Bibliographic record
Abstract
The classification of the unitary irreducible representations of symmetry groups is a cornerstone of modern quantum physics, as it provides the fundamental building blocks for constructing the Hilbert spaces of theories admitting these symmetries. In the context of gravitational theories, several arguments point toward the existence of a universal symmetry group associated with corners, whose structure is the same for every diffeomorphism-invariant theory in any dimension. Recently, the representations of the maximal central extension of this group in the two-dimensional case have been classified using purely algebraic techniques. In this work, we present a complementary and independent derivation based on Kirillov’s orbit method. We study the coadjoint orbits of the group <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mover accent="true"> <a:mrow> <a:mi>SL</a:mi> <a:mo stretchy="false">(</a:mo> <a:mn>2</a:mn> <a:mo>,</a:mo> <a:mi mathvariant="double-struck">R</a:mi> <a:mo stretchy="false">)</a:mo> </a:mrow> <a:mrow> <a:mo stretchy="true">˜</a:mo> </a:mrow> </a:mover> <a:mo>⋉</a:mo> <a:msub> <a:mrow> <a:mi mathvariant="double-struck">H</a:mi> </a:mrow> <a:mrow> <a:mn>3</a:mn> </a:mrow> </a:msub> </a:mrow> </a:math> , where <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi mathvariant="double-struck">H</i:mi> <i:mn>3</i:mn> </i:msub> </i:math> is the Heisenberg group of a quantum particle in one dimension. Our main result is that, despite the non-Abelian nature of the normal subgroup in the semidirect product, these orbits admit a simple description. In a coordinate system associated with modified Lie algebra generators, the orbits factorize into a product of coadjoint orbits of <l:math xmlns:l="http://www.w3.org/1998/Math/MathML" display="inline"> <l:mi>SL</l:mi> <l:mo stretchy="false">(</l:mo> <l:mn>2</l:mn> <l:mo>,</l:mo> <l:mi mathvariant="double-struck">R</l:mi> <l:mo stretchy="false">)</l:mo> </l:math> and <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:msub> <q:mi mathvariant="double-struck">H</q:mi> <q:mn>3</q:mn> </q:msub> </q:math> . The subsequent geometric quantization of these factorized orbits successfully reproduces the known representations.
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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 | 0.001 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| 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 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".