Anomaly of $(2+1)$-dimensional symmetry-enriched topological order from $(3+1)$-dimensional topological quantum field theory
Bibliographic record
Abstract
Symmetry acting on a (2+1) D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> </mml:math> topological order can be anomalous in the sense that they possess an obstruction to being realized as a purely (2+1) D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> </mml:math> on-site symmetry. In this paper, we develop a (3+1) D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> </mml:math> topological quantum field theory to calculate the anomaly indicators of a (2+1) D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> </mml:math> topological order with a general symmetry group G <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>G</mml:mi> </mml:math> , which may be discrete or continuous, Abelian or non-Abelian, contain anti-unitary elements or not, and permute anyons or not. These anomaly indicators are partition functions of the (3+1) D <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>D</mml:mi> </mml:math> topological quantum field theory on a specific manifold equipped with some G <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>G</mml:mi> </mml:math> -bundle, and they are expressed using the data characterizing the topological order and the symmetry actions. Our framework is applied to derive the anomaly indicators for various symmetry groups, including \mathbb{Z}_2\times\mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>×</mml:mo> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:mrow> </mml:math> , \mathbb{Z}_2^T\times\mathbb{Z}_2^T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , SO(N) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , O(N)^T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>O</mml:mi> <mml:msup> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mi>T</mml:mi> </mml:msup> </mml:mrow> </mml:math> , SO(N)\times \mathbb{Z}_2^T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow> <mml:mi>S</mml:mi> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo stretchy="true" form="prefix">(</mml:mo> <mml:mi>N</mml:mi> <mml:mo stretchy="true" form="postfix">)</mml:mo> </mml:mrow> <mml:mo>×</mml:mo> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> , etc, where \mathbb{Z}_2 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msub> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> and \mathbb{Z}_2^T <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msubsup> <mml:mi>ℤ</mml:mi> <mml:mn>2</mml:mn> <mml:mi>T</mml:mi> </mml:msubsup> </mml:math> denote a unitary and anti-unitary order-2 group, respectively, and <
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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 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".