Bibliographic record
Abstract
We introduce a new type of duality structure for $A_\infty$-categories called a relative weak Calabi-Yau pairing which generalizes Kontsevich and Soibelman's notion of a weak (proper) Calabi-Yau structure. We prove the existence of a relative weak Calabi-Yau pairing on Biran and Cornea's Fukaya category of Lagrangian cobordisms $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$. Here $M$ is a symplectic manifold which is closed or tame at infinity. This duality structure on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ extends the relative Poincaré duality satisfied by Floer complexes for pairs of Lagrangian cobordisms. Moreover, we show that the relative weak Calabi-Yau pairing on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ satisfies a compatibility condition with respect to the usual weak Calabi-Yau structure on the monotone Fukaya category of $M$. The construction of the relative weak Calabi-Yau pairing on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ is based on counts of curves in $\mathbb{C}\times M$ satisfying an inhomogeneous nonlinear Cauchy-Riemann equation. In order to prove the existence of this duality structure and to verify its properties, we extend the methods of Biran and Cornea to establish regularity and compactness results for the relevant moduli spaces. We also consider the implications of the existence of the relative weak Calabi-Yau pairing on $\mathcal{F}uk_{\mathit{cob}}(\mathbb{C}\times M)$ for the cone decomposition in the derived Fukaya category of $M$ associated to a Lagrangian cobordism, and we present an example involving Lagrangian surgery.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".