90:
4013:
5859:
compactifications where their presence leads to greater control of gauge symmetry breaking, giving a better handle on the final unbroken gauge group and also providing a mechanism for controlling the number of matter multiplets left after compactification. These properties make Wilson lines important
5673:
of the underlying Yang–Mills theory fields and couplings does not prevent the Wilson loops from requiring additional renormalization corrections. In a renormalized Yang–Mills theory, the particular way that the Wilson loops get renormalized depends on the geometry of the loop under consideration. The
398:
while the latter consists of vectors that are perpendicular to the fiber. This allows for the comparison of fiber values at different spacetime points by connecting them with curves in the principal bundle whose tangent vectors always live in the horizontal subspace, so the curve is always
4160:, but in this case it does not factorize and instead leads to expectation values of products of Wilson loops, rather than the product of their expectation values. This gives rise to an infinite chain of coupled equations for different Wilson loop expectation values, analogous to the
1760:
of the gauge group. Wilson loops satisfy the reconstruction property where knowing the set of Wilson loops for all possible loops allows for the reconstruction of all gauge invariant information about the gauge connection. Formally the set of all Wilson loops forms an
3805:
2970:
4229:
matrices have Wilson loops that satisfy a set of identities called the
Mandelstam identities, with these identities reflecting the particular properties of the underlying gauge group. The identities apply to loops formed from two or more subloops, with
3503:
2287:
of the quark separation. Since spacelike Wilson loops are not fundamentally different from the temporal ones, the quark potential is really directly related to the pure Yang–Mills theory structure and is a phenomenon independent of the matter content.
1080:
2294:
ensures that local non-gauge invariant operators cannot have a non-zero expectation values. Instead one must use non-local gauge invariant operators as order parameters for confinement. The Wilson loop is exactly such an order parameter in pure
5839:
theories, zero mode gauge field states that are locally pure gauge configurations but are globally inequivalent to the vacuum are parameterized by closed Wilson lines in the compact direction. The presence of these on a compactified
2274:
904:
5822:
limit, up to finite terms. While this duality was initially only suggested in the maximum helicity violating case, there are arguments that it can be extended to all helicity configurations by defining appropriate
3790:
1595:, which can be integrated out, yielding the Wilson line as the world-line of the test particle. This works in quantum field theory whether or not there actually is any matter content in the theory. However, the
1117:
theories. The horizontal lift starting at some initial fiber point other than the identity merely requires multiplication by the initial element of the original horizontal lift. More generally, it holds that if
3313:
1256:
1186:
2684:. Four links around a single square are known as a plaquette, with their trace forming the smallest Wilson loop. It is these plaquettes that are used to construct the lattice gauge action known as the
5066:
1587:
gauge invariant. It allows for the comparison of the matter field at different points in a gauge invariant way. Alternatively, the Wilson lines can also be introduced by adding an infinitely heavy
4068:
4274:
1585:
1455:
3188:
784:
5337:
2530:
2367:
46:
707:
4008:{\displaystyle \partial _{\mu }^{x}{\frac {\delta }{\delta \sigma _{\mu \nu }(x)}}\langle W\rangle =g^{2}N\oint _{\gamma }dy_{\nu }\delta ^{(D)}(x-y)\langle W\rangle \langle W\rangle .}
3543:
353:
2871:
4595:
2798:
1999:
1739:
3350:
656:
531:
374:, which is equivalent to introducing a gauge field. This is because a connection is a way to separate out the tangent space of the principal bundle into two subspaces known as the
3665:
4193:
919:
2437:
5790:
1607:
must have a corresponding particle of that charge be present in the theory. Decoupling these particles by taking the infinite mass limit no longer works since this would form
3613:
3068:
5368:
3099:
3014:
2863:
1843:
1107:
5914:
with the expectation value of a product of loops depending only on the manifold structure and on how the loops are tied together. This led to the famous connection made by
5186:
2161:
4098:
1808:
4328:
4301:
2682:
5152:
618:
5655:
4621:
4227:
1487:
5575:
5507:
4732:
3563:
3148:
2706:
2457:
2035:
1282:
460:
5732:
4681:
1943:
585:
558:
487:
427:
239:
212:
185:
158:
5276:
3633:
3248:
3228:
2646:
2119:
1317:
5908:
5610:
4650:
4461:
4415:
5820:
5712:
4435:
4158:
4138:
4118:
3583:
3342:
3208:
3034:
2626:
2582:
2562:
2399:
2075:
2055:
1912:
1863:
396:
310:
290:
267:
131:
111:
5758:
where a set of dualities between them and special types of scattering amplitudes has been found. These have first been suggested at strong coupling using the
5792:
5678:
Smooth non-intersecting curve: This can only have linear divergences proportional to the contour which can be removed through multiplicative renormalization.
2608:. The smallest Wilson lines on the lattice, those between two adjacent lattice points, are known as links, with a single link starting from a lattice point
2816:
2017:
Since temporal Wilson lines correspond to the configuration created by infinitely heavy stationary quarks, Wilson loop associated with a rectangular loop
2169:
5923:
358:
The issue that Wilson lines resolve is how to compare points on fibers at two different spacetime points. This is analogous to parallel transport in
2584:
is some constant. The area law of Wilson loops can be used to demonstrate confinement in certain low dimensional theories directly, such as for the
796:
6035:
5872:, the expectation value of Wilson loops does not change under smooth deformations of the loop since the field theory does not depend on the
3673:
1881:. Since the Hilbert space lives on a single time slice, the only Wilson loops that can act as operators on this space are ones formed using
3253:
2401:. This is motivated from the potential between infinitely heavy test quarks which in the confinement phase is expected to grow linearly
7526:
7498:
7473:
7410:
7326:
7019:
6913:
6685:
6652:
6571:
6489:
6456:
6431:
6398:
6287:
6010:
6522:
6252:
6060:
1191:
1121:
4417:, with this holding for any gauge group in any dimension. Mandelstam identities of the second kind are acquired by noting that in
709:
and that its tangent vectors always lie in the horizontal subspace. The fiber bundle formulation of gauge theory reveals that the
370:
at different points. For principal bundles there is a natural way to compare different fiber points through the introduction of a
5869:
7603:
3344:
is defined through the same idea as the usual derivative, as the normalized difference between the functional of the two loops
375:
4740:
241:. Curves on the spacetime are uplifted to curves in the principal bundle whose tangent vectors lie in the horizontal subspace.
1780:
4021:
3799:, the Wilson loop vacuum expectation value satisfies a closed functional form equation called the Makeenko–Migdal equation
7608:
6640:
5737:
Self-intersections: This leads to operator mixing between the Wilson loops associated with the full loop and the subloops.
4233:
1496:
1325:
4161:
3153:
7074:
6863:
6832:
1604:
1460:
This gauge transformation property is often used to directly introduce the Wilson line in the presence of matter fields
34:
5799:
2279:
making the Wilson loop useful for calculating the potential between quark pairs. This potential must necessarily be a
716:
5798:
factorize into a tree-level component and a loop level correction. This loop level correction does not depend on the
5288:
20:
6274:. World Scientific Lecture Notes in Physics: Volume 43. Vol. 82. World Scientific Publishing. pp. 95–108.
2469:
7613:
5836:
2309:
1490:
246:
661:
2965:{\displaystyle \sigma =-{\frac {1}{a^{2}}}\ln {\bigg (}{\frac {\beta }{18}}{\bigg )}(1+{\mathcal {O}}(\beta )),}
5666:
3511:
3123:
2082:
371:
7258:
6973:"Feynman Rules for Electromagnetic and Yang–Mills Fields from the Gauge-Independent Field-Theoretic Formalism"
5919:
3498:{\displaystyle {\frac {\delta F}{\delta \sigma _{\mu \nu }(x)}}={\frac {1}{\delta \sigma _{\mu \nu }(x)}}-F].}
2834:
of traces of plaquettes, where the first non-vanishing term in the expectation value of the Wilson loop in an
1877:
An alternative view of Wilson loops is to consider them as operators acting on the
Hilbert space of states in
323:
7598:
4469:
2822:
The lattice formulation of the Wilson loops also allows for an analytic demonstration of confinement in the
2714:
1948:
1752:
of the fiber into itself upon horizontal lift along a closed loop. The set of all holonomies itself forms a
1627:
1075:{\displaystyle g_{f}(t_{f})=W={\mathcal {P}}\exp {\bigg (}i\int _{x_{i}}^{x_{f}}A_{\mu }dx^{\mu }{\bigg )},}
623:
492:
355:
although globally it can have some twisted structure depending on how different fibers are glued together.
6079:
5759:
3638:
2827:
1765:
1615:
6418:. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. pp. 117–118.
5612:
holds for all gauge groups in the fundamental representations, for unitary groups it moreover holds that
4167:
2296:
6100:
4623:
3119:
2804:
787:
6901:
3070:, leaving the theory deconfined at weak coupling. Such a phase transition is not believed to exist for
2404:
5765:
7558:
7364:
7280:
7220:
7201:; Radu, R. (2008). "Scattering Amplitudes, Wilson Loops and the String/Gauge Theory Correspondence".
7172:
7129:
7090:
7048:
6984:
6942:
6875:
6806:
6764:
6718:
6606:
6551:
6355:
6316:
6206:
6157:
5975:
5873:
5755:
4463:
3588:
3047:
3036:
is the lattice spacing. While this argument holds for both the abelian and non-abelian case, compact
2601:
2291:
1776:
588:
66:
26:
6509:. Cambridge Monographs on Mathematical Physics. Cambridge: Cambridge University Press. p. 105.
5342:
3073:
2978:
2837:
1813:
1088:
7254:
7198:
5160:
2128:
1753:
1596:
4073:
3130:
called the area derivative and the perimeter derivative. To define the former, consider a contour
1786:
65:
in 1974, they were used to construct links and plaquettes which are the fundamental parameters in
7574:
7380:
7354:
7296:
7270:
7236:
7210:
7156:
6968:
6702:
6622:
6596:
6528:
6222:
6196:
6116:
5682:
4306:
4279:
2808:
2651:
2280:
1288:
359:
70:
38:
5077:
594:
6587:
Chen, Y.; et al. (2006). "Glueball spectrum and matrix elements on anisotropic lattices".
5615:
4600:
4206:
4140:, with the two points however close to each other. The equation can also be written for finite
7522:
7494:
7469:
7461:
7406:
7398:
7322:
7015:
6909:
6748:
6681:
6648:
6567:
6518:
6485:
6452:
6427:
6394:
6371:
6283:
6248:
6056:
6031:
6006:
5959:
2823:
1463:
317:
62:
58:
6477:
5802:
of the particles, but it was found to be dual to certain polygonal Wilson loops in the large
5519:
5376:
4686:
3548:
3133:
2691:
2442:
2020:
1261:
432:
7566:
7443:
7427:
7372:
7288:
7228:
7180:
7137:
7098:
7056:
6992:
6950:
6883:
6844:
6814:
6772:
6726:
6673:
6614:
6559:
6510:
6419:
6363:
6324:
6275:
6214:
6165:
6126:
5983:
5927:
5841:
3041:
2284:
2090:
1882:
1762:
74:
5717:
4659:
4656:
yields a set of identities between Wilson loops. These can be written in terms the objects
1921:
563:
536:
465:
405:
217:
190:
163:
136:
7514:
6473:
6187:; Tachikawa, Yuji (2013). "Reading between the lines of four-dimensional gauge theories".
