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Artículos de revistas sobre el tema "Yang-Mills instantons"

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Autor: Grafiati
Publicado: 14 de marzo de 2023

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1

Lai, Sheng-Hong, Jen-Chi Lee y I.-Hsun Tsai. "Extended complex Yang–Mills instanton sheaves". International Journal of Geometric Methods in Modern Physics 17, n.º 04 (marzo de 2020): 2050061. http://dx.doi.org/10.1142/s0219887820500619.

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In the search of Yang–Mills (YM) instanton sheaves with topological charge two, the rank of [Formula: see text] matrix in the monad construction can be dropped from the bundle case with rank [Formula: see text] to either rank [Formula: see text] [S. H. Lai, J. C. Lee and I. H. Tsai, Yang–Mills instanton sheaves, Ann. Phys. 377 (2017) 446] or 0 on some points of [Formula: see text] of the sheaf cases. In this paper, we first show that the sheaf case with rank [Formula: see text] does not exist for the previous construction of [Formula: see text] complex YM instantons [S. H. Lai, J. C. Lee and I. H. Tsai, Biquaternions and ADHM construction of concompact [Formula: see text] Yang–Mills instantons, Ann. Phys. 361 (2015) 14]. We then show that in the new “extended complex YM instantons” discovered in this paper, rank [Formula: see text] can be either 2 on the whole [Formula: see text] (bundle) with some given ADHM data or 1, 0 on some points of [Formula: see text] with other ADHM data (sheaves). These extended [Formula: see text] complex YM instantons have no real instanton counterparts. Finally, the potential applications to real physics systems are noted in the end of the paper.
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2

Kim, Hongsu y Yongsung Yoon. "Yang–Mills instantons in the gravitational instanton backgrounds". Physics Letters B 495, n.º 1-2 (diciembre de 2000): 169–75. http://dx.doi.org/10.1016/s0370-2693(00)01224-7.

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3

MIYAGI, SAYURI. "YANG–MILLS INSTANTONS ON SEVEN-DIMENSIONAL MANIFOLD OF G2 HOLONOMY". Modern Physics Letters A 14, n.º 37 (7 de diciembre de 1999): 2595–604. http://dx.doi.org/10.1142/s0217732399002728.

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We investigate Yang–Mills instantons on a seven-dimensional manifold of G2 holonomy. By proposing a spherically symmetric ansatz for the Yang–Mills connection, we have ordinary differential equations as the reduced instanton equation, and give some explicit and numerical solutions.
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4

Kronheimer, Peter B. y Hiraku Nakajima. "Yang-Mills instantons on ALE gravitational instantons". Mathematische Annalen 288, n.º 1 (diciembre de 1990): 263–307. http://dx.doi.org/10.1007/bf01444534.

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5

OH, JOHN J. y HYUN SEOK YANG. "EINSTEIN MANIFOLDS AS YANG–MILLS INSTANTONS". Modern Physics Letters A 28, n.º 21 (7 de julio de 2013): 1350097. http://dx.doi.org/10.1142/s0217732313500971.

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It is well known that Einstein gravity can be formulated as a gauge theory of Lorentz group where spin connections play a role of gauge fields and Riemann curvature tensors correspond to their field strengths. One can then pose an interesting question: What is the Einstein equation from the gauge theory point of view? Or equivalently, what is the gauge theory object corresponding to Einstein manifolds? We show that the Einstein equations in four dimensions are precisely self-duality equations in Yang–Mills gauge theory and so Einstein manifolds correspond to Yang–Mills instantons in SO (4) = SU (2)L × SU (2)R gauge theory. Specifically, we prove that any Einstein manifold with or without a cosmological constant always arises as the sum of SU (2)L instantons and SU (2)R anti-instantons. This result explains why an Einstein manifold must be stable because two kinds of instantons belong to different gauge groups, instantons in SU (2)L and anti-instantons in SU (2)R, and so they cannot decay into a vacuum. We further illuminate the stability of Einstein manifolds by showing that they carry nontrivial topological invariants.
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6

Belitsky, A. V., S. Vandoren y P. van Nieuwenhuizen. "Yang-Mills and D -instantons". Classical and Quantum Gravity 17, n.º 17 (23 de agosto de 2000): 3521–70. http://dx.doi.org/10.1088/0264-9381/17/17/305.

