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1 969 729 53C07 53C26 81T20

Etesi, Gabor (H-AOS) ; Hausel, Tamas (1-CA)

On Yang-Mills instantons over multi-centered gravitationalinstantons. (English. English summary)Comm. Math. Phys. 235 (2003), no. 2, 275–288.In this paper, the authors explicitly compute the analogue of the’t Hooft SU(2) instantons over the multi-Eguchi-Hanson and multi-Taub-NUT gravitational instantons X (also known as hyper-KahlerAk-ALE spaces and Ak-ALF spaces, respectively). The reducible U(1)instantons are particularly interesting: their curvature defines L2-harmonic 2-forms, which form a basis for the L2 cohomology H2(X).

Marcos Jardim (1-MA)

[References]

1. Aragone, C., Colaiacomo, G.: Gravity-matter instantons. Lett.Nuovo Cim. 31, 135–139 (1981) MR0621592 (82f:83043)

2. Atiyah, M.F., Hitchin, N., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. London A362,425–461 (1978)

3. Bianchi, M., Fucito, F., Rossi, G., Martenilli, M.: Explicit con-struction of Yang-Mills instantons on ALE spaces. Nucl. Phys.B473, 367–404 (1996)

4. Boutaleb-Joutei, H., Chakrabarti, A., Comtet, A.: Gauge fieldconfigurations in curved space-times I-V, Part I: Phys. Rev. D20,1884–1897 (1979); Part II: D20, 1898–1908 (1979); Part III: D21,979–983 (1980); Part IV: D21, 2280–2284 (1980); Part V: D21,2285–2290 (1980)

5. Chakrabarti, A.: Classical solutions of Yang-Mills fields. Fortschr.Phys. 35, 1–64 (1987)

6. Charap, J.M., Duff, M.J.: Space-time topology and a new class ofYang-Mills instanton. Phys. Lett. B71, 219–221 (1977)

7. Cherkis, S.A., Kapustin, A.: Singular monopoles and supersym-metric gauge theories in three dimensions. Nucl. Phys. B525,215–234 (1998)

8. Eguchi, T., Gilkey, P.B., Hanson, A.J.: Gravity, gauge theoriesand differential geometry. Phys. Rep. 66, 213–393 (1980)

9. Eguchi, T., Hanson, A.J.: Self-dual solutions to Euclidean gravity.Ann. Phys. 120, 82–106 (1979)

10. Etesi, G., Hausel, T.: Geometric interpretation of Schwarzschildinstantons. J. Geom. Phys. 37, 126–136 (2001)

11. Etesi, G., Hausel, T.: Geometric construction of new Yang-Mills


instantons over Taub-NUT space. Phys. Lett. B514, 189–199(2001)

12. Freed, D.S., Uhlenbeck, K.K.: Instantons and four-manifolds.MSRI Publications 1, Berlin: Springer-Verlag, 1984

13. Gibbons, G.W., Hawking, S.W.: Gravitational multi-instantons.Phys. Lett. B78, 430–432 (1978)

14. Hausel, T., Hunsicker, E., Mazzeo, R.: Hodge cohomology ofgravitational instantons. Preprint, arXiv:math.DG/0207169

15. Hawking, S.W.: Gravitational instantons. Phys. Lett. A60, 81–83(1977)

16. Jackiw, R., Nohl, C., Rebbi, C.: Conformal properties of pseudo-particle configurations. Phys. Rev. D15, 1642–1646 (1977)

17. Kim, H., Yoon, Y.: Yang-Mills instantons in the gravitationalinstanton background. Phys. Lett. B495, 169–175 (2000)

18. Kronheimer, P.B.: Monopoles and Taub-NUT metrics. Diplomathesis. Merton College, Oxford, 1985 (unpublished)

19. Kronheimer, P.B., Nakajima, H.: Yang-Mills instantons on ALEgravitational instantons. Math. Ann. 288, 263–307 (1990)

20. Newman, E.T., Tamburino, L., Unti, T.J.: Empty space general-ization of the Schwarzschild metric. J. Math. Phys. 4, 915–923(1963)