5670:
5191:
4653:
3618:
3233:
3213:
3037:
2631:
2605:
2585:
2095:
1878:
1600:
1293:
363:
42:
7342:
6083:
5887:
5855:, whose separations are determined by the Wilson lines. Wilson lines also play a role in
5580:
4629:
4440:
4336:
786:
is equivalent to the connection that defines the horizontal subspace, so this leads to a
7562:
7368:
7292:
7284:
7224:
7176:
7133:
7094:
7052:
6988:
6946:
6879:
6810:
6768:
6722:
6610:
6555:
6359:
6320:
6210:
6161:
5979:
1779:
of the gauge group. This can be reformulated in terms of Lie algebra language using the
6795:"Existence proof of a nonconfining phase in four-dimensional U(1) lattice gauge theory"
6240:
6184:
6104:
5939:
5805:
5795:
5688:
5513:
4420:
4143:
4123:
4103:
3568:
3318:
3193:
3019:
2611:
2567:
2538:
2375:
2300:
2060:
2040:
2002:
1888:
1848:
381:
295:
275:
252:
116:
96:
54:
7160:
7078:
7036:
6930:
6706:
6269:
4597:. In the fundamental representation, the holonomies used to form the Wilson loops are
2830:
where quark loops are neglected. This is done by expanding out the Wilson action as a
378:
and horizontal subspaces. The former consists of all vectors pointing along the fiber
89:
7592:
7542:
7447:
7384:
7240:
7184:
7102:
7060:
6954:
6887:
6848:
6776:
6744:
6730:
6532:
6226:
5915:
5844:
5824:
3796:
3102:
2685:
2122:
2006:
1915:
1592:
1588:
1114:
1110:
367:
78:
7578:
7300:
6972:
6794:
6626:
6343:
6145:
5963:
2604:, Wilson lines and loops play a fundamental role in formulating gauge fields on the
1493:
of the gauge group, where the Wilson line is an operator that makes the combination
45:. They encode all gauge information of the theory, allowing for the construction of
7314:
7117:
6304:
2831:
2269:{\displaystyle \langle W\rangle \sim e^{-TV(r)}(1+{\mathcal {O}}(e^{-T\Delta E})),}
2086:
1749:
313:
50:
7232:
2688:. Larger Wilson loops are expressed as products of link variables along some loop
7546:
5685:: Each cusp results in an additional local multiplicative renormalization factor
1603:, every Wilson line and 't Hooft line of a particular charge consistent with the
7376:
6218:
5911:
5282:
2460:
710:
6618:
6130:
1810:. In this case the types of Wilson loops are in one-to-one correspondence with
6677:
5877:
3127:
1866:
1608:
6996:
6818:
6514:
6423:
6367:
6169:
5987:
61:, where they satisfy what is known as the area law. Originally formulated by
7141:
6790:
6328:
5848:
2589:
2163:
are exponentially suppressed with time and so the expectation value goes as
1772:
899:{\displaystyle i{\frac {dg(t)}{dt}}=A_{\mu }(x){\frac {dx^{\mu }}{dt}}g(t).}
270:
3105:, instead they exhibit confinement at all values of the coupling constant.
2811:
of Wilson loops are also used as interpolating operators that give rise to
1599:
known as the completeness conjecture claims that in a consistent theory of
462:, then to see how this changes when moving to another spacetime coordinate
6375:
5740:
Lightlike segments: These give rise to additional logarithmic divergences.
4164:. The Makeenko–Migdal equation has been solved exactly in two dimensional
5881:
5856:
2865:
gauge theory gives rise to an area law with a string tension of the form
2812:
1757:
1745:
245:
To properly define Wilson loops in gauge theory requires considering the
6601:
5880:
on in these theories and are used to calculate global properties of the
19:"Wilson line" redirects here. For the Wilson Line shipping company, see
7570:
5852:
3785:{\displaystyle \partial _{\mu }^{x}F={\frac {1}{\delta x_{\mu }}}-F].}
2803:
These Wilson loops are used to study confinement and quark potentials
1591:
charged under the gauge group. Its charge forms a quantized internal
7431:
6752:
53:
in terms of these loops. In pure gauge theory they play the role of
6670:
3308:{\displaystyle \delta \sigma _{\mu \nu }=dx_{\mu }\wedge dx_{\nu }}
7359:
7345:(2011). "Notes on the scattering amplitudes/Wilson loop duality".
7275:
7215:
6707:"Strong coupling and mean field methods in lattice gauge theories"
6201:
6121:
5071:
In this notation the
Mandelstam identities of the second kind are
2078:
1618:
of closed Wilson lines is a gauge invariant quantity known as the
88:
6866:(1979). "Exact Equation for the Loop Average in Multicolor QCD".
6563:
6279:
3040:
only exhibits confinement at strong coupling, with there being a
4203:
Gauge groups that admit fundamental representations in terms of
587:. The corresponding curve in the principal bundle, known as the
1918:, which can be seen by noting that the electric field operator
7161:"Renormalization of the Wilson loops beyond the leading order"
6107:(2011). "Symmetries and Strings in Field Theory and Gravity".
7403:
String Theory Volume I: An
Introduction to the Bosonic String
1251:{\displaystyle {\tilde {\gamma }}'(t)={\tilde {\gamma }}(t)g}
1181:{\displaystyle {\tilde {\gamma }}'(0)={\tilde {\gamma }}(0)g}
5771:
2942:
2230:
1660:
1094:
989:
7261:(2007). "Gluon scattering amplitudes at strong coupling".
6672:. Lecture Notes in Physics 788. Springer. pp. 58–62.
5665:
Since Wilson loops are operators of the gauge fields, the
3508:
The perimeter derivative is similarly defined whereby now
3190:
which is the same contour but with an extra small loop at
2081:-antiquark pair at fixed separation. Over large times the
6931:"Non-linear strings in two-dimensional U(∞) gauge theory"
1744:
Mathematically the term within the trace is known as the
6030:. Lecture Notes in Physics. Springer. pp. 220–222.
5061:{\displaystyle (K+1)M_{K+1}=WM_{K}-M_{K}-\cdots -M_{K}.}
5281:
If the fundamental representation are matrices of unit
7466:
String Theory Volume II: Superstring Theory and Beyond
6146:"Reconstruction of gauge potentials from Wilson loops"
4333:
The
Mandelstam identity of the first kind states that
6449:
5890:
5808:
5768:
5720:
5691:
5618:
5583:
5522:
5379:
5345:
5291:
5194:
5163:
5080:
4743:
4689:
4662:
4632:
4603:
4472:
4443:
4423:
4339:
4309:
4282:
4236:
4209:
4170:
4146:
4126:
4106:
4076:
4063:{\displaystyle \gamma =\gamma _{xy}\cup \gamma _{yx}}
4024:
3808:
3676:
3641:
3635:
direction and of zero area. The perimeter derivative
3621:
3591:
3571:
3551:
3514:
3353:
3321:
3256:
3236:
3216:
3196:
3156:
3136:
3076:
3050:
3022:
2981:
2874:
2840:
2815:. The glueball masses can then be extracted from the
2717:
2694:
2654:
2634:
2614:
2570:
2541:
2472:
2445:
2407:
2378:
2312:
2172:
2131:
2098:
2063:
2043:
2023:
1951:
1924:
1891:
1851:
1816:
1789:
1630:
1499:
1466:
1328:
1296:
1264:
1194:
1124:
1091:
922:
799:
719:
664:
626:
597:
566:
539:
495:
468:
435:
408:
384:
326:
298:
278:
255:
220:
193:
166:
139:
119:
99:
4269:{\displaystyle \gamma =\gamma _{2}\circ \gamma _{1}}
1580:{\displaystyle \phi (x_{i})^{\dagger }W\phi (x_{f})}
1450:{\displaystyle W\rightarrow g(x_{f})Wg^{-1}(x_{i}).}
3183:{\displaystyle \gamma _{\delta \sigma _{\mu \nu }}}
2085:of the Wilson loop projects out the state with the
69:. Wilson loops fall into the broader class of loop
6753:"Phase Transition in Four-Dimensional Compact QED"
5902:
5814:
5784:
5726:
5706:
5649:
5604:
5569:
5501:
5362:
5331:
5270:
5180:
5146:
5060:
4726:
4675:
4644:
4615:
4589:
4455:
4429:
4409:
4322:
4295:
4268:
4221:
4187:
4152:
4132:
4112:
4092:
4062:
4007:
3784:
3659:
3627:
3607:
3577:
3557:
3537:
3497:
3336:
3315:. Then the area derivative of the loop functional
3307:
3242:
3222:
3202:
3182:
3142:
3093:
3062:
3028:
3008:
2964:
2857:
2792:
2700:
2676:
2640:
2620:
2576:
2556:
2524:
2459:is known as the string tension. Meanwhile, in the
2451:
2431:
2393:
2361:
2268:
2155:
2113:
2069:
2049:
2029:
1993:
1937:
1906:
1857:
1837:
1802:
1733:
1579:
1481:
1449:
1311:
1276:
1250:
1180:
1101:
1074:
898:
778:
701:
650:
612:
579:
552:
525:
481:
454:
421:
390:
347:
304:
284:
261:
233:
206:
179:
152:
125:
105:
7118:"Renormalization of loop functions for all loops"
6550:. World Scientific Publishing. pp. 232–233.
2926:
2909:
2782:
2740:
1723:
1716:
1673:
1653:
1064:
1002:
81:, which are the thermal version of Wilson loops.
7468:. Cambridge University Press. pp. 288–290.
7405:. Cambridge University Press. pp. 263–268.
6908:. Cambridge University Press. pp. 469–472.
6055:. Cambridge University Press. pp. 488–493.
2463:the expectation value follows the perimeter law
779:{\displaystyle A_{\mu }(x)=A_{\mu }^{a}(x)T^{a}}
77:, which are magnetic duals to Wilson loops, and
7547:"Quantum Field Theory and the Jones Polynomial"
5332:{\displaystyle M_{N}(\gamma ,\dots ,\gamma )=1}
133:separates out the tangent space at every point
7491:Quarks and Leptons From Orbifolded Superstring
6305:"Upper bound on the color-confining potential"
5860:in compactifications of superstring theories.
2525:{\displaystyle \langle W\rangle \sim e^{-bL},}
2005:it follows that the spatial loop measures the
249:of gauge theories. Here for each point in the
7035:Korchemskaya, I.A.; Korchemsky, G.P. (1992).
2362:{\displaystyle \langle W\rangle \sim e^{-aA}}
909:This has a unique formal solution called the
489:, one needs to consider some spacetime curve
8:
7519:Quantum Field Theory: An Integrated Approach
6835:(1983). "Loop Equations and 1/N Expansion".
3999:
3974:
3971:
3946:
3873:
3858:
2488:
2473:
2328:
2313:
2188:
2173:
1982:
7521:. Princeton University Press. p. 697.
6451:. Cambridge University Press. p. 168.
6053:Quantum Field Theory and the Standard Model
6005:(2 ed.). CRC Press. pp. 374–418.
2303:its expectation value follows the area law
1768:of solutions to the Gauss' law constraint.
702:{\displaystyle {\tilde {\gamma }}(0)=g_{i}}
6645:The Theory of Quark and Gluon Interactions
6548:Lattice Methods for Quantum Chromodynamics
6391:An Introduction to the Confinement Problem
5754:Wilson loops play a role in the theory of
5512:Fundamental representations consisting of
4276:being a loop formed by first going around
3667:of the loop functional is then defined as
7358:
7274:
7214:
7116:Brandt, R.A.; Neri, F.; Sato, M. (1981).