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7

Etesi, G�bor y Tam�s Hausel. "On Yang-Mills Instantons over Multi-Centered Gravitational Instantons". Communications in Mathematical Physics 235, n.º 2 (1 de abril de 2003): 275–88. http://dx.doi.org/10.1007/s00220-003-0806-8.

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8

Colladay, Don y Patrick McDonald. "Yang–Mills instantons with Lorentz violation". Journal of Mathematical Physics 45, n.º 8 (agosto de 2004): 3228–38. http://dx.doi.org/10.1063/1.1767624.

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9

Segert, Jan. "Frobenius manifolds from Yang-Mills instantons". Mathematical Research Letters 5, n.º 3 (1998): 327–44. http://dx.doi.org/10.4310/mrl.1998.v5.n3.a6.

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10

Groisser, David y Thomas H. Parker. "Semiclassical Yang-Mills theory I: Instantons". Communications in Mathematical Physics 135, n.º 1 (diciembre de 1990): 101–40. http://dx.doi.org/10.1007/bf02097659.

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11

Lled�, M., I. Mart�n, A. Restuccia y A. Mendoza. "Yang?Mills instantons over Riemann surfaces". Letters in Mathematical Physics 24, n.º 4 (abril de 1992): 275–81. http://dx.doi.org/10.1007/bf00420487.

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12

Ivanova, Tatiana A. "Scattering of instantons, monopoles and vortices in higher dimensions". International Journal of Geometric Methods in Modern Physics 13, n.º 03 (marzo de 2016): 1650032. http://dx.doi.org/10.1142/s0219887816500328.

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In this paper, we consider Yang–Mills theory on manifolds [Formula: see text] with a [Formula: see text]-dimensional Riemannian manifold [Formula: see text] of special holonomy admitting gauge instanton equations. Instantons are considered as particle-like solutions in [Formula: see text] dimensions whose static configurations are concentrated on [Formula: see text]. We study how they evolve in time when considered as solutions of the Yang–Mills equations on [Formula: see text] with moduli depending on time [Formula: see text]. It is shown that in the adiabatic limit, when the metric in the [Formula: see text] direction is scaled down, the classical dynamics of slowly moving instantons corresponds to a geodesic motion in the moduli space [Formula: see text] of gauge instantons on [Formula: see text]. Similar results about geodesic motion in the moduli space of monopoles and vortices in higher dimensions are briefly discussed.
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13

JIA, DUOJE y YISHI DUAN. "TOPOLOGICAL EFFECTS OF INSTANTON DUE TO DEFECTS". Modern Physics Letters A 16, n.º 29 (21 de septiembre de 2001): 1863–69. http://dx.doi.org/10.1142/s0217732301005163.

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A new doublet variable is proposed to decompose non-Abelian gauge field for describing the topological effects of instantons due to the defects in appropriate phase of SU(2) Yang–Mills theory. It is shown that the instanton number can be directly related to the isospin defects of the doublet order parameter and contributed from topological charges of these defects. The θ-term in instanton action is found to be the delta-function form of the doublet and the Lagrangian of instantons in terms of new variables is also presented.
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14

Papoulias, Vasileios Ektor. "Spin(7) Instantons and Hermitian Yang–Mills Connections for the Stenzel Metric". Communications in Mathematical Physics 384, n.º 3 (17 de mayo de 2021): 2009–66. http://dx.doi.org/10.1007/s00220-021-04055-5.

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AbstractWe use the highly symmetric Stenzel Calabi–Yau structure on $$T^{\star }S^{4}$$ T ⋆ S 4 as a testing ground for the relationship between the Spin(7) instanton and Hermitian–Yang–Mills (HYM) equations. We reduce both problems to tractable ODEs and look for invariant solutions. In the abelian case, we establish local equivalence and prove a global nonexistence result. We analyze the nonabelian equations with structure group SO(3) and construct the moduli space of invariant Spin(7) instantons in this setting. This is comprised of two 1-parameter families—one of them explicit—of irreducible Spin(7) instantons. Each carries a unique HYM connection. We thus negatively resolve the question regarding the equivalence of the two gauge theoretic PDEs. The HYM connections play a role in the compactification of this moduli space, exhibiting a removable singularity phenomenon that we aim to further examine in future work.
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15

Verbin, Y. "Yang-Mills instantons in closed Friedmann universes". Physics Letters B 223, n.º 3-4 (junio de 1989): 296–99. http://dx.doi.org/10.1016/0370-2693(89)91605-5.

