Center Dominance and Z2 Vortices in SU(2) Lattice Gauge Theory

J. Greensite; L. Del Debbio; M. Faber; S. Olejnik

REVIEW

Center Dominance and Z2 Vortices in SU(2) Lattice Gauge Theory

L. Del Debbio, M. Faber, J. Greensite, S. Olejnik

High Energy Physics - Lattice · Submitted 2 Oct 1996 · 17 references

Pith's verdict: The paper that introduced maximal center gauge and P-vortices; the correlation analysis is a real result, but the gauge-copy question is never addressed.

T0 · AI review

How this review was made: An AI referee (selfhosted/deepseek-v4.1-flash) read the paper on November 13, 2026 and answered a fixed list of questions about it. T0 is the starting level every paper gets. The levels describe how deeply Pith has checked the paper, never the authors.

T0 AI review An AI read the paper and answered a fixed list of questions.
T1 Audit Automatic checks that the references and files the paper relies on exist and match.
T2 Compute The paper's code or calculations re-run with the same results.
T3 Formal statement The claims written out in a formal math language a computer can read.
T4 Proof Those claims proved, and the proof checked by computer.

How Pith checks papers →

· 4 major objections · 7 minor · Ask about this paper · The compiled record · Read the full review →

Challenge this review Comment (0) Read on arXiv ↗

Cite ▾

arXiv IDhep-lat/9610005 (version 1)

Pith IDOOIXIISD, Pith's permanent record of this paper and its review

CitationJ. Greensite, L. Del Debbio, M. Faber, S. Olejnik. "Center Dominance and Z2 Vortices in SU(2) Lattice Gauge Theory." arXiv:hep-lat/9610005 (1996). https://pith.science/paper/OOIXIISD

Cite this reviewBibTeX for Pith's review of this paper

Claim paper

Why Pith says this

Highlight any text to comment on it.

Editor's note

read the letter

This is the paper that introduced maximal center gauge and P-vortices, and it's worth your time even though the authors spend the last section arguing against their own interpretation. What's new: the two-step fixing (maximal abelian, then maximize cos^2 theta) and the projection U -> Z. The center-dominance plot in Fig. 1 is striking: Creutz ratios from projected links sit on the scaling line from R=2 onward, which the full theory only reaches asymptotically. The real content is section 3. Selecting subensembles by P-vortex number and then measuring the full unprojected loops is a clean idea, and W1/W0 -> -1, W2/W0 -> +1 is a genuine prediction of the vortex picture, not an identity forced by the projection. The even/odd decomposition in eqs. (19)-(22) is well done, and Fig. 10, the QED3 control where projection overshoots by roughly 40%, is the kind of honest check people usually skip. Soft spots, in order. First, Gribov copies: both maximization steps have many local maxima, and the paper never says how the algorithm is initialized or whether a different copy moves the P-vortices and reshuffles the W0/W1/W2 ensembles. That is a real gap and it sits underneath the whole correlation analysis. I'd call it a gap rather than a refutation, since the effect is large enough that it would take a lot of copy-to-copy wandering to erase it, but the check should have been done. Second, most figures carry no error bars (footnote 3 acknowledges this without quantifying), and the W0 ensemble is a small minority at large loop area. Third, 'the asymptotic string tension vanishes' is an extrapolation from Creutz ratios out to R=5 on one lattice size per beta. Statistics are modest by modern standards, which is minor. The authors are candid about the adjoint-tension and Casimir-scaling objections in section 4 and they don't oversell; that counts for a lot. Who should read it: anyone working on confinement mechanisms, vortex models, or gauge-fixing systematics. It's a foundational reference, and the SU(3) extension in eq. (26) is a clean falsifiable target. Send it out.

