25:
34:
45:
9982:
54:
8533:
expectation values of the matter scalar fields can be chosen to completely break the gauge symmetry at weak coupling, allowing a reliable semi-classical saddle point calculation to proceed. By then considering perturbations by various mass terms they were able to calculate the superpotential in the presence of arbitrary numbers of colors and flavors, valid even when the theory is no longer weakly coupled.
4525:
6045:
3468:
4189:, dramatically changing the picture of the vacuum structure of this quantum mechanical system. In fact the naive perturbation theory has to be supplemented by boundary conditions, and these supply the nonperturbative effect, as is evident from the above explicit formula and analogous calculations for other potentials such as a cosine potential (cf.
3793:)) in the Euclidean path integral (pictorially speaking – in the Euclidean picture – this transition corresponds to a particle rolling from one hill of a double-well potential standing on its head to the other hill). This classical solution of the Euclidean equations of motion is often named "kink solution" and is an example of an
2163:
4959:
6211:(QFT), the vacuum structure of a theory may draw attention to instantons. Just as a double-well quantum mechanical system illustrates, a naĂŻve vacuum may not be the true vacuum of a field theory. Moreover, the true vacuum of a field theory may be an "overlap" of several topologically inequivalent sectors, so called "
4330:
5307:
2355:
5631:
4689:
8500:
Field theoretic techniques for instanton calculations in supersymmetric theories were extensively studied in the 1980s by multiple authors. Because supersymmetry guarantees the cancellation of fermionic vs. bosonic non-zero modes in the instanton background, the involved 't Hooft computation of the
6187:
2839:
has been given by MĂŒllerâKirsten with derivation by both a perturbation method (plus boundary conditions) applied to the Schrödinger equation, and explicit derivation from the path integral (and WKB). The result is the following. Defining parameters of the Schrödinger equation and the potential by
655:
Instantons are the tool to understand why this happens within the semi-classical approximation of the path-integral formulation in
Euclidean time. We will first see this by using the WKB approximation that approximately computes the wave function itself, and will move on to introduce instantons by
229:, families of instantons permit that instantons, i.e. different critical points of the equation of motion, be related to one another. In physics instantons are particularly important because the condensation of instantons (and noise-induced anti-instantons) is believed to be the explanation of the
8397:
Instantons play a central role in the nonperturbative dynamics of gauge theories. The kind of physical excitation that yields an instanton depends on the number of dimensions of the spacetime, but, surprisingly, the formalism for dealing with these instantons is relatively dimension-independent.
3755:
may be Wick-rotated back and give the same physical results as would be obtained by appropriate treatment of the (potentially divergent) Minkowskian path integral. As can be seen from this example, calculating the transition probability for the particle to tunnel through a classically forbidden
8532:
than the number of colors in the special unitary gauge group, because in the presence of fewer flavors an unbroken nonabelian gauge symmetry leads to an infrared divergence and in the case of more flavors the contribution is equal to zero. For this special choice of chiral matter, the vacuum
4267:
The stability of these pseudoclassical configurations can be investigated by expanding the
Lagrangian defining the theory around the pseudoparticle configuration and then investigating the equation of small fluctuations around it. For all versions of quartic potentials (double-well, inverted
8301:
3808:
field solution of the (Euclidean, i. e., with imaginary time) (1 + 1)-dimensional field theory – first quantized quantum mechanical description – allows to be interpreted as a tunneling effect between the two vacua (ground states – higher states require periodic
3166:
4255:
or pseudoclassical configurations. The "instanton" (kink) solution is accompanied by another solution known as "anti-instanton" (anti-kink), and instanton and anti-instanton are distinguished by "topological charges" +1 and −1 respectively, but have the same
Euclidean action.
1453:
1600:
9077:
3705:
8151:
4963:
The path integral is then approximated via a steepest descent integration, which takes into account only the contributions from the classical solutions and quadratic fluctuations around them. This yields for the rate constant expression in mass-weighted coordinates
1953:
8572:. These results were later extended for various gauge groups and matter contents, and the direct gauge theory derivation was also obtained in most cases. For gauge theories with gauge group U(N) the SeibergâWitten geometry has been derived from gauge theory using
4751:
1225:
5561:
618:
7560:
4280:
In the context of reaction rate theory, periodic instantons are used to calculate the rate of tunneling of atoms in chemical reactions. The progress of a chemical reaction can be described as the movement of a pseudoparticle on a high dimensional
6455:
is a field configuration fulfilling the classical equations of motion in
Euclidean spacetime, which is interpreted as a tunneling effect between these different topological vacua. It is again labelled by an integer number, its Pontryagin index,
3098:
4520:{\displaystyle k(\beta )=-{\frac {2}{\hbar }}{\text{Im}}\mathrm {F} ={\frac {2}{\beta \hbar }}{\text{Im}}\ {\text{ln}}(Z_{k})\approx {\frac {2}{\hbar \beta }}{\frac {{\text{Im}}Z_{k}}{{\text{Re}}Z_{k}}},\ \ {\text{Re}}Z_{k}\gg {\text{Im}}Z_{k}}
3930:
4969:
4242:
In one-dimensional field theory or quantum mechanics one defines as "instanton" a field configuration which is a solution of the classical (Newton-like) equation of motion with
Euclidean time and finite Euclidean action. In the context of
4263:
which are periodic functions (effectively generalisations of trigonometrical functions). In the limit of infinite period these periodic instantons – frequently known as "bounces", "bubbles" or the like – reduce to instantons.
2169:
6040:{\displaystyle E={\frac {1}{2}}q_{0}h^{2}-{\frac {3c^{2}}{4h^{4}}}(q_{0}^{2}+1)-{\frac {q_{0}c^{4}}{h^{10}}}(4q_{0}^{2}+29)+O({\frac {1}{h^{16}}})\pm i{\frac {2^{q_{0}}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{(2\pi )^{1/2}!}}e^{-h^{6}/6c^{2}}.}
6681:
4561:
8830:
2707:
799:
9348:
1019:
8162:
7660:
3463:{\displaystyle E_{\pm }(q_{0},h^{2})=-{\frac {h^{8}}{2^{5}c^{2}}}+{\frac {1}{\sqrt {2}}}q_{0}h^{2}-{\frac {c^{2}(3q_{0}^{2}+1)}{2h^{4}}}-{\frac {{\sqrt {2}}c^{4}q_{0}}{8h^{10}}}(17q_{0}^{2}+19)+O({\frac {1}{h^{16}}})}
7822:
8976:
6813:
7101:
4010:
9120:
7919:
1269:
6142:
4145:
7180:
1464:
3474:
2158:{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+{\sqrt {2}}\int _{\tau _{a}}^{\tau _{b}}d\tau {dx \over d\tau }{\sqrt {V(x)}}}
623:
to identify the energy eigenstates. If we do this, we will find only the unique lowest-energy state instead of two states. The ground-state wave function localizes at both of the classical minima
917:
2949:
8337:
and the chromodynamic sector, however, their existence has not yet been experimentally confirmed. Instanton effects are important in understanding the formation of condensates in the vacuum of
7941:
7027:
5377:
As for the double-well potential one can derive the eigenvalues for the inverted double-well potential. In this case, however, the eigenvalues are complex. Defining parameters by the equations
4954:{\displaystyle Z_{k}=\oint {\mathcal {D}}\mathbf {x} (\tau )e^{-S_{E}/\hbar },\ \ \ S_{E}=\int _{0}^{\beta \hbar }\left({\frac {\dot {\mathbf {x} }}{2}}^{2}+V(\mathbf {x} (\tau ))\right)d\tau }
2399:
2461:
449:
1780:
922:
This means that if the energy of the particle is smaller than the potential energy, one obtains an exponentially decreasing function. The associated tunneling amplitude is proportional to
855:
286:
that absolutely minimize the energy functional within their topological type. The first such solutions were discovered in the case of four-dimensional
Euclidean space compactified to the
9432:, the curves intersect each other orthogonally (in the yellow points) as in 4D. All curves are circles: the curves that intersect <0,0,0,1> have infinite radius (= straight line).
1054:
5338:
5367:
3741:
4744:
8971:
7330:
on 2-forms. Such solutions usually exist, although their precise character depends on the dimension and topology of the base space M, the principal bundle P, and the gauge group G.
1650:
5383:
8893:
8689:
6854:
1261:
8719:
2584:
230:
4208:
Therefore, the perturbative approach may not completely describe the vacuum structure of a physical system. This may have important consequences, for example, in the theory of
2617:
492:
8650:
8573:
7723:
7407:
5624:
6974:
6902:
3158:
2825:
2506:
6400:
2548:
2960:
480:, and these are called classical minima because the particle tends to lie in one of them in classical mechanics. There are two lowest energy states in classical mechanics.
8724:
7263:
7221:
6560:
9177:
6526:
5302:{\displaystyle k(\beta )={\frac {2}{\beta \hbar }}\left({\frac {{\text{det}}\left}{{\text{det}}\left}}\right)^{\frac {1}{2}}{\exp \left({\frac {-S_{E}}{\hbar }}\right)}}
6428:
3815:
2737:
7328:
6692:. He showed that zero modes of the Dirac equation in the instanton background lead to a non-perturbative multi-fermion interaction in the low energy effective action.
6376:
6339:
4177:
4049:
2350:{\displaystyle \quad =\int _{\tau _{a}}^{\tau _{b}}d\tau {1 \over 2}\left({dx \over d\tau }-{\sqrt {2V(x)}}\right)^{2}+\int _{-1}^{1}dx{1 \over {\sqrt {2}}}(1-x^{2}).}
1835:
650:
478:
686:
9248:
8857:
7392:
6304:
6269:
6179:
4554:
1709:
7360:
3783:
2766:
1919:
1864:
1679:
9223:
7292:
2792:
1945:
1890:
1806:
3785:) with the Minkowskian path integral corresponds to calculating the transition probability to tunnel through a classically allowed region (with potential −
9243:
9197:
6946:
6926:
6878:
6498:
6474:
6449:
4684:{\displaystyle Z_{k}={\text{Tr}}(e^{-\beta {\hat {H}}})=\int d\mathbf {x} \left\langle \mathbf {x} \left|e^{-\beta {\hat {H}}}\right|\mathbf {x} \right\rangle }
4323:
4303:
8459:
6592:
361:
can be used to calculate the transition probability for a quantum mechanical particle tunneling through a potential barrier. One example of a system with an
9672:
2625:
698:
4272:. The eigenvalues of these equations are known and permit in the case of instability the calculation of decay rates by evaluation of the path integral.
9484:
Interactions
Between Charged Particles in a Magnetic Field. By Hrachya Nersisyan, Christian Toepffer, GĂŒnter Zwicknagel. Springer, Apr 19, 2007. Pg 23
8296:{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} \geq {\frac {1}{2}}\left|\int _{\mathbb {R} ^{4}}\operatorname {Tr} \right|.}
4251:. In view of their analogy with the behaviour of classical particles such configurations or solutions, as well as others, are collectively known as
928:
9618:
Harald J.W. MĂŒller-Kirsten, Introduction to
Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific (Singapore, 2012).
6583:
369:. In contrast to a classical particle, there is non-vanishing probability that it crosses a region of potential energy higher than its own energy.
1048:
interpretation and the same result can be obtained with this approach. In path integral formulation, the transition amplitude can be expressed as
10414:
8589:
7616:
10016:
7753:
8540: = 2 supersymmetric gauge theories the superpotential receives no quantum corrections. However the correction to the metric of the
8652:. The ansatz gives explicit expressions for the gauge field and can be used to construct solutions with arbitrarily large instanton number.
8599: = 4 supersymmetric gauge theories the instantons do not lead to quantum corrections for the metric on the moduli space of vacua.
6733:
7692:. An instanton is a configuration where, for example, the arrows point away from a central point (i.e., a "hedgehog" state). In Euclidean
330:
7035:
4556:
is the canonical partition function, which is calculated by taking the trace of the
Boltzmann operator in the position representation.
8490:
1448:{\displaystyle K_{E}(a,b;\tau )=\langle x=a|e^{-{\frac {\mathbb {H} \tau }{\hbar }}}|x=b\rangle =\int de^{-{\frac {S_{E}}{\hbar }}},}
9581:
H.J.W. MĂŒller-Kirsten, Introduction to
Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed. (World Scientific, 2012),
9082:
8493:, which restrict the kinds of quantum corrections which are allowed. Many of these theorems only apply to corrections calculable in
9598:
H.J.W. MĂŒller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral, 2nd ed., World Scientific, 2012,
3809:
instantons) of the physical (1-dimensional space + real time) Minkowskian system. In the case of the double-well potential written
7860:
9380: â effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems
9377:
9350:
All solutions of instanton number 1 or 2 are of this form, but for larger instanton number there are solutions not of this form.
8307:
7737:
6053:
4057:
3941:
8420:, which vanishes in the case of the gauge group U(1). If the gauge symmetry is an orthogonal group then this class is the first
1595:{\displaystyle S_{E}=\int _{\tau _{a}}^{\tau _{b}}\left({\frac {1}{2}}m\left({\frac {dx}{d\tau }}\right)^{2}+V(x)\right)d\tau .}
9072:{\displaystyle A_{\mu }=\sigma _{\mu \nu }{\frac {\partial _{\nu }\rho }{\rho }}=\sigma _{\mu \nu }\partial _{\nu }\log(\rho )}
7131:
3700:{\displaystyle \mp {\frac {2^{q_{0}+1}h^{2}(h^{6}/2c^{2})^{q_{0}/2}}{{\sqrt {\pi }}2^{q_{0}/4}!}}e^{-h^{6}/6{\sqrt {2}}c^{2}}.}
9951:
9686:
9603:
9586:
9392:
7669:
derives from the fact that these fields are localized in space and (Euclidean) time â in other words, at a specific instant.
4185:
that a naĂŻve perturbation theory around one of those two vacua alone (of the Minkowskian description) would never show this
9958:
8146:{\displaystyle 0\leq {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} =\int _{\mathbb {R} ^{4}}\operatorname {Tr} }
866:
689:
2846:
9829:
Corrigan, E.; Fairlie, D.B. (March 1977). "Scalar field theory and exact solutions to a classical SU (2) gauge theory".
9733:
Bender, Carl M.; Wu, Tai Tsun (1973-03-15). "Anharmonic Oscillator. II. A Study of Perturbation Theory in Large Order".
9493:
Large-Order Behaviour of Perturbation Theory. Edited by J.C. Le Guillou, J. Zinn-Justin. Elsevier, Dec 2, 2012. Pg. 170.
6979:
6689:
first performed the field theoretic computation of the effects of the BPST instanton in a theory coupled to fermions in
2361:
10009:
6196:
2407:
8401:
In 4-dimensional gauge theories, as described in the previous section, instantons are gauge bundles with a nontrivial
9967:
9934:
9909:
9865:
8544:
of vacua from instantons was calculated in a series of papers. First, the one instanton correction was calculated by
8432:
346:
1220:{\displaystyle K(a,b;t)=\langle x=a|e^{-{\frac {i\mathbb {H} t}{\hbar }}}|x=b\rangle =\int de^{\frac {iS}{\hbar }}.}
380:
10381:
1714:
810:
7268:
is automatically also a solution of the YangâMills equation. This simplification occurs on 4 manifolds with :
7201:, and if the Minkowski current vector does not vanish, the zero on the rhs. of the second equation is replaced by
11214:
10611:
10346:
9628:
Liang, Jiu-Qing; MĂŒller-Kirsten, H.J.W.; Tchrakian, D.H. (1992). "Solitons, bounces and sphalerons on a circle".
8467:
8497:
and so instantons, which are not seen in perturbation theory, provide the only corrections to these quantities.
5314:
349:. This is because magnetic monopoles arise as solutions of a dimensional reduction of the YangâMills equations.