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16

Taniguchi, Tadashi. "ADHM construction of super Yang–Mills instantons". Journal of Geometry and Physics 59, n.º 9 (septiembre de 2009): 1199–209. http://dx.doi.org/10.1016/j.geomphys.2009.06.003.

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17

IVANOVA, TATIANA A., OLAF LECHTENFELD y HELGE MÜLLER-EBHARDT. "NONCOMMUTATIVE MODULI FOR MULTI-INSTANTONS". Modern Physics Letters A 19, n.º 32 (20 de octubre de 2004): 2419–30. http://dx.doi.org/10.1142/s0217732304015464.

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There exists a recursive algorithm for constructing BPST-type multi-instantons on commutative ℝ4. When deformed noncommutatively, however, it becomes difficult to write down non-singular instanton configurations with topological charge greater than one in explicit form. We circumvent this difficulty by allowing for the translational instanton moduli to become noncommutative as well. Such a scenario is natural in the self-dual Yang–Mills hierarchy of integrable equations where the moduli of solutions are seen as extended spacetime coordinates associated with higher flows. By judicious adjustment of the moduli-noncommutativity we achieve the ADHM construction of generalized 't Hooft multi-instanton solutions with everywhere self-dual field strengths on noncommutative ℝ4.
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18

OCHIAI, TETSUYUKI. "MULTI-INSTANTON EFFECT IN TWO-DIMENSIONAL QCD". Modern Physics Letters A 10, n.º 21 (10 de julio de 1995): 1549–63. http://dx.doi.org/10.1142/s0217732395001678.

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We analyze multi-instanton sector in two-dimensional U(N) Yang-Mills theory on a sphere. We obtain a contour integral representation of the multi-instanton amplitude and find “neutral” configurations of the even number instantons dominate in the large-N limit. Using this representation, we calculate the 1-, 2-, 3-, 4-body interactions and the free energies for N=3, 4, 5 numerically and find that the multi-instanton interaction effect essentially contributes to the large-N phase transition discovered by Douglas and Kazakov.
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19

Correia, Filipe Paccetti. "Hermitian Yang-Mills instantons on Calabi-Yau cones". Journal of High Energy Physics 2009, n.º 12 (1 de diciembre de 2009): 004. http://dx.doi.org/10.1088/1126-6708/2009/12/004.

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20

Figueroa-O'Farrill, J. M., C. Kohl y B. Spence. "Supersymmetric Yang-Mills, octonionic instantons and triholomorphic curves". Nuclear Physics B 521, n.º 3 (junio de 1998): 419–43. http://dx.doi.org/10.1016/s0550-3213(98)00285-5.

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21

Ivanova, Tatiana A. y Olaf Lechtenfeld. "Yang–Mills instantons and dyons on group manifolds". Physics Letters B 670, n.º 1 (diciembre de 2008): 91–94. http://dx.doi.org/10.1016/j.physletb.2008.10.027.

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22

Minasian, Ruben, Samson L. Shatashvili y Pierre Vanhove. "Closed strings from SO(8) Yang–Mills instantons". Nuclear Physics B 613, n.º 1-2 (octubre de 2001): 87–104. http://dx.doi.org/10.1016/s0550-3213(01)00369-8.

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23

Etesi, Gábor. "Spin(7)-manifolds and symmetric Yang–Mills instantons". Physics Letters B 521, n.º 3-4 (noviembre de 2001): 391–99. http://dx.doi.org/10.1016/s0370-2693(01)01239-4.

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24

Ivanova, Tatiana A., Olaf Lechtenfeld, Alexander D. Popov y Thorsten Rahn. "Instantons and Yang–Mills Flows on Coset Spaces". Letters in Mathematical Physics 89, n.º 3 (septiembre de 2009): 231–47. http://dx.doi.org/10.1007/s11005-009-0336-1.