6600:
6200:
6120:
5889:
5807:
5770:
5769:
5767:
5719:
5690:
5636:
5619:
5617:
5582:
5555:
5542:
5521:
5487:
5474:
5449:
5444:
5431:
5409:
5390:
5378:
5346:
5344:
5339:. For example, applying this identity to
5296:
5290:
5259:
5246:
5224:
5205:
5193:
5164:
5162:
5123:
5104:
5085:
5079:
5040:
5027:
5008:
4995:
4982:
4960:
4941:
4922:
4909:
4896:
4880:
4861:
4848:
4829:
4801:
4782:
4763:
4742:
4694:
4688:
4667:
4661:
4631:
4602:
4567:
4562:
4546:
4541:
4526:
4521:
4514:
4509:
4497:
4492:
4485:
4477:
4471:
4442:
4422:
4398:
4385:
4363:
4350:
4338:
4314:
4308:
4287:
4281:
4260:
4247:
4235:
4208:
4171:
4169:
4145:
4125:
4105:
4081:
4075:
4051:
4035:
4023:
3987:
3959:
3919:
3909:
3896:
3883:
3837:
3824:
3818:
3813:
3807:
3750:
3742:
3720:
3707:
3686:
3681:
3675:
3651:
3646:
3640:
3620:
3599:
3590:
3570:
3550:
3538:{\displaystyle \gamma _{\delta x_{\mu }}}
3527:
3519:
3513:
3460:
3452:
3418:
3405:
3381:
3354:
3352:
3320:
3299:
3283:
3264:
3255:
3235:
3215:
3195:
3169:
3161:
3155:
3135:
3077:
3075:
3049:
3021:
3000:
2991:
2980:
2941:
2940:
2925:
2924:
2914:
2908:
2907:
2893:
2884:
2873:
2841:
2839:
2781:
2780:
2765:
2749:
2739:
2738:
2733:
2716:
2693:
2659:
2653:
2633:
2613:
2569:
2540:
2498:
2471:
2444:
2406:
2377:
2338:
2311:
2242:
2229:
2228:
2198:
2171:
2130:
2097:
2062:
2042:
2022:
1974:
1956:
1950:
1929:
1923:
1890:
1850:
1827:
1821:
1815:
1794:
1788:
1722:
1721:
1715:
1714:
1708:
1695:
1685:
1672:
1671:
1659:
1658:
1652:
1651:
1646:
1629:
1568:
1549:
1536:
1520:
1510:
1498:
1465:
1435:
1419:
1406:
1393:
1374:
1352:
1339:
1327:
1295:
1263:
1225:
1224:
1197:
1196:
1193:
1155:
1154:
1127:
1126:
1123:
1093:
1092:
1090:
1063:
1062:
1056:
1043:
1031:
1026:
1019:
1014:
1001:
1000:
988:
987:
975:
962:
940:
927:
921:
864:
854:
839:
803:
798:
770:
751:
746:
724:
718:
693:
666:
665:
663:
628:
627:
625:
596:
571:
565:
544:
538:
494:
473:
467:
440:
434:
413:
407:
383:
333:
329:
328:
325:
320:. Locally the resulting space looks like
297:
277:
254:
225:
219:
198:
192:
171:
165:
144:
138:
118:
98:
73:, with some other notable examples being
6668:Gattringer, C.; Lang, C.B. (2009). "3".
6484:. Oxford University Press. p. 720.
6393:(2 ed.). Springer. pp. 37–40.
5876:. For this reason, Wilson loops are key
3126:, functions of loops admit two types of
2564:is the perimeter length of the loop and
1756:, which for principal bundles must be a
348:{\displaystyle \mathbb {R} ^{d}\times G}
7319:Scattering Amplitudes in Gauge Theories
6505:Montvay, I.; Munster, G. (1994). "43".
6344:"Concavity of the quarkonium potential"
6271:Lattice Gauge Theories: An Introduction
6245:An Introduction to Quantum Field Theory
5951:
5910:dimensions they are closely related to
5796:maximally helicity violating amplitudes
3545:is a slight deformation of the contour
2037:with two temporal components of length
2001:but it vanishes everywhere else. Using
402:If the starting fiber is at coordinate
312:forming what's known as a fiber of the
6074:
6072:
4590:{\displaystyle \delta _{}^{a_{N+1}}=0}
4100:being a line that does not close from
2793:{\displaystyle L={\text{tr}}{\bigg }.}
1994:{\displaystyle E^{i}W|0\rangle \neq 0}
1734:{\displaystyle W={\text{tr}}{\bigg }.}
429:with a starting point of the identity
6647:(4 ed.). Springer. p. 383.
6546:DeGrand, T.; DeTar, C. (2006). "11".
6243:; Schroeder, Daniel V. (1995). "15".
3585:has a small extruding loop of length
3016:is the inverse coupling constant and
2057:and two spatial components of length
651:{\displaystyle {\tilde {\gamma }}(t)}
526:{\displaystyle \gamma :\rightarrow M}
7:
6929:Kazakov, V.A.; Kostov, I.K. (1980).
6906:Introduction to Quantum Field Theory
6416:Methods of Contemporary Gauge Theory
5827:generalizations of the Wilson loop.
3660:{\displaystyle \partial _{\mu }^{x}}
4188:{\displaystyle {\text{U}}(\infty )}
292:there is a copy of the gauge group
93:A connection on a principal bundle
5577:. Furthermore, while the equality
4179:
3810:
3678:
3643:
2249:
2147:
1818:
1791:
1771:The set of all Wilson lines is in
399:perpendicular to any given fiber.
14:
6028:Topology and Geometry for Physics
4626:of the gauge groups. Contracting
2432:{\displaystyle V(r)\sim \sigma r}
2372:for a loop that encloses an area
316:. These fiber bundles are called
41:of gauge variables around closed
6482:From Classical to Quantum Fields
5851:to a theory with non-coincident
5793:supersymmetric Yang–Mills theory
5785:{\displaystyle {\mathcal {N}}=4}
7079:"Gauge fields as rings of glue"
6247:. Westview Press. p. 492.
5831:String theory compactifications
5714:that depends on the cusp angle
4466:indices vanishes, meaning that
3608:{\displaystyle \delta x_{\mu }}
3063:{\displaystyle \beta \sim 1.01}
2588:whose confinement is driven by
7489:Choi, K.S.; Kim, J.E. (2020).
7321:. Springer. pp. 153–158.
6051:Schwartz, M. D. (2014). "25".
6003:Geometry, Topology and Physics
5918:where he used Wilson loops in
5701:
5695:
5637:
5633:
5627:
5620:
5593:
5587:
5564:
5548:
5532:
5526:
5493:
5467:
5458:
5424:
5415:
5402:
5396:
5383:
5363:{\displaystyle {\text{SU}}(2)}
5357:
5351:
5320:
5302:
5265:
5239:
5230:
5217:
5211:
5198:
5175:
5169:
5135:
5097:
5052:
4988:
4966:
4902:
4886:
4854:
4841:
4822:
4813:
4775:
4756:
4744:
4721:
4715:
4706:
4700:
4558:
4478:
4404:
4378:
4369:
4343:
4182:
4176:
3996:
3980:
3968:
3952:
3943:
3931:
3926:
3920:
3870:
3864:
3852:
3846:
3776:
3773:
3767:
3758:
3735:
3729:
3701:
3695:
3489:
3486:
3480:
3471:
3445:
3439:
3433:
3427:
3396:
3390:
3369:
3363:
3331:
3325:
3094:{\displaystyle {\text{SU}}(N)}
3088:
3082:
3009:{\displaystyle \beta =6/g^{2}}
2956:
2953:
2947:
2931:
2858:{\displaystyle {\text{SU}}(3)}
2852:
2846:
2777:
2771:
2727:
2721:
2671:
2665:
2551:
2545:
2514:
2508:
2485:
2479:
2417:
2411:
2388:
2382:
2354:
2348:
2325:
2319:
2260:
2257:
2235:
2219:
2214:
2208:
2185:
2179:
2141:
2135:
2108:
2102:
1975:
1971:
1965:
1901:
1895:
1838:{\displaystyle \Lambda _{w}/W}
1640:
1634:
1574:
1561:
1555:
1529:
1517:
1503:
1476:
1470:
1441:
1428:
1412:
1386:
1380:
1367:
1361:
1358:
1332:
1319:the Wilson line transforms as
1306:
1300:
1242:
1236:
1230:
1218:
1212:
1202:
1172:
1166:
1160:
1148:
1142:
1132:
1102:{\displaystyle {\mathcal {P}}}
981:
955:
946:
933:
890:
884:
851:
845:
818:
812:
763:
757:
736:
730:
683:
677:
671:
645:
639:
633:
607:
601:
517:
514:
502:
1:
7293:10.1088/1126-6708/2007/06/064
7233:10.1016/j.physrep.2008.08.002
6088:Lecture Notes on Gauge Theory
5181:{\displaystyle {\text{U}}(1)}
3114:Makeenko–Migdal loop equation
2819:between these interpolators.
2156:{\displaystyle V(r)+\Delta E}
7448:10.1016/0370-2693(87)90066-9
7432:"Orbifolds and Wilson Lines"
7426:Ibanez, L.E.; Nilles, H.P.;
7185:10.1016/0550-3213(87)90277-X
7103:10.1016/0550-3213(80)90507-6
7061:10.1016/0370-2693(92)91895-G
7037:"On light-like Wilson loops"
7012:Loops, Knots, Gauge Theories
6955:10.1016/0550-3213(80)90072-3
6888:10.1016/0370-2693(79)90131-X
6849:10.1016/0370-1573(83)90076-5
6777:10.1016/0370-2693(80)90400-1
6731:10.1016/0370-1573(83)90034-0
5681:Non-intersecting curve with
4683:defined iteratively so that
4437:dimensions, any object with
4093:{\displaystyle \gamma _{xy}}
1803:{\displaystyle \Lambda _{w}}
1605:Dirac quantization condition
21:Thomas Wilson Sons & Co.
6507:Quantum Fields on a Lattice
6447:Paranjape, M. (2017). "9".
6389:Greensite, J. (2020). "4".
6001:Nakahara, M. (2003). "10".
4323:{\displaystyle \gamma _{2}}
4296:{\displaystyle \gamma _{1}}
2677:{\displaystyle U_{\mu }(n)}
1113:, which is unnecessary for
37:operators arising from the
7630:
6619:10.1103/PhysRevD.73.014516
6414:Makeenko, Y. (2002). "6".
6131:10.1103/PhysRevD.83.084019
5285:, then it also holds that
5147:{\displaystyle M_{N+1}=0.}
2077:, can be interpreted as a
1491:fundamental representation
1289:local gauge transformation
613:{\displaystyle \gamma (t)}
214:and a horizontal subspace
18:
7010:Gambini, R. (2008). "3".
6678:10.1007/978-3-642-01850-3
6268:Rothe, H.J. (2005). "7".
6026:Eschrig, H. (2011). "7".