Referee report

Significance

If it holds, the result is a substantial contribution: it establishes a center-projection analogue of abelian dominance and, more importantly, supplies a concrete, falsifiable correlation between an object defined in a gauge-fixed projected configuration and the confining properties of the full unprojected theory. The sign predictions W₁/W₀ → −1 and W₂/W₀ → +1 (Eq. 9) are sharp and testable, and the proposed SU(3) generalization (Eq. 26) is a genuine experimental discriminator. The paper is also unusually candid: Section 4 states the adjoint/Casimir-scaling objection in the authors' own voice, Section 5 states plainly that none of the evidence is conclusive, and the QED₃ control in Fig. 10 is included precisely because it weakens the case. That combination of a falsifiable prediction with a self-imposed control is a real strength and should be weighed in the assessment.

Major objections (4)

[§2 (Eqs. 1, 4) and §3] The entire correlation analysis of Figs. 4–9 rests on selecting subensembles by the number and location of P-vortices, and P-vortices are defined only after fixing to maximal center gauge. Both fixing steps, Eq. (1) and Eq. (4), are lattice maximization problems with many local maxima, and the manuscript nowhere discusses Gribov copies, the initialization of the maximization, or the stability of the P-vortex locations under a change of gauge copy. If a different copy moves the P-vortices, the ensembles W₀, W₁, W₂ are not well-defined and the sign tests measure a property of the gauge-fixing algorithm rather than of the Yang-Mills vacuum. This is logically prior to the assumption flagged in §3 ('assume for the moment that to each P-vortex ... there corresponds a Z₂ vortex in the full ... configuration'), because it questions whether the P-vortices used to define the subensembles are well-defined at all. The authors' awareness that gauge choice matters, evidenced by the alternative gauge of Eqs. (7)–(8), makes the absence of any copy-dependence test within the chosen gauge more conspicuous. At minimum, a repeat of Figs. 4–6 with a different initialization or a different maximization algorithm is needed to show that the subensembles, and not merely the average string tension, are stable.
[§3, Figs. 4 and 8] The conclusion that the asymptotic string tension 'vanishes altogether' for the zero-vortex (Fig. 4) and even-vortex (Fig. 8) ensembles is stronger than the data support. No figure in the paper carries error bars; footnote 3 asserts that errors are smaller for the no-vortex data but gives no number, and no binning or autocorrelation analysis is reported despite data being taken every tenth sweep. The quoted Creutz ratios extend only to R = 5, and a flattening over three or four points is not the same as a vanishing asymptotic slope. Since this is the load-bearing evidence for the correlation claim, the manuscript should report errors on χ₀ and χev, state the fit or criterion by which 'vanishing' is judged, and show the largest loops for which the subensemble still contains a statistically usable number of configurations.
[§3, Eqs. (9)–(13), and §4, Eq. (25)] The sign tests W₁/W₀ → −1 and W₂/W₀ → +1 are presented as evidence that thin P-vortices are associated with thick Z₂ vortices in the unprojected vacuum, but the test as constructed does not by itself discriminate that reading from a weaker one: if the full loop tracks the sign of the center-projected loop (which is what center dominance already asserts), the ratio Wn/W₀ ≈ (−1)ⁿ follows without any thick vortex being present. The decomposition of Eqs. (11)–(13) is the physical content that would license the stronger reading, and it is asserted rather than tested. The authors themselves identify the discriminating quantity in §4, Eq. (25): if σcp > σfund the first term of Eq. (22) becomes negligible asymptotically and the area law must come from ΔW(C) instead. The manuscript should state explicitly what measurement would distinguish the two readings and, if possible, report the σcp versus σfund comparison at the level of precision needed to settle Eq. (25), rather than deferring it to future work.
[§2, Fig. 1] The claim that center-projected Creutz ratios at fixed β ≥ 2.1 'all fall on top of one another' from two lattice spacings onward is made on 10⁴ lattices for β ≤ 2.3, with R ranging to 5. On a 10⁴ lattice, χ(5,5) is close to the maximal loop and is subject to serious finite-volume contamination, so the apparent collapse onto the scaling line of Eq. (6) may be partly a finite-size artifact rather than a demonstration of precocious scaling. The manuscript acknowledges finite-size effects only for the Polyakov lines (footnote 2) and does not repeat the Creutz-ratio analysis on the larger volumes used at β = 2.4 and 2.5. A comparison of χ(R,R) at fixed β across two volumes, or a restriction of the scaling claim to R values safely below the lattice extent, is needed before the 'sweeps away the short-distance 1/r potential' conclusion can be regarded as established.

