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Uniform Graded Local Correlators

Abstract

Actual graded local fields have a common finite homogeneous numerator.

Let V be a complex vector space, Omega a vacuum, and A_i actual VertexOperator maps with nonnegative integer weights h_i. An energy endomorphism H annihilates Omega and satisfies [H,A_i[-e-1]]=(h_i+e) A_i[-e-1] at every integer exponent. Creation means A_i[n] Omega=0 for n at least zero. The output selector P_d is an idempotent endomorphism satisfying H P_d=P_d H=d P_d and fixing every vector of energy d. The uniform order k is supplied by actual pairwise locality of the operator commutators, independently of the state on which they act.

Theorem 1.1 (A finite numerator for every insertion length and every ordering).

Lean statement: D5/S3/VertexAlgebra/UniformGradedLocalCorrelator.uniform_graded_local_correlator

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

Source. Repository-derived.

Acknowledgement. Atsushi Matsuo; Kiyokazu Nagatomo (1997). On axioms for a vertex algebra and the locality of quantum fields. URL: https://arxiv.org/abs/hep-th/9706118v1.

Commentary.

For any list containing each label exactly once, the coefficient at exponent e is P_d applied to the successive actual modes A_i[-e_i-1] acting on Omega. Keep the labels fixed and put Q=product over i<j of (z_i-z_j)^k. Multiplication by Q is finite convolution in the full coefficient-distribution space. Pairwise locality clears adjacent exchanges, so Q times the coefficient distribution is the same for every permutation. No cancellation of Q in that distribution space is used.

Moving label i to the rightmost position and applying creation eliminates every negative exponent in coordinate i. Each polynomial difference preserves this property. Energy covariance gives total exponent d minus the sum of the weights; the clearing factors raise it to D=d-sum h_i+k times the number of labelled pairs. Each nonzero numerator coefficient therefore has a nonnegative exponent vector of total degree D. Every coordinate is bounded by D, giving a finite box. The numerator is constructed as a Finsupp on natural exponent vectors with coefficients in V, rather than assuming its support.

If every coefficient of one ordering lies in a fixed complex submodule W, each numerator coefficient lies in W: it is the finite sum of scalar multiples prescribed by Q. When D is negative the numerator is zero. For no insertions the word is the identity and Q is one; for one insertion there are no pair factors. Zero weights and locality order zero are included without exceptions.

This statement applies to any actual family with these laws. Its application to the moonshine VOA requires that carrier, its field maps, energy and output projections. Rational collision expansions, associativity with spectators, Casimir expressions and the identification of Monster invariants are additional mathematical obligations.

References