11209:
10973:
116:
6586:. The true vacuum of the theory is labelled by an "angle" theta and is an overlap of the topological sectors:
6233:. For a YangâMills theory these inequivalent sectors can be (in an appropriate gauge) classified by the third
5343:
3713:
10553:
10059:
10002:
8528:. More precisely, they were only able to perform the calculation when the theory contains one less flavor of
7395:
7194:
5556:{\displaystyle {\frac {d^{2}y}{dz^{2}}}+y(z)=0,\;\;\;V(z)={\frac {1}{4}}h^{4}z^{2}-{\frac {1}{2}}c^{2}z^{4},}
4696:
4248:
333:
over a given four-dimensional differentiable manifold as a new invariant of the manifold that depends on its
8898:
6050:
The imaginary part of this expression agrees with the well known result of Bender and Wu. In their notation
4259:"Periodic instantons" are a generalization of instantons. In explicit form they are expressible in terms of
11219:
10998:
10323:
10054:
8562:
Electric â magnetic duality, monopole condensation, and confinement in N=2 supersymmetric YangâMills theory
8440:
6575:
1608:
183:
9507:
613:{\displaystyle -{\hbar ^{2} \over 2m}{\partial ^{2} \over \partial x^{2}}\psi (x)+V(x)\psi (x)=E\psi (x),}
8862:
8658:
7555:{\displaystyle DF=dF+A\wedge F-F\wedge A=d(dA+A\wedge A)+A\wedge (dA+A\wedge A)-(dA+A\wedge A)\wedge A=0}
6821:
5369:
is the trivial solution of the pseudoparticle at rest which represents the reactant state configuration.
4268:
double-well) and periodic (Mathieu) potentials these equations were discovered to be Lamé equations, see
4260:
1237:
10578:
8694:
2553:
11199:
10499:
10204:
10151:
9404:
2589:
1808:
just for simplicity of computation. Since we want to know how the two classically lowest energy states
250:
234:
8626:
7699:
5569:
4746:, one obtains a path integral representation for the partition function in mass-weighted coordinates:
3093:{\displaystyle V(z)=-{\frac {1}{4}}z^{2}h^{4}+{\frac {1}{2}}c^{2}z^{4},\;\;\;c^{2}>0,\;h^{4}>0,}
10793:
8565:
6955:
6883:
4221:
3106:
2797:
2466:
1041:
10838:
9776:
Amoroso, Simone; Kar, Deepak; Schott, Matthias (2021). "How to discover QCD Instantons at the LHC".
8466:. They are responsible for many nonperturbative effects in string theory, playing a central role in
6382:
2511:
804:
If the potential were constant, the solution would be a plane wave, up to a proportionality factor,
11018:
10938:
10753:
10687:
10049:
9454:
9442:
9398:
6713:
6191:
4282:
4225:
4198:
334:
260:
246:
127:
10898:
7233:
7204:
6531:
3925:{\displaystyle V(\phi )={\frac {m^{4}}{2g^{2}}}\left(1-{\frac {g^{2}\phi ^{2}}{m^{2}}}\right)^{2}}
11204:
11158:
10968:
10682:
10520:
10494:
10366:
10235:
10124:
10066:
9386:
8444:
8334:
4217:
283:
10525:
9506:
VaÄnshteÄn, A. I.; Zakharov, Valentin I.; Novikov, Viktor A.; Shifman, Mikhail A. (1982-04-30).
9136:
6503:
484:
318:
10923:
10664:
10470:
10361:
10333:
10156:
8619:
provides a solution to the anti-self dual YangâMills equations with gauge group SU(2) from any
8338:
7693:
6410:
4202:
2715:
211:
169:
9389: â Four-dimensional complete Riemannian manifold satisfying the vacuum Einstein equations
7297:
6345:
6317:
4153:
4018:
24:
10778:
10718:
10659:
10626:
10621:
10419:
10409:
10376:
10240:
10117:
10112:
10107:
10092:
10082:
9130:
8413:
6272:
6230:
3798:
2836:
1811:
626:
454:
366:
218:
111:(bottom left). A visual representation of the field strength of a BPST instanton with center
33:
11073:
671:
377:
Consider the quantum mechanics of a single particle motion inside the double-well potential
99:
coefficient (top right). These coefficients determine the restriction of the BPST instanton
11163:
10948:
10161:
10146:
10102:
9795:
9742:
9637:
9126:
8835:
8405:
7569:
7365:
6686:
6407:
6282:
6247:
6208:
6157:
4532:
1687:
226:
203:
165:
149:
7336:
3759:
2742:
1895:
1840:
1655:
8:
11078:
10963:
10616:
10515:
10141:
9202:
8494:
7840:
7677:
7673:
7589:
7271:
4237:
2771:
1924:
1869:
1785:
268:
153:
9799:
9746:
9641:
6676:{\displaystyle |\theta \rangle =\sum _{N=-\infty }^{N=+\infty }e^{i\theta N}|N\rangle .}
345:
four-manifolds. Many methods developed in studying instantons have also been applied to
11123:
11043:
10943:
10903:
10833:
10783:
10748:
10583:
10460:
10356:
10097:
9888:
9811:
9785:
9228:
9182:
8327:
7848:
6931:
6911:
6863:
6483:
6459:
6434:
4308:
4288:
10510:
9807:
652:
instead of only one of them because of the quantum interference or quantum tunneling.
11088:
10993:
10828:
10738:
10708:
10504:
10399:
10351:
10245:
9963:
9947:
9930:
9905:
9861:
9842:
9815:
9758:
9682:
9653:
9649:
9599:
9582:
9545:
9527:
8825:{\displaystyle \sigma _{ij}=\epsilon _{ijk}T_{k}\,,\sigma _{i4}=-\sigma _{4i}=T_{i},}
8620:
8568:. They extended their calculation to SU(2) gauge theories with fundamental matter in
8529:
8478:
8474:
8402:
7607:
6579:
6571:
6226:
665:
314:
310:
276:
195:
191:
161:
9986:
9675:(2020). "Instanton Theory to Calculate Tunnelling Rates and Tunnelling Splittings".
9523:
182:
In such quantum theories, solutions to the equations of motion may be thought of as
11098:
11033:
11003:
10883:
10823:
10788:
10733:
10723:
10636:
10588:
10546:
10451:
10444:
10437:
10430:
10423:
10341:
10131:
10039:
9892:
9838:
9803:
9750:
9715:
9645:
9519:
9475:
Instantons in Gauge Theories. Edited by Mikhail A. Shifman. World Scientific, 1994.
8421:
7844:
7741:
7399:
7122:
6701:
6276:
4190:
2702:{\displaystyle x(\tau )=\tanh \left({1 \over {\sqrt {2}}}(\tau -\tau _{0})\right).}
794:{\displaystyle {\frac {d^{2}\psi }{dx^{2}}}={\frac {2m(V(x)-E)}{\hbar ^{2}}}\psi .}
264:
187:
10703:
6271:). A certain topological vacuum (a "sector" of the true vacuum) is labelled by an
4269:
4194:
11178:
11133:
11083:
11068:
11058:
10953:
10918:
10743:
10313:
10087:
9359:
8585:
8581:
8577:
8342:
7733:
7581:
4252:
3743:) degenerate as expected as a consequence of the harmonic part of the potential.
322:
301:
YangâMills instantons have been explicitly constructed in many cases by means of
173:
11023:
6690:
11153:
11148:
11108:
11048:
10878:
10868:
10863:
10858:
10773:
10768:
10763:
10728:
10713:
10641:
10318:
10181:
9918:
9678:
Tunnelling in Molecules: Nuclear Quantum Effects from Bio to Physical Chemistry
9371:
8553:
8545:
8521:
8509:
8346:
7925:
7593:
6567:
6314:
6234:
342:
302:
70:
11038:
10958:
9343:{\displaystyle \rho (x)=\sum _{p=1}^{N}{\frac {\lambda _{p}}{|x-x_{p}|^{2}}}.}
2739:
is an arbitrary constant. Since this solution jumps from one classical vacuum
1014:{\displaystyle e^{-{\frac {1}{\hbar }}\int _{a}^{b}{\sqrt {2m(V(x)-E)}}\,dx},}
217:
they can be used to study the tunneling behavior in various systems such as a
44:
11193:
11143:
11128:
11103:
11093:
11063:
11008:
10983:
10913:
10908:
10873:
10848:
10808:
10598:
10250:
10166:
10044:
10025:
9762:
9657:
9569:
9531:
9429:
8616:
8557:
8409:
8366:
7681:
7585:
3752:
1231:
338:
306:
214:
as the leading quantum corrections to the classical behavior of a system, and
10928:
9754:
8552:. The full set of corrections for SU(2) YangâMills theory was calculated by
7676:
may be easier to visualise because it admits the simplest case of the gauge
2835:
The explicit formula for the eigenenergies of the Schrödinger equation with
126:(bottom right). The BPST instanton is a classical instanton solution to the
11173:
11013:
10893:
10843:
10813:
10798:
10606:
10573:
10465:
10391:
10371:
10308:
10171:
8541:
8517:
8448:
7729:
7728:
The field configuration of an instanton is very different from that of the
7689:
6949:
326:
279:
199:
157:
89:
8524:
calculated the instanton corrections to the superpotential in their paper
8341:(QCD) and in explaining the mass of the so-called 'eta-prime particle', a
7847:
invariant can be defined at the 3-space boundary. This is equivalent, via
7831:. If we insist that the solutions to the YangâMills equations have finite
7655:{\displaystyle \mathbf {F} =d\mathbf {A} +\mathbf {A} \wedge \mathbf {A} }
11168:
11138:
11118:
10978:
10933:
10888:
10853:
10803:
10568:
10537:
10298:
10255:
8612:
8570:
Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD
8513:
8428:
8417:
8362:
7817:{\displaystyle {\frac {1}{2}}\int _{\mathbb {R} ^{4}}\operatorname {Tr} }
7596:
7107:
6905:
6857:
6212:
287:
9981:
9676:
8832:
where Greek indices run from 1 to 4, Latin indices run from 1 to 3, and
8333:
In the Standard Model instantons are expected to be present both in the
664:
One way to calculate this probability is by means of the semi-classical
11113:
11053:
10988:
10651:
10631:
10530:
10489:
10303:
8508: = 1 supersymmetric gauge theories instantons can modify the
7828:
7603:
7224:
6406:
there are infinitely many topologically inequivalent vacua, denoted by
4201:) and irrespective of whether one uses the Schrödinger equation or the
272:
9719:
8569:
8561:
8436:
8365:
and, of course, the vacuum structure of QCD. For example, in oriented
6808:{\displaystyle S_{YM}=\int _{M}\left|F\right|^{2}d\mathrm {vol} _{M},}
1652:
under the Wick rotation and the minima transform into maxima, thereby
321:), a geometric invariant theory procedure. The groundbreaking work of
11028:
10818:
10758:
10278:
10199:
10194:
10189:
9407: â Study of vector bundles, principal bundles, and fibre bundles
9401: â Partial differential equations whose solutions are instantons
8549:
8525:
7932:
7836:
6720:
176:
168:. More precisely, it is a solution to the equations of motion of the
9703:
8353:
in a theory with one additional space dimension. Recent research on
6186:
2550:. Such solutions exist, and the solution takes the simple form when
290:, and turned out to be localized in space-time, prompting the names
10674:
10563:
10558:
10288:
10283:
10220:
10136:
9790:
8501:
instanton saddle point reduces to an integration over zero modes.
8358:
7852:
7577:
7227:. Thus a solution to the simpler first order (non-linear) equation
6342:
6242:
6225:
and its interpretation can be found in the context of a QFT with a
9962:, S.K. Donaldson, P.B. Kronheimer, Oxford University Press, 1990,
7096:{\displaystyle F\wedge *F=\langle F,F\rangle d\mathrm {vol} _{M}.}
10293:
10273:
10230:
10225:
9896:
9614:
9612:
9365:
8350:
6379:
6307:
4244:
53:
9994:
9505:
3751:
Results obtained from the mathematically well-defined Euclidean
107:
to this slice. The corresponding field strength centered around
9929:
p. 265, Sidney Coleman, Cambridge University Press, 1985,
9704:"Theory and Simulation of Atom Tunneling in Chemical Reactions"
8608:
8463:
8452:
7832:
7223:. But notice how similar these equations are; they differ by a
7118:
3801:, turn into hills in the Euclideanized version of the problem.
3797:. In this example, the two "vacua" (i.e. ground states) of the
9609:
9115:{\displaystyle \rho :\mathbb {R} ^{4}\rightarrow \mathbb {R} }
8369:, a Dp brane is a gauge theory instanton in the world volume (
4693:
Using a Wick rotation and identifying the Euclidean time with
2619:. The explicit formula for the instanton solution is given by
1684:
Let us now consider the local minimum of the Euclidean action
1035:
10265:
6238:
6215:
4305:
can then be related to the imaginary part of the free energy
4209:
2830:
9627:
8564:," in the process creating a subject that is today known as
7914:{\displaystyle \int _{\mathbb {R} ^{4}}\operatorname {Tr} .}
1032:
are the beginning and endpoint of the tunneling trajectory.
7110:
6137:{\displaystyle \hbar =1,q_{0}=2K+1,h^{6}/2c^{2}=\epsilon .}
4140:{\displaystyle \phi _{c}(\tau )={\frac {m}{g}}\tanh \left,}
4005:{\displaystyle {\frac {d^{2}\phi }{d\tau ^{2}}}=V'(\phi ),}
8349:
of QCD. Note that there is sometimes also a corresponding
7175:{\displaystyle \mathrm {d} F=0,\quad \mathrm {d} {*F}=0.}
7732:. Because of this instantons cannot be studied by using
9681:. London: Royal Society of Chemistry. p. 245-260.
372:
282:. Instantons are topologically nontrivial solutions of
8526:
Dynamical Supersymmetry Breaking in Supersymmetric QCD
9887:, a compilation of articles on instantons, edited by
9572:'s paper "Self-Duality Equations on Riemann Surface".
9251:
9231:
9205:
9185:
9139:
9085:
8979:
8901:
8865:
8838:
8727:
8697:
8661:
8629:
8443:
demonstrated that instanton effects in 3-dimensional
8165:
7944:
7863:
7756:
7702:
7619:
7410:
7368:
7339:
7300:
7274:
7236:
7207:
7134:
7038:
6982:
6958:
6934:
6914:
6886:
6866:
6824:
6736:
6595:
6534:
6506:
6486:
6462:
6437:
6413:
6385:
6348:
6320:
6285:
6250:
6160:
6056:
5634:
5572:
5386:
5346:
5317:
4972:
4754:
4699:
4564:
4535:
4333:
4311:
4291:
4275:
4156:
4060:
4021:
3944:
3818:
3762:
3716:
3477:
3169:
3109:
2963:
2849:
2800:
2774:
2745:
2718:
2628:
2592:
2556:
2514:
2469:
2410:
2404:
The above inequality is saturated by the solution of
2364:
2172:
1956:
1927:
1898:
1872:
1843:
1814:
1788:
1717:
1690:
1658:
1611:
1467:
1272:
1240:
1057:
931:
869:
813:
701:
674:
629:
495:
457:
383:
9382:
Pages displaying wikidata descriptions as a fallback
9362: â Non-perturbative path integral approximation
5566:
the eigenvalues as given by MĂŒller-Kirsten are, for
912:{\displaystyle k={\frac {\sqrt {2m(E-V)}}{\hbar }}.}
8484:
8306:If this bound is saturated, then the solution is a
7924:This is a homotopy invariant and it tells us which
7580:nontrivial field configuration in four-dimensional
4212:where the non-trivial QCD vacuum effects (like the
2944:{\displaystyle {\frac {d^{2}y(z)}{dz^{2}}}+y(z)=0,}
9342:
9237:
9217:
9191:
9171:
9114:
9071:
8965:
8887:
8851:
8824:
8713:
8683:
8644:
8295:
8145:
7913:
7816:
7717:
7654:
7554:
7386:
7354:
7322:
7286:
7257:
7215:
7174:
7095:
7022:{\displaystyle \int _{M}\mathrm {Tr} (F\wedge *F)}
7021:
6968:
6940:
6920:
6896:
6872:
6848:
6807:
6675:
6554:
6520:
6492:
6468:
6443:
6422:
6394:
6370:
6333:
6298:
6263:
6173:
6136:
6039:
5618:
5555:
5361:
5332:
5301:
4953:
4738:
4683:
4548:
4519:
4317:
4297:
4171:
4139:
4043:
4004:
3924:
3777:
3735:
3699:
3462:
3152:
3092:
2943:
2819:
2786:
2760:
2731:
2701:
2611:
2578:
2542:
2500:
2455:
2394:{\displaystyle \quad \geq {2{\sqrt {2}} \over 3}.}
2393:
2349:
2157:
1939:
1913:
1884:
1858:
1829:
1800:
1774:
1703:
1673:
1644:
1594:
1447:
1255:
1219:
1013:
911:
849:
793:
680:
644:
612:
472:
443:
9904:, R. Rajaraman (Amsterdam: North Holland, 1987),
9741:(6). American Physical Society (APS): 1620â1636.