Minor objections (7)

[§2, Eq. (6)] The scaling line of Eq. (6) is described as though the data fall on it independently, but √σ/Λ = 67 is a fitted parameter; the statement should make clear which quantities are fitted and which are predicted, and should give the fit range and χ².
[Figs. 1–10] None of the figures display error bars. Even where the authors judge errors to be small (footnote 3), showing them would let the reader assess the claims about flatness and about the sign of Wodd in Fig. 9.
[§3, Fig. 7] The statement that even and odd P-vortex fractions approach one another for large loops is central to the cancellation argument of Eq. (22); the figure should be accompanied by the numerical values of Pevn and Podd at the largest areas shown, not only the graphical trend.
[§2, Eqs. (7)–(8)] The alternative maximal center gauge is introduced and then set aside as numerically difficult. Since the paper's conclusions depend on the specific gauge choice, a sentence on why the two definitions are expected to give the same P-vortex structure, or an explicit statement that this is an open question, would help the reader calibrate the result.
[§4, Fig. 10] The QED₃ comparison is a useful control, but the text reports only that projected Creutz ratios are 'approximately 40% larger' without giving the loop sizes or errors; the quantitative basis for the comparison should be stated.
[§3] There are several typographical errors, including 'the number of P-vortices piercing any given loop are a will fluctuate' and 'as far as a possible' in the discussion of Fig. 5; these should be corrected.
[References] Reference [4] and [9] are listed as 'to appear' in the LATTICE 96 proceedings; the authors should confirm at proof stage that these have appeared, or supply archive numbers, so that the reader can locate them.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

Stripping SU(2) links down to their Z2 sign keeps the confining string tension, and the vortices left behind track where confinement happens.

The reading

This paper asks whether confinement in $SU(2)$ lattice gauge theory can be traced to the $Z_2$ center of the gauge group rather than to an abelian $U(1)$ subgroup. The authors fix a gauge, maximal center gauge, that pushes each link as close as possible to $+I$ or $-I$, then discard everything except the sign, leaving a $Z_2$ link variable. The stripped-down theory reproduces the full theory's string tension, a phenomenon they call center dominance, and the vortices visible in the stripped configurations turn out to be strongly correlated with whether the untouched configurations confine. If that correlation is real, the Yang-Mills vacuum behaves like a condensate of thick vortex lines and confinement is a center effect rather than a monopole effect.

What carries the argument

The carrier is the pair (maximal center gauge, center projection). Starting from maximal abelian gauge, which maximizes $sum_x sum_mu Tr[sigma_3 U_mu(x) sigma_3 U^dagger_mu(x)]$, the authors use the leftover $U(1)$ freedom to maximize $sum_x sum_mu cos^2 theta_mu(x)$, leaving a residual $Z_2$ symmetry; center projection then replaces each link by $Z = sign(cos theta) = plus or minus 1$. A plaquette equal to $-1$ in the projected configuration marks a thin projected vortex, or P-vortex, piercing the minimal surface of a loop. The diagnostic is the subensemble Wilson loop $W_n(C)$, evaluated on full links but selected by counting P-vortices through the loop; if P-vortices correspond to real thick vortices, then $W_n(C)/W_0(C) -> (-1)^n$, and the full loop's area law comes from the weighted sum $W(C) = P_{even} W_{even}(C) + P_{odd} W_{odd}(C)$ in which the two components have opposite signs and nearly cancel.