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25

Reinhardt, H., K. Langfeld y L. v. Smekal. "Instantons in field strength formulated Yang-Mills theories". Physics Letters B 300, n.º 1-2 (febrero de 1993): 111–17. http://dx.doi.org/10.1016/0370-2693(93)90756-8.

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26

Bianchi, Massimo, Stefano Kovacs, Giancarlo Rossi y Michael B. Green. "Instantons in supersymmetric Yang-Mills and D-instantons in IIB superstring theory". Journal of High Energy Physics 1998, n.º 08 (30 de agosto de 1998): 013. http://dx.doi.org/10.1088/1126-6708/1998/08/013.

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27

Poppitz, Erich. "Notes on Confinement on R3 × S1: From Yang–Mills, Super-Yang–Mills, and QCD (adj) to QCD(F)". Symmetry 14, n.º 1 (17 de enero de 2022): 180. http://dx.doi.org/10.3390/sym14010180.

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This is a pedagogical introduction to the physics of confinement on R3×S1, using SU(2) Yang–Mills with massive or massless adjoint fermions as the prime example; we also add fundamental flavours to conclude. The small-S1 limit is remarkable, allowing for controlled semiclassical determination of the nonperturbative physics in these, mostly non-supersymmetric, theories. We begin by reviewing the Polyakov confinement mechanism on R3. Moving on to R3×S1, we show how introducing adjoint fermions stabilizes center symmetry, leading to abelianization and semiclassical calculability. We explain how monopole–instantons and twisted monopole–instantons arise. We describe the role of various novel topological excitations in extending Polyakov’s confinement to the locally four-dimensional case, discuss the nature of the confining string, and the θ-angle dependence. We study the global symmetry realization and, when available, present evidence for the absence of phase transitions as a function of the S1 size. As our aim is not to cover all work on the subject, but to prepare the interested reader for its study, we also include brief descriptions of topics not covered in detail: the necessity for analytic continuation of path integrals, the study of more general theories, and the ’t Hooft anomalies involving higher-form symmetries.
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28

Shulyakovsky, Roman G., Alexander S. Gribowsky, Alexander S. Garkun, Maxim N. Nevmerzhitsky, Alexei O. Shaplov y Denis A. Shohonov. "Classical instanton solutions in quantum field theory". Journal of the Belarusian State University. Physics, n.º 2 (7 de junio de 2020): 78–85. http://dx.doi.org/10.33581/2520-2243-2020-2-78-85.

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Instantons are non-trivial solutions of classical Euclidean equations of motion with a finite action. They provide stationary phase points in the path integral for tunnel amplitude between two topologically distinct vacua. It make them useful in many applications of quantum theory, especially for describing the wave function of systems with a degenerate vacua in the framework of the path integrals formalism. Our goal is to introduce the current situation about research on instantons and prepare for experiments. In this paper we give a review of instanton effects in quantum theory. We find in stanton solutions in some quantum mechanical problems, namely, in the problems of the one-dimensional motion of a particle in two-well and periodic potentials. We describe known instantons in quantum field theory that arise, in particular, in the two-dimensional Abelian Higgs model and in SU(2) Yang – Mills gauge fields. We find instanton solutions of two-dimensional scalar field models with sine-Gordon and double-well potentials in a limited spatial volume. We show that accounting of instantons significantly changes the form of the Yukawa potential for the sine-Gordon model in two dimensions.
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29

Volkov, Boris O. "Lévy Laplacians and instantons on manifolds". Infinite Dimensional Analysis, Quantum Probability and Related Topics 23, n.º 02 (junio de 2020): 2050008. http://dx.doi.org/10.1142/s0219025720500083.

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The equivalence of the anti-selfduality Yang–Mills equations on the four-dimensional orientable Riemannian manifold and the Laplace equations for some infinite-dimensional Laplacians is proved. A class of modified Lévy Laplacians parameterized by the choice of a curve in the group [Formula: see text] is introduced. It is shown that a connection is an instanton (a solution of the anti-selfduality Yang–Mills equations) if and only if the parallel transport generalized by this connection is a solution of the Laplace equations for some three modified Lévy Laplacians from this class.
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30

Lee, Jen-Chi. "Biquaternion Construction of SL(2,C) Yang-Mills Instantons". Journal of Physics: Conference Series 670 (25 de enero de 2016): 012032. http://dx.doi.org/10.1088/1742-6596/670/1/012032.