6221:A well understood and illustrative example of an
2456:{\displaystyle {dx \over d\tau }={\sqrt {2V(x)}}}
11191:
9860:. Oxford: Oxford University Press. p. 123.
9775:
8550:Supersymmetry and Nonperturbative beta Functions
8392:
6500:to quantify tunneling between topological vacua
4247:theory the corresponding solution is known as a
1234:(analytic continuation) to Euclidean spacetime (
451:The potential energy takes its minimal value at
9828:
9670:
8512:, sometimes lifting all of the vacua. In 1984,
8435:play the role of instantons. In his 1977 paper
5372:
8437:Quark Confinement and Topology of Gauge Groups
3710:Clearly these eigenvalues are asymptotically (
444:{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}.}
10010:
9664:
8481:, which is invisible in perturbation theory.
8416:then this characteristic class is the second
1775:{\displaystyle V(x)={1 \over 4}(x^{2}-1)^{2}}
850:{\displaystyle \psi =\exp(-\mathrm {i} kx)\,}
9695:
7185:The first of these is an identity, because d
7125:, the YangâMills equations follow. They are
7066:
7054:
6667:
6604:
6549:
6515:
6451:is their corresponding Pontryagin index. An
6417:
6150:
1365:
1307:
1146:
1085:
8602:
1036:Path integral interpretation via instantons
190:. The critical points of the action may be
148:) is a notion appearing in theoretical and
10017:
10003:
5471:
5470:
5469:
5333:{\displaystyle \mathbf {x} _{\text{Inst}}}
3070:
3050:
3049:
3048:
2831:Explicit formula for double-well potential
152:. An instanton is a classical solution to
9789:
9108:
9094:
8770:
8632:
8489:Supersymmetric gauge theories often obey
8248:
8183:
8078:
7968:
7871:
7774:
7705:
7254:
6391:
6387:
4193:) or other periodic potentials (cf. e.g.
1947:, we can rewrite the Euclidean action as
1336:
1117:
999:
846:
9946:. M. Dunajski, Oxford University Press.
9855:
9849:
9732:
8458:In 2-dimensional abelian gauge theories
7839:of the solution at infinity (taken as a
6185:
5362:{\displaystyle \mathbf {x} _{\text{RS}}}
3736:{\displaystyle h^{2}\rightarrow \infty }
1681:exhibits two "hills" of maximal energy.
9923:Proc. Int. School of Subnuclear Physics
9701:
9411:
6146:
4739:{\displaystyle \hbar \beta =1/(k_{b}T)}
11192:
9374: â American physicist (1937â2007)
8966:{\displaystyle =-\epsilon _{ijk}T_{k}}
7843:) has to be zero. This means that the
7740:effects. Instantons are fundamentally
7193:= 0, but the second is a second-order
4231:
337:and applied it to the construction of
9998:
9395: â Chemical reaction rate theory
9393:Semiclassical transition state theory
8427:In 3-dimensional gauge theories with
7725:, abelian instantons are impossible.
6700:The classical YangâMills action on a
1645:{\displaystyle V(x)\rightarrow -V(x)}
690:time independent Schrödinger equation
656:using the path integral formulation.
325:, for which he was later awarded the
10415:Bogomol'nyiâPrasadâSommerfield bound
9501:
9499:
9368: â Finite temperature instanton
8345:which has acquired mass through the
7931:Since the integral of a nonnegative
6695:
659:
373:Motivation of considering instantons
352:
8888:{\displaystyle {\mathfrak {su}}(2)}
8871:
8868:
8684:{\displaystyle {\mathfrak {su}}(2)}
8667:
8664:
7333:In nonabelian YangâMills theories,
6961:
6889:
6849:{\displaystyle d\mathrm {vol} _{M}}
1256:{\displaystyle it\rightarrow \tau }
483:In quantum mechanics, we solve the
13:
9858:Solitons, instantons, and twistors
9045:
9010:
8714:{\displaystyle \sigma _{\mu \nu }}
7747:The YangâMills energy is given by
7154:
7136:
7117:will be the electromagnetic field
7080:
7077:
7074:
6997:
6994:
6836:
6833:
6830:
6792:
6789:
6786:
6638:
6624:
6199:(blue) and hypermeridians (green).
5124:
5114:
5037:
5027:
4773:
4368:
4276:Instantons in reaction rate theory
4220:explicitly and transform massless
3730:
2606:
2579:{\displaystyle \tau _{a}=-\infty }
2573:
1605:The potential energy changes sign
833:
534:
524:
305:, which relates them to algebraic
14:
11231:
10024:
9974:
9944:Solitons, Instantons and Twistors
9496:
6279:. As the third homotopy group of
5289:
4997:
4878:
4835:
4384:
4357:
4285:(PES). The thermal rate constant
4187:non-perturbative tunneling effect
2612:{\displaystyle \tau _{b}=\infty }
1435:
1344:
1208:
1125:
945:
901:
259:is a self-dual or anti-self-dual
9980:
8645:{\displaystyle \mathbb {R} ^{4}}
8485:4d supersymmetric gauge theories
8310:state. For such states, either â
8278:
8270:
8216:
8208:
8136:
8128:
8111:
8103:
8058:
8034:
8020:
7996:
7901:
7893:
7807:
7799:
7718:{\displaystyle \mathbb {R} ^{4}}
7648:
7640:
7632:
7621:
7610:. This means the field strength
7592:). Specifically, it refers to a
7209:
7106:For example, in the case of the
6566:= 1, the configuration is named
6306:has been found to be the set of
5619:{\displaystyle q_{0}=1,3,5,...,}
5349:
5320:
5145:
5058:
4924:
4894:
4814:
4779:
4672:
4634:
4624:
3935:the instanton, i.e. solution of
317:, or hyperkÀhler reduction (see
271:that plays the role of physical
52:
43:
32:
23:
10612:Eleven-dimensional supergravity
9822:
9808:10.1140/epjc/s10052-021-09412-1
9778:The European Physical Journal C
9769:
9726:
9621:
9524:10.1070/PU1982v025n04ABEH004533
9447:
9435:
9422:
9245:-soliton solutions of the form
8156:for all real Ξ. So, this means
7665:vanishes at infinity. The name
7152:
6969:{\displaystyle {\mathfrak {g}}}
6897:{\displaystyle {\mathfrak {g}}}
3153:{\displaystyle q_{0}=1,3,5,...}
2820:{\displaystyle \tau =\tau _{0}}
2501:{\displaystyle x(\tau _{a})=-1}
2365:
2173:
1711:with the double-well potential
202:. Instantons are important in
16:Solitons in Euclidean spacetime
9959:The Geometry of Four-Manifolds
9592:
9575:
9562:
9546:"Yang-Mills instanton in nLab"
9538:
9487:
9478:
9469:
9324:
9302:
9261:
9255:
9156:
9141:
9104:
9066:
9060:
8928:
8902:
8882:
8876:
8678:
8672:
8282:
8266:
8220:
8201:
8140:
8096:
8065:
8062:
8030:
8024:
7989:
7986:
7905:
7889:
7811:
7792:
7688:can be visualised as simply a
7672:The case of instantons on the
7537:
7516:
7510:
7489:
7477:
7456:
7123:principle of stationary action
7016:
7001:
6976:, then this may be denoted as
6948:takes values, is given by the
6660:
6597:
6536:
6508:
6395:{\displaystyle \mathbb {Z} \,}
6362:
6349:
5987:
5976:
5957:
5954:
5937:
5927:
5901:
5869:
5830:
5810:
5801:
5774:
5732:
5708:
5481:
5475:
5457:
5451:
5445:
5442:
5436:
5424:
5285:
5282:
5276:
5263:
5247:
5241:
5228:
5175:
5172:
5166:
5153:
5088:
5085:
5079:
5066:
4982:
4976:
4937:
4934:
4928:
4920:
4827:
4824:
4818:
4810:
4789:
4783:
4733:
4717:
4659:
4611:
4603:
4583:
4416:
4403:
4343:
4337:
4126:
4107:
4077:
4071:
3996:
3990:
3828:
3822:
3772:
3766:
3727:
3640:
3629:
3610:
3607:
3549:
3517:
3457:
3437:
3428:
3401:
3329:
3302:
3206:
3180:
2973:
2967:
2929:
2923:
2917:
2914:
2908:
2896:
2872:
2866:
2688:
2669:
2638:
2632:
2543:{\displaystyle x(\tau _{b})=1}
2531:
2518:
2486:
2473:
2448:
2442:
2341:
2322:
2265:
2259:
2150:
2144:
2058:
2052:
1763:
1743:
1727:
1721:
1668:
1662:
1639:
1633:
1624:
1621:
1615:
1575:
1569:
1431:
1428:
1422:
1416:
1392:
1389:
1383:
1377:
1352:
1320:
1301:
1283:
1247:
1204:
1201:
1195:
1189:
1173:
1170:
1164:
1158:
1133:
1098:
1079:
1061:
994:
985:
979:
973:
897:
885:
843:
826:
769:
760:
754:
748:
668:, which requires the value of
604:
598:
586:
580:
574:
568:
559:
553:
429:
409:
393:
387:
240:
1:
10060:Second superstring revolution
9985:The dictionary definition of
9636:(1â2). Elsevier BV: 105â110.
8408:. If the gauge symmetry is a
8393:Various numbers of dimensions
8381:) gauge theory on a stack of
8357:links them to topics such as
8326:depending on the sign of the
7396:exterior covariant derivative
7195:partial differential equation
6241:(whose group manifold is the
2827:, it is called an instanton.
10554:Generalized complex manifold
10055:First superstring revolution
9939:Instantons in Gauge Theories
9885:Instantons in Gauge Theories
9843:10.1016/0370-2693(77)90808-5
9650:10.1016/0370-2693(92)90486-n
9589:; formula (18.175b), p. 525.
8574:Nekrasov partition functions
8373: + 5)-dimensional
7258:{\displaystyle {*F}=\pm F\,}
7216:{\displaystyle \mathbf {J} }
6555:{\displaystyle |N+Q\rangle }
5373:Inverted double-well formula
5340:is a periodic instanton and
2768:to another classical vacuum
7:
9428:Because this projection is
9353:
8655:Defining the antisymmetric
8491:nonrenormalization theorems
8433:'t HooftâPolyakov monopoles
4261:Jacobian elliptic functions
4226:pseudo-NambuâGoldstone ones
231:noise-induced chaotic phase
10:
11236:
10152:Non-critical string theory
9702:KĂ€stner, Johannes (2014).
9405:Gauge theory (mathematics)
9172:{\displaystyle |x-y|^{-2}}
7928:the instanton belongs to.
7680:, namely U(1), that is an
6880:. If the inner product on
6723:(YangâMills field tensor)
6521:{\displaystyle |N\rangle }
4235:
3746:
1837:are connected, let us set
1458:with the Euclidean action
1040:Alternatively, the use of
365:effect is a particle in a
331:moduli space of instantons
251:Gauge theory (mathematics)
244:
235:self-organized criticality
10696:
10673:
10650:
10597:
10482:
10390:
10332:
10264:
10213:
10180:
10075:
10032:
9856:Dunajski, Maciej (2010).
9079:is a solution as long as
7684:. In this case the field
6423:{\displaystyle N\rangle }
6153:
2732:{\displaystyle \tau _{0}}
1230:Following the process of
10688:Introduction to M-theory
10382:WessâZuminoâWitten model
10324:HananyâWitten transition
10050:History of string theory
9925:, (Erice, 1977); and in
9443:Non-abelian gauge theory
9125:In four dimensions, the
8389: + 4)-branes.
7323:{\displaystyle *^{2}=+1}
6371:{\displaystyle (S^{3})=}
6334:{\displaystyle \pi _{3}}
6192:Stereographic projection
4283:potential energy surface
4199:spheroidal wave function
4172:{\displaystyle \tau =it}
4044:{\displaystyle E_{cl}=0}
335:differentiable structure
267:over a four-dimensional
10367:Vertex operator algebra
10067:String theory landscape
9902:Solitons and Instantons
9755:10.1103/physrevd.7.1620
9387:Gravitational instanton
9378:HolsteinâHerring method
8603:Explicit solutions on R
8451:lead to a mass for the
7935:is always nonnegative,
6227:non-abelian gauge group
4179:is the Euclidean time.