Extended reading notes

Core claim

The paper reports three results. First, center dominance: Creutz ratios computed from center-projected $Z_2$ links, once the loop reaches two lattice spacings, all collapse onto a single scaling line $s a^2 = s Lambda^2 (61 pi^2 beta/121)^{102/121} exp[-61 pi^2 beta/121]$ with $sqrt(s)/Lambda = 67$, in contrast to the full theory where only the envelope of Creutz ratios scales. Second, correlation: when Wilson loops are evaluated in the full, unprojected links but restricted to configurations in which no projected vortex pierces the loop, the asymptotic string tension appears to vanish, and the same happens for the larger ensemble with an even number of such vortices. Third, sign tests: the ratios $W_1(C)/W_0(C)$ and $W_2(C)/W_0(C)$ approach $-1$ and $+1$ respectively as loop area grows, as they would if one or two real $Z_2$ vortices threaded the loop. The authors then argue against their own result in section 4, noting that adjoint loops are blind to $Z_N$ vortices yet show Casimir-scaled string tensions at intermediate distances, and they conclude that none of the evidence is conclusive.

Load-bearing premise

The reading rests on the premise the authors state explicitly in section 3: that each P-vortex piercing a loop corresponds to a real thick $Z_2$ vortex in the full, unprojected field configuration, so that the sign tests measure the Yang-Mills vacuum rather than an artifact of the projection.

Editorial extensions

If this is right

If center dominance holds with $s_{cp} = s_{fund}$, the asymptotic string tension can be read off from a $Z_2$ gauge theory, and the confining degrees of freedom live in the center of the gauge group.
If P-vortices mark real thick vortices, then a Wilson loop's area law arises from a near-cancellation between a larger positive even-vortex component and a larger negative odd-vortex component, neither of which confines on its own.
The same machinery transfers to $SU(3)$: a ratio $W_1(C)/W_0(C) -> e^{2 pi i/3}$ for one unit of center flux would extend the evidence to the physically relevant gauge group.
A precise numerical comparison of $s_{cp}$ and $s_{fund}$ at larger $beta$ and on larger lattices becomes the decisive follow-up, since the present data were taken on workstations rather than supercomputers.
Alternate definitions of maximal center gauge, such as maximizing $sum_x sum_mu {Tr U_mu(x)}^2$ with $Z = sign(Tr U)$, remain to be tested and could give a cleaner projection.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper's own comparison with compact QED$_3$, where projected Creutz ratios run about 40 percent above unprojected ones, suggests a useful diagnostic: the ratio $s_{cp}/s_{fund}$ may measure how much of the projected string tension is manufactured by the gauge choice rather than inherited from the vacuum.
If thick vortices are real, they should leave a detectable trace in the adjoint sector at intermediate distances, for instance a modification of Casimir scaling near vortex cores; the authors' objection in section 4 is really a prediction that such a trace exists or that the vortex picture is incomplete.
Because the even-vortex and odd-vortex ensembles each appear non-confining while their weighted sum confines, the confinement mechanism here is closer to a phase interference than to a classical condensate, which suggests looking at the sign of the loop rather than its magnitude as the order parameter.
The correlation test used here, selecting configurations by a property of the projected field and then measuring the unprojected loop, is a general template that could be applied to monopole-based projections to see whether the same subensembles lose their string tension.

