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One-point functions for $C_2$-cofinite VOAs: pseudo-traces and trace spaces of projective modules

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Pith's one-line read Symmetric functions on the endomorphism algebra of a projective generator map surjectively onto one-point functions for any C2-cofinite vertex operator algebra.

desk verdict This paper gives a complementary representation-theoretic proof of surjectivity for the Gainutdinov-Runkel map on C2-cofinite VOAs via pseudo-traces, plus conditional injectivity. read the letter →

arxiv 2606.19622 v1 pith:N63CR5OR submitted 2026-06-17 math.QA math.RT

Max-Niklas Steffen This is my paper
classification math.QAmath.RT
keywords vertexoperatoralgebrasC2-cofiniteone-pointfunctionsprojectivemodulestracespacespseudo-tracessymmetric
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper connects the space of one-point functions on the torus for possibly nonrational C2-cofinite vertex operator algebras to trace objects in their representation categories. It shows that the dual of the trace space of projective modules is given by symmetric functions on the endomorphism algebra of a projective generator. A map from these symmetric functions to the one-point functions is proven to be surjective in all cases. Under the extra condition that conformal weights are separated modulo the integers, the map is injective.

What carries the argument

the trace object in the subcategory of projective objects, identified in duality with symmetric functions on the endomorphism algebra E of a projective generator

What would settle it

Constructing a C2-cofinite vertex operator algebra for which there exists a one-point function not arising from any symmetric function on E via the corresponding map.

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Extended reading notes

Core claim

The central result is that the map sending symmetric functions on the endomorphism algebra E of a projective generator to one-point functions is surjective for every C2-cofinite vertex operator algebra. Injectivity holds additionally when the conformal weights are separated modulo Z. The proof relies on identifying the dual trace space with the symmetric functions and using pseudo-trace constructions on projective modules.

Load-bearing premise

The vertex operator algebra must be C2-cofinite so that its projective modules form a subcategory admitting a trace object dual to the symmetric functions on the endomorphism algebra of a projective generator.

Editorial extensions

If this is right

  • The one-point functions are spanned by pseudo-traces associated to symmetric functions on E.
  • For VOAs with separated weights, the dimension of the one-point function space equals the dimension of the space of symmetric functions on E.
  • This gives a way to determine one-point functions from the structure of projective modules.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Similar trace objects might be used to study multi-point functions on higher genus surfaces.
  • The result suggests that representation-theoretic data from projectives fully determines the torus one-point functions without needing rationality.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper develops a representation-theoretic approach to the space of one-point functions on the torus for possibly non-rational C_2-cofinite vertex operator algebras V. It relates this space to a trace object on the subcategory of projective modules, identifies the dual of the trace space with the space of symmetric functions on the endomorphism algebra E of a projective generator, and uses Arike-Nagatomo pseudo-traces to prove surjectivity of the Gainutdinov-Runkel map. Injectivity is established under the additional hypothesis that conformal weights are separated modulo Z, employing projective-cover techniques.

Significance. If the derivations hold, the work supplies a valuable complementary proof of the surjectivity part of the Gainutdinov-Runkel conjecture (recently obtained by Gui-Zhang via different methods) that stays within the framework of pseudo-traces and the projective subcategory. The explicit isolation of the separated-weights-mod-Z hypothesis for injectivity and the clean identification of the dual trace space with symmetric functions on End(P) strengthen the structural understanding of one-point functions beyond the rational case. The approach builds directly on established tools without introducing free parameters or ad-hoc identifications.

minor comments (3)
  1. [§3.2] §3.2, after Definition 3.4: the construction of the trace object on the projective subcategory is stated abstractly; a brief explicit computation for a known example (e.g., a simple current extension) would clarify how the dual identification with symmetric functions on E is realized concretely.
  2. [§5.1] §5.1, paragraph following Theorem 5.3: the argument for injectivity invokes the separated-weights condition to ensure that the pseudo-trace pairing is non-degenerate, but the precise step where the condition rules out kernel elements is only sketched; expanding this paragraph with a short diagram chase would improve readability.
  3. [Introduction] Notation: the symbol E is used both for the endomorphism algebra and (occasionally) for an extension; a single clarifying sentence in the introduction would prevent any momentary confusion.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work, the clear summary of its contributions, and the recommendation for minor revision. The report correctly identifies the complementary nature of our representation-theoretic approach to the surjectivity result of Gui-Zhang and the role of the separated-weights-mod-Z hypothesis for injectivity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The paper establishes surjectivity of the Gainutdinov-Runkel map from symmetric functions on the endomorphism algebra E of a projective generator to one-point functions, via the dual of a trace object on the projective subcategory and Arike-Nagatomo pseudo-traces. Injectivity is shown under the separated conformal weights mod Z hypothesis using projective-cover techniques. The Gainutdinov-Runkel conjecture is referenced as already proved by Gui-Zhang with independent methods, serving only as motivation; the present argument is a complementary representation-theoretic derivation that does not reduce any load-bearing step to a self-citation, fitted parameter, or definitional tautology. All identifications are stated explicitly from category theory and prior external results on pseudo-traces, rendering the derivation self-contained against external benchmarks.

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

The paper rests on the standard axioms and definitions of vertex operator algebra theory, C2-cofiniteness, and the representation category of modules, without introducing new free parameters or invented entities.

assumptions (1)
  • standard math Standard axioms of vertex operator algebras and their module categories, including C2-cofiniteness and existence of projective covers.
    Invoked throughout to define the representation category, projective generator, and pseudo-traces.

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Cite this review

Pith. "Pith review of One-point functions for $C_2$-cofinite VOAs: pseudo-traces and trace spaces of projective modules." pith.science (2026). https://pith.science/paper/N63CR5OR

@misc{pith2026260619622,
  author       = {Pith},
  title        = {Pith review of: One-point functions for $C_2$-cofinite VOAs: pseudo-traces and trace spaces of projective modules},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N63CR5OR}},
  note         = {Machine review of arXiv:2606.19622}
}
abstract

We study the space of one-point functions on the torus for a possibly nonrational $C_2$-cofinite vertex operator algebra $V$ by relating it to a trace object of the subcategory of projective objects in the representation category of $V$. We identify the dual of the trace space with symmetric functions on the endomorphism algebra $E$ of a projective generator. Motivated by the Gainutdinov-Runkel conjecture, recently established using different methods by Gui and Zhang, we present a complementary representation-theoretic approach based on Arike-Nagatomo pseudo-traces. In this framework, we prove surjectivity of the Gainutdinov-Runkel map from symmetric functions on $E$ to one-point functions. Under the additional assumption of separated conformal weights modulo $\mathbb{Z}$, we also prove injectivity, using projective-cover techniques inspired by Huang.

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