21. Pope, C.N. Yuille, A.L.: A Yang-Mills instanton in Taub-NUTspace. Phys. Lett. B78, 424–426 (1978)

22. Ruback, P.: The motion of Kaluza-Klein monopoles. Commun.Math. Phys. 107, 93–102 (1986)

23. Taub, A.H.: Empty space-times admitting a three parametergroup of motions. Ann. Math. 53, 472–490 (1951)

24. Uhlenbeck, K.K.: Removable singularities in Yang-Mills fields.Commun. Math. Phys. 83, 11–29 (1982)

2003k:53025 53C07 53C80 81T13

Buniy, Roman V. (1-VDB-PA) ;Kephart, Thomas W. (1-VDB-PA)

Construction of multi-instantons in eight dimensions.(English. English summary)Phys. Lett. B 548 (2002), no. 1-2, 97–101.Instantons have played a fundamental role in the understanding ofnon-perturbative aspects of quantum field theory, while also openinga number of doors in mathematics. Given that M-theory and stringtheory live in higher dimensions, generalizing these intrinsically four-dimensional objects is of considerable interest.

In this paper, the authors consider a generalization of instantons


to dimension 8. Let x be an octonion parametrizing a point in R8.An involution � on 2-forms Λ2R8 is defined by declaring that thecomponents of dx∧ dx are self-dual, while the components of dx∧dx are anti-self-dual. Octonionic instantons, that is, connections overR8 whose curvature is (anti-)self-dual with respect to �, are thenconstructed via a clever adaptation of the usual ADHM constructionof 4-dimensional instantons. It is conjectured that all octonionicinstantons can be obtained in this way, and that the moduli space ofoctonionic instantons of charge k has dimension 16k− 7.


2003h:81132 81T13 53C80

Pomeroy, N. B. (4-DRHM-P)

The U(N) ADHM two-instanton. (English. Englishsummary)Phys. Lett. B 547 (2002), no. 1-2, 85–94.From the introduction: “In this paper we solve the ADHM constraints(in commutative spacetime) for topological charge two units andgauge group U(N), for any value of N ≥ 2. This constitutes the firstexact and explicit general multi-instanton configuration for the gaugegroup U(N).” Marcos Jardim (1-MA)

2003f:81161 81T13 58E50

Ford, C. [Ford, Christopher] (NL-LEID-N) ;Pawlowski, J. M. (D-ERL-T3)

Constituents of doubly periodic instantons. (English. Englishsummary)Phys. Lett. B 540 (2002), no. 1-2, 153–158.The authors investigate SU(2) instantons on T 2×R2 (that is, doubly-periodic instantons) of charge one, with radial symmetry in the non-compact direction. Using the Nahm transform, a constituent picturefor such instantons is developed (for the case of calorons and theirconstituent monopoles, see [T. C. Kraan and P. van Baal, Phys. Lett.B 435 (1998), no. 3-4, 389–395].

Due to the radial symmetry, such constituents are expected to lieon the torus over the origin of R2, and the instanton’s action density isexplicitly computed within this two-dimensional slice. An interestingnumerical analysis of the expression obtained is given. In particular,it is observed that, for certain values of the relevant parameters,the action density contains two lumps, which can be interpreted asperiodic monopoles. Marcos Jardim (1-MA)


2003f:81163 81T13 58E50

Grigoriev, D. Yu. [Grigoriev, Dmitri] (RS-AOS-NR) ;Sutcliffe, P. M. (4-KENT-IM) ; Tchrakian, D. H. (IRL-MNTH-MP)

Skyrmed monopoles. (English. English summary)Phys. Lett. B 540 (2002), no. 1-2, 146–152.Summary: “We investigate multi-monopole solutions of a modifiedversion of the BPS Yang-Mills-Higgs model in which a term quarticin the covariant derivatives of the Higgs field (a Skyrme term) isincluded in the Lagrangian. Using numerical methods we find thatthis modification leads to multi-monopole bound states. We computeaxially symmetric monopoles up to charge five and also monopoleswith Platonic symmetry for charges three, four and five. The numericalevidence suggests that, in contrast to Skyrmions. Skyrmed monopolesof minimal energy are axially symmetric.” Marcos Jardim (1-MA)

2003e:81196 81T75 81R60 81T13

Hamanaka, Masashi (J-TOKYO-P)

