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Bibliographic record
Abstract
We consider unitary, modular invariant, two-dimensional CFTs which are invariant under the parity transformation P <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>P</mml:mi> </mml:math> . Combining P <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>P</mml:mi> </mml:math> with modular inversion S <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>S</mml:mi> </mml:math> leads to a continuous family of fixed points of the SP <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> </mml:math> transformation. A particular subset of this locus of fixed points exists along the line of positive left- and right-moving temperatures satisfying \beta_L \beta_R = 4\pi^2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>L</mml:mi> </mml:msub> <mml:msub> <mml:mi>β</mml:mi> <mml:mi>R</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>4</mml:mn> <mml:msup> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> . We use this fixed locus to prove a conjecture of Hartman, Keller, and Stoica that the free energy of a large- c <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>c</mml:mi> </mml:math> CFT _2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mn>2</mml:mn> </mml:msub> </mml:math> with a suitably sparse low-lying spectrum matches that of AdS _3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi/> <mml:mn>3</mml:mn> </mml:msub> </mml:math> gravity at all temperatures and all angular potentials. We also use the fixed locus to generalize the modular bootstrap equations, obtaining novel constraints on the operator spectrum and providing a new proof of the statement that the twist gap is smaller than (c-1)/12 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>c</mml:mi> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false" form="postfix">)</mml:mo> <mml:mi>/</mml:mi> <mml:mn>12</mml:mn> </mml:mrow> </mml:math> when c>1 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>c</mml:mi> <mml:mo>></mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . At large c <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>c</mml:mi> </mml:math> we show that the operator dimension of the first excited primary lies in a region in the (h,\overline{h}) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mo stretchy="false" form="prefix">(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>,</mml:mo> <mml:mover> <mml:mi>h</mml:mi> <mml:mo accent="true">¯</mml:mo> </mml:mover> <mml:mo stretchy="false" form="postfix">)</mml:mo> </mml:mrow> </mml:math> -plane that is significantly smaller than h+\overline{h}<c/6 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>+</mml:mo> <mml:mover> <mml:mi>h</mml:mi> <mml:mo accent="true">¯</mml:mo> </mml:mover> <mml:mo>></mml:mo> <mml:mi>c</mml:mi> <mml:mi>/</mml:mi> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> . Our results for the free energy and constraints on the operator spectrum extend to theories without parity symmetry through the construction of an auxiliary parity-invariant partition function.
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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) | 0.000 | 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