An affine quantum cohomology ring for flag manifolds and the periodic Toda lattice
Bibliographic record
Abstract
Consider the generalized flag manifold G / B and the corresponding affine flag manifold F ℓ G . In this paper we use curve neighborhoods for Schubert varieties in F ℓ G to construct certain affine Gromov–Witten invariants of F ℓ G , and to obtain a family of ‘affine quantum Chevalley’ operators Λ 0 , … , Λ n indexed by the simple roots in the affine root system of G. These operators act on the cohomology ring H ∗ ( F ℓ G ) with coefficients in Z [ q 0 , … , q n ] . By analyzing commutativity and invariance properties of these operators we deduce the existence of two quantum cohomology rings, which satisfy properties conjectured earlier by Guest and Otofuji for G = SL n ( C ) . The first quantum ring is a deformation of the subalgebra of H ∗ ( F ℓ G ) generated by divisors. The second ring, denoted QH aff ∗ ( G / B ) , deforms the ordinary quantum cohomology ring QH ∗ ( G / B ) by adding an affine quantum parameter q 0 . We prove that QH aff ∗ ( G / B ) is a Frobenius algebra, and that the new quantum product determines a flat Dubrovin connection. Further, we develop an analogue of Givental and Kim formalism for this ring and we deduce a presentation of QH aff ∗ ( G / B ) by generators and relations. The ideal of relations is generated by the integrals of motion for the periodic Toda lattice associated to the dual of the extended Dynkin diagram of G.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.000 |
| 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".