Scattering equations: from projective spaces to tropical grassmannians
Bibliographic record
Abstract
A bstract We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ℂℙ 1 , to higher-dimensional projective spaces ℂℙ k − 1 . The standard, k = 2 Mandelstam invariants, s ab , are generalized to completely symmetric tensors $$ {\mathrm{s}}_{a_1{a}_2\dots {a}_k} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>s</mml:mi> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>…</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:mrow> </mml:msub> </mml:math> subject to a ‘massless’ condition $$ {\mathrm{s}}_{a_1{a}_2\dots {a}_{k-2}bb}=0 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>s</mml:mi> <mml:mrow> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:msub> <mml:mi>a</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>…</mml:mo> <mml:msub> <mml:mi>a</mml:mi> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>−</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msub> <mml:mi>b</mml:mi> <mml:mi>b</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> and to ‘momentum conservation’. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the k = 3 case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all ‘biadjoint amplitudes’ for ( k, n ) = (3 , 6) and find a direct connection to the tropical Grassmannian. This leads to the notion of k = 3 Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for k = 2, and provides analytic solutions analogous to the MHV ones.
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How this classification was reachedexpand
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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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".