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Charge Characters and the Full Peirce Spectral Sign

Abstract

The explicit charge character preserves every actual lattice mode and equals the norm-four Peirce spectral sign on the whole fixed energy-two space.

For an even integral lattice D and an arbitrary charge a, define chi_a on a state single(b,p) by multiplication by (-1) to the integral pairing B(a,b). The polynomial p is unchanged. The map is the finite-support linear sum of these sector maps, and chi is its involutive linear equivalence. Neither positivity, rootlessness, norm four nor a choice of sign is required for this construction.

Theorem 1.1 (The exact charge-sector action).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.chi_single

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.chi_single (✓ 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.

Commentary.

Every oscillator polynomial in sector b is multiplied by the same pairing character; no oscillator coordinate is reflected or removed.

Theorem 1.2 (The square is the identity on every state).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.chi_involutive

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.chi_involutive (✓ 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.

Commentary.

The square of the pairing sign is one. Equality on all single-sector states extends to the full carrier. This includes the identity character and rank zero, and asserts no exact order.

Theorem 1.3 (The actual vacuum is fixed).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.chi_vacuum

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.chi_vacuum (✓ 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.

Commentary.

The vacuum has zero charge, whose pairing character is one.

Theorem 1.4 (All states and all integer modes).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.stateField_chi

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.stateField_chi (✓ 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 actual u,v and every integer n, chi(Y(u)[n]v)=Y(chi(u))[n]chi(v). The existing LatticeSectorIntertwiner.charge_homogeneity identifies each actual coefficient of Y(single(b,p))[n]single(c,q) with a literal single(b+c,r). Multiplicativity of the pairing character then proves covariance on these states. Two extensions of linear maps from Finsupp singles prove the statement for arbitrary oscillator polynomials and all states. Charge homogeneity is reused from the production fields, without an assumed graded vertex algebra or a new normal-product construction.

Theorem 1.5 (The whole literal fixed energy-two space).

Lean statement: D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.actual_fixed_energy_two_spectral_sign

Proof. Machine-checked in Lean as D5/S3/VertexAlgebra/Lattice/NormFourPeirceInvolution.actual_fixed_energy_two_spectral_sign (✓ 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.

Assume the real Gram matrix is positive definite, every integral charge has norm different from two, B(a,a)=4, and the complex scalar s has square one. Let L be the production NormFourVirasoroVector.mode(D,a,s,0). On every vector in the literal intersection of fixedSpace(D) with the eigenvalue-two space of the canonical inverse-Gram Sugawara zero mode, chi equals I-(8192/217)L(L-I/2)(L-2I), as an equality in the ambient actual carrier.

The complete actualFixedEnergyTwoBasis supplies unordered neutral colour pairs and all norm-four charge sign orbits. Neutral states have character one. The imported actual Q,R,A,E zero-mode actions give the cubic L(L-I/2)(L-2I)=0 on neutral quadratics and on the matching A,E block. This reduction includes repeated colours; it is not inferred from a quartic annihilator.

For distinct sign orbits, positivity and rootlessness bound B(a,b) between -2 and 2. Pairing zero gives LE(b)=0. Pairing plus or minus one gives LE(b)=E(b)/16 and character minus one. At 1/16 the cubic is 217/4096, so the polynomial value is exactly minus one. Pairing -2 uses the actual return shift a-(a+b)=-b and the existing cocycle identities: the two-state block has diagonal 1/4 and off-diagonal c=s epsilon(a,b)/4 with c squared equal to 1/16. Consequently L squared E(b)=LE(b)/2, so the cubic vanishes and the polynomial is plus one. Negation gives pairing two. Equal and opposite charges have even pairing and use the matching block. The pairing character is unchanged by negating b. Equality of linear maps on the complete actual basis proves the stated whole-space equality.

The production norm-four formula is the classical formula attributed by Shimakura, arXiv:1202.6107, equation (1.3), to Dong-Mason-Zhu, Theorem 6.3. The lattice construction follows Bakalov-Kac, arXiv:math/0402315, section 4.1, and Frenkel-Lepowsky-Meurman. The actual Peirce actions and fixed-space basis are imported mathematical suppliers. The result identifies the explicit all-mode character with the spectral polynomial; it makes no claim of Ising simplicity or fusion, occurrence of all four eigenvalues, numerical Leech multiplicities or Monster moonshine.

References