Bosonization and Anomaly Indicators of (2+1)-D Fermionic Topological Orders
Bibliographic record
Abstract
Abstract We provide a mathematical proposal for the anomaly indicators of symmetries of (2+1)-D fermionic topological orders, and work out the consequences of our proposal in several nontrivial examples. Our proposal is an invariant of a super modular tensor category with a fermionic group action, which gives a (3+1)-D topological field theory (TFT) that we conjecture to be invertible; the anomaly indicators are partition functions of this TFT on 4-manifolds generating the corresponding twisted spin bordism group. Our construction relies on a bosonization construction due to Gaiotto–Kapustin and Tata–Kobayashi–Bulmash–Barkeshli, together with a “bosonization conjecture” which we explain in detail. In the second half of the paper, we discuss several examples of our invariants relevant to condensed-matter physics. The most important example we consider is $$\mathbb {Z}/4^T\times \mathbb {Z}/2^f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mn>4</mml:mn> <mml:mi>T</mml:mi> </mml:msup> <mml:mo>×</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mn>2</mml:mn> <mml:mi>f</mml:mi> </mml:msup> </mml:mrow> </mml:math> time-reversal symmetry with symmetry algebra $${\mathcal {T}}^2 = (-1)^FC$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>F</mml:mi> </mml:msup> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> , which many fermionic topological orders enjoy, including the $$\textrm{U}(1)_5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>U</mml:mtext> <mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>5</mml:mn> </mml:msub> </mml:mrow> </mml:math> spin Chern–Simons theory. Using newly developed tools involving the Smith long exact sequence, we calculate the cobordism group that classifies its anomaly, present the generating manifold, and calculate the partition function on the generating manifold which serves as our anomaly indicator. Our approach allows us to reproduce anomaly indicators known in the literature with simpler proofs, including $$\mathbb {Z}/4^{Tf}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>/</mml:mo> <mml:msup> <mml:mn>4</mml:mn> <mml:mrow> <mml:mi>Tf</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> time-reversal symmetry with symmetry algebra $$\mathcal T^2 = (-1)^F$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>T</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>F</mml:mi> </mml:msup> </mml:mrow> </mml:math> , and other symmetry groups in the 10-fold way involving Lie group symmetries.
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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 | 0.000 | 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 | 0.000 |
| 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 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".