2794:instantaneously around
1830:{\displaystyle x=\pm 1}
692:for the particle reads
645:{\displaystyle x=\pm 1}
473:{\displaystyle x=\pm 1}
288:four-dimensional sphere
11215:Quantum chromodynamics
10665:AdS/CFT correspondence
10420:Exceptional Lie groups
10362:Superconformal algebra
10334:Conformal field theory
10205:MontonenâOlive duality
10157:Non-linear sigma model
9915:The Uses of Instantons
9708:WIREs Comput. Mol. Sci
9512:Soviet Physics Uspekhi
9455:Pseudo-Goldstone boson
9344:
9287:
9239:
9219:
9193:
9173:
9116:
9073:
8967:
8889:
8853:
8826:
8715:
8685:
8646:
8477:, instantons describe
8339:quantum chromodynamics
8297:
8147:
7915:
7818:
7719:
7656:
7556:
7388:
7356:
7324:
7288:
7259:
7217:
7176:
7097:
7023:
6970:
6942:
6922:
6898:
6874:
6850:
6809:
6677:
6642:
6570:after its discoverers
6556:
6522:
6494:
6470:
6445:
6424:
6396:
6372:
6335:
6300:
6265:
6200:
6175:
6138:
6041:
5620:
5557:
5363:
5334:
5303:
4955:
4740:
4685:
4550:
4521:
4319:
4299:
4222:NambuâGoldstone bosons
4173:
4141:
4045:
4006:
3926:
3779:
3737:
3701:
3464:
3154:
3094:
2945:
2821:
2788:
2762:
2733:
2703:
2613:
2580:
2544:
2502:
2457:
2395:
2351:
2159:
1941:
1915:
1886:
1860:
1831:
1802:
1776:
1705:
1675:
1646:
1596:
1449:
1257:
1221:
1015:
913:
851:
795:
682:
681:{\displaystyle \hbar }
646:
614:
474:
445:
170:classical field theory
11210:Differential geometry
10660:Holographic principle
10627:Type IIB supergravity
10622:Type IIA supergravity
10474:-form electrodynamics
10093:Bosonic string theory
9345:
9267:
9240:
9220:
9194:
9174:
9117:
9074:
8968:
8890:
8854:
8852:{\displaystyle T_{i}}
8827:
8716:
8686:
8647:
8584:and independently by
8566:SeibergâWitten theory
8460:worldsheet instantons
8414:special unitary group
8347:axial current anomaly
8298:
8148:
7916:
7819:
7736:, which only include
7720:
7674:two-dimensional space
7657:
7557:
7389:
7387:{\displaystyle D*F=0}
7357:
7325:
7289:
7260:
7218:
7177:
7098:
7024:
6971:
6943:
6923:
6899:
6875:
6851:
6810:
6704:with structure group
6678:
6610:
6557:
6523:
6495:
6476:. One can imagine an
6471:
6446:
6425:
6397:
6373:
6336:
6301:
6299:{\displaystyle S^{3}}
6266:
6264:{\displaystyle S^{3}}
6189:
6176:
6174:{\displaystyle S^{3}}
6139:
6042:
5621:
5558:
5364:
5335:
5304:
4956:
4741:
4686:
4551:
4549:{\displaystyle Z_{k}}
4522:
4320:
4300:
4218:PecceiâQuinn symmetry
4174:
4142:
4046:
4007:
3927:
3799:double-well potential
3780:
3738:
3702:
3465:
3155:
3095:
2946:
2837:double-well potential
2822:
2789:
2763:
2734:
2704:
2614:
2581:
2545:
2503:
2458:
2396:
2352:
2160:
1942:
1916:
1887:
1861:
1832:
1803:
1777:
1706:
1704:{\displaystyle S_{E}}
1676:
1647:
1597:
1450:
1258:
1222:
1016:
914:
852:
796:
683:
647:
615:
475:
446:
367:double-well potential
10579:HoĆavaâWitten theory
10526:HyperkÀhler manifold
10214:Particles and fields
10162:Tachyon condensation
10147:Matrix string theory
9412:References and notes
9399:YangâMills equations
9249:
9229:
9203:
9183:
9137:
9127:fundamental solution
9083:
8977:
8899:
8863:
8836:
8725:
8695:
8659:
8627:
8406:characteristic class
8163:
7942:
7861:
7754:
7700:
7617:
7570:quantum field theory
7408:
7366:
7355:{\displaystyle DF=0}
7337:
7298:
7272:
7234:
7205:
7132:
7036:
6980:
6956:
6932:
6912:
6884:
6864:
6822:
6734:
6593:
6532:
6504:
6484:
6460:
6435:
6411:
6383:
6346:
6318:
6283:
6248:
6209:quantum field theory
6158:
6147:Quantum field theory
6054:
5632:
5570:
5384:
5344:
5315:
4970:
4752:
4697:
4562:
4533:
4331:
4309:
4289:
4154:
4058:
4019:
3942:
3816:
3778:{\displaystyle V(x)}
3760:
3714:
3475:
3167:
3107:
3103:the eigenvalues for
2961:
2847:
2798:
2772:
2761:{\displaystyle x=-1}
2743:
2716:
2626:
2590:
2554:
2512:
2467:
2408:
2362:
2170:
1954:
1925:
1914:{\displaystyle a=-1}
1896:
1870:
1859:{\displaystyle a=-1}
1841:
1812:
1786:
1715:
1688:
1674:{\displaystyle V(x)}
1656:
1609:
1465:
1270:
1238:
1055:
929:
867:
811:
699:
672:
627:
493:
485:Schrödinger equation
455:
381:
319:hyperkÀhler manifold
284:YangâMills equations
257:YangâMills instanton
247:YangâMills equations
204:quantum field theory
166:quantum field theory
150:mathematical physics
128:YangâMills equations
11220:Anomalies (physics)
10617:Type I supergravity
10521:CalabiâYau manifold
10516:Ricci-flat manifold
10495:KaluzaâKlein theory
10236:RamondâRamond field
10142:String field theory
9927:Aspects of Symmetry
9800:2021EPJC...81..624A
9747:1973PhRvD...7.1620B
9642:1992PhLB..282..105L
9568:See, for instance,
9508:"ABC of instantons"
9218:{\displaystyle N+1}
8495:perturbation theory
7590:Minkowski spacetime
7584:(considered as the
7398:. Furthermore, the
7287:{\displaystyle s=1}
7197:for the connection
6273:unaltered transform
5794:
5725:
4882:
4238:Periodic instantons
4232:Periodic instantons
3421:
3322:
2787:{\displaystyle x=1}
2463:with the condition
2301:
2205:
2112:
1998:
1940:{\displaystyle b=1}
1885:{\displaystyle b=1}
1801:{\displaystyle m=1}
1509:
964:
269:Riemannian manifold
210:they appear in the
154:equations of motion
10584:K-theory (physics)
10461:ADE classification
10098:Superstring theory
9889:Mikhail A. Shifman
9671:Zaverkin, Viktor;
9340:
9235:
9215:
9189:
9169:
9131:Laplace's equation
9112:
9069:
8963:
8885:
8849:
8822:
8711:
8681:
8642:
8441:Alexander Polyakov
8335:electroweak sector
8328:homotopy invariant
8293:
8143:
7911:
7814:
7715:
7652:
7552:
7384:
7352:
7320:
7284:
7255:
7213:
7172:
7093:
7019:
6966:
6938:
6918:
6894:
6870:
6846:
6805:
6673:
6576:Alexander Polyakov
6552:
6518:
6490:
6466:
6441:
6420:
6392:
6368:
6331:
6296:
6261:
6201:
6171:
6134:
6037:
5780:
5711:
5616:
5553:
5359:
5330:
5299:
4951:
4865:
4736:
4681:
4546:
4517:
4315:
4295:
4169:
4137:
4041:
4015:(i.e. with energy
4002:
3922:
3775:
3733:
3697:
3460:
3407:
3308:
3150:
3090:
2941:
2817:
2784:
2758:
2729:
2699:
2609:
2576:
2540:
2498:
2453:
2391:
2347:
2284:
2177:
2155:
2084:
1970:
1937:
1911:
1882:
1856:
1827:
1798:
1772:
1701:
1671:
1642:
1592:
1481:
1445:
1253:
1217:
1011:
950:
909:
847:
791:
678:
642:
610:
470:
441:
311:algebraic surfaces
255:Mathematically, a
11200:Quantum mechanics
11187:
11186:
10969:van Nieuwenhuizen
10505:Why 10 dimensions
10410:ChernâSimons form
10377:KacâMoody algebra
10357:Conformal algebra
10352:Conformal anomaly
10246:Magnetic monopole
10241:KalbâRamond field
10083:NambuâGoto action
9952:978-0-19-857063-9
9831:Physics Letters B
9735:Physical Review D
9720:10.1002/wcms.1165
9688:978-1-83916-037-0
9673:KĂ€stner, Johannes
9630:Physics Letters B
9604:978-981-4397-73-5
9587:978-981-4397-73-5
9335:
9238:{\displaystyle N}
9192:{\displaystyle y}
9026:
8621:harmonic function
8475:quantum mechanics
8473:In 1-dimensional
8234:
8174:
7959:
7765:
7602:which approaches
6941:{\displaystyle F}
6921:{\displaystyle G}
6873:{\displaystyle M}
6696:YangâMills theory
6580:Albert S. Schwarz
6572:Alexander Belavin
6493:{\displaystyle Q}
6469:{\displaystyle Q}
6444:{\displaystyle N}
6231:YangâMills theory
6205:
6204:
6195:Parallels (red),
5994:
5828:
5772:
5706:
5649:
5528:
5495:
5419:
5356:
5327:
5292:
5273:
5238:
5198:
5184:
5163:
5138:
5101:
5076:
5051:
5014:
5001:
4906:
4901:
4851:
4848:
4845:
4662:
4606:
4581:
4505:
4487:
4483:
4480:
4473:
4460:
4443:
4435:
4401:
4397:
4393:
4388:
4365:
4360:
4318:{\displaystyle F}
4298:{\displaystyle k}
4091:
3977:
3909:
3861:
3680:
3647:
3580:
3455:
3399:
3361:
3348:
3264:
3263:
3249:
3160:are found to be:
3023:
2990:
2891:
2667:
2665:
2451:
2429:
2386:
2380:
2320:
2318:
2268:
2246:
2220:
2153:
2137:
2082:
2061:
2039:
2013:
1741:
1551:
1523:
1438:
1347:
1211:
1128:
997:
948:
904:
900:
783:
734:
688:to be small. The
666:WKB approximation
660:WKB approximation
548:
519:
407:
353:Quantum mechanics
315:ADHM construction
219:YangâMills theory
162:quantum mechanics
69:coefficient of a
11227:
10697:String theorists
10637:Lie superalgebra
10589:Twisted K-theory
10547:Spin(7)-manifold
10500:Compactification
10342:Virasoro algebra
10125:Heterotic string
10019:
10012:
10005:
9996:
9995:
9984:
9872:
9871:
9853:
9847:
9846:
9826:
9820:
9819:
9793:
9773:
9767:
9766:
9730:
9724:
9723:
9699:
9693:
9692:
9668:
9662:
9661:
9625:
9619:
9616:
9607:
9596:
9590:
9579:
9573:
9566:
9560:
9559:
9557:
9556:
9542:
9536:
9535:
9503:
9494:
9491:
9485:
9482:
9476:
9473:
9457:
9451:
9445:
9439:
9433:
9426:
9383:
9349:
9347:
9346:
9341:
9336:
9334:
9333:
9332:
9327:
9321:
9320:
9305:
9299:
9298:
9289:
9286:
9281:
9244:
9242:
9241:
9236:
9224:
9222:
9221:
9216:
9198:
9196:
9195:
9190:
9178:
9176:
9175:
9170:
9168:
9167:
9159:
9144:
9121:
9119:
9118:
9113:
9111:
9103:
9102:
9097:
9078:
9076:
9075:
9070:
9053:
9052:
9043:
9042:
9027:
9022:
9018:
9017:
9007:
9005:
9004:
8989:
8988:
8972:
8970:
8969:
8964:
8962:
8961:
8952:
8951:
8927:
8926:
8914:
8913:
8894:
8892:
8891:
8886:
8875:
8874:
8858:
8856:
8855:
8850:
8848:
8847:
8831:
8829:
8828:
8823:
8818:
8817:
8805:
8804:
8786:
8785:
8769:
8768:
8759:
8758:
8740:
8739:
8720:
8718:
8717:
8712:
8710:
8709:
8691:-valued objects
8690:
8688:
8687:
8682:
8671:
8670:
8651:
8649:
8648:
8643:
8641:
8640:
8635:
8422:Pontrjagin class
8302:
8300:
8299:
8294:
8289:
8285:
8281:
8273:
8259:
8258:
8257:
8256:
8251:
8235:
8227:
8219:
8211:
8194:
8193:
8192:
8191:
8186:
8175:
8167:
8152:
8150:
8149:
8144:
8139:
8131:
8114:
8106:
8089:
8088:
8087:
8086:
8081:
8061:
8053:
8052:
8037:
8023:
8018:
8017:
7999:
7979:
7978:
7977:
7976:
7971:
7960:
7952:
7920:
7918:
7917:
7912:
7904:
7896:
7882:
7881:
7880:
7879:
7874:
7851:, to taking the
7823:
7821:
7820:
7815:
7810:
7802:
7785:
7784:
7783:
7782:
7777:
7766:
7758:
7742:non-perturbative
7734:Feynman diagrams
7724:
7722:
7721:
7716:
7714:
7713:
7708:
7661:
7659:
7658:
7653:
7651:
7643:
7635:
7624:
7608:spatial infinity
7561:
7559:
7558:
7553:
7400:Bianchi identity
7393:
7391:
7390:
7385:
7361:
7359:
7358:
7353:
7329:
7327:
7326:
7321:
7310:
7309:
7293:
7291:
7290:
7285:
7264:
7262:
7261:
7256:
7244:
7222:
7220:
7219:
7214:
7212:
7181:
7179:
7178:
7173:
7165:
7157:
7139:
7102:
7100:
7099:
7094:
7089:
7088:
7083:
7028:
7026:
7025:
7020:
7000:
6992:
6991:
6975:
6973:
6972:
6967:
6965:
6964:
6947:
6945:
6944:
6939:
6927:
6925:
6924:
6919:
6903:
6901:
6900:
6895:
6893:
6892:
6879:
6877:
6876:
6871:
6855:
6853:
6852:
6847:
6845:
6844:
6839:
6814:
6812:
6811:
6806:
6801:
6800:
6795:
6780:
6779:
6774:
6762:
6761:
6749:
6748:
6702:principal bundle
6682:
6680:
6679:
6674:
6663:
6658:
6657:
6641:
6627:
6600:
6561:
6559:
6558:
6553:
6539:
6527:
6525:
6524:
6519:
6511:
6499:
6497:
6496:
6491:
6475:
6473:
6472:
6467:
6450:
6448:
6447:
6442:
6429:
6427:
6426:
6421:
6401:
6399:
6398:
6393:
6390:
6377:
6375:
6374:
6369:
6361:
6360:
6340:
6338:
6337:
6332:
6330:
6329:
6305:
6303:
6302:
6297:
6295:
6294:
6277:Pontryagin index
6270:
6268:
6267:
6262:
6260:
6259:
6180:
6178:
6177:
6172:
6170:
6169:
6151:
6143:
6141:
6140:
6135:
6124:
6123:
6111:
6106:
6105:
6078:
6077:
6046:
6044:
6043:
6038:
6033:
6032:
6031:
6030:
6018:
6013:
6012:
5995:
5993:
5983:
5969:
5968:
5953:
5952:
5948:
5925:
5924:
5923:
5919:
5914:
5913:
5899:
5898:
5886:
5881:
5880:
5868:
5867:
5858:
5857:
5856:
5855:
5840:
5829:
5827:
5826:
5814:
5793:
5788:
5773:
5771:
5770:
5761:
5760:
5759:
5750:
5749:
5739:
5724:
5719:
5707:
5705:
5704:
5703:
5690:
5689:
5688:
5675:
5670:
5669:
5660:
5659:
5650:
5642:
5625:
5623:
5622:
5617:
5582:
5581:
5562:
5560:
5559:
5554:
5549:
5548:
5539:
5538:
5529:
5521:
5516:
5515:
5506:
5505:
5496:
5488:
5420:
5418:
5417:
5416:
5403:
5399:
5398:
5388:
5368:
5366:
5365:
5360:
5358:
5357:
5354:
5352:
5339:
5337:
5336:
5331:
5329:
5328:
5325:
5323:
5308:
5306:
5305:
5300:
5298:
5297:
5293:
5288:
5275:
5274:
5271:
5262:
5261:
5240:
5239:
5236:
5227:
5226:
5213:
5200:
5199:
5191:
5189:
5185:
5183:
5182:
5178:
5165:
5164:
5161:
5152:
5148:
5139:
5137:
5136:
5135:
5122:
5121:
5112:
5102:
5099:
5096:
5095:
5091:
5078:
5077:
5074:
5065:
5061:
5052:
5050:
5049:
5048:
5035:
5034:
5025:
5015:
5012:
5009:
5002:
5000:
4989:
4960:
4958:
4957:
4952:
4944:
4940:
4927:
4913:
4912:
4907:
4902:
4897:
4892:
4890:
4881:
4873:
4861:
4860:
4849:
4846:
4843:
4839:
4838:
4834:
4817:
4809:
4808:
4782:
4777:
4776:
4764:
4763:
4745:
4743:
4742:
4737:
4729:
4728:
4716:
4690:
4688:
4687:
4682:
4680:
4676:
4675:
4670:
4666:
4665:
4664:
4663:
4655:
4637:
4627:
4610:
4609:
4608:
4607:
4599:
4582:
4579:
4574:
4573:
4555:
4553:
4552:
4547:
4545:
4544:
4526:
4524:
4523:
4518:
4516:
4515:
4506:
4503:
4498:
4497:
4488:
4485:
4481:
4478:
4474:
4472:
4471:
4470:
4461:
4458:
4455:
4454:
4453:
4444:
4441:
4438:
4436:
4434:
4423:
4415:
4414:
4402:
4399:
4395:
4394:
4391:
4389:
4387:
4376:
4371:
4366:
4363:
4361:
4353:
4324:
4322:
4321:
4316:
4304:
4302:
4301:
4296:
4191:Mathieu function
4178:
4176:
4175:
4170:
4146:
4144:
4143:
4138:
4133:
4129:
4125:
4124:
4092:
4084:
4070:
4069:
4050:
4048:
4047:
4042:
4034:
4033:
4011:
4009:
4008:
4003:
3989:
3978:
3976:
3975:
3974:
3961:
3957:
3956:
3946:
3931:
3929:
3928:
3923:
3921:
3920:
3915:
3911:
3910:
3908:
3907:
3898:
3897:
3896:
3887:
3886:
3876:
3862:
3860:
3859:
3858:
3845:
3844:
3835:
3784:
3782:
3781:
3776:
3742:
3740:
3739:
3734:
3726:
3725:
3706:
3704:
3703:
3698:
3693:
3692:
3691:
3690:
3681:
3676:
3671:
3666:
3665:
3648:
3646:
3636:
3622:
3621:
3606:
3605:
3601:
3596:
3595:
3581:
3576:
3573:
3572:
3571:
3567:
3562:
3561:
3547:
3546:
3534:
3529:
3528:
3516:
3515:
3506:
3505:
3498:
3497:
3482:
3469:
3467:
3466:
3461:
3456:
3454:
3453:
3441:
3420:
3415:
3400:
3398:
3397:
3396:
3383:
3382:
3381:
3372:
3371:
3362:
3357:
3354:
3349:
3347:
3346:
3345:
3332:
3321:
3316:
3301:
3300:
3290:
3285:
3284:
3275:
3274:
3265:
3259:
3255:
3250:
3248:
3247:
3246:
3237:
3236:
3226:
3225:
3216:
3205:
3204:
3192:
3191:
3179:
3178:
3159:
3157:
3156:
3151:
3119:
3118:
3099:
3097:
3096:
3091:
3080:
3079:
3060:
3059:
3044:
3043:
3034:
3033:
3024:
3016:
3011:
3010:
3001:
3000:
2991:
2983:
2950:
2948:
2947:
2942:
2892:
2890:
2889:
2888:
2875:
2862:
2861:
2851:
2826:
2824:
2823:
2818:
2816:
2815:
2793:
2791:
2790:
2785:
2767:
2765:
2764:
2759:
2738:
2736:
2735:
2730:
2728:
2727:
2708:
2706:
2705:
2700:
2695:
2691:
2687:
2686:
2668:
2666:
2661:
2656:
2618:
2616:
2615:
2610:
2602:
2601:
2585:
2583:
2582:
2577:
2566:
2565:
2549:
2547:
2546:
2541:
2530:
2529:
2507:
2505:
2504:
2499:
2485:
2484:
2462:
2460:
2459:
2454:
2452:
2435:
2430:
2428:
2420:
2412:
2400:
2398:
2397:
2392:
2387:
2382:
2381:
2376:
2370:
2356:
2354:
2353:
2348:
2340:
2339:
2321:
2319:
2314:
2309:
2300:
2295:
2280:
2279:
2274:
2270:
2269:
2252:
2247:
2245:
2237:
2229:
2221:
2213:
2204:
2203:
2202:
2192:
2191:
2190:
2164:
2162:
2161:
2156:
2154:
2140:
2138:
2136:
2128:
2120:
2111:
2110:
2109:
2099:
2098:
2097:
2083:
2078:
2073:
2072:
2067:
2063:
2062:
2045:
2040:
2038:
2030:
2022:
2014:
2006:
1997:
1996:
1995:
1985:
1984:
1983:
1966:
1965:
1946:
1944:
1943:
1938:
1920:
1918:
1917:
1912:
1891:
1889:
1888:
1883:
1865:
1863:
1862:
1857:
1836:
1834:
1833:
1828:
1807:
1805:
1804:
1799:
1781:
1779:
1778:
1773:
1771:
1770:
1755:
1754:
1742:
1734:
1710:
1708:
1707:
1702:
1700:
1699:
1680:
1678:
1677:
1672:
1651:
1649:
1648:
1643:
1601:
1599:
1598:
1593:
1582:
1578:
1562:
1561:
1556:
1552:
1550:
1542:
1534:
1524:
1516:
1508:
1507:
1506:
1496:
1495:
1494:
1477:
1476:
1454:
1452:
1451:
1446:
1441:
1440:
1439:
1434:
1415:
1414:
1404:
1355:
1350:
1349:
1348:
1343:
1339:
1333:
1323:
1282:
1281:
1262:
1260:
1259:
1254:
1226:
1224:
1223:
1218:
1213:
1212:
1207:
1181:
1136:
1131:
1130:
1129:
1124:
1120:
1111:
1101:
1020:
1018:
1017:
1012:
1007:
1006:
998:
966:
963:
958:
949:
941:
918:
916:
915:
910:
905:
878:
877:
856:
854:
853:
848:
836:
800:
798:
797:
792:
784:
782:
781:
772:
740:
735:
733:
732:
731:
718:
714:
713:
703:
687:
685:
684:
679:
651:
649:
648:
643:
619:
617:
616:
611:
549:
547:
546:
545:
532:
531:
522:
520:
518:
510:
509:
500:
479:
477:
476:
471:
450:
448:
447:
442:
437:
436:
421:
420:
408:
400:
265:principal bundle
117:compactification
92:(top left). The
56:
47:
36:
27:
11235:
11234:
11230:
11229:
11228:
11226:
11225:
11224:
11190:
11189:
11188:
11183:
10692:
10669:
10646:
10593:
10541:
10511:KĂ€hler manifold
10478:
10455:
10448:
10441:
10434:
10427:
10386:
10347:Mirror symmetry
10328:
10314:Brane cosmology
10260:
10209:
10176:
10132:N=2 superstring
10118:Type IIB string
10113:Type IIA string
10088:Polyakov action
10071:
10028:
10023:
9977:
9876:
9875:
9868:
9854:
9850:
9827:
9823:
9774:
9770:
9731:
9727:
9700:
9696:
9689:
9669:
9665:
9626:
9622:
9617:
9610:
9597:
9593:
9580:
9576:
9567:
9563:
9554:
9552:
9544:
9543:
9539:
9504:
9497:
9492:
9488:
9483:
9479:
9474:
9470:
9460:
9452:
9448:
9440:
9436:
9427:
9423:
9414:
9381:
9360:Instanton fluid
9356:
9328:
9323:
9322:
9316:
9312:
9301:
9300:
9294:
9290:
9288:
9282:
9271:
9250:
9247:
9246:
9230:
9227:
9226:
9225:of these gives
9204:
9201:
9200:
9184:
9181:
9180:
9160:
9155:
9154:
9140:
9138:
9135:
9134:
9107:
9098:
9093:
9092:
9084:
9081:
9080:
9048:
9044:
9035:
9031:
9013:
9009:
9008:
9006:
8997:
8993:
8984:
8980:
8978:
8975:
8974:
8957:
8953:
8941:
8937:
8922:
8918:
8909:
8905:
8900:
8897:
8896:
8867:
8866:
8864:
8861:
8860:
8843:
8839:
8837:
8834:
8833:
8813:
8809:
8797:
8793:
8778:
8774:
8764:
8760:
8748:
8744:
8732:
8728:
8726:
8723:
8722:
8702:
8698:
8696:
8693:
8692:
8663:
8662:
8660:
8657:
8656:
8636:
8631:
8630:
8628:
8625:
8624:
8605:
8586:Hiraku Nakajima
8582:Andrei Okounkov
8578:Nikita Nekrasov
8487:
8468:mirror symmetry
8395:
8367:string theories
8343:Goldstone-boson
8277:
8269:
8252:
8247:
8246:
8245:
8241:
8240:
8236:
8226:
8215:
8207:
8187:
8182:
8181:
8180:
8176:
8166:
8164:
8161:
8160:
8135:
8127:
8110:
8102:
8082:
8077:
8076:
8075:
8071:
8057:
8045:
8041:
8033:
8019:
8007:
8003:
7995:
7972:
7967:
7966:
7965:
7961:
7951:
7943:
7940:
7939:
7900:
7892:
7875:
7870:
7869:
7868:
7864:
7862:
7859:
7858:
7849:Stokes' theorem
7827:where â is the
7806:
7798:
7778:
7773:
7772:
7771:
7767:
7757:
7755:
7752:
7751:
7709:
7704:
7703:
7701:
7698:
7697:
7694:four dimensions
7647:
7639:
7631:
7620:
7618:
7615:
7614:
7582:Euclidean space
7409:
7406:
7405:
7394:where D is the
7367:
7364:
7363:
7338:
7335:
7334:
7305:
7301:
7299:
7296:
7295:
7273:
7270:
7269:
7237:
7235:
7232:
7231:
7208:
7206:
7203:
7202:
7158:
7153:
7135:
7133:
7130:
7129:
7084:
7073:
7072:
7037:
7034:
7033:
6993:
6987:
6983:
6981:
6978:
6977:
6960:
6959:
6957:
6954:
6953:
6933:
6930:
6929:
6913:
6910:
6909:
6888:
6887:
6885:
6882:
6881:
6865:
6862:
6861:
6840:
6829:
6828:
6823:
6820:
6819:
6796:
6785:
6784:
6775:
6764:
6763:
6757:
6753:
6741:
6737:
6735:
6732:
6731:
6698:
6687:Gerard 't Hooft
6659:
6647:
6643:
6628:
6614:
6596:
6594:
6591:
6590:
6535:
6533:
6530:
6529:
6507:
6505:
6502:
6501:
6485:
6482:
6481:
6461:
6458:
6457:
6436:
6433:
6432:
6412:
6409:
6408:
6386:
6384:
6381:
6380:
6356:
6352:
6347:
6344:
6343:
6325:
6321:
6319:
6316:
6315:
6290:
6286:
6284:
6281:
6280:
6255:
6251:
6249:
6246:
6245:
6194:
6165:
6161:
6159:
6156:
6155:
6149:
6119:
6115:
6107:
6101:
6097:
6073:
6069:
6055:
6052:
6051:
6026:
6022:
6014:
6008:
6004:
6000:
5996:
5979:
5964:
5960:
5944:
5940:
5936:
5926:
5915:
5909:
5905:
5904:
5900:
5894:
5890:
5882:
5876:
5872:
5863:
5859:
5851:
5847:
5846:
5842:
5841:
5839:
5822:
5818:
5813:
5789:
5784:
5766:
5762:
5755:
5751:
5745:
5741:
5740:
5738:
5720:
5715:
5699:
5695:
5691:
5684:
5680:
5676:
5674:
5665:
5661:
5655:
5651:
5641:
5633:
5630:
5629:
5577:
5573:
5571:
5568:
5567:
5544:
5540:
5534:
5530:
5520:
5511:
5507:
5501:
5497:
5487:
5412:
5408:
5404:
5394:
5390:
5389:
5387:
5385:
5382:
5381:
5375:
5353:
5348:
5347:
5345:
5342:
5341:
5324:
5319:
5318:
5316:
5313:
5312:
5270:
5266:
5257:
5253:
5235:
5231:
5222:
5218:
5214:
5212:
5208:
5201:
5190:
5160:
5156:
5144:
5143:
5131:
5127:
5123:
5117:
5113:
5111:
5107:
5103:
5098:
5097:
5073:
5069:
5057:
5056:
5044:
5040:
5036:
5030:
5026:
5024:
5020:
5016:
5011:
5010:
5008:
5004:
5003:
4993:
4988:
4971:
4968:
4967:
4923:
4908:
4893:
4891:
4889:
4888:
4887:
4883:
4874:
4869:
4856:
4852:
4830:
4813:
4804:
4800:
4796:
4792:
4778:
4772:
4771:
4759:
4755:
4753:
4750:
4749:
4724:
4720:
4712:
4698:
4695:
4694:
4671:
4654:
4653:
4646:
4642:
4638:
4633:
4632:
4628:
4623:
4598:
4597:
4590:
4586:
4578:
4569:
4565:
4563:
4560:
4559:
4540:
4536:
4534:
4531:
4530:
4511:
4507:
4502:
4493:
4489:
4484:
4466:
4462:
4457:
4456:
4449:
4445:
4440:
4439:
4437:
4427:
4422:
4410:
4406:
4398:
4390:
4380:
4375:
4367:
4362:
4352:
4332:
4329:
4328:
4310:
4307:
4306:
4290:
4287:
4286:
4278:
4253:pseudoparticles
4240:
4234:
4155:
4152:
4151:
4120:
4116:
4103:
4099:
4083:
4065:
4061:
4059:
4056:
4055:
4026:
4022:
4020:
4017:
4016:
3982:
3970:
3966:
3962:
3952:
3948:
3947:
3945:
3943:
3940:
3939:
3916:
3903:
3899:
3892:
3888:
3882:
3878:
3877:
3875:
3868:
3864:
3863:
3854:
3850:
3846:
3840:
3836:
3834:
3817:
3814:
3813:
3761:
3758:
3757:
3749:
3721:
3717:
3715:
3712:
3711:
3686:
3682:
3675:
3667:
3661:
3657:
3653:
3649:
3632:
3617:
3613:
3597:
3591:
3587:
3586:
3582:
3575:
3574:
3563:
3557:
3553:
3552:
3548:
3542:
3538:
3530:
3524:
3520:
3511:
3507:
3493:
3489:
3488:
3484:
3483:
3481:
3476:
3473:
3472:
3449:
3445:
3440:
3416:
3411:
3392:
3388:
3384:
3377:
3373:
3367:
3363:
3356:
3355:
3353:
3341:
3337:
3333:
3317:
3312:
3296:
3292:
3291:
3289:
3280:
3276:
3270:
3266:
3254:
3242:
3238:
3232:
3228:
3227:
3221:
3217:
3215:
3200:
3196:
3187:
3183:
3174:
3170:
3168:
3165:
3164:
3114:
3110:
3108:
3105:
3104:
3075:
3071:
3055:
3051:
3039:
3035:
3029:
3025:
3015:
3006:
3002:
2996:
2992:
2982:
2962:
2959:
2958:
2884:
2880:
2876:
2857:
2853:
2852:
2850:
2848:
2845:
2844:
2833:
2811:
2807:
2799:
2796:
2795:
2773:
2770:
2769:
2744:
2741:
2740:
2723:
2719:
2717:
2714:
2713:
2682:
2678:
2660:
2655:
2654:
2650:
2627:
2624:
2623:
2597:
2593:
2591:
2588:
2587:
2561:
2557:
2555:
2552:
2551:
2525:
2521:
2513:
2510:
2509:
2480:
2476:
2468:
2465:
2464:
2434:
2421:
2413:
2411:
2409:
2406:
2405:
2375:
2371:
2369:
2363:
2360:
2359:
2335:
2331:
2313:
2308:
2296:
2288:
2275:
2251:
2238:
2230:
2228:
2227:
2223:
2222:
2212:
2198:
2194:
2193:
2186:
2182:
2181:
2171:
2168:
2167:
2139:
2129:
2121:
2119:
2105:
2101:
2100:
2093:
2089:
2088:
2077:
2068:
2044:
2031:
2023:
2021:
2020:
2016:
2015:
2005:
1991:
1987:
1986:
1979:
1975:
1974:
1961:
1957:
1955:
1952:
1951:
1926:
1923:
1922:
1897:
1894:
1893:
1871:
1868:
1867:
1842:
1839:
1838:
1813:
1810:
1809:
1787:
1784:
1783:
1766:
1762:
1750:
1746:
1733:
1716:
1713:
1712:
1695:
1691:
1689:
1686:
1685:
1657:
1654:
1653:
1610:
1607:
1606:
1557:
1543:
1535:
1533:
1529:
1528:
1515:
1514:
1510:
1502:
1498:
1497:
1490:
1486:
1485:
1472:
1468:
1466:
1463:
1462:
1410:
1406:
1405:
1403:
1399:
1395:
1351:
1335:
1334:
1332:
1328:
1324:
1319:
1277:
1273:
1271:
1268:
1267:
1239:
1236:
1235:
1182:
1180:
1176:
1132:
1116:
1112:
1110:
1106:
1102:
1097:
1056:
1053:
1052:
1038:
965:
959:
954:
940:
936:
932:
930:
927:
926:
876:
868:
865:
864:
832:
812:
809:
808:
777:
773:
741:
739:
727:
723:
719:
709:
705:
704:
702:
700:
697:
696:
673:
670:
669:
662:
628:
625:
624:
541:
537:
533:
527:
523:
521:
511:
505:
501:
499:
494:
491:
490:
456:
453:
452:
432:
428:
416:
412:
399:
382:
379:
378:
375:
355:
323:Simon Donaldson
253:
243:
194:of the action,
184:critical points
158:non-zero action
156:with a finite,
138:
137:
136:
135:
97:
86:
67:
59:
58:
57:
49:
48:
39:
38:
37:
29:
28:
17:
12:
11:
5:
11233:
11223:
11222:
11217:
11212:
11207:
11205:Gauge theories
11202:
11185:
11184:
11182:
11181:
11176:
11171:
11166:
11161:
11156:
11151:
11146:
11141:
11136:
11131:
11126:
11121:
11116:
11111:
11106:
11101:
11096:
11091:
11086:
11081:
11076:
11071:
11066:
11061:
11056:
11051:
11046:
11041:
11036:
11031:
11026:
11021:
11019:Randjbar-Daemi
11016:
11011:
11006:
11001:
10996:
10991:
10986:
10981:
10976:
10971:
10966:
10961:
10956:
10951:
10946:
10941:
10936:
10931:
10926:
10921:
10916:
10911:
10906:
10901:
10896:
10891:
10886:
10881:
10876:
10871:
10866:
10861:
10856:
10851:
10846:
10841:
10836:
10831:
10826:
10821:
10816:
10811:
10806:
10801:
10796:
10791:
10786:
10781:
10776:
10771:
10766:
10761:
10756:
10751:
10746:
10741:
10736:
10731:
10726:
10721:
10716:
10711:
10706:
10700:
10698:
10694:
10693:
10691:
10690:
10685:
10679:
10677:
10671:
10670:
10668:
10667:
10662:
10656:
10654:
10648:
10647:
10645:
10644:
10642:Lie supergroup
10639:
10634:
10629:
10624:
10619:
10614:
10609:
10603:
10601:
10595:
10594:
10592:
10591:
10586:
10581:
10576:
10571:
10566:
10561:
10556:
10551:
10550:
10549:
10544:
10539:
10535:
10534:
10533:
10523:
10513:
10508:
10502:
10497:
10492:
10486:
10484:
10480:
10479:
10477:
10476:
10468:
10463:
10458:
10453:
10446:
10439:
10432:
10425:
10417:
10412:
10407:
10402:
10396:
10394:
10388:
10387:
10385:
10384:
10379:
10374:
10369:
10364:
10359:
10354:
10349:
10344:
10338:
10336:
10330:
10329:
10327:
10326:
10321:
10319:Quiver diagram
10316:
10311:
10306:
10301:
10296:
10291:
10286:
10281:
10276:
10270:
10268:
10262:
10261:
10259:
10258:
10253:
10248:
10243:
10238:
10233:
10228:
10223:
10217:
10215:
10211:
10210:
10208:
10207:
10202:
10197:
10192:
10186:
10184:
10182:String duality
10178:
10177:
10175:
10174:
10169:
10164:
10159:
10154:
10149:
10144:
10139:
10134:
10129:
10128:
10127:
10122:
10121:
10120:
10115:
10108:Type II string
10105:
10095:
10090:
10085:
10079:
10077:
10073:
10072:
10070:
10069:
10064:
10063:
10062:
10057:
10047:
10045:Cosmic strings
10042:
10036:
10034:
10030:
10029:
10022:
10021:
10014:
10007:
9999:
9993:
9992:
9976:
9975:External links
9973:
9972:
9971:
9955:
9941:
9919:Sidney Coleman
9912:
9899:
9881:
9880:
9874:
9873:
9866:
9848:
9821:
9768:
9725:
9694:
9687:
9663:
9620:
9608:
9591:
9574:
9561:
9537:
9495:
9486:
9477:
9467:
9466:
9465:
9464:
9459:
9458:
9446:
9434:
9420:
9419:
9418:
9413:
9410:
9409:
9408:
9402:
9396:
9390:
9384:
9375:
9372:Sidney Coleman
9369:
9363:
9355:
9352:
9339:
9331:
9326:
9319:
9315:
9311:
9308:
9304:
9297:
9293:
9285:
9280:
9277:
9274:
9270:
9266:
9263:
9260:
9257:
9254:
9234:
9214:
9211:
9208:
9199:. Superposing
9188:
9179:for any fixed
9166:
9163:
9158:
9153:
9150:
9147:
9143:
9110:
9106:
9101:
9096:
9091:
9088:
9068:
9065:
9062:
9059:
9056:
9051:
9047:
9041:
9038:
9034:
9030:
9025:
9021:
9016:
9012:
9003:
9000:
8996:
8992:
8987:
8983:
8960:
8956:
8950:
8947:
8944:
8940:
8936:
8933:
8930:
8925:
8921:
8917:
8912:
8908:
8904:
8884:
8881:
8878:
8873:
8870:
8859:is a basis of
8846:
8842:
8821:
8816:
8812:
8808:
8803:
8800:
8796:
8792:
8789:
8784:
8781:
8777:
8773:
8767:
8763:
8757:
8754:
8751:
8747:
8743:
8738:
8735:
8731:
8708:
8705:
8701:
8680:
8677:
8674:
8669:
8666:
8639:
8634:
8604:
8601:
8554:Nathan Seiberg
8546:Nathan Seiberg
8522:Nathan Seiberg
8510:superpotential
8486:
8483:
8394:
8391:
8304:
8303:
8292:
8288:
8284:
8280:
8276:
8272:
8268:
8265:
8262:
8255:
8250:
8244:
8239:
8233:
8230:
8225:
8222:
8218:
8214:
8210:
8206:
8203:
8200:
8197:
8190:
8185:
8179:
8173:
8170:
8154:
8153:
8142:
8138:
8134:
8130:
8126:
8123:
8120:
8117:
8113:
8109:
8105:
8101:
8098:
8095:
8092:
8085:
8080:
8074:
8070:
8067:
8064:
8060:
8056:
8051:
8048:
8044:
8040:
8036:
8032:
8029:
8026:
8022:
8016:
8013:
8010:
8006:
8002:
7998:
7994:
7991:
7988:
7985:
7982:
7975:
7970:
7964:
7958:
7955:
7950:
7947:
7926:homotopy class
7922:
7921:
7910:
7907:
7903:
7899:
7895:
7891:
7888:
7885:
7878:
7873:
7867:
7825:
7824:
7813:
7809:
7805:
7801:
7797:
7794:
7791:
7788:
7781:
7776:
7770:
7764:
7761:
7712:
7707:
7663:
7662:
7650:
7646:
7642:
7638:
7634:
7630:
7627:
7623:
7565:is satisfied.
