A/8+s g_i E(a). The exceptional block is LA=A+4s E(a), LE(a)=s A/4+E(a). These equalities prove L(L-1/2)(L-2)Q(i,j)=0 directly, including repeated colours. No transverse decomposition is assumed.

For norm-four charges b outside the sign orbit of a, rootlessness and positivity applied to a-b and a+b bound the integral pairing between -2 and 2. Pairing zero gives LE(b)=0; pairing plus or minus one gives LE(b)=E(b)/16. For pairing -2, the partner a+b has norm four and pairing two with a. The actual cocycle obeys epsilon(a,a+b)=epsilon(a,b), since its diagonal norm-four phase is one. The return shift a-(a+b)=-b gives the same orbit. The resulting two-state block has diagonal 1/4 and off-diagonal s epsilon(a,b)/4; the square of the off-diagonal coefficient is 1/16. Thus L squared E(b)=LE(b)/2. Negating b gives the pairing-two case. Equal and opposite charges use the exceptional A,E(a) block.

The complete actualFixedEnergyTwoBasis consists of all unordered neutral colour pairs and all norm-four charge sign orbits. The neutral coordinate identification and the actual charged basis vectors identify their ambient states. Equality of linear maps on this basis then proves the quartic for every vector of the full literal fixed energy-two space. Rank zero is included; the norm-four premise may then have no witness.

The norm-four state formula is classical: Shimakura, arXiv:1202.6107, equation (1.3), attributes the Ising-vector assertion to Dong-Mason-Zhu, Theorem 6.3. The lattice-field construction and contractions follow Bakalov-Kac, arXiv:math/0402315, section 4.1, and Frenkel-Lepowsky-Meurman. The normal-product and residue sources retain Carnahan, Matsuo-Nagatomo and Kytola attribution and their Apache-2.0 licenses. The whole-space argument combines the neutral cubic, the actual charge blocks and the complete fixed-space basis.

This polynomial identity does not establish Ising simplicity or fusion, spectral-sign automorphisms, Miyamoto involutions, numerical Leech multiplicities, twisted sectors, Monster moonshine, CFT anomalies, string theory or AdS/CFT.

References