Atiyah-Drinfeld-Hitchin-Manin and Nahm constructionsof localized solitons in noncommutative gauge theories.(English. English summary)Phys. Rev. D (3) 65 (2002), no. 8, 085022, 13 pp.Summary: “We study the relationship between Atiyah-Drinfeld-Hitchin-Manin (ADHM) and Nahm constructions and the ‘solutionsgenerating technique’ of Bogomol′nyı-Prasad-Sommerfield (BPS)solitons in noncommutative gauge theories. ADHM and Nahm con-structions and the solution generating technique are the strongestways to construct exact BPS solitons. Localized solitons are the soli-tons which are generated by the solution generating technique. Theshift operators which play crucial roles in the solution generatingtechnique naturally appear in the ADHM and Nahm constructionsand we can construct various exact localized solitons including newsolitons: localized periodic instantons (= localized calorons) and lo-calized doubly periodic instantons. Nahm’s construction also givesrise to BPS fluxons straightforwardly from the appropriate inputNahm data which is expected from the D-brane picture of BPSfluxons. We also show that the Fourier-transformed soliton of the lo-calized caloron in the zero-period limit exactly coincides with theBPS fluxon.” Marcos Jardim (1-MA)


2003g:53069 53C26 53C07 53C80

Cherkis, Sergey A. (1-UCLA-P) ; Kapustin, Anton (1-CAIT)

Hyper-Kahler metrics from periodic monopoles. (English.English summary)Phys. Rev. D (3) 65 (2002), no. 8, 084015, 10 pp.The authors study the relative moduli spaces of periodic monopoles(solutions of the Bogomol′nyı equation on R2 × S1). In particular,when such a moduli space is four-dimensional, it is shown that it isan asymptotically locally flat hyper-Kahler manifold which asymptot-ically has a triholomorphic T 2 action. This new class of gravitationalinstantons is called the class of ALG manifolds. Five topologicallydistinct ALG manifolds are constructed in this way.


2002k:53039 53C07 14H70 53C28

Murray, Michael K. (5-ADLD)

Monopoles. (English. English summary)Geometric analysis and applications to quantum field theory(Adelaide, 1998/1999), 119–135, Progr. Math., 205, BirkhauserBoston, Boston, MA, 2002.This paper is a short, easy-to-read survey of 25 years of research onmonopoles. After introducing the Bogomol′nyı equations on R3 andthe relevant asymptotic behaviour, the author presents the variousapproaches to the construction of solutions: the spectral curve method,the twistor description, Nahm transform, and rational maps.

The survey is very informative, and it is a good introduction to theextensive literature on monopoles.{For the entire collection see 2002i:00011} Marcos Jardim (1-MA)

2002k:81087 81R12 14C05 81R60 81T13

Braden, H. W. (4-EDIN-MS) ; Nekrasov, N. A. (F-IHES)

Instantons, Hilbert schemes and integrability. (English.English summary)Integrable structures of exactly solvable two-dimensional models ofquantum field theory (Kiev, 2000), 35–54, NATO Sci. Ser. II Math.Phys. Chem., 35, Kluwer Acad. Publ., Dordrecht, 2001.This review paper aims at describing various aspects of the deformedADHM equations. These describe an integrable system closely relatedto the Calogero-Moser system, the moduli space of “freckled” instan-tons, the Hilbert scheme of points on a complex plane, and the modulispace of noncommutative instantons. In particular, it is shown that


the deformed ADHM equations can be used to construct a U(1) con-nection on a blow-up of C2 which is smooth and anti-self-dual withrespect to the so-called Burns metric.{For the entire collection see 2002g:81003}


2002i:53039 53C07 39A12 39A13 81T13

Nakamula, Atsushi (J-KITAS-P)

Selfdual solution of classical Yang-Mills fields through a q-analog of ADHM construction. (English. English summary)Proceedings of the XXXII Symposium on Mathematical Physics(Torun, 2000).Rep. Math. Phys. 48 (2001), no. 1-2, 195–202.Inspired by the quantum Knizhnik-Zamolodchikov equation in con-formal field theory [see I. B. Frenkel and N. Yu. Reshetikhin, Comm.Math. Phys. 146 (1992), no. 1, 1–60; MR 94c:17024] the author con-siders a version of the ADHM construction of instantons over thealgebra of square integrable functions on the q-interval:

Iq ={±1

2,±1

2q,±1

2q2, . . .

}, 0 < q < 1.

In the limit q → 1, the self-dual gauge field constructed in thispaper approaches the usual BPS monopole. On the other hand, in thelimit q → 0, the instanton approaches zero.