Original abstract

We find, in close analogy to abelian dominance in maximal abelian gauge, the phenomenon of center dominance in maximal center gauge for $SU(2)$ lattice gauge theory. Maximal center gauge is a gauge-fixing condition that preserves a residual $Z_2$ gauge symmetry; ``center projection'' is the projection of $SU(2)$ link variables onto $Z_2$ center elements, and ``center dominance'' is the fact that the center-projected link elements carry most of the information about the string tension of the full theory. We present numerical evidence that the thin $Z_2$ vortices of the projected configurations are associated with ``thick'' $Z_2$ vortices in the unprojected configurations. The evidence also suggests that the thick $Z_2$ vortices may play a significant role in the confinement process.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. 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.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,748 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.12 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.84 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We find, in close analogy to abelian dominance in maximal abelian gauge, the phenomenon of center dominance in maximal center gauge for $SU(2)$ lattice gauge theory. Maximal center gauge is a gauge-fixing condition that preserves a residual $Z_2$ gauge symmetry; ``center projection'' is the projection of $SU(2)$ link variables onto $Z_2$ center elements, and ``center dominance'' is the fact that the center-projected link elements carry most of the information about the string tension of the full theory. We present numerical evidence that the thin $Z_2$ vortices of the projected configurations are associated with ``thick'' $Z_2$ vortices in the unprojected configurations. The evidence also suggests that the thick $Z_2$ vortices may play a significant role in the confinement process.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We find, in close analogy to abelian dominance in maximal abelian gauge, the phenomenon of center dominance in maximal center gauge for $SU(2)$ lattice gauge theory.	result 0.99	0.95	0.70	asserted 1.00
s1	Maximal center gauge is a gauge-fixing condition that preserves a residual $Z_2$ gauge symmetry; ``center projection'' is the projection of $SU(2)$ link variables onto $Z_2$ center elements, and ``center dominance'' is the fact that the center-projected link elements carry most of the information about the string tension of the full theory.	background 0.99	0.08	0.29	asserted 0.99
s2	We present numerical evidence that the thin $Z_2$ vortices of the projected configurations are associated with ``thick'' $Z_2$ vortices in the unprojected configurations.	result 0.99	0.93	0.20	asserted 0.79
s3	The evidence also suggests that the thick $Z_2$ vortices may play a significant role in the confinement process.	interpretation 0.99	0.82	0.30	hedged possibility 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Creutz ratios from center-projected lattice configura

**Figure 2.** Creutz ratios χ(R, R) vs. R at β = 2.4, for full, center-projected, and U(1)/Z2-projected lattice configurations. 6

**Figure 3.** Polyakov lines vs. β in center projection, for a 63 × 2 lattice. with center projection defined by Z = sign(TrU) (8) This version is more difficult to implement numerically, and has not yet been studied. In any case, center dominance in maximal center gauge, as displayed in the figures above, does not necessarily imply that confinement is due to vortex condensation. In fact our initial view, expounded in ref. [11], was that since center dominance would appear to support a theory, namely vortex condensation, which is “obviously wrong” (for reasons discussed in section 4), its success…

**Figure 4.** Creutz ratios χ0(R, R) extracted from loops with no P-vortices, as compared to the usual Creutz ratios χ(R, R), at β = 2.3. projected configurations is strongly correlated with the presence or absence of Confiners in the unprojected configurations. The next question is whether we can exclude the possibility that these Confiners are actually Z2 vortices. To address this question, assume for the moment that to each P-vortex piercing a given loop, there corresponds a Z2 vortex in the full, unprojected field configuration piercing that loop. This assumption has the consequence that, in…

**Figure 5.** Ratio of the 1-Vortex to the 0-Vortex Wilson loops,

**Figure 6.** Ratio of the 2-Vortex to the 0-Vortex Wilson loops,

**Figure 7.** Fraction of link configurations containing even/odd numbe

**Figure 8.** Creutz ratios χev(R, R) extracted from Wilson loops Wevn(C), taken from configurations with even numbers of P-vortices piercing the loop. The standard Creutz ratios χ(R, R) at this coupling (β = 2.3) are also shown. loop. If this is the case, then neither Wevn(C) alone, nor Wodd(C) alone, would have an area law, but only the weighted sum W(C) = Pevn(C)Wevn(C) + Podd(C)Wodd(C) (14) where Pevn and Podd are the fractions of configurations, shown in