5650:{\displaystyle |W|\leq N}
4616:{\displaystyle N\times N}
4222:{\displaystyle N\times N}
4162:Schwinger–Dyson equations
2299:, where in the confining
1773:one-to-one correspondence
187:into a vertical subspace
16:Gauge field loop operator
6997:10.1103/PhysRev.175.1580
6819:10.1103/PhysRevD.21.2291
6515:10.1017/CBO9780511470783
6424:10.1017/CBO9780511535147
6368:10.1103/PhysRevD.33.2723
6170:10.1103/PhysRevD.24.2160
5988:10.1103/PhysRevD.10.2445
5870:topological field theory
5864:Topological field theory
3044:to the Coulomb phase at
2281:monotonically increasing
2121:between the quarks. The
2083:vacuum expectation value
1914:create a closed loop of
1482:{\displaystyle \phi (x)}
790:for the horizontal lift
247:fiber bundle formulation
7377:10.1007/JHEP07(2011)058
7142:10.1103/PhysRevD.24.879
6329:10.1103/PhysRevD.18.482
6219:10.1007/JHEP08(2013)115
5964:"Confinement of quarks"
5745:Additional applications
5570:{\displaystyle W=W^{*}}
5502:{\displaystyle WW=W+W.}
5188:gauge group this gives
4727:{\displaystyle M_{1}=W}
3558:{\displaystyle \gamma }
3143:{\displaystyle \gamma }
2701:{\displaystyle \gamma }
2452:{\displaystyle \sigma }
2030:{\displaystyle \gamma }
1945:is nonzero on the loop
1873:Hilbert space operators
1277:{\displaystyle t\geq 0}
913:between the two points
455:{\displaystyle g_{i}=e}
7604:Quantum chromodynamics
5904:
5816:
5786:
5760:AdS/CFT correspondence
5728:
5708:
5651:
5606:
5571:
5503:
5364:
5333:
5272:
5182:
5148:
5062:
4728:
4677:
4646:
4624:matrix representations
4617:
4591:
4457:
4431:
4411:
4324:
4303:and then going around
4297:
4270:
4223:
4189:
4154:
4134:
4114:
4094:
4064:
4009:
3786:
3661:
3629:
3609:
3579:
3559:
3539:
3499:
3338:
3309:
3244:
3224:
3204:
3184:
3144:
3124:functions of functions
3095:
3064:
3030:
3010:
2966:
2859:
2828:quenched approximation
2794:
2702:
2678:
2642:
2622:
2578:
2558:
2526:
2453:
2433:
2395:
2363:
2270:
2157:
2115:
2071:
2051:
2031:
1995:
1939:
1908:
1885:loops. Such operators
1859:
1839:
1804:
1735:
1581:
1483:
1451:
1313:
1278:
1252:
1182:
1111:path-ordering operator
1103:
1076:
900:
780:
703:
652:
614:
581:
554:
527:
483:
456:
423:
392:
349:
306:
286:
263:
242:
235:
208:
181:
154:
127:
107:
5905:
5817:
5787:
5756:scattering amplitudes
5750:Scattering amplitudes
5729:
5727:{\displaystyle \phi }
5709:
5652:
5607:
5572:
5504:
5365:
5334:
5273:
5183:
5149:
5063:
4729:
4678:
4676:{\displaystyle M_{K}}
4647:
4618:
4592:
4464:totally antisymmetric
4458:
4432:
4412:
4325:
4298:
4271:
4224:
4199:Mandelstam identities
4190:
4155:
4135:
4115:
4095:
4065:
4010:
3787:
3662:
3630:
3610:
3580:
3560:
3540:
3500:
3339:
3310:
3245:
3225:
3205:
3185:
3145:
3120:functional derivative
3096:
3065:
3031:
3011:
2967:
2860:
2795:
2703:
2679:
2648:direction denoted by
2643:
2623:
2579:
2559:
2527:
2454:
2434:
2396:
2364:
2271:
2158:
2116:
2072:
2052:
2032:
1996:
1940:
1938:{\displaystyle E^{i}}
1909:
1860:
1840:
1805:
1736:
1582:
1484:
1452:
1314:
1279:
1253:
1183:
1104:
1077:
901:
788:differential equation
781:
704:
653:
615:
582:
580:{\displaystyle x_{f}}
555:
553:{\displaystyle x_{i}}
528:
484:
482:{\displaystyle x_{f}}
457:
424:
422:{\displaystyle x_{i}}
393:
350:
307:
287:
264:
236:
234:{\displaystyle H_{p}}
209:
207:{\displaystyle V_{p}}
182:
180:{\displaystyle G_{p}}
155:
153:{\displaystyle x_{p}}
128:
108:
92:
49:which fully describe
7609:Lattice field theory
5888:
5847:is equivalent under
5806:
5766:
5718:
5689:
5616:
5581:
5520:
5377:
5343:
5289:
5271:{\displaystyle WW=W}
5192:
5161:
5078:
4741:
4687:
4660:
4652:holonomies with the
4630:
4601:
4470:
4441:
4421:
4337:
4307:
4280:
4234:
4207:
4168:
4144:
4124:
4104:
4074:
4022:
3806:
3674:
3639:
3628:{\displaystyle \mu }
3619:
3589:
3569:
3549:
3512:
3351:
3319:
3254:
3243:{\displaystyle \nu }
3234:
3223:{\displaystyle \mu }
3214:
3194:
3154:
3150:and another contour
3134:
3074:
3048:
3020:
2979:
2872:
2838:
2826:phase, assuming the
2817:correlation function
2715:
2692:
2652:
2641:{\displaystyle \mu }
2632:
2612:
2602:lattice field theory
2568:
2539:
2470:
2443:
2405:
2376:
2310:
2170:
2129:
2114:{\displaystyle V(r)}
2096:
2061:
2041:
2021:
1949:
1922:
1889:
1849:
1814:
1787:
1748:, which describes a
1628:
1597:swampland conjecture
1497:
1489:transforming in the
1464:
1326:
1312:{\displaystyle g(x)}
1294:
1262:
1192:
1122:
1089:
920:
797:
717:
662:
624:
595:
564:
537:
493:
466:
433:
406:
382:
324:
296:
276:
253:
218:
191:
164:
137:
117:
97:
67:lattice gauge theory
47:loop representations
27:quantum field theory
7563:1989CMaPh.121..351W
7369:2011JHEP...07..058C
7285:2007JHEP...06..064A
7225:2008PhR...468..153A
7177:1987NuPhB.283..342K
7134:1981PhRvD..24..879B
7095:1980NuPhB.164..171P
7053:1992PhLB..287..169K
6989:1968PhRv..175.1580M
6947:1980NuPhB.176..199K
6880:1979PhLB...88..135M
6811:1980PhRvD..21.2291G
6769:1980PhLB...95...63L
6723:1983PhR...102....1D
6611:2006PhRvD..73a4516C
6556:2006lmqc.book.....D
6360:1986PhRvD..33.2723B
6342:Bachas, C. (1986).
6321:1978PhRvD..18..482S
6303:Seiler, E. (1978).
6211:2013JHEP...08..115A
6162:1981PhRvD..24.2160G
5980:1974PhRvD..10.2445W
5920:Chern–Simons theory
5903:{\displaystyle 2+1}
5605:{\displaystyle W=N}
5457:
5157:For example, for a
4645:{\displaystyle N+1}
4580:
4533:
4504:
4456:{\displaystyle N+1}
4410:{\displaystyle W=W}
3823:
3691:
3656:
2809:Linear combinations
2596:Lattice formulation
1879:Minkowski signature
1783:of the gauge group
1038:
756:
713:valued gauge field
7571:10.1007/BF01217730
7551:Commun. Math. Phys
7155:Korchemsky, G.P.;
7014:. pp. 63–67.
6241:Peskin, Michael E.
6144:Giles, R. (1981).
5924:partition function
5900:
5812:
5782:
5762:. For example, in
5724:
5704:
5674:main features are
5647:
5602:
5567:
5499:
5440:
5360:
5329:
5268:
5178:
5144:
5058:
4724:
4673:
4642:
4613:
4587:
4537:
4505:
4473:
4453:
4427:
4407:
4320:
4293:
4266:
4219:
4185:
4150:
4130:
4110:
4090:
4060:
4005:
3809:
3782:
3677:
3657:
3642:
3625:
3605:
3575:
3565:which at position
3555:
3535:
3495:
3334:
3305:
3240:
3220:
3200:
3180:
3140:
3101:gauge theories at
3091:
3060:
3026:
3006:
2962:
2855:
2790:
2760:
2698:
2674:
2638:
2618:
2574:
2554:
2522:
2449:
2429:
2391:
2359:
2266:
2153:
2111:
2067:
2047:
2027:
2009:through the loop.
1991:
1935:
1904:
1855:
1835:
1800:
1731:
1577:
1479:
1447:
1309:
1274:
1248:
1178:
1099:
1072:
1010:
896:
776:
742:
699:
648:
610:
577:
550:
523:
479:
452:
419:
388:
360:general relativity
345:
302:
282:
259:
243:
231:
204:
177:
150:
123:
103:
39:parallel transport
7614:Phase transitions
7165:Nuclear Physics B
7083:Nuclear Physics B
7041:Physics Letters B
6935:Nuclear Physics B
6037:978-3-642-14699-2
5928:Jones polynomials
5815:{\displaystyle N}
5707:{\displaystyle Z}
5349:
5167:
4430:{\displaystyle N}
4174:
4153:{\displaystyle N}
4133:{\displaystyle y}
4113:{\displaystyle x}
3856:
3727:
3578:{\displaystyle x}
3437:
3400:
3337:{\displaystyle F}
3203:{\displaystyle x}
3118:Similarly to the
3080:
3029:{\displaystyle a}
2922:
2899:
2844:
2745:
2736:
2621:{\displaystyle n}
2577:{\displaystyle b}
2557:{\displaystyle L}
2394:{\displaystyle A}
2297:Yang–Mills theory
2292:Elitzur's theorem
2070:{\displaystyle r}
2050:{\displaystyle T}
1907:{\displaystyle W}
1858:{\displaystyle W}
1649:
1233:
1205:
1163:
1135:
879:
830:
674:
636:
391:{\displaystyle G}
366:that live in the
318:principal bundles
305:{\displaystyle G}
285:{\displaystyle M}
262:{\displaystyle d}
126:{\displaystyle M}
106:{\displaystyle P}
63:Kenneth G. Wilson
7621:
7583:
7582:
7539:
7533:
7532:
7511:
7505:
7504:
7486:
7480:
7479:
7458:
7452:
7451:
7423:
7417:
7416:
7395:
7389:
7388:
7362:
7339:
7333:
7332:
7311:
7305:
7304:
7278:
7251:
7245:
7244:
7218:
7195:
7189:
7188:
7157:Radyushkin, A.V.
7152:
7146:
7145:
7113:
7107:
7106:
7071:
7065:
7064:
7032:
7026:
7025:
7007:
7001:
7000:
6983:(5): 1580–1603.
6965:
6959:
6958:
6926:
6920:
6919:
6898:
6892:
6891:
6874:(1–2): 135–137.
6862:Makeenko, Y.M.;
6859:
6853:
6852:
6829:
6823:
6822:
6805:(8): 2291–2307.
6787:
6781:
6780:
6741:
6735:
6734:
6698:
6692:
6691:
6665:
6659:
6658:
6637:
6631:
6630:
6604:
6584:
6578:
6577:
6543:
6537:
6536:
6502:
6496:
6495:
6469:
6463:
6462:
6444:
6438:
6437:
6411:
6405:
6404:
6386:
6380:
6379:
6354:(9): 2723–2725.
6339:
6333:
6332:
6300:
6294:
6293:
6265:
6259:
6258:
6237:
6231:
6230:
6204:
6180:
6174:
6173:
6156:(8): 2160–2168.
6141:
6135:
6134:
6124:
6097:
6091:
6090:
6076:
6067:
6066:
6048:
6042:
6041:
6023:
6017:
6016:
5998:
5992:
5991:
5974:(8): 2445–2459.
5956:
5930:of knot theory.
