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arxiv: 2606.07960 · v1 · pith:DSPCZNU3new · submitted 2026-06-06 · 🧮 math.AG · math-ph· math.MP· math.SG

B-model Categorical Enumerative Invariants and holomorphic anomaly equations

Yefeng Shen , Junwu Tu This is my paper

Pith reviewed 2026-06-27 19:26 UTC · model grok-4.3

classification 🧮 math.AG math-phmath.MPmath.SG
keywords categorical enumerative invariantsholomorphic anomaly equationsCalabi-Yau threefoldsderived categoriesB-modelGivental quantizationtopological strings
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The pith

B-model categorical enumerative invariants for Calabi-Yau threefolds satisfy the BCOV holomorphic anomaly equations.

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

The paper proves that categorical enumerative invariants extracted from the derived category of coherent sheaves on a smooth projective Calabi-Yau threefold obey the holomorphic anomaly equations first written down by Bershadsky, Cecotti, Ooguri and Vafa. The authors first establish that these invariants satisfy general versions of the dilaton, string and divisor equations. They then feed those relations into the Givental quantization procedure to conclude that the full generating function, for any miniversal deformation family, satisfies the anomaly equations. A sympathetic reader would care because this supplies an algebraic definition of the B-model topological string partition function that does not rely on choosing a mirror or on geometric constructions of the invariants.

Core claim

The B-model CEI for any miniversal family of smooth projective Calabi-Yau 3-folds satisfies the holomorphic anomaly equations introduced by Bershadsky-Cecotti-Ooguri-Vafa. This provides strong evidence that CEI may be taken as a rigorous mathematical definition of the B-model topological string partition function.

What carries the argument

Givental quantization formula applied after establishing the analogs of the dilaton, string, and divisor equations for categorical enumerative invariants on the derived category.

If this is right

  • The categorical enumerative invariants satisfy the analog of the dilaton equation in general.
  • The categorical enumerative invariants satisfy the analog of the string equation in general.
  • The categorical enumerative invariants satisfy the analog of the divisor equation in general.
  • The full generating function built from these invariants obeys the BCOV holomorphic anomaly equations for every miniversal family.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result would let the B-model partition function be defined directly from the derived category, without selecting a symplectic mirror or a geometric cycle class.
  • It supplies a route to comparing A-model and B-model invariants entirely inside categorical language, rather than through geometric mirror symmetry.
  • Explicit calculations of CEI on particular threefolds could produce independent checks or new values for higher-genus Gromov-Witten invariants.

Load-bearing premise

The categorical enumerative invariants are well-defined on the derived category of coherent sheaves and the analogs of the dilaton, string, and divisor equations hold in sufficient generality to apply the Givental quantization formula without additional assumptions.

What would settle it

A concrete computation, for a specific Calabi-Yau threefold and a point in its miniversal family, showing that the CEI generating function violates one of the BCOV holomorphic anomaly equations at a fixed genus.

read the original abstract

In this paper, we study the B-model categorical enumerative invariants (CEI) associated with derived categories of coherent sheaves on smooth projective Calabi-Yau $3$-folds. We first prove the analogs of the dilaton, string, and divisor equations of CEI in a general context. Then we use these equations and the Givental quantization formula to prove that the B-model CEI for any miniversal family of smooth projective Calabi-Yau $3$-folds satisfies the holomorphic anomaly equations introduced by Bershadsky-Cecotti-Ooguri-Vafa. This provides strong evidence that CEI may be taken as a rigorous mathematical definition of the B-model topological string partition function.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims to prove analogs of the dilaton, string, and divisor equations for B-model categorical enumerative invariants (CEI) associated to derived categories of coherent sheaves on smooth projective Calabi-Yau 3-folds in a general context. It then applies the Givental quantization formula to these equations to show that the B-model CEI for any miniversal family of such threefolds satisfies the holomorphic anomaly equations of Bershadsky-Cecotti-Ooguri-Vafa, positioning CEI as a candidate rigorous definition of the B-model topological string partition function.

Significance. If the proofs of the equation analogs and the subsequent application of quantization hold in the stated generality, the result would supply a concrete mathematical link between categorical enumerative invariants and the B-model holomorphic anomaly equations. This constitutes a substantive contribution to the program of giving rigorous foundations to B-model topological strings via derived-category invariants, and the strategy of first establishing the three key equations before invoking Givental's formalism is a direct and appropriate route to the main claim.

minor comments (3)
  1. [Introduction] The introduction would benefit from a brief explicit statement of the precise generality (e.g., which classes of Calabi-Yau threefolds and which stability conditions) under which the analogs of the dilaton, string, and divisor equations are established before the quantization step is applied.
  2. [§1] Notation for the CEI and the associated generating functions should be introduced with a short table or list of symbols in §1 to improve readability when the quantization formula is invoked later.
  3. A short remark clarifying whether the miniversal family assumption is used only for the final statement or also in the proof of the equation analogs would help readers track the logical dependencies.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its significance in linking categorical enumerative invariants to the B-model holomorphic anomaly equations, and recommendation of minor revision. No specific major comments appear in the provided report, so we have no individual points requiring detailed rebuttal or clarification at this stage. We will address any minor suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external quantization formula after proving independent equations

full rationale

The paper first establishes the analogs of the dilaton, string, and divisor equations for CEI in a general derived-category context on Calabi-Yau 3-folds. It then applies the external Givental quantization formula to derive the holomorphic anomaly equations. No step reduces a claimed prediction or central result to a fitted parameter, self-definition, or load-bearing self-citation; the cited quantization formula is independent and the proofs of the preparatory equations are presented as new content. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of derived categories and the applicability of Givental quantization; no free parameters or new entities are mentioned in the abstract.

axioms (1)
  • domain assumption Standard properties of derived categories of coherent sheaves on smooth projective Calabi-Yau 3-folds allow definition of CEI and the stated equations.
    Invoked to establish the dilaton/string/divisor analogs before quantization.

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Reference graph

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