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The Actual Arbitrary-Matrix Quadratic State

Abstract

An arbitrary matrix quadratic state has precisely the actual Sugawara field.

Let D be any finite-rank ordinary lattice: its Gram matrix G is integral and symmetric with even diagonal. Rank zero is included. No positivity, nondegeneracy or unimodularity is assumed. Charges are Fin(rank(D)) to Z, oscillators are complex multivariate polynomials indexed by Fin(rank(D)) times N, and V is the finite-support charge direct sum of that polynomial algebra. Write B for the original integral bilinear form. Normalized coefficient q means the Laurent coefficient at -q-1. The vacuum is single(0,1), Y is the constructed actual state-field map, T is the charge-sensitive translation, and mu(a,q,b)=(Y(a))_q b.

Let H be any complex rank-by-rank matrix, with no inverse, symmetry or positivity premise. Define omega as single(0,(1/2) sum_i,j H_ij X(i,0)X(j,0)). Define L_m to be sugawaraMode(D,H,m), namely the coefficient m+1 of the actual Sugawara field. This convention is used in every state law.

Theorem 1.1 (The genuine quadratic state evaluates to the normal current product).

Lean statement: D5/S3/VertexAlgebra/LatticeActualConformalState.quadratic_state_field

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

Citation. 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 all i,j, Y(single(0,X(i,0)X(j,0))) equals normalMinusOne(neutralField(i),neutralField(j)).operator. The actual current state, its minus-one product, and proved normal state-product compatibility give the result independently of occurrence order.

Theorem 1.2 (The field of the actual omega is the Sugawara field).

Lean statement: D5/S3/VertexAlgebra/LatticeActualConformalState.Y_omega

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

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

Commentary.

Y(omega(D,H))=sugawaraField(D,H) for arbitrary H. Finite double-sum linearity and the actual quadratic identity prove this equality.

Bakalov-Kac, arXiv math/0402315v1, section 4.1, equations (4.12)-(4.16), DOI 10.1142/9789812702562_0001, supplies the lattice field, ordered-product, translation and conformal construction. Equation numbers refer to arXiv v1.

Matsuo-Nagatomo, hep-th/9706118v1, Proposition 1.5.5 and Theorem 5.4.1, supplies residue locality and reconstruction by creative local fields, divided derivatives and nested normal products.

Finite normal-product and integer residue kernels retain Scott Carnahan attribution: vertexAlg revision 4453e34ec390e82a0c789c731ada8f9a6e86bdea, VertexAlg/VertexBasic/VertexOperator.lean, Apache-2.0. Actual coefficient proofs are re-elaborated on V; no polynomial Fock theorem is transferred between carriers.

The consumed Sugawara normal-ordering and commutator architecture retains Kalle Kytola, VirasoroProject revision 5ff4245383b2cdd4eea7a0524bc1274c32041eb4, Apache-2.0 attribution. The actual carrier and matrix contractions are explicit. The complete Virasoro theorem is a separately delivered supplier; these state laws do not replace it with a partial proof.

The carrier and formal-series interfaces use pinned Mathlib revision db584cd6d46c92f209a44c0f1c829460d327499d and Lean 4.33.0.

This is algebraic ungraded vertex-algebra mathematics. Finite graded pieces, positivity, PCT, Leech specialization, twisted extensions, the Monster, anomaly, fusion categories, string theory, AdS/CFT and physical completion are not proved here.

References