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 preserving the holomorphic volume form.We define integers n g,h () which give a virtual count of the number of genus g curves C on X, in the class H 2 (X), which are invariant under , and whose quotient C/ has genus h.We give two definitions of n g,h () 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 (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. Ordinary GV invariantsLet X be a Calabi-Yau threefold (CY3), by which we mean a smooth quasi-projective variety over C of dimension 3 with K X = O X .In 1998 [GV98], Gopakumar and Vafa (GV) defined via physics integer invariants n g () which give a virtual count of curves C X of genus g and classMathematically, there are two conjecturally equivalent sheaf-theoretic approaches to defining n g (), one by Pandharipande and Thomas (PT) via their stable pair invariants [PT10], and one more recently given by Maulik and Toda (MT) using perverse sheaves [MTo18].We begin by reviewing ordinary GV theory, and then we develop in a parallel fashion a theory of GV invariants for CY3s with an involution. GV invariants via PT theoryLet PT ,n (X) be the moduli space of PT pairs [PT09]: PT ,n (X) = (F, s) : s H 0 (X, F ), [supp(F )] = , (F ) = n ,
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 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 teacher head, 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".