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31

HELMKE, U. "Linear Dynamical Systems and Instantons in Yang-Mills Theory". IMA Journal of Mathematical Control and Information 3, n.º 2-3 (1986): 151–66. http://dx.doi.org/10.1093/imamci/3.2-3.151.

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32

Loginov, E. K. y E. D. Loginova. "Yang-Mills like instantons in eight and seven dimensions". Journal of Mathematical Physics 55, n.º 10 (octubre de 2014): 101701. http://dx.doi.org/10.1063/1.4897444.

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33

Bianchi, Massimo, Francesco Fucito, Giancarlo Rossi y Maurizio Martellini. "Explicit construction of Yang-Mills instantons on ALE spaces". Nuclear Physics B 473, n.º 1-2 (agosto de 1996): 367–404. http://dx.doi.org/10.1016/0550-3213(96)00240-4.

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34

Bonini, G. F., S. Habib, E. Mottola, C. Rebbi, R. Singleton y P. G. Tinyakov. "Periodic instantons in SU(2) Yang–Mills–Higgs theory". Physics Letters B 474, n.º 1-2 (febrero de 2000): 113–21. http://dx.doi.org/10.1016/s0370-2693(00)00022-8.

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35

Paban, Sonia, Savdeep Sethi y Mark Stern. "Summing up instantons in three-dimensional Yang–Mills theories". Advances in Theoretical and Mathematical Physics 3, n.º 2 (1999): 343–61. http://dx.doi.org/10.4310/atmp.1999.v3.n2.a6.

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36

Yatsun, V. A. "Integrable model of Yang-Mills theory and quasi-instantons". Letters in Mathematical Physics 11, n.º 2 (febrero de 1986): 153–59. http://dx.doi.org/10.1007/bf00398427.

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37

MAEDA, TAKASHI y TOSHIO NAKATSU. "AMOEBAS AND INSTANTONS". International Journal of Modern Physics A 22, n.º 05 (20 de febrero de 2007): 937–83. http://dx.doi.org/10.1142/s0217751x07034970.

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We study a statistical model of random plane partitions. The statistical model has interpretations as five-dimensional [Formula: see text] supersymmetric SU (N) Yang–Mills on ℝ4 × S1 and as Kähler gravity on local SU (N) geometry. At the thermodynamic limit a typical plane partition called the limit shape dominates in the statistical model. The limit shape is linked with a hyperelliptic curve, which is a five-dimensional version of the SU (N) Seiberg–Witten curve. Amoebas and the Ronkin functions play intermediary roles between the limit shape and the hyperelliptic curve. In particular, the Ronkin function realizes an integration of thermodynamical density of the main diagonal partitions, along one-dimensional slice of it and thereby is interpreted as the counting function of gauge instantons. The radius of S1 can be identified with the inverse temperature of the statistical model. The large radius limit of the five-dimensional Yang–Mills is the low temperature limit of the statistical model, where the statistical model is frozen to a ground state that is associated with the local SU (N) geometry. We also show that the low temperature limit corresponds to a certain degeneration of amoebas and the Ronkin functions known as tropical geometry.
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38

Kim, Seok, Ki-Myeong Lee y Sungjay Lee. "Dyonic instantons in 5-dim Yang-Mills Chern-Simons theories". Journal of High Energy Physics 2008, n.º 08 (19 de agosto de 2008): 064. http://dx.doi.org/10.1088/1126-6708/2008/08/064.

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39

Deser, Andreas, Olaf Lechtenfeld y Alexander D. Popov. "Sigma-model limit of Yang–Mills instantons in higher dimensions". Nuclear Physics B 894 (mayo de 2015): 361–73. http://dx.doi.org/10.1016/j.nuclphysb.2015.03.009.

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40

Lechtenfeld, Olaf y Warren Siegel. "N = 2 worldsheet instantons yield cubic self-dual Yang-Mills". Physics Letters B 405, n.º 1-2 (julio de 1997): 49–54. http://dx.doi.org/10.1016/s0370-2693(97)00595-9.