**Figure 9.** Wilson loops Wevn(C), Wodd(C) and W(C) at larger loop areas, taken from configurations with even numbers of P-vortices, odd numbers of P-vortices, and any number of P-vortices, respectively piercing the loop. Again β = 2.3. then the area law falloff of a Wilson loop W(C) is due to a very delicate cancellation between much larger positive and negative components, associated with even and odd numbers of P-vortices respectively. Neither Wevn(C) nor Wodd(C), by itself, would appear to have an area law. 4 Against Vortices The data presented in the previous section suggests that ZN…

**Figure 10.** Test of Z2 dominance in compact QED3: Z2 Projected vs. Full Creutz ratios at β = 2.2 on a 303 lattice. an analogous calculation in compact QED3. In QED3 it is also possible to define a “maximal Z2 gauge,” via eq. (4), and a “Z2 projection” (the term “center projection” would be a misnomer here) according to eq. (5). A sample result for lattice QED3 is shown below in

Figures reproduced from arXiv:hep-lat/9610005 by the authors.

References

Works cited (17)

[1]

G. ’t Hooft, Nucl. Phys. B190 [FS3] (1981) 455

View on Pith 1981
[2]

T. Suzuki and I. Yotsuyanagi, Phys. Rev. D42 (1990) 4257; S. Hioki et al., Phys. Lett. B272 (1991) 326

View on Pith 1990
[3]

A. Kronfeld, M. Laursen, G. Schierholz, and U.-J. Wiese, Phys. L ett. B198 (1987) 516

View on Pith 1987
[4]

M. Polikarpov, to appear in the Proceedings of LATTICE 96, arch ive: hep- lat/9609020

View on Pith
[5]

H. B. Nielsen and P. Olesen, Nucl. Phys. B160 (1979) 380

View on Pith 1979
[6]

G. ’t Hooft, Nucl. Phys. B153 (1979) 141

View on Pith 1979
[7]

G. Mack, in Recent Developments in Gauge Theories , edited by G. ’t Hooft et al. (Plenum, New York, 1980)

View on Pith 1980
[8]

T. Yoneya, Nucl. Phys. B144 (1978) 195

View on Pith 1978
[9]

T. Kov´ acs and E. Tomboulis, to appear in the Proceedings of LAT TICE 96, archive: hep-lat/9607068

View on Pith Pith review arXiv
[10]

J. Ambjørn and P. Olesen, Nucl. Phys. B170 [FS1] (1980) 265

View on Pith 1980
[11]

L. Del Debbio, M. Faber, J. Greensite, and ˇS. Olejn ´ ık, to appear in the Proceed- ings of LATTICE 96, archive: hep-lat/9607053

View on Pith Pith review arXiv
[12]

L. Del Debbio, M. Faber, and J. Greensite, Nucl. Phys. B414 (1 994) 594

View on Pith
[13]

L. Del Debbio, M. Faber, J. Greensite, and ˇS. Olejn ´ ık, Phys. Rev. D53 (96) 5891

View on Pith
[14]

J. Greensite and M. B. Halpern, Phys. Rev. D27 (1983) 2545

View on Pith 1983
[15]

J. Ambjørn, P. Olesen, and C. Peterson, Nucl. Phys. B240 [FS1 2] (1984) 189; 533

View on Pith 1984
[16]

G. Poulis and H. Trottier, archive: hep-lat/9504015; C. Michael, Nucl. Phys. B (Proc. Suppl.) 26 (1992) 417; Nucl. Phys. B 259 (1985) 58. 20

View on Pith Pith review arXiv 1992
[17]

N. Cambell, I. Jorysz, and C. Michael, Phys. Lett. B167 (1986) 91; M. Faber and H. Markum, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 204; M. M¨ uller, W. Beirl, M. Faber, and H. Markum, Nucl. Phys. B (Proc. S uppl.) 26 (1992) 423. 21