5922:to relate their
5909:
5907:
5906:
5901:
5821:
5819:
5818:
5813:
5791:
5789:
5788:
5783:
5775:
5774:
5733:
5731:
5730:
5725:
5713:
5711:
5710:
5705:
5656:
5654:
5653:
5648:
5640:
5623:
5611:
5609:
5608:
5603:
5576:
5574:
5573:
5568:
5563:
5562:
5547:
5546:
5514:unitary matrices
5508:
5506:
5505:
5500:
5492:
5491:
5479:
5478:
5456:
5448:
5436:
5435:
5414:
5413:
5395:
5394:
5369:
5367:
5366:
5361:
5350:
5347:
5338:
5336:
5335:
5330:
5301:
5300:
5277:
5275:
5274:
5269:
5264:
5263:
5251:
5250:
5229:
5228:
5210:
5209:
5187:
5185:
5184:
5179:
5168:
5165:
5153:
5151:
5150:
5145:
5134:
5133:
5109:
5108:
5096:
5095:
5067:
5065:
5064:
5059:
5051:
5050:
5032:
5031:
5013:
5012:
5000:
4999:
4987:
4986:
4965:
4964:
4946:
4945:
4933:
4932:
4914:
4913:
4901:
4900:
4885:
4884:
4866:
4865:
4853:
4852:
4840:
4839:
4812:
4811:
4787:
4786:
4774:
4773:
4733:
4731:
4730:
4725:
4699:
4698:
4682:
4680:
4679:
4674:
4672:
4671:
4651:
4649:
4648:
4643:
4622:
4620:
4619:
4614:
4596:
4594:
4593:
4588:
4579:
4578:
4577:
4561:
4557:
4556:
4532:
4531:
4530:
4520:
4519:
4518:
4503:
4502:
4501:
4491:
4490:
4489:
4462:
4460:
4459:
4454:
4436:
4434:
4433:
4428:
4416:
4414:
4413:
4408:
4403:
4402:
4390:
4389:
4368:
4367:
4355:
4354:
4329:
4327:
4326:
4321:
4319:
4318:
4302:
4300:
4299:
4294:
4292:
4291:
4275:
4273:
4272:
4267:
4265:
4264:
4252:
4251:
4228:
4226:
4225:
4220:
4194:
4192:
4191:
4186:
4175:
4172:
4159:
4157:
4156:
4151:
4139:
4137:
4136:
4131:
4119:
4117:
4116:
4111:
4099:
4097:
4096:
4091:
4089:
4088:
4069:
4067:
4066:
4061:
4059:
4058:
4043:
4042:
4014:
4012:
4011:
4006:
3995:
3994:
3967:
3966:
3930:
3929:
3914:
3913:
3901:
3900:
3888:
3887:
3857:
3855:
3845:
3844:
3825:
3822:
3817:
3791:
3789:
3788:
3783:
3757:
3756:
3755:
3754:
3728:
3726:
3725:
3724:
3708:
3690:
3685:
3666:
3664:
3663:
3658:
3655:
3650:
3634:
3632:
3631:
3626:
3614:
3612:
3611:
3606:
3604:
3603:
3584:
3582:
3581:
3576:
3564:
3562:
3561:
3556:
3544:
3542:
3541:
3536:
3534:
3533:
3532:
3531:
3504:
3502:
3501:
3496:
3470:
3469:
3468:
3467:
3438:
3436:
3426:
3425:
3406:
3401:
3399:
3389:
3388:
3372:
3355:
3343:
3341:
3340:
3335:
3314:
3312:
3311:
3306:
3304:
3303:
3288:
3287:
3272:
3271:
3250:plane with area
3249:
3247:
3246:
3241:
3229:
3227:
3226:
3221:
3209:
3207:
3206:
3201:
3189:
3187:
3186:
3181:
3179:
3178:
3177:
3176:
3149:
3147:
3146:
3141:
3103:zero temperature
3100:
3098:
3097:
3092:
3081:
3078:
3069:
3067:
3066:
3061:
3042:phase transition
3035:
3033:
3032:
3027:
3015:
3013:
3012:
3007:
3005:
3004:
2995:
2971:
2969:
2968:
2963:
2946:
2945:
2930:
2929:
2923:
2915:
2913:
2912:
2900:
2898:
2897:
2885:
2864:
2862:
2861:
2856:
2845:
2842:
2824:strongly coupled
2799:
2797:
2796:
2791:
2786:
2785:
2770:
2769:
2759:
2744:
2743:
2737:
2734:
2707:
2705:
2704:
2699:
2683:
2681:
2680:
2675:
2664:
2663:
2647:
2645:
2644:
2639:
2627:
2625:
2624:
2619:
2583:
2581:
2580:
2575:
2563:
2561:
2560:
2555:
2531:
2529:
2528:
2523:
2518:
2517:
2458:
2456:
2455:
2450:
2438:
2436:
2435:
2430:
2400:
2398:
2397:
2392:
2368:
2366:
2365:
2360:
2358:
2357:
2285:concave function
2275:
2273:
2272:
2267:
2256:
2255:
2234:
2233:
2218:
2217:
2162:
2160:
2159:
2154:
2120:
2118:
2117:
2112:
2076:
2074:
2073:
2068:
2056:
2054:
2053:
2048:
2036:
2034:
2033:
2028:
2000:
1998:
1997:
1992:
1978:
1961:
1960:
1944:
1942:
1941:
1936:
1934:
1933:
1913:
1911:
1910:
1905:
1864:
1862:
1861:
1856:
1844:
1842:
1841:
1836:
1831:
1826:
1825:
1809:
1807:
1806:
1801:
1799:
1798:
1740:
1738:
1737:
1732:
1727:
1726:
1720:
1719:
1713:
1712:
1700:
1699:
1690:
1689:
1677:
1676:
1664:
1663:
1657:
1656:
1650:
1647:
1586:
1584:
1583:
1578:
1573:
1572:
1554:
1553:
1541:
1540:
1525:
1524:
1515:
1514:
1488:
1486:
1485:
1480:
1456:
1454:
1453:
1448:
1440:
1439:
1427:
1426:
1411:
1410:
1398:
1397:
1379:
1378:
1357:
1356:
1344:
1343:
1318:
1316:
1315:
1310:
1283:
1281:
1280:
1275:
1257:
1255:
1254:
1249:
1235:
1234:
1226:
1211:
1207:
1206:
1198:
1187:
1185:
1184:
1179:
1165:
1164:
1156:
1141:
1137:
1136:
1128:
1108:
1106:
1105:
1100:
1098:
1097:
1081:
1079:
1078:
1073:
1068:
1067:
1061:
1060:
1048:
1047:
1037:
1036:
1035:
1025:
1024:
1023:
1006:
1005:
993:
992:
980:
979:
967:
966:
945:
944:
932:
931:
905:
903:
902:
897:
880:
878:
870:
869:
868:
855:
844:
843:
831:
829:
821:
804:
785:
783:
782:
777:
775:
774:
755:
750:
729:
728:
708:
706:
705:
700:
698:
697:
676:
675:
667:
657:
655:
654:
649:
638:
637:
629:
619:
617:
616:
611:
586:
584:
583:
578:
576:
575:
559:
557:
556:
551:
549:
548:
532:
530:
529:
524:
488:
486:
485:
480:
478:
477:
461:
459:
458:
453:
445:
444:
428:
426:
425:
420:
418:
417:
397:
395:
394:
389:
354:
352:
351:
346:
338:
337:
332:
311:
309:
308:
303:
291:
289:
288:
283:
268:
266:
265:
260:
240:
238:
237:
232:
230:
229:
213:
211:
210:
205:
203:
202:
186:
184:
183:
178:
176:
175:
160:along the fiber
159:
157:
156:
151:
149:
148:
132:
130:
129:
124:
112:
110:
109:
104:
7629:
7628:
7624:
7623:
7622:
7620:
7619:
7618:
7589:
7588:
7587:
7586:
7541:
7540:
7536:
7529:
7513:
7512:
7508:
7501:
7488:
7487:
7483:
7476:
7460:
7459:
7455:
7425:
7424:
7420:
7413:
7397:
7396:
7392:
7341:
7340:
7336:
7329:
7313:
7312:
7308:
7259:Maldacena, J.M.
7253:
7252:
7248:
7197:
7196:
7192:
7154:
7153:
7149:
7115:
7114:
7110:
7073:
7072:
7068:
7034:
7033:
7029:
7022:
7009:
7008:
7004:
6967:
6966:
6962:
6928:
6927:
6923:
6916:
6900:
6899:
6895:
6861:
6860:
6856:
6831:
6830:
6826:
6789:
6788:
6784:
6743:
6742:
6738:
6711:Physics Reports
6701:Drouffe, J.M.;
6700:
6699:
6695:
6688:
6667:
6666:
6662:
6655:
6639:
6638:
6634:
6602:hep-lat/0510074
6586:
6585:
6581:
6574:
6545:
6544:
6540:
6525:
6504:
6503:
6499:
6492:
6471:
6470:
6466:
6459:
6446:
6445:
6441:
6434:
6413:
6412:
6408:
6401:
6388:
6387:
6383:
6341:
6340:
6336:
6302:
6301:
6297:
6290:
6267:
6266:
6262:
6255:
6239:
6238:
6234:
6182:
6181:
6177:
6143:
6142:
6138:
6099:
6098:
6094:
6078:
6077:
6070:
6063:
6050:
6049:
6045:
6038:
6025:
6024:
6020:
6013:
6000:
5999:
5995:
5958:
5957:
5953:
5948:
5936:
5886:
5885:
5866:
5833:
5804:
5803:
5764:
5763:
5752:
5747:
5716:
5715:
5687:
5686:
5671:renormalization
5663:
5661:Renormalization
5614:
5613:
5579:
5578:
5551:
5538:
5518:
5517:
5483:
5470:
5427:
5405:
5386:
5375:
5374:
5341:
5340:
5292:
5287:
5286:
5255:
5242:
5220:
5201:
5190:
5189:
5159:
5158:
5119:
5100:
5081:
5076:
5075:
5036:
5023:
5004:
4991:
4978:
4956:
4937:
4918:
4905:
4892:
4876:
4857:
4844:
4825:
4797:
4778:
4759:
4739:
4738:
4690:
4685:
4684:
4663:
4658:
4657:
4654:delta functions
4628:
4627:
4599:
4598:
4563:
4542:
4522:
4510:
4493:
4481:
4468:
4467:
4439:
4438:
4419:
4418:
4394:
4381:
4359:
4346:
4335:
4334:
4310:
4305:
4304:
4283:
4278:
4277:
4256:
4243:
4232:
4231:
4205:
4204:
4201:
4166:
4165:
4142:
4141:
4122:
4121:
4102:
4101:
4077:
4072:
4071:
4047:
4031:
4020:
4019:
3983:
3955:
3915:
3905:
3892:
3879:
3833:
3829:
3804:
3803:
3746:
3738:
3716:
3712:
3672:
3671:
3637:
3636:
3617:
3616:
3595:
3587:
3586:
3567:
3566:
3547:
3546:
3523:
3515:
3510:
3509:
3456:
3448:
3414:
3410:
3377:
3373:
3356:
3349:
3348:
3317:
3316:
3295:
3279:
3260:
3252:
3251:
3232:
3231:
3212:
3211:
3192:
3191:
3165:
3157:
3152:
3151:
3132:
3131:
3116:
3111:
3072:
3071:
3046:
3045:
3038:electrodynamics
3018:
3017:
2996:
2977:
2976:
2889:
2870:
2869:
2836:
2835:
2813:glueball states
2761:
2713:
2712:
2690:
2689:
2655:
2650:
2649:
2630:
2629:
2610:
2609:
2598:
2586:Schwinger model
2566:
2565:
2537:
2536:
2494:
2468:
2467:
2441:
2440:
2403:
2402:
2374:
2373:
2334:
2308:
2307:
2238:
2194:
2168:
2167:
2127:
2126:
2094:
2093:
2089:, which is the
2059:
2058:
2039:
2038:
2019:
2018:
2015:
1952:
1947:
1946:
1925:
1920:
1919:
1887:
1886:
1875:
1847:
1846:
1817:
1812:
1811:
1790:
1785:
1784:
1777:representations
1742:
1704:
1691:
1681:
1626:
1625:
1601:quantum gravity
1564:
1545:
1532:
1516:
1506:
1495:
1494:
1462:
1461:
1431:
1415:
1402:
1389:
1370:
1348:
1335:
1324:
1323:
1292:
1291:
1260:
1259:
1195:
1190:
1189:
1125:
1120:
1119:
1087:
1086:
1052:
1039:
1027:
1015:
971:
958:
936:
923:
918:
917:
871:
860:
856:
835:
822:
805:
795:
794:
766:
720:
715:
714:
689:
660:
659:
622:
621:
620:, is the curve
593:
592:
589:horizontal lift
567:
562:
561:
540:
535:
534:
491:
490:
469:
464:
463:
436:
431:
430:
409:
404:
403:
380:
379:
364:tangent vectors
362:which compares
327:
322:
321:
294:
293:
274:
273:
251:
250:
221:
216:
215:
194:
189:
188:
167:
162:
161:
140:
135:
134:
115:
114:
113:with spacetime
95:
94:
87:
55:order operators
35:gauge invariant
23:
17:
12:
11:
5:
7627:
7625:
7617:
7616:
7611:
7606:
7601:
7599:Gauge theories
7591:
7590:
7585:
7584:
7557:(3): 351–399.