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41

Wo-Jun, Zhong y Duan Yi-Shi. "Topological Quantization of Instantons in SU (2) Yang–Mills Theory". Chinese Physics Letters 25, n.º 5 (mayo de 2008): 1534–37. http://dx.doi.org/10.1088/0256-307x/25/5/004.

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42

Kovalev, A. G. "On the reduction of Yang-Mills instantons to Nahm's equations". Russian Mathematical Surveys 52, n.º 6 (31 de diciembre de 1997): 1305–6. http://dx.doi.org/10.1070/rm1997v052n06abeh002170.

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43

Castro, Carlos. "Wgravity,N=2 strings, and 2+2SU*(∞) Yang–Mills instantons". Journal of Mathematical Physics 35, n.º 6 (junio de 1994): 3013–24. http://dx.doi.org/10.1063/1.530500.

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44

Yang, Hyun Seok y Sangheon Yun. "Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry". Advances in High Energy Physics 2017 (2017): 1–27. http://dx.doi.org/10.1155/2017/7962426.

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We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.
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45

Morris, T. R., D. A. Ross y C. T. Sachrajda. "Instantons and the renormalisation group in supersymmetric Yang-Mills theories". Nuclear Physics B 264 (enero de 1986): 111–53. http://dx.doi.org/10.1016/0550-3213(86)90476-1.

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46

Chimento, Samuele, Tomás Ortín y Alejandro Ruipérez. "Yang–Mills instantons in Kähler spaces with one holomorphic isometry". Physics Letters B 778 (marzo de 2018): 371–76. http://dx.doi.org/10.1016/j.physletb.2018.01.046.

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47

Bergvelt, M. J. y E. A. De Kerf. "The Hamiltonian structure of Yang-Mills theories and instantons I". Physica A: Statistical Mechanics and its Applications 139, n.º 1 (noviembre de 1986): 101–24. http://dx.doi.org/10.1016/0378-4371(86)90007-5.

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48

Bergvelt, M. J. y E. A. De Kerf. "The Hamiltonian structure of Yang-Mills theories and instantons II". Physica A: Statistical Mechanics and its Applications 139, n.º 1 (noviembre de 1986): 125–48. http://dx.doi.org/10.1016/0378-4371(86)90008-7.

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49

Billó, Marco, Pietro Fré, Riccardo D'auria, Sergio Ferrara, Paolo Soriani y Antoine Van Proeyen. "R Symmetry and the Topological Twist of N = 2 Effective Supergravities of Heterotic Strings". International Journal of Modern Physics A 12, n.º 02 (20 de enero de 1997): 379–418. http://dx.doi.org/10.1142/s0217751x97000475.

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We discuss R symmetries in locally supersymmetric N = 2 gauge theories coupled to hypermultiplets which can be thought of as effective theories of heterotic superstring models. In this type of supergravities a suitable R symmetry exists and can be used to topologically twist the theory: the vector multiplet containing the dilaton–axion field has different R charge assignments with respect to the other vector multiplets. Correspondingly a system of coupled instanton equations emerges, mixing gravitational and Yang–Mills instantons with triholomorphic hyperinstantons and axion instantons. For the tree level classical special manifolds ST(n) = SU(1,1)/U(1) × SO(2,n)/[SO(2) × SO(n)], R symmetry with the specified properties is a continuous symmetry, but for the quantum-corrected manifolds [Formula: see text] a discrete R group of electric–magnetic duality rotations is sufficient and we argue that it exists.
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DALLAGNOL, LLOHANN y MARCOS JARDIM. "NONSINGULAR COMPLEX INSTANTONS ON EUCLIDEAN SPACETIME". International Journal of Geometric Methods in Modern Physics 05, n.º 06 (septiembre de 2008): 963–71. http://dx.doi.org/10.1142/s0219887808003132.

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Building on a variation of 't Hooft's harmonic function ansatz for SU(2) instantons on ℝ4, we provide new explicit nonsingular solutions of the Yang–Mills anti-self-duality equations on Euclidean spacetime with gauge group SL(2, ℂ) and SL(3, ℝ).
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