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A further extension to the group of Ginsparg-Wilson (overlap) chiral symmetries
High Energy Physics - Lattice · Submitted 24 Jan 2011
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim Working through the same renormalisation group procedure to map this action to the lattice overlap action yields additional Ginsparg-Wilson relations satisfied by the overlap operator, and more (non-commuting) lattice chiral symmetries.
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The abstract, sentence by sentence
As shown by Mandula, the Ginsparg-Wilson lattice realisation of chiral symmetry has a possible ambiguity: there is no unique lattice chiral symmetry, but an infinite group of symmetries with non-commuting generators. The physical implications of this abundance of symmetry remain unclear. In recent work, it has been shown how these chiral symmetries for overlap fermions can be derived from a renormalisation group blocking in the continuum, transforming the action from the standard continuum action to an equivalent to the lattice overlap action. There is no unique blocking, and different blockings lead to different chiral symmetries. The group of symmetries found by Mandula immediately follows. In this way, the excess chiral symmetry on the lattice can be explained in terms of different renormalisation schemes. The previous work suffered from one technical challenge: there is no continuum analogue of the lattice chiral eigenvectors at eigenvalue $2/a$. As the construction of the overlap operator required a mapping between lattice and continuum eigenvalues, the lack of a continuum counterpart to the doublers of the zero modes creates an ambiguity in the construction. Although the lattice chiral symmetry can still be defined, this leads to difficulties when considering $\mathcal{CP}$ symmetry on the lattice. In this work, we investigate the possibility of resolving this ambiguity by adding a second fermion field to the original continuum action used as a basis of the renormalisation group blockings. This second fermion field has a mass of the order of the momentum cut-off, to simulate the effects of the fermion doublers. Working through the same renormalisation group procedure to map this action to the lattice overlap action yields additional Ginsparg-Wilson relations satisfied by the overlap operator, and more (non-commuting) lattice chiral symmetries.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | As shown by Mandula, the Ginsparg-Wilson lattice realisation of chiral symmetry has a possible ambiguity: there is no unique lattice chiral symmetry, but an infinite group of symmetries with non-commuting generators. | background 1.00 | 0.04 | 0.07 | asserted 0.71 |
| s1 | The physical implications of this abundance of symmetry remain unclear. | gap 0.69 | 0.06 | 0.02 | asserted 0.84 |
| s2 | In recent work, it has been shown how these chiral symmetries for overlap fermions can be derived from a renormalisation group blocking in the continuum, transforming the action from the standard continuum action to an equivalent to the lattice overlap action. | background 0.96 | 0.12 | 0.07 | asserted 1.00 |
| s3 | There is no unique blocking, and different blockings lead to different chiral symmetries. | background 0.56 | 0.18 | 0.06 | asserted 1.00 |
| s4 | The group of symmetries found by Mandula immediately follows. | result 0.64 | 0.14 | 0.08 | asserted 1.00 |
| s5 | In this way, the excess chiral symmetry on the lattice can be explained in terms of different renormalisation schemes. | interpretation 0.93 | 0.21 | 0.16 | asserted 0.96 |
| s6 | The previous work suffered from one technical challenge: there is no continuum analogue of the lattice chiral eigenvectors at eigenvalue $2/a$. | gap 0.52 | 0.11 | 0.02 | asserted 1.00 |
| s7 | As the construction of the overlap operator required a mapping between lattice and continuum eigenvalues, the lack of a continuum counterpart to the doublers of the zero modes creates an ambiguity in the construction. | gap 0.58 | 0.11 | 0.03 | asserted 0.99 |
| s8 | Although the lattice chiral symmetry can still be defined, this leads to difficulties when considering $\mathcal{CP}$ symmetry on the lattice. | background 0.40 | 0.13 | 0.03 | asserted 0.92 |
| s9 | In this work, we investigate the possibility of resolving this ambiguity by adding a second fermion field to the original continuum action used as a basis of the renormalisation group blockings. | contribution 0.91 | 0.65 | 0.13 | hedged possibility 0.95 |
