An introduction to modern enumerative geometry with applications to the banana manifold
Bibliographic record
Abstract
In this masters thesis we survey some foundational material needed for the presentation (in the final chapter) of new results on the Donaldson-Thomas and Gromov-Witten theories of the formal banana manifold. The first half is geared towards understanding stable vector bundles and instantons in Yang-Mills theory, stability conditions on coherent sheaves, applications to D-branes, as well as the standard curve-counting invariants in modern enumerative geometry and their physical interpretations. Much of the second half is devoted to introducing equivariant localization, automorphic forms, and arithmetic lifts, all of which appear in our new results. These are vast topics, but we are particularly interested in equivariant elliptic genera, Jacobi forms and Siegel modular forms, as well as the Maass and Borcherds lifts. In the final chapter, using a theorem of J. Bryan, we show that the Donaldson-Thomas partition function of the formal banana manifold is the formal Borcherds lift of a weight zero weak Jacobi form of matrix index. This weak Jacobi form is the equivariant elliptic genus of ℂ², and the formal Borcherds lift is closely related to a partition function of rank one instantons on ℂ². Assuming the GW/DT correspondence, we then prove that for g ≥ 2, the Gromov-Witten potentials Fg are genus two meromorphic Siegel modular forms of weight 2g - 2 arising as the Maass lift of an explicit weak Jacobi form. With this identification, we observe that the invariances of such Siegel modular forms precisely encode geometric symmetries of the formal banana manifold, as well as a flop symmetry. The Gopakumar-Vafa invariants are encoded non-trivially into the equivariant elliptic genus of ℂ², and depend only on a quadratic form. We observe that they share a number of features with those coming from the Katz-Klemm-Vafa formula.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.004 |
| Insufficient payload (model declined to judge) | 0.024 | 0.006 |
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".