7534:
7528:978-0691149080
7527:
7517:(2021). "22".
7506:
7500:978-3030540043
7499:
7493:(2 ed.).
7481:
7475:978-1551439761
7474:
7464:(1998). "16".
7462:Polchinski, J.
7453:
7442:(1–2): 25–32.
7418:
7412:978-0143113799
7411:
7399:Polchinski, J.
7390:
7343:Caron-Huot, S.
7334:
7328:978-3642540219
7327:
7306:
7246:
7209:(5): 153–211.
7190:
7147:
7128:(4): 879–902.
7108:
7075:Polyakov, A.M.
7066:
7047:(1): 169–175.
7027:
7021:978-0521654753
7020:
7002:
6969:Mandelstam, S.
6960:
6941:(1): 199–215.
6921:
6915:978-1108493994
6914:
6904:(2019). "50".
6893:
6854:
6843:(4): 199–290.
6824:
6782:
6736:
6693:
6687:978-3642018497
6686:
6660:
6654:978-3540332091
6653:
6641:Yndurain, F.J.
6632:
6579:
6573:978-9812567277
6572:
6538:
6523:
6497:
6491:978-0198788409
6490:
6480:(2017). "25".
6474:Iliopoulos, J.
6464:
6458:978-1107155473
6457:
6439:
6433:978-0521809115
6432:
6406:
6400:978-3030515621
6399:
6381:
6334:
6315:(2): 482–483.
6295:
6289:978-9814365857
6288:
6260:
6253:
6232:
6175:
6136:
6092:
6068:
6061:
6043:
6036:
6018:
6012:978-0750306065
6011:
5993:
5950:
5949:
5947:
5944:
5943:
5942:
5940:Winding number
5935:
5932:
5899:
5896:
5893:
5865:
5862:
5832:
5829:
5825:supersymmetric
5811:
5781:
5778:
5773:
5751:
5748:
5746:
5743:
5742:
5741:
5738:
5735:
5723:
5703:
5700:
5697:
5694:
5679:
5667:regularization
5662:
5659:
5646:
5643:
5639:
5635:
5632:
5629:
5626:
5622:
5601:
5598:
5595:
5592:
5589:
5586:
5566:
5561:
5558:
5554:
5550:
5545:
5541:
5537:
5534:
5531:
5528:
5525:
5510:
5509:
5498:
5495:
5490:
5486:
5482:
5477:
5473:
5469:
5466:
5463:
5460:
5455:
5452:
5447:
5443:
5439:
5434:
5430:
5426:
5423:
5420:
5417:
5412:
5408:
5404:
5401:
5398:
5393:
5389:
5385:
5382:
5359:
5356:
5353:
5328:
5325:
5322:
5319:
5316:
5313:
5310:
5307:
5304:
5299:
5295:
5267:
5262:
5258:
5254:
5249:
5245:
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5238:
5235:
5232:
5227:
5223:
5219:
5216:
5213:
5208:
5204:
5200:
5197:
5177:
5174:
5171:
5155:
5154:
5143:
5140:
5137:
5132:
5129:
5126:
5122:
5118:
5115:
5112:
5107:
5103:
5099:
5094:
5091:
5088:
5084:
5069:
5068:
5057:
5054:
5049:
5046:
5043:
5039:
5035:
5030:
5026:
5022:
5019:
5016:
5011:
5007:
5003:
4998:
4994:
4990:
4985:
4981:
4977:
4974:
4971:
4968:
4963:
4959:
4955:
4952:
4949:
4944:
4940:
4936:
4931:
4928:
4925:
4921:
4917:
4912:
4908:
4904:
4899:
4895:
4891:
4888:
4883:
4879:
4875:
4872:
4869:
4864:
4860:
4856:
4851:
4847:
4843:
4838:
4835:
4832:
4828:
4824:
4821:
4818:
4815:
4810:
4807:
4804:
4800:
4796:
4793:
4790:
4785:
4781:
4777:
4772:
4769:
4766:
4762:
4758:
4755:
4752:
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4723:
4720:
4717:
4714:
4711:
4708:
4705:
4702:
4697:
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4670:
4666:
4641:
4638:
4635:
4612:
4609:
4606:
4586:
4583:
4576:
4573:
4570:
4566:
4560:
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4552:
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4540:
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4529:
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4508:
4500:
4496:
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4484:
4480:
4476:
4452:
4449:
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4426:
4406:
4401:
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4388:
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4380:
4377:
4374:
4371:
4366:
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4358:
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4349:
4345:
4342:
4317:
4313:
4290:
4286:
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4255:
4250:
4246:
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4239:
4218:
4215:
4212:
4200:
4197:
4184:
4181:
4178:
4149:
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4109:
4087:
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4080:
4057:
4054:
4050:
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4038:
4034:
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4027:
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3993:
3990:
3986:
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3970:
3965:
3962:
3958:
3954:
3951:
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3936:
3933:
3928:
3925:
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3918:
3912:
3908:
3904:
3899:
3895:
3891:
3886:
3882:
3878:
3875:
3872:
3869:
3866:
3863:
3860:
3854:
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3848:
3843:
3840:
3836:
3832:
3828:
3821:
3816:
3812:
3793:
3792:
3781:
3778:
3775:
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3769:
3766:
3763:
3760:
3753:
3749:
3745:
3741:
3737:
3734:
3731:
3723:
3719:
3715:
3711:
3706:
3703:
3700:
3697:
3694:
3689:
3684:
3680:
3654:
3649:
3645:
3624:
3602:
3598:
3594:
3574:
3554:
3530:
3526:
3522:
3518:
3506:
3505:
3494:
3491:
3488:
3485:
3482:
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3476:
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3459:
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3451:
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3444:
3441:
3435:
3432:
3429:
3424:
3421:
3417:
3413:
3409:
3404:
3398:
3395:
3392:
3387:
3384:
3380:
3376:
3371:
3368:
3365:
3362:
3359:
3333:
3330:
3327:
3324:
3302:
3298:
3294:
3291:
3286:
3282:
3278:
3275:
3270:
3267:
3263:
3259:
3239:
3219:
3199:
3175:
3172:
3168:
3164:
3160:
3139:
3122:which acts on
3115:
3112:
3110:
3107:
3090:
3087:
3084:
3059:
3056:
3053:
3025:
3003:
2999:
2994:
2990:
2987:
2984:
2973:
2972:
2961:
2958:
2955:
2952:
2949:
2944:
2939:
2936:
2933:
2928:
2921:
2918:
2911:
2906:
2903:
2896:
2892:
2888:
2883:
2880:
2877:
2854:
2851:
2848:
2801:
2800:
2789:
2784:
2779:
2776:
2773:
2768:
2764:
2758:
2755:
2752:
2748:
2742:
2732:
2729:
2726:
2723:
2720:
2697:
2673:
2670:
2667:
2662:
2658:
2637:
2617:
2597:
2594:
2573:
2553:
2550:
2547:
2544:
2533:
2532:
2521:
2516:
2513:
2510:
2507:
2504:
2501:
2497:
2493:
2490:
2487:
2484:
2481:
2478:
2475:
2448:
2428:
2425:
2422:
2419:
2416:
2413:
2410:
2390:
2387:
2384:
2381:
2370:
2369:
2356:
2353:
2350:
2347:
2344:
2341:
2337:
2333:
2330:
2327:
2324:
2321:
2318:
2315:
2277:
2276:
2265:
2262:
2259:
2254:
2251:
2248:
2245:
2241:
2237:
2232:
2227:
2224:
2221:
2216:
2213:
2210:
2207:
2204:
2201:
2197:
2193:
2190:
2187:
2184:
2181:
2178:
2175:
2152:
2149:
2146:
2143:
2140:
2137:
2134:
2123:excited states
2110:
2107:
2104:
2101:
2087:minimum energy
2066:
2046:
2026:
2014:
2013:Order operator
2011:
2003:Stokes theorem
1990:
1987:
1984:
1981:
1977:
1973:
1970:
1967:
1964:
1959:
1955:
1932:
1928:
1903:
1900:
1897:
1894:
1874:
1871:
1854:
1834:
1830:
1824:
1820:
1797:
1793:
1781:weight lattice
1730:
1725:
1718:
1711:
1707:
1703:
1698:
1694:
1688:
1684:
1680:
1675:
1670:
1667:
1662:
1655:
1645:
1642:
1639:
1636:
1633:
1623:
1576:
1571:
1567:
1563:
1560:
1557:
1552:
1548:
1544:
1539:
1535:
1531:
1528:
1523:
1519:
1513:
1509:
1505:
1502:
1478:
1475:
1472:
1469:
1458:
1457:
1446:
1443:
1438:
1434:
1430:
1425:
1422:
1418:
1414:
1409:
1405:
1401:
1396:
1392:
1388:
1385:
1382:
1377:
1373:
1369:
1366:
1363:
1360:
1355:
1351:
1347:
1342:
1338:
1334:
1331:
1308:
1305:
1302:
1299:
1273:
1270:
1267:
1247:
1244:
1241:
1238:
1232:
1229:
1223:
1220:
1217:
1214:
1210:
1204:
1201:
1177:
1174:
1171:
1168:
1162:
1159:
1153:
1150:
1147:
1144:
1140:
1134:
1131:
1096:
1083:
1082:
1071:
1066:
1059:
1055:
1051:
1046:
1042:
1034:
1030:
1022:
1018:
1013:
1009:
1004:
999:
996:
991:
986:
983:
978:
974:
970:
965:
961:
957:
954:
951:
948:
943:
939:
935:
930:
926:
907:
906:
895:
892:
889:
886:
883:
877:
874:
867:
863:
859:
853:
850:
847:
842:
838:
834:
828:
825:
820:
817:
814:
811:
808:
802:
773:
769:
765:
762:
759:
754:
749:
745:
741:
738:
735:
732:
727:
723:
696:
692:
688:
685:
682:
679:
673:
670:
647:
644:
641:
635:
632:
609:
606:
603:
600:
574:
570:
547:
543:
522:
519:
516:
513:
510:
507:
504:
501:
498:
476:
472:
451:
448:
443:
439:
416:
412:
387:
368:tangent spaces
344:
341:
336:
331:
301:
281:
258:
228:
224:
201:
197:
174:
170:
147:
143:
122:
102:
86:
83:
79:Polyakov loops
75:'t Hooft loops
51:gauge theories
15:
13:
10:
9:
6:
4:
3:
2:
7626:
7615:
7612:
7610:
7607:
7605:
7602:
7600:
7597:
7596:
7594:
7580:
7576:
7572:
7568:
7564:
7560:
7556:
7552:
7548:
7544:
7538:
7535:
7530:
7524:
7520:
7516:
7510:
7507:
7502:
7496:
7492:
7485:
7482:
7477:
7471:
7467:
7463:
7457:
7454:
7449:
7445:
7441:
7437:
7436:Phys. Lett. B
7433:
7429:
7422:
7419:
7414:
7408:
7404:
7401:(1998). "8".