7563:
7562:
7551:
7548:
7545:
7542:
7539:
7536:
7533:
7530:
7527:
7524:
7521:
7518:
7515:
7512:
7509:
7506:
7503:
7500:
7497:
7494:
7491:
7488:
7485:
7482:
7479:
7476:
7473:
7470:
7467:
7464:
7461:
7458:
7455:
7452:
7449:
7446:
7443:
7440:
7437:
7434:
7431:
7428:
7425:
7422:
7419:
7416:
7413:
7383:
7380:
7377:
7374:
7371:
7351:
7348:
7345:
7342:
7319:
7316:
7313:
7308:
7304:
7283:
7280:
7277:
7266:
7265:
7253:
7250:
7247:
7243:
7240:
7211:
7183:
7182:
7171:
7168:
7164:
7161:
7156:
7151:
7148:
7145:
7142:
7138:
7104:
7103:
7092:
7087:
7082:
7079:
7076:
7071:
7068:
7065:
7062:
7059:
7056:
7053:
7050:
7047:
7044:
7041:
7018:
7015:
7012:
7009:
7006:
7003:
6999:
6996:
6990:
6986:
6963:
6937:
6917:
6891:
6869:
6843:
6838:
6835:
6832:
6827:
6816:
6815:
6804:
6799:
6794:
6791:
6788:
6783:
6778:
6773:
6770:
6767:
6760:
6756:
6752:
6747:
6744:
6740:
6697:
6694:
6684:
6683:
6672:
6669:
6666:
6662:
6656:
6653:
6650:
6646:
6640:
6637:
6634:
6631:
6626:
6623:
6620:
6617:
6613:
6609:
6606:
6603:
6599:
6584:Yu. S. Tyupkin
6568:BPST instanton
6551:
6548:
6545:
6542:
6538:
6517:
6514:
6510:
6489:
6465:
6440:
6419:
6416:
6404:
6403:
6389:
6367:
6364:
6359:
6355:
6351:
6328:
6324:
6293:
6289:
6258:
6254:
6235:homotopy group
6203:
6202:
6182:
6181:
6168:
6164:
6148:
6145:
6133:
6130:
6127:
6122:
6118:
6114:
6110:
6104:
6100:
6096:
6093:
6090:
6087:
6084:
6081:
6076:
6072:
6068:
6065:
6062:
6059:
6048:
6047:
6036:
6029:
6025:
6021:
6017:
6011:
6007:
6003:
5999:
5992:
5989:
5986:
5982:
5978:
5975:
5972:
5967:
5963:
5959:
5956:
5951:
5947:
5943:
5939:
5935:
5932:
5929:
5922:
5918:
5912:
5908:
5903:
5897:
5893:
5889:
5885:
5879:
5875:
5871:
5866:
5862:
5854:
5850:
5845:
5838:
5835:
5832:
5825:
5821:
5817:
5812:
5809:
5806:
5803:
5800:
5797:
5792:
5787:
5783:
5779:
5776:
5769:
5765:
5758:
5754:
5748:
5744:
5737:
5734:
5731:
5728:
5723:
5718:
5714:
5710:
5702:
5698:
5694:
5687:
5683:
5679:
5673:
5668:
5664:
5658:
5654:
5648:
5645:
5640:
5637:
5615:
5612:
5609:
5606:
5603:
5600:
5597:
5594:
5591:
5588:
5585:
5580:
5576:
5564:
5563:
5552:
5547:
5543:
5537:
5533:
5527:
5524:
5519:
5514:
5510:
5504:
5500:
5494:
5491:
5486:
5483:
5480:
5477:
5474:
5468:
5465:
5462:
5459:
5456:
5453:
5450:
5447:
5444:
5441:
5438:
5435:
5432:
5429:
5426:
5423:
5415:
5411:
5407:
5402:
5397:
5393:
5374:
5371:
5351:
5322:
5296:
5291:
5287:
5284:
5281:
5278:
5269:
5265:
5260:
5256:
5252:
5249:
5246:
5243:
5234:
5230:
5225:
5221:
5217:
5211:
5207:
5204:
5197:
5194:
5188:
5181:
5177:
5174:
5171:
5168:
5159:
5155:
5151:
5147:
5142:
5134:
5130:
5126:
5120:
5116:
5110:
5106:
5094:
5090:
5087:
5084:
5081:
5072:
5068:
5064:
5060:
5055:
5047:
5043:
5039:
5033:
5029:
5023:
5019:
5007:
4999:
4996:
4992:
4987:
4984:
4981:
4978:
4975:
4950:
4947:
4943:
4939:
4936:
4933:
4930:
4926:
4922:
4919:
4916:
4911:
4905:
4900:
4896:
4886:
4880:
4877:
4872:
4868:
4864:
4859:
4855:
4842:
4837:
4833:
4829:
4826:
4823:
4820:
4816:
4812:
4807:
4803:
4799:
4795:
4791:
4788:
4785:
4781:
4775:
4770:
4767:
4762:
4758:
4735:
4732:
4727:
4723:
4719:
4715:
4711:
4708:
4705:
4702:
4679:
4674:
4669:
4661:
4658:
4652:
4649:
4645:
4641:
4636:
4631:
4626:
4622:
4619:
4616:
4613:
4605:
4602:
4596:
4593:
4589:
4585:
4577:
4572:
4568:
4543:
4539:
4514:
4510:
4501:
4496:
4492:
4477:
4469:
4465:
4452:
4448:
4433:
4430:
4426:
4421:
4418:
4413:
4409:
4405:
4386:
4383:
4379:
4374:
4370:
4359:
4356:
4351:
4348:
4345:
4342:
4339:
4336:
4314:
4294:
4277:
4274:
4236:Main article:
4233:
4230:
4168:
4165:
4162:
4159:
4148:
4147:
4136:
4132:
4128:
4123:
4119:
4115:
4112:
4109:
4106:
4102:
4098:
4095:
4090:
4087:
4082:
4079:
4076:
4073:
4068:
4064:
4040:
4037:
4032:
4029:
4025:
4013:
4012:
4001:
3998:
3995:
3992:
3988:
3985:
3981:
3973:
3969:
3965:
3960:
3955:
3951:
3933:
3932:
3919:
3914:
3906:
3902:
3895:
3891:
3885:
3881:
3874:
3871:
3867:
3857:
3853:
3849:
3843:
3839:
3833:
3830:
3827:
3824:
3821:
3774:
3771:
3768:
3765:
3748:
3745:
3732:
3729:
3724:
3720:
3708:
3707:
3696:
3689:
3685:
3679:
3674:
3670:
3664:
3660:
3656:
3652:
3645:
3642:
3639:
3635:
3631:
3628:
3625:
3620:
3616:
3612:
3609:
3604:
3600:
3594:
3590:
3585:
3579:
3570:
3566:
3560:
3556:
3551:
3545:
3541:
3537:
3533:
3527:
3523:
3519:
3514:
3510:
3504:
3501:
3496:
3492:
3487:
3480:
3470:
3459:
3452:
3448:
3444:
3439:
3436:
3433:
3430:
3427:
3424:
3419:
3414:
3410:
3406:
3403:
3395:
3391:
3387:
3380:
3376:
3370:
3366:
3360:
3352:
3344:
3340:
3336:
3331:
3328:
3325:
3320:
3315:
3311:
3307:
3304:
3299:
3295:
3288:
3283:
3279:
3273:
3269:
3262:
3258:
3253:
3245:
3241:
3235:
3231:
3224:
3220:
3214:
3211:
3208:
3203:
3199:
3195:
3190:
3186:
3182:
3177:
3173:
3149:
3146:
3143:
3140:
3137:
3134:
3131:
3128:
3125:
3122:
3117:
3113:
3101:
3100:
3089:
3086:
3083:
3078:
3074:
3069:
3066:
3063:
3058:
3054:
3047:
3042:
3038:
3032:
3028:
3022:
3019:
3014:
3009:
3005:
2999:
2995:
2989:
2986:
2981:
2978:
2975:
2972:
2969:
2966:
2952:
2951:
2940:
2937:
2934:
2931:
2928:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2904:
2901:
2898:
2895:
2887:
2883:
2879:
2874:
2871:
2868:
2865:
2860:
2856:
2840:the equations
2832:
2829:
2814:
2810:
2806:
2803:
2783:
2780:
2777:
2757:
2754:
2751:
2748:
2726:
2722:
2710:
2709:
2698:
2694:
2690:
2685:
2681:
2677:
2674:
2671:
2664:
2659:
2653:
2649:
2646:
2643:
2640:
2637:
2634:
2631:
2608:
2605:
2600:
2596:
2575:
2572:
2569:
2564:
2560:
2539:
2536:
2533:
2528:
2524:
2520:
2517:
2497:
2494:
2491:
2488:
2483:
2479:
2475:
2472:
2450:
2447:
2444:
2441:
2438:
2433:
2427:
2424:
2419:
2416:
2402:
2401:
2390:
2385:
2379:
2374:
2368:
2357:
2346:
2343:
2338:
2334:
2330:
2327:
2324:
2317:
2312:
2307:
2304:
2299:
2294:
2291:
2287:
2283:
2278:
2273:
2267:
2264:
2261:
2258:
2255:
2250:
2244:
2241:
2236:
2233:
2226:
2219:
2216:
2211:
2208:
2201:
2197:
2189:
2185:
2180:
2176:
2165:
2152:
2149:
2146:
2143:
2135:
2132:
2127:
2124:
2118:
2115:
2108:
2104:
2096:
2092:
2087:
2081:
2076:
2071:
2066:
2060:
2057:
2054:
2051:
2048:
2043:
2037:
2034:
2029:
2026:
2019:
2012:
2009:
2004:
2001:
1994:
1990:
1982:
1978:
1973:
1969:
1964:
1960:
1936:
1933:
1930:
1910:
1907:
1904:
1901:
1881:
1878:
1875:
1855:
1852:
1849:
1846:
1826:
1823:
1820:
1817:
1797:
1794:
1791:
1769:
1765:
1761:
1758:
1753:
1749:
1745:
1740:
1737:
1732:
1729:
1726:
1723:
1720:
1698:
1694:
1670:
1667:
1664:
1661:
1641:
1638:
1635:
1632:
1629:
1626:
1623:
1620:
1617:
1614:
1603:
1602:
1591:
1588:
1585:
1581:
1577:
1574:
1571:
1568:
1565:
1560:
1555:
1549:
1546:
1541:
1538:
1532:
1527:
1522:
1519:
1513:
1505:
1501:
1493:
1489:
1484:
1480:
1475:
1471:
1456:
1455:
1444:
1437:
1433:
1430:
1427:
1424:
1421:
1418:
1413:
1409:
1402:
1398:
1394:
1391:
1388:
1385:
1382:
1379:
1376:
1373:
1370:
1367:
1364:
1361:
1358:
1354:
1346:
1342:
1338:
1331:
1327:
1322:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1280:
1276:
1252:
1249:
1246:
1243:
1228:
1227:
1216:
1210:
1206:
1203:
1200:
1197:
1194:
1191:
1188:
1185:
1179:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1135:
1127:
1123:
1119:
1115:
1109:
1105:
1100:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1042:path integrals
1037:
1034:
1022:
1021:
1010:
1005:
1002:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
962:
957:
953:
947:
944:
939:
935:
920:
919:
908:
903:
899:
896:
893:
890:
887:
884:
881:
875:
872:
858:
857:
845:
842:
839:
835:
831:
828:
825:
822:
819:
816:
802:
801:
790:
787:
780:
776:
771:
768:
765:
762:
759:
756:
753:
750:
747:
744:
738:
730:
726:
722:
717:
712:
708:
677:
661:
658:
641:
638:
635:
632:
621:
620:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
552:
544:
540:
536:
530:
526:
517:
514:
508:
504:
498:
469:
466:
463:
460:
440:
435:
431:
427:
424:
419:
415:
411:
406:
403:
398:
395:
392:
389:
386:
374:
371:
354:
351:
313:, and via the
307:vector bundles
303:twistor theory
292:pseudoparticle
242:
239:
223:
222:
215:
146:pseudoparticle
95:
84:
71:BPST instanton
65:
61:
60:
51:
50:
42:
41:
40:
31:
30:
22:
21:
20:
19:
18:
15:
9:
6:
4:
3:
2:
11232:
11221:
11218:
11216:
11213:
11211:
11208:
11206:
11203:
11201:
11198:
11197:
11195:
11180:
11177:
11175:
11172:
11170:
11167:
11165:
11164:Zamolodchikov
11162:
11160:
11159:Zamolodchikov
11157:
11155:
11152:
11150:
11147:
11145:
11142:
11140:
11137:
11135:
11132:
11130:
11127:
11125:
11122:
11120:
11117:
11115:
11112:
11110:
11107:
11105:
11102:
11100:
11097:
11095:
11092:
11090:
11087:
11085:
11082:
11080:
11077:
11075:
11072:
11070:
11067:
11065:
11062:
11060:
11057:
11055:
11052:
11050:
11047:
11045:
11042:
11040:
11037:
11035:
11032:
11030:
11027:
11025:
11022:
11020:
11017:
11015:
11012:
11010:
11007:
11005:
11002:
11000:
10997:
10995:
10992:
10990:
10987:
10985:
10982:
10980:
10977:
10975:
10972:
10970:
10967:
10965:
10962:
10960:
10957:
10955:
10952:
10950:
10947:
10945:
10942:
10940:
10937:
10935:
10932:
10930:
10927:
10925:
10922:
10920:
10917:
10915:
10912:
10910:
10907:
10905:
10902:
10900:
10897:
10895:
10892:
10890:
10887:
10885:
10882:
10880:
10877:
10875:
10872:
10870:
10867:
10865:
10862:
10860:
10857:
10855:
10852:
10850:
10847:
10845:
10842:
10840:
10837:
10835:
10832:
10830:
10827:
10825:
10822:
10820:
10817:
10815:
10812:
10810:
10807:
10805:
10802:
10800:
10797:
10795:
10792:
10790:
10787:
10785:
10782:
10780:
10777:
10775:
10772:
10770:
10767:
10765:
10762:
10760:
10757:
10755:
10752:
10750:
10747:
10745:
10742:
10740:
10737:
10735:
10732:
10730:
10727:
10725:
10722:
10720:
10717:
10715:
10712:
10710:
10707:
10705:
10702:
10701:
10699:
10695:
10689:
10686:
10684:
10683:Matrix theory
10681:
10680:
10678:
10676:
10672:
10666:
10663:
10661:
10658:
10657:
10655:
10653:
10649:
10643:
10640:
10638:
10635:
10633:
10630:
10628:
10625:
10623:
10620:
10618:
10615:
10613:
10610:
10608:
10605:
10604:
10602:
10600:
10599:Supersymmetry
10596:
10590:
10587:
10585:
10582:
10580:
10577:
10575:
10572:
10570:
10567:
10565:
10562:
10560:
10557:
10555:
10552:
10548:
10545:
10543:
10536:
10532:
10529:
10528:
10527:
10524:
10522:
10519:
10518:
10517:
10514:
10512:
10509:
10506:
10503:
10501:
10498:
10496:
10493:
10491:
10488:
10487:
10485:
10481:
10475:
10473:
10469:
10467:
10464:
10462:
10459:
10456:
10449:
10442:
10435:
10428:
10421:
10418:
10416:
10413:
10411:
10408:
10406:
10403:
10401:
10398:
10397:
10395:
10393:
10389:
10383:
10380:
10378:
10375:
10373:
10370:
10368:
10365:
10363:
10360:
10358:
10355:
10353:
10350:
10348:
10345:
10343:
10340:
10339:
10337:
10335:
10331:
10325:
10322:
10320:
10317:
10315:
10312:
10310:
10307:
10305:
10302:
10300:
10297:
10295:
10292:
10290:
10287:
10285:
10282:
10280:
10277:
10275:
10272:
10271:
10269:
10267:
10263:
10257:
10254:
10252:
10251:Dual graviton
10249:
10247:
10244:
10242:
10239:
10237:
10234:
10232:
10229:
10227:
10224:
10222:
10219:
10218:
10216:
10212:
10206:
10203:
10201:
10198:
10196:
10193:
10191:
10188:
10187:
10185:
10183:
10179:
10173:
10170:
10168:
10167:RNS formalism
10165:
10163:
10160:
10158:
10155:
10153:
10150:
10148:
10145:
10143:
10140:
10138:
10135:
10133:
10130:
10126:
10123:
10119:
10116:
10114:
10111:
10110:
10109:
10106:
10104:
10103:Type I string
10101:
10100:
10099:
10096:
10094:
10091:
10089:
10086:
10084:
10081:
10080:
10078:
10074:
10068:
10065:
10061:
10058:
10056:
10053:
10052:
10051:
10048:
10046:
10043:
10041:
10038:
10037:
10035:
10031:
10027:
10026:String theory
10020:
10015:
10013:
10008:
10006:
10001:
10000:
9997:
9991:at Wiktionary
9990:
9989:
9983:
9979:
9978:
9969:
9968:0-19-853553-8
9965:
9961:
9960:
9956:
9953:
9949:
9945:
9942:
9940:
9936:
9935:0-521-31827-0
9932:
9928:
9924:
9920:
9916:
9913:
9911:
9910:0-444-87047-4
9907:
9903:
9900:
9898:
9894:
9890:
9886:
9883:
9882:
9878:
9877:
9869:
9867:9780198570639
9863:
9859:
9852:
9844:
9840:
9836:
9832:
9825:
9817:
9813:
9809:
9805:
9801:
9797:
9792:
9787:
9783:
9779:
9772:
9764:
9760:
9756:
9752:
9748:
9744:
9740:
9736:
9729:
9721:
9717:
9713:
9709:
9705:
9698:
9690:
9684:
9680:
9679:
9674:
9667:
9659:
9655:
9651:
9647:
9643:
9639:
9635:
9631:
9624:
9615:
9613:
9605:
9601:
9595:
9588:
9584:
9578:
9571:
9570:Nigel Hitchin
9565:
9551:
9547:
9541:
9533:
9529:
9525:
9521:
9517:
9513:
9509:
9502:
9500:
9490:
9481:
9472:
9468:
9462:
9461:
9456:
9450:
9444:
9438:
9431:
9425:
9421:
9416:
9415:
9406:
9403:
9400:
9397:
9394:
9391:
9388:
9385:
9379:
9376:
9373:
9370:
9367:
9364:
9361:
9358:
9357:
9351:
9337:
9329:
9317:
9313:
9309:
9306:
9295:
9291:
9283:
9278:
9275:
9272:
9268:
9264:
9258:
9252:
9232:
9212:
9209:
9206:
9186:
9164:
9161:
9151:
9148:
9145:
9132:
9128:
9123:
9122:is harmonic.
9099:
9089:
9086:
9063:
9057:
9054:
9049:
9039:
9036:
9032:
9028:
9023:
9019:
9014:
9001:
8998:
8994:
8990:
8985:
8981:
8958:
8954:
8948:
8945:
8942:
8938:
8934:
8931:
8923:
8919:
8915:
8910:
8906:
8879:
8844:
8840:
8819:
8814:
8810:
8806:
8801:
8798:
8794:
8790:
8787:
8782:
8779:
8775:
8771:
8765:
8761:
8755:
8752:
8749:
8745:
8741:
8736:
8733:
8729:
8706:
8703:
8699:
8675:
8653:
8637:
8622:
8618:
8614:
8610:
8600:
8598:
8593:
8591:
8590:Kota Yoshioka
8587:
8583:
8579:
8575:
8571:
8567:
8563:
8559:
8558:Edward Witten
8555:
8551:
8547:
8543:
8539:
8534:
8531:
8530:chiral matter
8527:
8523:
8519:
8515:
8511:
8507:
8502:
8498:
8496:
8492:
8482:
8480:
8476:
8471:
8469:
8465:
8462:are magnetic
8461:
8456:
8454:
8450:
8447:coupled to a
8446:
8442:
8438:
8434:
8430:
8425:
8423:
8419:
8415:
8411:
8410:unitary group
8407:
8404:
8399:
8390:
8388:
8384:
8380:
8376:
8372:
8368:
8364:
8360:
8356:
8352:
8348:
8344:
8340:
8336:
8331:
8329:
8325:
8321:
8317:
8313:
8309:
8290:
8286:
8274:
8263:
8260:
8253:
8242:
8237:
8231:
8228:
8223:
8212:
8204:
8198:
8195:
8188:
8177:
8171:
8168:
8159:
8158:
8157:
8132:
8124:
8121:
8118:
8115:
8107:
8099:
8093:
8090:
8083:
8072:
8068:
8054:
8049:
8046:
8042:
8038:
8027:
8014:
8011:
8008:
8004:
8000:
7992:
7983:
7980:
7973:
7962:
7956:
7953:
7948:
7945:
7938:
7937:
7936:
7934:
7929:
7927:
7908:
7897:
7886:
7883:
7876:
7865:
7857:
7856:
7855:
7854:
7850:
7846:
7842:
7838:
7834:
7830:
7803:
7795:
7789:
7786:
7779:
7768:
7762:
7759:
7750:
7749:
7748:
7745:
7743:
7739:
7735:
7731:
7726:
7710:
7695:
7691:
7687:
7683:
7682:abelian group
7679:
7675:
7670:
7668:
7644:
7636:
7628:
7625:
7613:
7612:
7611:
7609:
7605:
7601:
7598:
7595:
7591:
7587:
7586:Wick rotation
7583:
7579:
7578:topologically
7575:
7571:
7566:
7549:
7546:
7543:
7540:
7534:
7531:
7528:
7525:
7522:
7519:
7513:
7507:
7504:
7501:
7498:
7495:
7492:
7486:
7483:
7480:
7474:
7471:
7468:
7465:
7462:
7459:
7453:
7450:
7447:
7444:
7441:
7438:
7435:
7432:
7429:
7426:
7423:
7420:
7417:
7414:
7411:
7404:
7403:
7402:
7401:
7397:
7381:
7378:
7375:
7372:
7369:
7349:
7346:
7343:
7340:
7331:
7317:
7314:
7311:
7306:
7302:
7281:
7278:
7275:
7251:
7248:
7245:
7241:
7238:
7230:
7229:
7228:
7226:
7200:
7196:
7192:
7188:
7169:
7166:
7162:
7159:
7149:
7146:
7143:
7140:
7128:
7127:
7126:
7124:
7120:
7116:
7112:
7109:
7090:
7085:
7069:
7063:
7060:
7057:
7051:
7048:
7045:
7042:
7039:
7032:
7031:
7030:
7013:
7010:
7007:
7004:
6988:
6984:
6951:
6935:
6915:
6907:
6867:
6859:
6841:
6825:
6802:
6797:
6781:
6776:
6771:
6768:
6765:
6758:
6754:
6750:
6745:
6742:
6738:
6730:
6729:
6728:
6726:
6722:
6718:
6715:
6711:
6707:
6703:
6693:
6691:
6688:
6670:
6664:
6654:
6651:
6648:
6644:
6635:
6632:
6629:
6621:
6618:
6615:
6611:
6607:
6601:
6589:
6588:
6587:
6585:
6581:
6577:
6573:
6569:
6565:
6546:
6543:
6540:
6512:
6487:
6479:
6463:
6454:
6438:
6430:
6414:
6402:
6378:
6365:
6357:
6353:
6341:
6326:
6322:
6313:
6312:
6311:
6309:
6291:
6287:
6278:
6274:
6256:
6252:
6244:
6240:
6236:
6232:
6228:
6224:
6219:
6217:
6214:
6210:
6198:
6193:
6188:
6184:
6183:
6166:
6162:
6152:
6144:
6131:
6128:
6125:
6120:
6116:
6112:
6108:
6102:
6098:
6094:
6091:
6088:
6085:
6082:
6079:
6074:
6070:
6066:
6063:
6060:
6057:
6034:
6027:
6023:
6019:
6015:
6009:
6005:
6001:
5997:
5990:
5984:
5980:
5973:
5970:
5965:
5961:
5949:
5945:
5941:
5933:
5930:
5920:
5916:
5910:
5906:
5895:
5891:
5887:
5883:
5877:
5873:
5864:
5860:
5852:
5848:
5843:
5836:
5833:
5823:
5819:
5815:
5807:
5804:
5798:
5795:
5790:
5785:
5781:
5777:
5767:
5763:
5756:
5752:
5746:
5742:
5735:
5729:
5726:
5721:
5716:
5712:
5700:
5696:
5692:
5685:
5681:
5677:
5671:
5666:
5662:
5656:
5652:
5646:
5643:
5638:
5635:
5628:
5627:
5626:
5613:
5610:
5607:
5604:
5601:
5598:
5595:
5592:
5589:
5586:
5583:
5578:
5574:
5550:
5545:
5541:
5535:
5531:
5525:
5522:
5517:
5512:
5508:
5502:
5498:
5492:
5489:
5484:
5478:
5472:
5466:
5463:
5460:
5454:
5448:
5439:
5433:
5430:
5427:
5421:
5413:
5409:
5405:
5400:
5395:
5391:
5380:
5379:
5378:
5370:
5309:
5294:
5279:
5267:
5258:
5254:
5250:
5244:
5232:
5223:
5219:
5215:
5209:
5205:
5202:
5195:
5192:
5186:
5179:
5169:
5157:
5149:
5140:
5132:
5128:
5118:
5108:
5104:
5092:
5082:
5070:
5062:
5053:
5045:
5041:
5031:
5021:
5017:
5005:
4994:
4990:
4985:
4979:
4973:
4965:
4961:
4948:
4945:
4941:
4931:
4917:
4914:
4909:
4903:
4898:
4884:
4875:
4870:
4866:
4862:
4857:
4853:
4840:
4831:
4821:
4805:
4801:
4797:
4793:
4786:
4768:
4765:
4760:
4756:
4747:
4730:
4725:
4721:
4713:
4709:
4706:
4703:
4700:
4691:
4677:
4667:
4656:
4650:
4647:
4643:
4639:
4629:
4620:
4617:
4614:
4600:
4594:
4591:
4587:
4575:
4570:
4566:
4557:
4541:
4537:
4527:
4512:
4508:
4499:
4494:
4490:
4475:
4467:
4463:
4450:
4446:
4431:
4428:
4424:
4419:
4411:
4407:
4381:
4377:
4372:
4354:
4349:
4346:
4340:
4334:
4326:
4312:
4292:
4284:
4273:
4271:
4270:Lamé function
4265:
4262:
4257:
4254:
4250:
4246:
4239:
4229:
4227:
4224:into massive
4223:
4219:
4215:
4211:
4206:
4204:
4203:path integral
4200:
4196:
4195:Lamé function
4192:
4188:
4184:
4180:
4166:
4163:
4160:
4157:
4134:
4130:
4121:
4117:
4113:
4110:
4104:
4100:
4096:
4093:
4088:
4085:
4080:
4074:
4066:
4062:
4054:
4053:
4052:
4038:
4035:
4030:
4027:
4023:
3999:
3993:
3986:
3983:
3979:
3971:
3967:
3963:
3958:
3953:
3949:
3938:
3937:
3936:
3917:
3912:
3904:
3900:
3893:
3889:
3883:
3879:
3872:
3869:
3865:
3855:
3851:
3847:
3841:
3837:
3831:
3825:
3819:
3812:
3811:
3810:
3807:
3802:
3800:
3796:
3792:
3788:
3769:
3763:
3754:
3753:path integral
3744:
3722:
3718:
3694:
3687:
3683:
3677:
3672:
3668:
3662:
3658:
3654:
3650:
3643:
3637:
3633:
3626:
3623:
3618:
3614:
3602:
3598:
3592:
3588:
3583:
3577:
3568:
3564:
3558:
3554:
3543:
3539:
3535:
3531:
3525:
3521:
3512:
3508:
3502:
3499:
3494:
3490:
3485:
3478:
3471:
3450:
3446:
3442:
3434:
3431:
3425:
3422:
3417:
3412:
3408:
3404:
3393:
3389:
3385:
3378:
3374:
3368:
3364:
3358:
3350:
3342:
3338:
3334:
3326:
3323:
3318:
3313:
3309:
3305:
3297:
3293:
3286:
3281:
3277:
3271:
3267:
3260:
3256:
3251:
3243:
3239:
3233:
3229:
3222:
3218:
3212:
3209:
3201:
3197:
3193:
3188:
3184:
3175:
3171:
3163:
3162:
3161:
3147:
3144:
3141:
3138:
3135:
3132:
3129:
3126:
3123:
3120:
3115:
3111:
3087:
3084:
3081:
3076:
3072:
3067:
3064:
3061:
3056:
3052:
3045:
3040:
3036:
3030:
3026:
3020:
3017:
3012:
3007:
3003:
2997:
2993:
2987:
2984:
2979:
2976:
2970:
2964:
2957:
2956:
2955:
2938:
2935:
2932:
2926:
2920:
2911:
2905:
2902:
2899:
2893:
2885:
2881:
2877:
2869:
2863:
2858:
2854:
2843:
2842:
2841:
2838:
2828:
2812:
2808:
2804:
2801:
2781:
2778:
2775:
2755:
2752:
2749:
2746:
2724:
2720:
2696:
