RECORD
Anomalies, gauge field topology, and the lattice
High Energy Physics - Lattice · Submitted 30 Jul 2010
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.
The compiled record
The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
Open the compiled record
The abstract, sentence by sentence
Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I use the fourth power of the naive Dirac operator to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally "fall through the lattice." Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I use the fourth power of the naive Dirac operator to define a local lattice measure of topological charge. | contribution 0.71 | 0.93 | 0.08 | asserted 0.99 |
| s1 | For smooth gauge fields this reduces to the usual topological density. | result 0.71 | 0.59 | 0.04 | asserted 1.00 |
| s2 | For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. | result 0.84 | 0.74 | 0.17 | asserted 0.86 |
| s3 | On cooling with respect to the Wilson gauge action, instanton like structures do emerge. | result 0.89 | 0.81 | 0.08 | asserted 0.99 |
| s4 | As cooling proceeds, these objects tend shrink and finally "fall through the lattice." Modifying the action can block the shrinking at the expense of a loss of reflection positivity. | result 0.95 | 0.65 | 0.09 | asserted 0.89 |
| s5 | The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility. | interpretation 0.91 | 0.81 | 0.61 | hedged possibility 0.96 |
What was examined, and what was not
- arXiv listing abstract v4
- full text
- figures and tables
- references and citing sentences
- relations to other papers
- P abstract
- S sections
- T body sentences
- C citations
- R cross-paper relations
The paper
Original abstract
Motivated by the connection between gauge field topology and the axial anomaly in fermion currents, I use the fourth power of the naive Dirac operator to define a local lattice measure of topological charge. For smooth gauge fields this reduces to the usual topological density. For typical gauge field configurations in a numerical simulation, however, quantum fluctuations dominate, and the sum of this density over the system does not generally give an integer winding. On cooling with respect to the Wilson gauge action, instanton like structures do emerge. As cooling proceeds, these objects tend shrink and finally "fall through the lattice." Modifying the action can block the shrinking at the expense of a loss of reflection positivity. The cooling procedure is highly sensitive to the details of the initial steps, suggesting that quantum fluctuations induce a small but fundamental ambiguity in the definition of topological susceptibility.
Human review
How the literature uses it
4 arXiv papers cite this in 4 sentences
Papers citing it · 2 sentences without a confident label
Cites it as background 2 papers
-
A similar discussion appears in the nice review by Creutz [ref] who also analyzes the effects of these types of action modifications to reflection positivity [ref] .
Instanton liquid properties from lattice QCD · Locating instantons · 0.84
-
There exist many definitions of the topological charge on the lattice[For a short review and discussion of different definitions and for further references, we refer to Ref. [ref] .] and there has been a debate in the literature about the validity of different approaches.
Topological susceptibility from the twisted mass Dirac operator spectrum · Introduction · 1.00
What it cites · 17 sentences without a confident label
Uses the method or tool of 1 paper
-
With an overlap formulation [ref] , calculating $Tr γ_5 e^D^2/Λ^2$ amounts to a simple counting of zero modes.
Exactly massless quarks on the lattice · A local lattice operator for topology · 0.95
Compares with 1 paper
-
This compares well with the “admissibility condition” of Luscher [ref] which, when obeyed, allows the gauge fields to be uniquely continued between lattice sites to form a smooth continuum field.
Weyl fermions on the lattice and the non-abelian gauge anomaly · Cooling · 1.00
Cites as background 8 papers
-
Although such issues are quite old [ref] , they continue to be of considerable interest [ref] .
Wilson mass dependence of the overlap topological charge density · Sensitivity to initial conditions · 0.86
-
Although such issues are quite old [ref] , they continue to be of considerable interest [ref] .
Comparing topological charge definitions using topology fixing actions · Sensitivity to initial conditions · 0.86
-
Although such issues are quite old [ref] , they continue to be of considerable interest [ref] .
Topological effects in QCD and the problem of short-distance singularities · Sensitivity to initial conditions · 0.86
Discussion
Know something about this paper? Leave a comment, or highlight any passage above to comment on it.