7400:
7394:
7391:
7386:
7382:
7378:
7374:
7370:
7366:
7361:
7356:
7352:
7348:
7344:
7338:
7335:
7330:
7324:
7320:
7317:(2014). "4".
7316:
7310:
7307:
7302:
7298:
7294:
7290:
7286:
7282:
7277:
7272:
7268:
7264:
7260:
7256:
7250:
7247:
7242:
7238:
7234:
7230:
7226:
7222:
7217:
7212:
7208:
7204:
7200:
7194:
7191:
7186:
7182:
7178:
7174:
7170:
7166:
7162:
7158:
7151:
7148:
7143:
7139:
7135:
7131:
7127:
7123:
7119:
7112:
7109:
7104:
7100:
7096:
7092:
7088:
7084:
7080:
7076:
7070:
7067:
7062:
7058:
7054:
7050:
7046:
7042:
7038:
7031:
7028:
7023:
7017:
7013:
7006:
7003:
6998:
6994:
6990:
6986:
6982:
6978:
6974:
6970:
6964:
6961:
6956:
6952:
6948:
6944:
6940:
6936:
6932:
6925:
6922:
6917:
6911:
6907:
6903:
6897:
6894:
6889:
6885:
6881:
6877:
6873:
6869:
6868:Phys. Lett. B
6865:
6858:
6855:
6850:
6846:
6842:
6838:
6834:
6828:
6825:
6820:
6816:
6812:
6808:
6804:
6800:
6796:
6792:
6786:
6783:
6778:
6774:
6770:
6766:
6762:
6758:
6757:Phys. Lett. B
6754:
6750:
6749:Nauenberg, M.
6746:
6745:Lautrup, B.E.
6740:
6737:
6732:
6728:
6724:
6720:
6716:
6712:
6708:
6704:
6697:
6694:
6689:
6683:
6679:
6675:
6671:
6664:
6661:
6656:
6650:
6646:
6643:(2006). "9".
6642:
6636:
6633:
6628:
6624:
6620:
6616:
6612:
6608:
6603:
6598:
6595:(1): 014516.
6594:
6590:
6583:
6580:
6575:
6569:
6565:
6561:
6557:
6553:
6549:
6542:
6539:
6534:
6530:
6526:
6524:9780511470783
6520:
6516:
6512:
6508:
6501:
6498:
6493:
6487:
6483:
6479:
6475:
6472:Baulieu, L.;
6468:
6465:
6460:
6454:
6450:
6443:
6440:
6435:
6429:
6425:
6421:
6417:
6410:
6407:
6402:
6396:
6392:
6385:
6382:
6377:
6373:
6369:
6365:
6361:
6357:
6353:
6349:
6345:
6338:
6335:
6330:
6326:
6322:
6318:
6314:
6310:
6306:
6299:
6296:
6291:
6285:
6281:
6277:
6273:
6272:
6264:
6261:
6256:
6254:9780201503975
6250:
6246:
6242:
6236:
6233:
6228:
6224:
6220:
6216:
6212:
6208:
6203:
6198:
6194:
6190:
6186:
6179:
6176:
6171:
6167:
6163:
6159:
6155:
6151:
6147:
6140:
6137:
6132:
6128:
6123:
6118:
6114:
6110:
6106:
6102:
6096:
6093:
6089:
6085:
6081:
6075:
6073:
6069:
6064:
6062:9781107034730
6058:
6054:
6047:
6044:
6039:
6033:
6029:
6022:
6019:
6014:
6008:
6004:
5997:
5994:
5989:
5985:
5981:
5977:
5973:
5969:
5965:
5961:
5955:
5952:
5945:
5941:
5938:
5937:
5933:
5931:
5929:
5925:
5921:
5917:
5916:Edward Witten
5913:
5897:
5894:
5891:
5883:
5879:
5875:
5871:
5863:
5861:
5858:
5854:
5850:
5846:
5845:string theory
5843:
5838:
5830:
5828:
5826:
5809:
5801:
5797:
5794:
5779:
5776:
5761:
5757:
5749:
5744:
5739:
5736:
5721:
5698:
5692:
5684:
5680:
5677:
5676:
5675:
5672:
5668:
5660:
5658:
5644:
5641:
5630:
5624:
5599:
5596:
5590:
5584:
5559:
5556:
5552:
5543:
5539:
5535:
5529:
5523:
5515:
5496:
5488:
5484:
5480:
5475:
5471:
5464:
5461:
5453:
5450:
5445:
5441:
5437:
5432:
5428:
5421:
5418:
5410:
5406:
5399:
5391:
5387:
5380:
5373:
5372:
5371:
5354:
5326:
5323:
5317:
5314:
5311:
5308:
5305:
5297:
5293:
5284:
5279:
5260:
5256:
5252:
5247:
5243:
5236:
5233:
5225:
5221:
5214:
5206:
5202:
5195:
5172:
5141:
5138:
5130:
5127:
5124:
5120:
5116:
5113:
5110:
5105:
5101:
5092:
5089:
5086:
5082:
5074:
5073:
5072:
5055:
5047:
5044:
5041:
5037:
5033:
5028:
5024:
5020:
5017:
5014:
5009:
5005:
5001:
4996:
4992:
4983:
4979:
4975:
4972:
4969:
4961:
4957:
4953:
4950:
4947:
4942:
4938:
4934:
4929:
4926:
4923:
4919:
4915:
4910:
4906:
4897:
4893:
4889:
4881:
4877:
4873:
4870:
4867:
4862:
4858:
4849:
4845:
4836:
4833:
4830:
4826:
4819:
4816:
4808:
4805:
4802:
4798:
4794:
4791:
4788:
4783:
4779:
4770:
4767:
4764:
4760:
4753:
4750:
4747:
4737:
4736:
4735:
4718:
4712:
4709:
4703:
4695:
4691:
4668:
4664:
4655:
4639:
4636:
4633:
4625:
4610:
4607:
4604:
4584:
4581:
4574:
4571:
4568:
4564:
4553:
4550:
4547:
4543:
4538:
4534:
4527:
4523:
4515:
4511:
4506:
4498:
4494:
4486:
4482:
4474:
4465:
4450:
4447:
4444:
4424:
4399:
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4391:
4386:
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4372:
4364:
4360:
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4351:
4347:
4340:
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4315:
4311:
4288:
4284:
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4257:
4253:
4248:
4244:
4240:
4237:
4216:
4213:
4210:
4198:
4196:
4163:
4147:
4127:
4107:
4085:
4082:
4078:
4055:
4052:
4048:
4044:
4039:
4036:
4032:
4028:
4025:
4002:
3991:
3988:
3984:
3977:
3963:
3960:
3956:
3949:
3940:
3937:
3934:
3923:
3916:
3910:
3906:
3902:
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3893:
3889:
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3876:
3867:
3861:
3849:
3841:
3838:
3834:
3830:
3826:
3819:
3814:
3802:
3801:
3800:
3798:
3797:large N-limit
3779:
3770:
3764:
3761:
3751:
3747:
3743:
3739:
3732:
3721:
3717:
3713:
3709:
3704:
3698:
3692:
3687:
3682:
3670:
3669:
3668:
3652:
3647:
3622:
3600:
3596:
3592:
3572:
3552:
3528:
3524:
3520:
3516:
3492:
3483:
3477:
3474:
3464:
3461:
3457:
3453:
3449:
3442:
3430:
3422:
3419:
3415:
3411:
3407:
3402:
3393:
3385:
3382:
3378:
3374:
3366:
3360:
3357:
3347:
3346:
3345:
3328:
3322:
3300:
3296:
3292:
3289:
3284:
3280:
3276:
3273:
3268:
3265:
3261:
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3237:
3217:
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3173:
3170:
3166:
3162:
3158:
3137:
3129:
3125:
3121:
3113:
3108:
3106:
3104:
3085:
3057:
3054:
3051:
3043:
3039:
3023:
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2997:
2992:
2988:
2985:
2982:
2959:
2950:
2937:
2934:
2919:
2916:
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2901:
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2890:
2886:
2881:
2878:
2875:
2868:
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2866:
2849:
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2829:
2825:
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2818:
2814:
2810:
2806:
2787:
2774:
2766:
2762:
2756:
2753:
2750:
2746:
2730:
2724:
2718:
2711:
2710:
2709:
2708:, denoted by
2695:
2687:
2686:Wilson action
2668:
2660:
2656:
2635:
2628:going in the
2615:
2607:
2603:
2595:
2593:
2591:
2587:
2571:
2548:
2542:
2519:
2511:
2505:
2502:
2499:
2495:
2491:
2482:
2476:
2466:
2465:
2464:
2462:
2446:
2426:
2423:
2420:
2414:
2408:
2385:
2379:
2351:
2345:
2342:
2339:
2335:
2331:
2322:
2316:
2306:
2305:
2304:
2302:
2298:
2293:
2289:
2286:
2282:
2263:
2252:
2246:
2243:
2239:
2225:
2222:
2211:
2205:
2202:
2199:
2195:
2191:
2182:
2176:
2166:
2165:
2164:
2150:
2144:
2138:
2132:
2124:
2105:
2099:
2092:
2088:
2084:
2080:
2064:
2044:
2024:
2012:
2010:
2008:
2007:magnetic flux
2004:
1988:
1985:
1979:
1968:
1962:
1957:
1953:
1930:
1926:
1917:
1916:electric flux
1898:
1892:
1884:
1880:
1872:
1870:
1868:
1852:
1832:
1828:
1822:
1795:
1782:
1778:
1774:
1769:
1767:
1764:
1759:
1755:
1751:
1747:
1741:
1728:
1709:
1705:
1701:
1696:
1692:
1686:
1682:
1678:
1668:
1665:
1643:
1637:
1631:
1622:
1621:
1617:
1612:
1610:
1606:
1602:
1598:
1594:
1593:Hilbert space
1590:
1589:test particle
1569:
1565:
1558:
1550:
1546:
1542:
1537:
1533:
1526:
1521:
1511:
1507:
1500:
1492:
1473:
1467:
1444:
1436:
1432:
1423:
1420:
1416:
1407:
1403:
1399:
1394:
1390:
1383:
1375:
1371:
1364:
1353:
1349:
1345:
1340:
1336:
1329:
1322:
1321:
1320:
1303:
1297:
1290:
1285:
1271:
1268:
1265:
1245:
1239:
1227:
1221:
1215:
1208:
1199:
1175:
1169:
1157:
1151:
1145:
1138:
1129:
1116:
1112:
1069:
1057:
1053:
1049:
1044:
1040:
1032:
1028:
1020:
1016:
1011:
1007:
997:
994:
984:
976:
972:
968:
963:
959:
952:
949:
941:
937:
928:
924:
916:
915:
914:
912:
893:
887:
881:
875:
872:
865:
861:
857:
848:
840:
836:
832:
826:
823:
815:
809:
806:
800:
793:
792:
791:
789:
771:
767:
760:
752:
747:
743:
739:
733:
725:
721:
712:
694:
690:
686:
680:
668:
642:
630:
604:
598:
590:
572:
568:
545:
541:
520:
511:
508:
505:
499:
496:
474:
470:
449:
446:
441:
437:
414:
410:
400:
385:
377:
373:
369:
365:
361:
356:
342:
339:
334:
319:
315:
299:
279:
272:
269:-dimensional
256:
248:
226:
222:
199:
195:
172:
168:
145:
141:
120:
100:
91:
84:
82:
80:
76:
72:
68:
64:
60:
56:
52:
48:
44:
40:
36:
32:
28:
22:
7554:
7550:
7537:
7518:
7509:
7490:
7484:
7465:
7456:
7439:
7435:
7421:
7402:
7393:
7350:
7346:
7337:
7318:
7309:
7266:
7262:
7249:
7206:
7202:
7193:
7168:
7164:
7150:
7125:
7122:Phys. Rev. D
7121:
7111:
7086:
7082:
7069:
7044:
7040:
7030:
7011:
7005:
6980:
6976:
6963:
6938:
6934:
6924:
6905:
6896:
6871:
6867:
6864:Migdal, A.A.
