Representation theory of 𝐿_{𝑘}(𝔬𝔰𝔭(1|2)) from vertex tensor categories and Jacobi forms
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Bibliographic record
Abstract
The purpose of this work is to illustrate in a family of interesting examples how to study the representation theory of vertex operator superalgebras by combining the theory of vertex algebra extensions and modular forms. Let <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript k Baseline left-parenthesis German o German s German p left-parenthesis 1 vertical-bar 2 right-parenthesis right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">o</mml:mi> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">L_k\left (\mathfrak {osp}(1 | 2)\right )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> be the simple affine vertex operator superalgebra of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="German o German s German p left-parenthesis 1 vertical-bar 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">o</mml:mi> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">p</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathfrak {osp}(1|2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> at an admissible level <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k"> <mml:semantics> <mml:mi>k</mml:mi> <mml:annotation encoding="application/x-tex">k</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We use a Jacobi form decomposition to see that this is a vertex operator superalgebra extension of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L Subscript k Baseline left-parenthesis German s German l Subscript 2 Baseline right-parenthesis circled-times Vir left-parenthesis p comma left-parenthesis p plus p Superscript prime Baseline right-parenthesis slash 2 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="fraktur">s</mml:mi> <mml:mi mathvariant="fraktur">l</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mo> ⊗ </mml:mo> <mml:mtext>Vir</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>,</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>p</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">L_k(\mathfrak {sl}_2)\otimes \text {Vir}(p, (p+p’)/2)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="k plus 3 slash 2 equals p slash left-parenthesis 2 p prime right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>p</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">k+3/2=p/(2p’)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="Vir left-parenthesis u comma v right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mtext>Vir</mml:mtext> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\text {Vir}(u, v)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denotes the regular Virasoro vertex operator algebra of central charge <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="c equals 1 minus 6 left-parenthesis u minus v right-parenthesis squared slash left-parenthesis u v right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mo> − </mml:mo> <mml:mn>6</mml:mn> <mml:mo stretchy="false">(</mml:mo> <mml:mi>u</mml:mi> <mml:mo> − </mml:mo> <mml:mi>v</mml:mi> <mml:msup> <mml:mo stretchy="fals
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.
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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| 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 it