Counting invariant curves: A theory of Gopakumar–Vafa invariants \n for Calabi–Yau threefolds with an involution
Bibliographic record
Abstract
We develop a theory of Gopakumar-Vafa (GV) invariants for a Calabi-Yau threefold (CY3) X which is equipped with an involution ı preserving the holomorphic volume form.We define integers n g,h (β) which give a virtual count of the number of genus g curves C on X, in the class β ∈ H 2 (X), which are invariant under ı, and whose quotient C/ı has genus h.We give two definitions of n g,h (β) which we conjecture to be equivalent: one in terms of a version of Pandharipande-Thomas theory and one in terms of a version of Maulik-Toda theory.We compute our invariants and give evidence for our conjecture in several cases.In particular, we compute our invariants when X = S × C, where S is an Abelian surface with ı(a) = -a or a K3 surface with a symplectic involution (a Nikulin K3 surface).For these cases, we give formulas for our invariants in terms of Jacobi modular forms.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.008 | 0.003 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.004 | 0.006 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".