6857:
6840:
6836:
6833:Migdal, A.A.
6827:
6802:
6799:Phys. Rev. D
6798:
6785:
6763:(1): 63–66.
6760:
6756:
6739:
6717:(1): 1–119.
6714:
6710:
6696:
6669:
6663:
6644:
6635:
6592:
6589:Phys. Rev. D
6588:
6582:
6564:10.1142/6065
6547:
6541:
6506:
6500:
6481:
6467:
6448:
6442:
6415:
6409:
6390:
6384:
6351:
6348:Phys. Rev. D
6347:
6337:
6312:
6309:Phys. Rev. D
6308:
6298:
6280:10.1142/8229
6270:
6263:
6244:
6235:
6192:
6188:
6178:
6153:
6150:Phys. Rev. D
6149:
6139:
6112:
6109:Phys. Rev. D
6108:
6095:
6087:
6052:
6046:
6027:
6021:
6002:
5996:
5971:
5968:Phys. Rev. D
5967:
5960:Wilson, K.G.
5954:
5867:
5837:compactified
5834:
5753:
5664:
5511:
5280:
5156:
5070:
4332:
4202:
4017:
3794:
3507:
3117:
2974:
2832:power series
2821:
2802:
2599:
2534:
2371:
2290:
2278:
2125:with energy
2016:
1876:
1770:
1763:overcomplete
1743:
1624:
1619:
1613:
1459:
1286:
1084:
910:
908:
401:
357:
314:fibre bundle
244:
31:Wilson loops
30:
24:
7515:Fradkin, E.
7428:Quevedo, F.
7255:Alday, L.F.
7199:Alday, L.F.
7171:: 342–364.
7089:: 171–188.
6902:Năstase, H.
6703:Zuber, J.B.
6185:Seiberg, N.
6105:Seiberg, N.
5912:knot theory
5878:observables
5283:determinant
3128:derivatives
2805:numerically
2461:Higgs phase
1620:Wilson loop
1609:black holes
911:Wilson line
711:Lie-algebra
59:confinement
7593:Categories
7543:Witten, E.
7315:Henn, J.M.
6791:Guth, A.H.
6478:Sénéor, R.
6195:(8): 115.
6183:Ofer, A.;
6115:: 084019.
5946:References
5800:helicities
3109:Properties
2590:instantons
1867:Weyl group
658:such that
372:connection
85:Definition
7385:118676335
7360:1010.1167
7353:(7): 58.
7276:0705.0303
7269:(6): 64.
7241:119220578
7216:0807.1889
7203:Phys. Rep
6977:Phys. Rev
6837:Phys. Rep
6533:118339104
6227:118572353
6202:1305.0318
6122:1011.5120
6101:Banks, T.
5849:T-duality
5722:ϕ
5699:ϕ
5642:≤
5631:γ
5557:−
5553:γ
5544:∗
5530:γ
5485:γ
5481:∘
5472:γ
5451:−
5442:γ
5438:∘
5429:γ
5407:γ
5388:γ
5318:γ
5312:…
5306:γ
5257:γ
5253:∘
5244:γ
5222:γ
5203:γ
5121:γ
5114:…
5102:γ
5038:γ
5034:∘
5025:γ
5018:…
5006:γ
4993:γ
4976:−
4973:⋯
4970:−
4958:γ
4951:…
4939:γ
4920:γ
4916:∘
4907:γ
4890:−
4878:γ
4871:…
4859:γ
4827:γ
4799:γ
4792:…
4780:γ
4719:γ
4704:γ
4608:×
4539:δ
4535:⋯
4507:δ
4475:δ
4396:γ
4392:∘
4383:γ
4361:γ
4357:∘
4348:γ
4312:γ
4285:γ
4258:γ
4254:∘
4245:γ
4238:γ
4214:×
4180:∞
4079:γ
4049:γ
4045:∪
4033:γ
4026:γ
4000:⟩
3985:γ
3975:⟨
3972:⟩
3957:γ
3947:⟨
3938:−
3917:δ
3911:ν
3898:γ
3894:∮
3874:⟩
3868:γ
3859:⟨
3842:ν
3839:μ
3835:σ
3831:δ
3827:δ
3815:μ
3811:∂
3771:γ
3762:−
3752:μ
3744:δ
3740:γ
3722:μ
3714:δ
3699:γ
3683:μ
3679:∂
3648:μ
3644:∂
3623:μ
3601:μ
3593:δ
3553:γ
3529:μ
3521:δ
3517:γ
3484:γ
3475:−
3465:ν
3462:μ
3458:σ
3454:δ
3450:γ
3423:ν
3420:μ
3416:σ
3412:δ
3386:ν
3383:μ
3379:σ
3375:δ
3367:γ
3358:δ
3329:γ
3301:ν
3290:∧
3285:μ
3269:ν
3266:μ
3262:σ
3258:δ
3238:ν
3218:μ
3174:ν
3171:μ
3167:σ
3163:δ
3159:γ
3138:γ
3055:∼
3052:β
2983:β
2951:β
2917:β
2905:
2882:−
2876:σ
2767:μ
2757:γ
2754:∈
2747:∏
2696:γ
2661:μ
2636:μ
2549:γ
2512:γ
2500:−
2492:∼
2489:⟩
2483:γ
2474:⟨
2447:σ
2424:σ
2421:∼
2386:γ
2352:γ
2340:−
2332:∼
2329:⟩
2323:γ
2314:⟨
2250:Δ
2244:−
2200:−
2192:∼
2189:⟩
2183:γ
2174:⟨
2148:Δ
2091:potential
2025:γ
1986:≠
1983:⟩
1969:γ
1899:γ
1883:spacelike
1819:Λ
1792:Λ
1775:with the
1710:μ
1697:μ
1687:γ
1683:∮
1669:
1638:γ
1559:ϕ
1522:†
1501:ϕ
1468:ϕ
1421:−
1362:→
1269:≥
1231:~
1228:γ
1203:~
1200:γ
1161:~
1158:γ
1133:~
1130:γ
1058:μ
1045:μ
1012:∫
998:
866:μ
841:μ
748:μ
726:μ
672:~
669:γ
634:~
631:γ
599:γ
518:→
497:γ
340:×
271:spacetime
71:operators
7579:14951363
7545:(1989).
7430:(1986).
7301:10711473
7159:(1987).
7077:(1980).
6971:(1968).
6793:(1980).
6751:(1980).
6705:(1983).
6627:15741174
6082:(2018),
6080:Tong, D.
5962:(1974).
5934:See also
5882:manifold
5857:orbifold
5853:D-branes
5516:satisfy
4195:theory.
1758:subgroup
1746:holonomy
1287:Under a
1258:for all
1209:′
1139:′
533:between
376:vertical
7559:Bibcode
7365:Bibcode
7281:Bibcode
7221:Bibcode
7173:Bibcode
7130:Bibcode
7091:Bibcode
7049:Bibcode
6985:Bibcode
6943:Bibcode
6876:Bibcode
6807:Bibcode
6765:Bibcode
6719:Bibcode
6607:Bibcode
6552:Bibcode
6376:9956963
6356:Bibcode
6317:Bibcode
6207:Bibcode
6158:Bibcode
5976:Bibcode
3795:In the
3615:in the
3210:in the
2606:lattice
1865:is the
1750:mapping
1115:abelian
1109:is the
7577:
7525:
7497:
7472:
7409:
7383:
7325:
7299:
7239:
7018:
6912:
6684:
6651:
6625:
6570:
6531:
6521:
6488:
6455:
6430:
6397:
6374:
6286:
6251:
6225:
6059:
6034:
6009:
5874:metric
5370:gives
2975:where
2535:where
2439:where
1845:where
1085:where
7575:S2CID
7381:S2CID
7355:arXiv
7297:S2CID
7271:arXiv
7237:S2CID
7211:arXiv
6623:S2CID
6597:arXiv
6529:S2CID
6223:S2CID
6197:arXiv
6117:arXiv
5884:. In
5868:In a
5683:cusps
4070:with
4018:Here
2301:phase
2079:quark
1766:basis
1754:group
1616:trace
1188:then
43:loops
7523:ISBN
7495:ISBN
7470:ISBN
7407:ISBN
7351:2011
7347:JHEP
7323:ISBN
7263:JHEP
7016:ISBN
6910:ISBN
6682:ISBN
6649:ISBN
6568:ISBN
6519:ISBN
6486:ISBN
6453:ISBN
6428:ISBN
6395:ISBN
6372:PMID
6284:ISBN
6249:ISBN
6193:2013
6189:JHEP
6057:ISBN
6032:ISBN
6007:ISBN
5842:open
5669:and
4734:and
3058:1.01
2283:and
1614:The
560:and
57:for
33:are
7567:doi
7555:121
7444:doi
7440:187
7373:doi
7289:doi
7229:doi
7207:468
7181:doi
7169:283
7138:doi
7099:doi
7087:164
7057:doi
7045:287
6993:doi
6981:175
6951:doi
6939:176
6884:doi
6845:doi
6841:102
6815:doi
6773:doi
6727:doi
6715:102
6674:doi
6615:doi
6560:doi
6511:doi
6420:doi
6364:doi
6325:doi
6276:doi
6215:doi
6166:doi
6127:doi
6084:"2"
5984:doi
5926:to
5835:In
4120:to
2600:In
1666:exp
995:exp
591:of
25:In
7595::
7573:.
7565:.
7553:.
7549:.
7438:.
7434:.
7379:.
7371:.
7363:.
7349:.
7295:.
7287:.
7279:.
7265:.
7257:;
7235:.
7227:.
7219:.
7205:.
7179:.
7167:.
7163:.
7136:.
7126:24
7124:.
7120:.
7097:.
7085:.
7081:.
7055:.
7043:.
7039:.
6991:.
6979:.
6975:.
6949:.
6937:.
6933:.
6882:.
6872:88
6870:.
6839:.
6813:.
6803:21
6801:.
6797:.
6771:.
6761:95
6759:.
6755:.
6747:;
6725:.
6713:.
6709:.
6680:.
6621:.
6613:.
6605:.
6593:73
6591:.
6566:.
6558:.
6527:.
6517:.
6476:;
6426:.
6370:.
6362:.
6352:33
6350:.
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6323:.
6313:18
6311:.
6307:.
6282:.
6221:.
6213:.
6205:.
6191:.
6164:.
6154:24
6152:.
6148:.
6125:.
6113:83
6111:.
6103:;
6086:,
6071:^
5982:.
5972:10
5970:.
5966:.
5657:.
5348:SU
5278:.
5142:0.
4330:.
3079:SU
2920:18
2902:ln
2843:SU
2807:.
2735:tr
2592:.
1869:.
1648:tr
1611:.
1284:.
29:,
7581:.
7569::
7561::
7531:.
7503:.
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7450:.
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7415:.
7387:.
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7367::
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7303:.
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7283::
7273::
7267:6
7243:.
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7223::
7213::
7187:.
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7144:.
7140::
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7101::
7093::
7063:.
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7051::
7024:.
6999:.
6995::
6987::
6957:.
6953::
6945::
6918:.
6890:.
6886::
6878::
6851:.
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6821:.
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6809::
6779:.
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6767::
6733:.
6729::
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6690:.
6676::
6657:.
6629:.
6617::
6609::
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6562::
6554::
6535:.
6513::
6494:.
6461:.
6436:.
6422::
6403:.
6378:.
6366::
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6331:.
6327::
6319::
6292:.
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6229:.
6217::
6209::
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