View on Pith 1986

Forward citations

Cited by 10 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Emergence and polymer physics of the center-vortex network: Confronting SU(4) lattice Yang-Mills theory
hep-lat 2026-11 conditional novelty 7.0 of 10

A Cartan-flux model of confinement predicts that doubly charged center-vortex chains in SU(4) have tension 4/3 that of singly charged chains; new lattice data give 1.366(2).
Monopoles, Center Vortices, Confinement in (3+1)d, and the Lens-Space Twisted Partition Function
hep-th 2026-06 unverdicted novelty 7.0 of 10

Proposes torus and lens-space twisted partition functions as criteria for center-vortex and monopole condensation and proves vortex condensation implies monopole condensation in gapped phases.
Center vortex and confinement in Yang-Mills theory and QCD with anomaly-preserving compactifications
hep-th 2022-01 conditional novelty 7.0 of 10

Compactifying 4d gauge theories on a small T^2 with 't Hooft flux gives a calculable center-vortex description that reproduces confinement, chiral symmetry breaking, and vacuum structure, conjecturally connected to R^4.
First steps towards gauge-independent vortex identification through machine learning
hep-lat 2026-05 unverdicted novelty 6.0 of 10

A neural network trained on 2D SU(2) lattices with inserted thin Z2 vortices, after random gauge transformations, noise, and cooling, can locate center vortices at moderate visibility levels and scales via tiling.
Detection of Center Vortices in the Lattice Yang-Mills Vacuum
hep-lat 1998-01 conditional novelty 6.0 of 10

Center vortices located by center projection in direct maximal center gauge account for the full asymptotic string tension of SU(2) lattice Yang-Mills, and configurations with no such vortices show no area law.

Show 5 more citing papers

Cartan Fluxes in $SU(3)$ Lattice Gauge Theory
hep-lat 2026-04 unverdicted novelty 5.0 of 10

A root-lattice based DeGrand-Toussaint algorithm detects SU(3) monopoles with a density about 15% lower than the standard U(1)^3 method and yields Weyl-symmetric charge counts.
Quark confinement: dual superconductor picture based on a non-Abelian Stokes theorem and reformulations of Yang-Mills theory
hep-th 2014-09 conditional novelty 4.0 of 10

A gauge-covariant change of variables plus a non-Abelian Stokes theorem yields magnetic monopoles whose lattice effects reproduce the confining string tension in SU(2) and SU(3) Yang-Mills theory.
Emergent Structure in QCD
hep-lat 2020-09 conditional novelty 3.0 of 10

Stereoscopic visualizations of lattice QCD vacuum fields show dense, percolating center vortices, frequent branching points, and an apparent correlation between branching points and topological charge density.
The Confinement Problem in Lattice Gauge Theory
hep-lat 2003-01 accept novelty 3.0 of 10

Confinement in SU(N) lattice gauge theory is organized by Z_N center symmetry, and center vortices, not abelian monopoles, are the configurations that reproduce the observed N-ality dependence of the string tension.
Dual superconductor models of color confinement
hep-ph 2003-10 accept novelty 2.0 of 10

A pedagogical review showing how a dual Landau-Ginzburg (Abelian Higgs) superconductor reproduces lattice-measured color flux tubes and a linear confining potential, with model masses fitted to lattice data.

How the literature uses it

40 arXiv papers cite this in 53 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 31 sentences without a confident label

Uses its method or tool 4 papers

These are identified by first fixing the gauge field to Maximal Centre Gauge [ref] and then projecting the links to the nearest centre element.

Emergent phenomena from centre vortices in dynamical QCD · Introduction · 0.95
Here, vortices are identified by gauge fixing the original Monte-Carlo generated configurations directly to Maximal Centre Gauge [ref] , without any preconditioning [ref] .

Visualizations of Centre Vortex Structure in Lattice Simulations · Centre Vortex Identification · 1.00
Here, we consider the original Monte-Carlo generated configurations and gauge transform them directly to Maximal Centre Gauge [ref] , avoiding any preconditioning [ref] .