2692:
2683:
2679:
2675:
2672:
2662:
2657:
2651:
2647:
2644:
2641:
2635:
2629:
2622:
2621:
2620:
2603:
2598:
2594:
2570:
2567:
2562:
2558:
2537:
2534:
2526:
2522:
2515:
2495:
2492:
2489:
2481:
2477:
2470:
2445:
2439:
2436:
2431:
2425:
2422:
2417:
2414:
2388:
2383:
2377:
2372:
2366:
2358:
2344:
2336:
2332:
2328:
2325:
2315:
2310:
2305:
2302:
2297:
2292:
2289:
2285:
2281:
2276:
2271:
2262:
2256:
2253:
2248:
2242:
2239:
2234:
2231:
2224:
2217:
2214:
2209:
2206:
2199:
2195:
2187:
2183:
2178:
2174:
2166:
2147:
2141:
2133:
2130:
2125:
2122:
2116:
2113:
2106:
2102:
2094:
2090:
2085:
2079:
2074:
2069:
2064:
2055:
2049:
2046:
2041:
2035:
2032:
2027:
2024:
2017:
2010:
2007:
2002:
1999:
1992:
1988:
1980:
1976:
1971:
1967:
1962:
1958:
1950:
1949:
1948:
1934:
1931:
1928:
1908:
1905:
1902:
1899:
1879:
1876:
1873:
1853:
1850:
1847:
1844:
1824:
1821:
1818:
1815:
1795:
1792:
1789:
1782:, and we set
1767:
1759:
1756:
1751:
1747:
1738:
1735:
1730:
1724:
1718:
1696:
1692:
1682:
1665:
1659:
1636:
1630:
1627:
1618:
1612:
1589:
1586:
1583:
1579:
1572:
1566:
1563:
1558:
1553:
1547:
1544:
1539:
1536:
1530:
1525:
1520:
1517:
1511:
1503:
1499:
1491:
1487:
1482:
1478:
1473:
1469:
1461:
1460:
1459:
1442:
1425:
1419:
1411:
1407:
1400:
1396:
1386:
1380:
1374:
1371:
1368:
1362:
1359:
1356:
1340:
1329:
1325:
1316:
1313:
1310:
1304:
1298:
1295:
1292:
1289:
1286:
1278:
1274:
1266:
1265:
1264:
1250:
1244:
1241:
1233:
1232:Wick rotation
1214:
1198:
1192:
1186:
1183:
1177:
1167:
1161:
1155:
1152:
1149:
1143:
1140:
1137:
1121:
1113:
1107:
1103:
1094:
1091:
1088:
1082:
1076:
1073:
1070:
1067:
1064:
1058:
1051:
1050:
1049:
1047:
1043:
1033:
1031:
1027:
1008:
1003:
1000:
991:
988:
982:
976:
970:
967:
960:
955:
951:
942:
937:
933:
925:
924:
923:
906:
894:
891:
888:
882:
879:
873:
870:
863:
862:
861:
840:
837:
829:
823:
820:
817:
814:
807:
806:
805:
788:
785:
778:
774:
766:
763:
757:
751:
745:
742:
736:
728:
724:
720:
715:
710:
706:
695:
694:
693:
691:
675:
667:
657:
653:
639:
636:
633:
630:
607:
601:
595:
592:
589:
583:
577:
571:
565:
562:
556:
550:
542:
538:
528:
515:
512:
506:
502:
496:
489:
488:
487:
486:
481:
467:
464:
461:
458:
438:
433:
425:
422:
417:
413:
404:
401:
396:
390:
384:
370:
368:
364:
360:
350:
348:
344:
343:diffeomorphic
340:
336:
332:
328:
324:
320:
316:
312:
308:
304:
299:
297:
293:
289:
285:
281:
278:
274:
270:
266:
262:
258:
252:
248:
238:
236:
232:
228:
220:
216:
213:
212:path integral
209:
208:
207:
205:
201:
200:saddle points
197:
193:
189:
185:
180:
178:
175:
171:
167:
163:
159:
155:
151:
147:
143:
133:
129:
125:
121:
118:
114:
110:
106:
102:
98:
91:
88:is the third
87:
80:
76:
72:
68:
55:
46:
35:
26:
10709:Arkani-Hamed
10607:Supergravity
10574:Moduli space
10471:
10466:Dirac string
10404:
10392:Gauge theory
10372:Loop algebra
10309:Black string
10172:GS formalism
9987:
9957:
9943:
9938:
9926:
9922:
9914:
9901:
9897:10.1142/2281
9884:
9857:
9851:
9837:(1): 69â71.
9834:
9830:
9824:
9781:
9777:
9771:
9738:
9734:
9728:
9711:
9707:
9697:
9677:
9666:
9633:
9629:
9623:
9594:
9577:
9564:
9553:. Retrieved
9549:
9540:
9515:
9511:
9489:
9480:
9471:
9449:
9437:
9424:
9124:
8654:
8611:provided by
8606:
8596:
8594:
8542:moduli space
8537:
8535:
8518:Michael Dine
8505:
8503:
8499:
8488:
8472:
8457:
8449:scalar field
8429:Higgs fields
8426:
8400:
8396:
8386:
8382:
8378:
8374:
8370:
8354:
8332:
8323:
8319:
8315:
8311:
8305:
8155:
7930:
7923:
7845:ChernâSimons
7826:
7746:
7738:perturbative
7727:
7690:vector field
7685:
7671:
7666:
7664:
7599:
7573:
7567:
7564:
7332:
7267:
7198:
7190:
7186:
7184:
7114:
7105:
6950:Killing form
6817:
6724:
6716:
6709:
6705:
6699:
6685:
6563:
6477:
6452:
6405:
6222:
6220:
6207:In studying
6206:
6190:Hypersphere
6154:Hypersphere
6049:
5565:
5376:
5310:
4966:
4962:
4748:
4692:
4558:
4528:
4327:
4279:
4266:
4258:
4241:
4216:) spoil the
4213:
4207:
4186:
4182:
4181:
4149:
4014:
3934:
3805:
3803:
3794:
3790:
3786:
3750:
3709:
3102:
2953:
2834:
2711:
2403:
1683:
1604:
1457:
1263:), one gets
1229:
1045:
1039:
1029:
1025:
1023:
921:
859:
803:
663:
654:
622:
482:
376:
362:
358:
356:
339:homeomorphic
327:Fields medal
300:
295:
291:
280:gauge theory
256:
254:
225:Relevant to
224:
196:local minima
192:local maxima
181:
160:, either in
145:
141:
139:
131:
123:
119:
112:
108:
104:
100:
93:
90:Pauli matrix
82:
78:
74:
63:
11069:Silverstein
10569:Orientifold
10304:Black holes
10299:Black brane
10256:Dual photon
9550:ncatlab.org
8895:satisfying
8576:in 2003 by
8514:Ian Affleck
8418:Chern class
8363:Black holes
7835:, then the
7597:gauge field
7121:. From the
7108:gauge group
6906:Lie algebra
6858:volume form
6480:with index
6213:topological
329:, used the
277:non-abelian
241:Mathematics
105:g=2,Ï=1,z=0
11194:Categories
11089:Strominger
11084:Steinhardt
11079:Staudacher
10994:Polchinski
10944:Nanopoulos
10904:Mandelstam
10884:Kontsevich
10724:Berenstein
10652:Holography
10632:Superspace
10531:K3 surface
10490:Worldsheet
10405:Instantons
10033:Background
9791:2012.09120
9784:(7): 624.
9555:2023-04-11
9518:(4): 195.
9453:See also:
9441:See also:
8355:instantons
8322:= −
7829:Hodge dual
7604:pure gauge
7594:YangâMills
7225:Hodge star
6714:connection
4214:instantons
3804:Thus, the
1044:allows an
273:space-time
261:connection
245:See also:
77:-slice of
11124:Veneziano
11004:Rajaraman
10899:Maldacena
10789:Gopakumar
10739:Dijkgraaf
10734:Curtright
10400:Anomalies
10279:NS5-brane
10200:U-duality
10195:S-duality
10190:T-duality
9988:instanton
9937:; and in
9816:229220708
9763:0556-2821
9658:0370-2693
9532:0038-5670
9463:Citations
9430:conformal
9310:−
9292:λ
9269:∑
9253:ρ
9162:−
9149:−
9105:→
9087:ρ
9064:ρ
9058:
9050:ν
9046:∂
9040:ν
9037:μ
9033:σ
9024:ρ
9020:ρ
9015:ν
9011:∂
9002:ν
8999:μ
8995:σ
8986:μ
8939:ϵ
8935:−
8795:σ
8791:−
8776:σ
8746:ϵ
8730:σ
8707:ν
8704:μ
8700:σ
8479:tunneling
8403:four-form
8275:∧
8264:
8243:∫
8224:≥
8213:∧
8205:∗
8199:
8178:∫
8133:∧
8125:θ
8122:
8108:∧
8100:∗
8094:
8073:∫
8055:∗
8050:θ
8028:∧
8015:θ
8009:−
7993:∗
7984:
7963:∫
7949:≤
7933:integrand
7898:∧
7887:
7866:∫
7837:curvature
7804:∧
7796:∗
7790:
7769:∫
7667:instanton
7645:∧
7574:instanton
7541:∧
7532:∧
7514:−
7505:∧
7487:∧
7472:∧
7445:∧
7439:−
7433:∧
7373:∗
7303:∗
7249:±
7239:∗
7160:∗
7067:⟩
7055:⟨
7046:∗
7043:∧
7011:∗
7008:∧
6985:∫
6928:in which
6755:∫
6721:curvature
6668:⟩
6652:θ
6639:∞
6625:∞
6622:−
6612:∑
6605:⟩
6602:θ
6550:⟩
6516:⟩
6478:instanton
6453:instanton
6418:⟩
6323:π
6223:instanton
6197:meridians
6129:ϵ
6058:ℏ
6002:−
5971:−
5934:π
5834:±
5736:−
5672:−
5518:−
5431:−
5290:ℏ
5280:τ
5245:τ
5216:−
5206:
5170:τ
5129:τ
5125:∂
5115:∂
5109:−
5083:τ
5042:τ
5038:∂
5028:∂
5022:−
4998:ℏ
4995:β
4980:β
4949:τ
4932:τ
4899:˙
4879:ℏ
4876:β
4867:∫
4836:ℏ
4822:τ
4798:−
4787:τ
4769:∮
4704:β
4701:ℏ
4660:^
4651:β
4648:−
4618:∫
4604:^
4595:β
4592:−
4500:≫
4432:β
4429:ℏ
4420:≈
4385:ℏ
4382:β
4358:ℏ
4350:−
4341:β
4158:τ
4118:τ
4114:−
4111:τ
4097:
4075:τ
4063:ϕ
3994:ϕ
3968:τ
3959:ϕ
3890:ϕ
3873:−
3826:ϕ
3806:instanton
3795:instanton
3731:∞
3728:→
3655:−
3624:−
3578:π
3479:∓
3351:−
3287:−
3213:−
3176:±
2980:−
2903:−
2809:τ
2802:τ
2753:−
2721:τ
2680:τ
2676:−
2673:τ
2648:
2636:τ
2607:∞
2595:τ
2574:∞
2571:−
2559:τ
2523:τ
2493:−
2478:τ
2426:τ
2367:≥
2329:−
2290:−
2286:∫
2249:−
2243:τ
2210:τ
2196:τ
2184:τ
2179:∫
2134:τ
2117:τ
2103:τ
2091:τ
2086:∫
2042:−
2036:τ
2003:τ
1989:τ
1977:τ
1972:∫
1906:−
1851:−
1822:±
1757:−
1628:−
1625:→
1587:τ
1548:τ
1500:τ
1488:τ
1483:∫
1436:ℏ
1426:τ
1401:−
1387:τ
1372:∫
1366:⟩
1345:ℏ
1341:τ
1330:−
1308:⟨
1299:τ
1251:τ
1248:→
1209:ℏ
1153:∫
1147:⟩
1126:ℏ
1108:−
1086:⟨
1046:instanton
989:−
952:∫
946:ℏ
938:−
902:ℏ
892:−
830:−
824:
815:ψ
786:ψ
775:ℏ
764:−
716:ψ
676:ℏ
637:±
596:ψ
578:ψ
551:ψ
535:∂
525:∂
503:ℏ
497:−
465:±
423:−
363:instanton
359:instanton
347:monopoles
296:instanton
233:known as
206:because:
177:spacetime
174:Euclidean
142:instanton
11179:Zwiebach
11134:Verlinde
11129:Verlinde
11104:Townsend
11099:Susskind
11034:Sagnotti
10999:Polyakov
10954:Nekrasov
10919:Minwalla
10914:Martinec
10879:Knizhnik
10874:Klebanov
10869:Kapustin
10834:'t Hooft
10769:Fischler
10704:AganagiÄ
10675:M-theory
10564:Conifold
10559:Orbifold
10542:manifold
10483:Geometry
10289:M5-brane
10284:M2-brane
10221:Graviton
10137:F-theory
9354:See also
8613:Corrigan
8464:vortices
8359:D-branes
7853:integral
7294:so that
7029:, since
6431:, where
6308:integers
6243:3-sphere
5150:″
5063:″
4678:⟩
4630:⟨
4529:whereby
4210:"axions"
3987:′
3756:region (
341:but not
227:dynamics
11109:Trivedi
11094:Sundrum
11059:Shenker
11049:Seiberg
11044:Schwarz
11014:Randall
10974:Novikov
10964:Nielsen
10949:NÄstase
10859:Kallosh
10844:Gibbons
10784:Gliozzi
10774:Friedan
10764:Ferrara
10749:Douglas
10744:Distler
10294:S-brane
10274:D-brane
10231:Tachyon
10226:Dilaton
10040:Strings
9879:General
9796:Bibcode
9743:Bibcode
9714:: 158.
9638:Bibcode
9366:Caloron
8973:. Then
8617:Fairlie
8351:soliton
6856:is the
6708:, base
4245:soliton
3747:Results
1892:. For
186:of the
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844:)
841:x
838:k
834:i
827:(
818:=
789:.
779:2
770:)
767:E
761:)
758:x
755:(
752:V
749:(
746:m
743:2
737:=
729:2
725:x
721:d
711:2
707:d
640:1
634:=
631:x
608:,
605:)
602:x
599:(
593:E
590:=
587:)
584:x
581:(
575:)
572:x
569:(
566:V
563:+
560:)
557:x
554:(
543:2
539:x
529:2
516:m
513:2
507:2
468:1
462:=
459:x
439:.
434:2
430:)
426:1
418:2
414:x
410:(
405:4
402:1
397:=
394:)
391:x
388:(
385:V
221:.
134:.
132:R
124:R
120:S
113:z
101:A
96:3
85:3
83:Ï
79:R
66:3
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_plot;_BPST_instanton.png)
_plot;_BPST_instanton.png)
