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Modularity of Vertex Operator Algebra Correlators with Zero Modes

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Pith's one-line read VOA correlators with zero modes inserted are proven to be quasi-modular (zero modes only) or quasi-Jacobi (mixed with vertex operators) forms, and a class of heavy Heisenberg algebras is introduced whose correlators have homogeneous weight.

arxiv 2411.08008 v2 pith:HXO4NISG submitted 2024-11-12 math.QA hep-th

Darlayne Addabbo , Christoph A. Keller This is my paper · ORCID
classification math.QAhep-th
keywords correlatorszeromodespropertiesformslikemodulartransform
verification ladder T0 review T1 audit T2 compute T3 formal

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The reading

Vertex operator algebras (VOAs) are algebraic structures used in conformal field theory. On a torus, one can compute traces that produce functions of the modular parameter tau. Zhu's theorem says ordinary correlators are Jacobi forms, meaning they transform nicely under the modular group. This paper asks what happens when you additionally insert zero modes, operators that measure conserved charges of inserted fields.

The authors derive recursion relations that reduce a correlator with several zero modes to correlators with fewer zero modes. Iterating, they find that a correlator containing only zero modes transforms like a quasi-modular form, while a correlator containing both zero modes and ordinary vertex operators transforms like a quasi-Jacobi form. Quasi-modular and quasi-Jacobi forms differ from ordinary ones by extra polynomial terms involving c/(c tau + d), which is the modular anomaly.

The paper then isolates conditions under which higher-weight fields reproduce the clean behavior of weight-1 fields. It defines a heavy Heisenberg algebra (HHA), a set of fields whose zero modes commute and whose mode algebra closes. For such fields, the zero-mode correlators have homogeneous weight and transform like quasi-modular forms with tails made of lower-point correlators. Examples are worked out for weight 1 and weight 2 fields, including explicit lattice VOA computations, and a shorter proof of Miyamoto's weight-1 transformation law is given.

A caveat: the main structural theorem is stated in terms of quasi-Jacobi forms, but the paper notes the general correlators need not have homogeneous weight, which conflicts with the fixed-weight definition given earlier. This is a precision issue that should be resolved.

Extended reading notes

Core claim

Theorem 3.4: For a holomorphic VOA of central charge 24p, the mixed correlator S(b_0^{(1,2,...,r)}; (a_1,z_1), ..., (a_n,z_n); tau) is a weakly holomorphic quasi-Jacobi form with a pole of at most order p at tau = i infinity. Corollary 3.5 then states that zero mode correlators are quasi-modular forms. If correct, this gives the exact modular transformation structure for correlators with zero mode insertions, extending Zhu's theorem for full correlators.

Load-bearing premise

The proof of Theorem 3.4 depends on the claim, proven in Appendix A, that g^i_j(z,tau) is a quasi-Jacobi form of weight i+j for every i,j, including i >= j. The argument for i >= j uses termwise integration of an infinite series in z and an induction for the elliptic anomaly; if that analytic step fails, the recursion no longer lands in the quasi-Jacobi category. The paper also assumes that the infinite sums over modes m in Proposition 3.3 converge so that the induction in Theorem 3.4 is legitimate.

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Assumptions & free parameters 0 free parameters · 6 assumptions · 1 invented entities

The central claim rests on standard VOA and modular form background, plus a new structural definition (HHA). No free parameters are fitted. The main fragility is the unproven convergence/closure of the recursion and the ambiguous meaning of quasi-Jacobi form for inhomogeneous weights.

assumptions (6)
  • domain assumption V is a holomorphic VOA, i.e., C2-cofinite and rational with a single irreducible module, with central charge c=24p
    Used throughout; the paper states it assumes this for simplicity and only conjectures the general rational C2-cofinite case via vector-valued forms.
  • standard math Zhu's modular invariance theorem (Zhu96, Theorem 1 here): full correlators of C2-cofinite rational VOAs transform like weakly holomorphic Jacobi forms
    This is the base case r=0 in the induction proving Theorem 3.4.
  • standard math The functions P_k and g^i_j have the stated convergence, q-expansion, and modular/elliptic transformation properties
    Used in the recursion and in computing anomalies; partially proven in Appendix A using convergence of Weierstrass series.
  • domain assumption The infinite sums over modes in Proposition 3.3 can be interchanged with the trace and converge in the relevant domain
    The recursion is derived by formal manipulation of sums over k and m; convergence is not addressed.
  • ad hoc to paper The HHA axioms (commuting zero modes and closure under m>=0 square-bracket modes) suffice to make the recursion stay inside the HHA and produce homogeneous-weight quasi-Jacobi forms
    Definition 4.1 is constructed specifically to mimic the Heisenberg algebra; examples are provided but no classification.
  • ad hoc to paper The class of quasi-Jacobi forms is understood to include finite sums of different weights (ring sense); otherwise Theorem 3.4 is inconsistent with Definition 2.3
    The paper says the resulting form need not have homogeneous weight, so the fixed-weight Definition 2.3 must be relaxed for the theorem to hold as stated.
invented entities (1)
  • Heavy Heisenberg algebra (HHA) independent evidence
    purpose: A finite set of states whose zero modes commute and whose square-bracket mode algebra closes; used to construct higher-weight fields whose zero-mode correlators transform like quasi-modular forms with homogeneous weight
    Existence is demonstrated by explicit examples (weight 1 fields, and weight 2 fields x=a[-1]^2 1 in lattice VOAs), so the structure is not merely postulated.

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Pith. "Pith review of Modularity of Vertex Operator Algebra Correlators with Zero Modes." pith.science (2026). https://pith.science/paper/HXO4NISG

@misc{pith2026241108008,
  author       = {Pith},
  title        = {Pith review of: Modularity of Vertex Operator Algebra Correlators with Zero Modes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HXO4NISG}},
  note         = {Machine review of arXiv:2411.08008}
}
abstract

It is known from Zhu's results that under modular transformations, correlators of rational $C_2$-cofinite vertex operator algebras transform like Jacobi forms. We investigate the modular transformation properties of VOA correlators that have zero modes inserted. We derive recursion relations for such correlators and use them to establish modular transformation properties. We find that correlators with only zero modes transform like quasi-modular forms, and mixed correlators with both zero modes and vertex operators transform like quasi-Jacobi forms. As an application of our results, we introduce algebras of higher weight fields whose zero mode correlators mimic the properties of those of weight 1 fields. We also give a simplified proof of the weight 1 transformation properties originally proven by Miyamoto.

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