Commutative algebras in Grothendieck–Verdier categories, rigidity, and vertex operator algebras
Bibliographic record
Abstract
Let A be a commutative algebra in a braided monoidal category [Formula: see text]. For example, A could be a vertex operator algebra (VOA) extension of a VOA V in a category [Formula: see text] of V-modules. We first find conditions for the category [Formula: see text] of A-modules in [Formula: see text] and its subcategory [Formula: see text] of local modules to inherit rigidity from [Formula: see text]. Second and more importantly, we prove a converse result, finding conditions under which [Formula: see text] and [Formula: see text] inherit rigidity from [Formula: see text]. For our first results, we assume that [Formula: see text] is a braided finite tensor category and identify mild conditions under which [Formula: see text] and [Formula: see text] are also rigid. These conditions are based on criteria due to Etingof and Ostrik for A to be an exact algebra in [Formula: see text]. As an application, we show that if A is a simple [Formula: see text]-graded VOA containing a strongly rational vertex operator subalgebra V, then A is also strongly rational, without requiring the dimension of A in the modular tensor category of V-modules to be non-zero. We also identify conditions under which the category of A-modules inherits rigidity from the module category of a [Formula: see text]-cofinite non-rational subalgebra V. For our converse result, we assume that [Formula: see text] is a Grothendieck–Verdier category, which means that [Formula: see text] admits a weaker duality structure than rigidity. We first show that [Formula: see text] is also a Grothendieck–Verdier category. Using this, we then prove that if [Formula: see text] is rigid, then so is [Formula: see text] under conditions that include a mild non-degeneracy assumption on [Formula: see text], as well as assumptions that every simple object of [Formula: see text] is local and that induction [Formula: see text] commutes with duality. These conditions are motivated by free field-like VOA extensions [Formula: see text] where A is often an indecomposable V-module, and thus our result will make it more feasible to prove rigidity for many vertex algebraic braided monoidal categories. In a follow-up work, our results are used to prove rigidity of the category of weight modules for the simple affine VOA of [Formula: see text] at any admissible level, which embeds by Adamović’s inverse quantum Hamiltonian reduction into a rational Virasoro VOA tensored with a half-lattice VOA.
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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 | 0.002 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.003 | 0.007 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".