Modular Exercises for Four-Point Blocks -- I
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Bibliographic record
Abstract
The well-known modular property of the torus characters and torus partition functions of (rational) vertex operator algebras (VOAs) and 2d conformal field theories (CFTs) has been an invaluable tool for studying this class of theories. In this work we prove that sphere four-point chiral blocks of rational VOAs are vector-valued modular forms for the groups $Γ(2)$, $Γ_0(2)$, or $\mathrm{SL}_2(\mathbb{Z})$. Moreover, we prove that the four-point correlators, combining the holomorphic and anti-holomorphic chiral blocks, are modular invariant. In particular, in this language the crossing symmetries are simply modular symmetries. This gives the possibility of exploiting the available techniques and knowledge about modular forms to determine or constrain the physically interesting quantities such as chiral blocks and fusion coefficients, which we illustrate with a few examples. We also highlight the existence of a sphere-torus correspondence equating the sphere quantities of certain theories ${\mathcal T}_s$ with the torus quantities of another family of theories ${\mathcal T}_t$. A companion paper will delve into more examples and explore more systematically this sphere-torus duality.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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 it