| s10 | This second fermion field has a mass of the order of the momentum cut-off, to simulate the effects of the fermion doublers. | method 0.92 | 0.63 | 0.04 | asserted 0.98 |
| s11 | Working through the same renormalisation group procedure to map this action to the lattice overlap action yields additional Ginsparg-Wilson relations satisfied by the overlap operator, and more (non-commuting) lattice chiral symmetries. | result 0.92 | 0.89 | 0.68 | asserted 1.00 |
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The paper
Original abstract
As shown by Mandula, the Ginsparg-Wilson lattice realisation of chiral symmetry has a possible ambiguity: there is no unique lattice chiral symmetry, but an infinite group of symmetries with non-commuting generators. The physical implications of this abundance of symmetry remain unclear. In recent work, it has been shown how these chiral symmetries for overlap fermions can be derived from a renormalisation group blocking in the continuum, transforming the action from the standard continuum action to an equivalent to the lattice overlap action. There is no unique blocking, and different blockings lead to different chiral symmetries. The group of symmetries found by Mandula immediately follows. In this way, the excess chiral symmetry on the lattice can be explained in terms of different renormalisation schemes. The previous work suffered from one technical challenge: there is no continuum analogue of the lattice chiral eigenvectors at eigenvalue $2/a$. As the construction of the overlap operator required a mapping between lattice and continuum eigenvalues, the lack of a continuum counterpart to the doublers of the zero modes creates an ambiguity in the construction. Although the lattice chiral symmetry can still be defined, this leads to difficulties when considering $\mathcal{CP}$ symmetry on the lattice. In this work, we investigate the possibility of resolving this ambiguity by adding a second fermion field to the original continuum action used as a basis of the renormalisation group blockings. This second fermion field has a mass of the order of the momentum cut-off, to simulate the effects of the fermion doublers. Working through the same renormalisation group procedure to map this action to the lattice overlap action yields additional Ginsparg-Wilson relations satisfied by the overlap operator, and more (non-commuting) lattice chiral symmetries.
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How the literature uses it
1 arXiv paper cite this in 2 sentences
What it cites · 17 sentences without a confident label
Uses the method or tool of 2 papers
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Following [ref] , we can construct the measure in terms of the eigenvectors $g^±$ of $γ_Rd$ and $H_±$ of $γ_5 D$, [equation] where we have suppressed the eigenvalue index.
A Ginsparg-Wilson approach to lattice CP symmetry, Weyl and Majoranna fermions, and the Higgs mechanism · Chiral gauge theories · 0.92
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The changes to the eigenvectors $H_+$ and $H_-$ after an infinitesimal change in the gauge field which induces a change $δD_d$ in the Dirac operator and $δα$ in $α$ are [equation] Thus, [equation] The change in the measure induced by this change in the basis is [ref] $ℒ = ∑_i (g^+_i,δg^+_i)$.
Abelian chiral gauge theories on the lattice with exact gauge invariance · Chiral gauge theories · 0.84
Cites as background 6 papers
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Chiral symmetry in massless lattice QCD has been established through the Ginsparg-Wilson relation [ref] and practically implementable Dirac operators that satisfy this symmetry to any desired precision, the overlap [ref] and similar [ref] Dirac operators.
New solutions to the Ginsparg-Wilson equation · Introduction · 1.00
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Chiral symmetry in massless lattice QCD has been established through the Ginsparg-Wilson relation [ref] and practically implementable Dirac operators that satisfy this symmetry to any desired precision, the overlap [ref] and similar [ref] Dirac operators.
Chiral Determinant as an Overlap of Two Vacua · Introduction · 1.00
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Chiral symmetry in massless lattice QCD has been established through the Ginsparg-Wilson relation [ref] and practically implementable Dirac operators that satisfy this symmetry to any desired precision, the overlap [ref] and similar [ref] Dirac operators.
Chiral Fermions on the Lattice · Introduction · 1.00
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