Impact of Dynamical Fermions on Centre Vortex Structure · Centre Vortex Identification · 0.97

Cites it as background 18 papers

Lattice gauge theory provides a computational formalism that allows for the direct study of center vortex phenomenology, established by significant pioneering work [ref] .

Numerical indication that center vortices drive dynamical mass generation in QCD · Introduction · 0.95
Over recent years, centre vortices have been shown to play a pivotal role in the generation of dynamical chiral symmetry breaking and quark confinement in the QCD vacuum [ref] .

Static quark potential from centre vortices in the presence of dynamical fermions · Introduction · 0.89
The most fundamental mechanism proposed to give rise to these phenomena are centre vortices [ref] within the ground-state vacuum fields.

Impact of Dynamical Fermions on Centre Vortex Structure · Introduction · 0.97

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

G. ’t Hooft, Nucl. Phys. B190 [FS3] (1981) 455

View on Pith 1981

[2] [2]

T. Suzuki and I. Yotsuyanagi, Phys. Rev. D42 (1990) 4257; S. Hioki et al., Phys. Lett. B272 (1991) 326

View on Pith 1990

[3] [3]

A. Kronfeld, M. Laursen, G. Schierholz, and U.-J. Wiese, Phys. L ett. B198 (1987) 516

View on Pith 1987

[4] [4]

M. Polikarpov, to appear in the Proceedings of LATTICE 96, arch ive: hep- lat/9609020

View on Pith

[5] [5]

H. B. Nielsen and P. Olesen, Nucl. Phys. B160 (1979) 380

View on Pith 1979

[6] [6]

G. ’t Hooft, Nucl. Phys. B153 (1979) 141

View on Pith 1979

[7] [7]

G. Mack, in Recent Developments in Gauge Theories , edited by G. ’t Hooft et al. (Plenum, New York, 1980)

View on Pith 1980

[8] [8]

T. Yoneya, Nucl. Phys. B144 (1978) 195

View on Pith 1978

[9] [9]

T. Kov´ acs and E. Tomboulis, to appear in the Proceedings of LAT TICE 96, archive: hep-lat/9607068

View on Pith Pith review arXiv

[10] [10]

J. Ambjørn and P. Olesen, Nucl. Phys. B170 [FS1] (1980) 265

View on Pith 1980

[11] [11]

L. Del Debbio, M. Faber, J. Greensite, and ˇS. Olejn ´ ık, to appear in the Proceed- ings of LATTICE 96, archive: hep-lat/9607053

View on Pith Pith review arXiv

[12] [12]

L. Del Debbio, M. Faber, and J. Greensite, Nucl. Phys. B414 (1 994) 594

View on Pith

[13] [13]

L. Del Debbio, M. Faber, J. Greensite, and ˇS. Olejn ´ ık, Phys. Rev. D53 (96) 5891

View on Pith

[14] [14]

J. Greensite and M. B. Halpern, Phys. Rev. D27 (1983) 2545

View on Pith 1983

[15] [15]

J. Ambjørn, P. Olesen, and C. Peterson, Nucl. Phys. B240 [FS1 2] (1984) 189; 533

View on Pith 1984

[16] [16]

G. Poulis and H. Trottier, archive: hep-lat/9504015; C. Michael, Nucl. Phys. B (Proc. Suppl.) 26 (1992) 417; Nucl. Phys. B 259 (1985) 58. 20

View on Pith Pith review arXiv 1992

[17] [17]

N. Cambell, I. Jorysz, and C. Michael, Phys. Lett. B167 (1986) 91; M. Faber and H. Markum, Nucl. Phys. B (Proc. Suppl.) 4 (1988) 204; M. M¨ uller, W. Beirl, M. Faber, and H. Markum, Nucl. Phys. B (Proc. S uppl.) 26 (1992) 423. 21

View on Pith 1986