The main idea in this paper is an interesting one, and deservesfurther attention. In particular, the relation with noncommutativegauge theories is yet to be explained. Marcos Jardim (1-MA)

2002f:53039 53C07 53C80 81T13 81T20

Etesi, Gabor (J-KYOT-TP) ; Hausel, Tamas (1-CA)

Geometric construction of new Yang-Mills instantons overTaub-NUT space. (English. English summary)Phys. Lett. B 514 (2001), no. 1-2, 189–199.Summary: “In this paper we exhibit a one-parameter family of newTaub-NUT instantons parameterized by a half-line. The endpoint ofthe half-line is the reducible Yang-Mills instanton corresponding to theEguchi-Hanson-Gibbons L2-harmonic 2-form, while at an inner pointwe recover the Pope-Yuille instanton constructed as a projection of theLevi-Civita connection onto the positive su(2)+ ⊂ so(4) subalgebra.Our method imitates the Jackiw-Nohl-Rebbi construction originallydesigned for flat R4. That is, we find a one-parameter family of


harmonic functions on the Taub-NUT space with a point singularity,rescale the metric and project the Levi-Civita connection obtainedonto the other negative su(2)− ⊂ so(4) part. Our solutions possess thefull U(2) symmetry, and thus provide more solutions to the recentlyproposed U(2) symmetric ansatz of H. Kim and Y. Yoon [Phys. Lett.B 495 (2000), no. 1-2, 169–175].” Marcos Jardim (1-MA)

2002c:81218 81T75 58B34 81R60

Lee, Kimyeong (KR-AIST-SP) ; Tong, David (4-LNDKC) ;Yi, Sangheon (KR-SNU-TP)

Moduli space of two U(1) instantons on noncommutative R4

and R3×S1. (English. English summary)Phys. Rev. D (3) 63 (2001), no. 6, 065017, 10 pp.The authors employ the Nekrasov-Schwarz noncommutative versionof the ADHM construction to describe the moduli space of two abelianinstantons on noncommutative spacetime. It is shown that the naturalmetric on the moduli space is the Eguchi-Hanson metric.

The moduli space of two noncommutative calorons is also discussed,and the asymptotic behaviour of the metric is given.

Finally, the moduli spaces of noncommutative instantons andcalorons are realised as the vacuum moduli spaces of certain three-dimensional gauge theories. Marcos Jardim (1-MA)

2002b:53034 53C07 58D27 81T13 81T30

Cherkis, Sergey (1-UCLA-TE) ; Kapustin, Anton (1-IASP)

Nahm transform for periodic monopoles and N = 2 superYang-Mills theory. (English. English summary)Comm. Math. Phys. 218 (2001), no. 2, 333–371.Following a long history of papers dealing with invariant instantons onR4 by both mathematicians and physicists, the authors study periodicmonopoles (that is, solutions of the Bogomol′nyı equations on R2 ×S1) for the first time.

After briefly describing the string theoretical motivation behindthe problem, the authors describe the Nahm transform of periodicmonopoles, showing that they correspond to solutions of Hitchin’sequations on a cylinder. More interestingly, periodic monopoles canalso be associated with a so-called spectral data, consisting of analgebraic curve plus a line bundle over it. It is shown that suchspectral data coincides with the usual spectral data coming fromthe Nahm transformed solution of Hitchin’s equations. Existence ofperiodic monopoles can be guaranteed via Nahm transform (at least


for unit monopole charge) and the paper concludes with a shortdiscussion of the moduli space. Marcos Jardim (1-MA)

[References]

1. Cherkis, S. and Kapustin, A.: New Hyperkahler Metrics FromPeriodic Monopoles. Work in progress

2. Chalmers, G. and Hanany, A.: Three Dimensional Gauge TheoriesAnd Monopoles. Nucl. Phys. B 489, 223 (1997) [hep-th/9608105]MR1443803 (98g:81154)

3. Hanany, A. and Witten, E.: Type IIB Superstrings, BPS Mono-poles, And Three-Dimensional Gauge Dynamics. Nucl. Phys. B492, 152 (1997) [hep-th/9611230] MR1451054 (98h:81096)

4. Witten, E.: Solutions of Four-Dimensional Field Theories Via M-Theory. Nucl. Phys. B 500, 3 (1997) [hep-th/9703166] MR1471647(99a:81158)

5. Cherkis, S. and Kapustin, A.: Periodic Monopoles With Singular-ities and N = 2 Super-QCD. To appear

6. Jaffe, A. and Taubes, C.: Vortices And Monopoles. Structureof Static Gauge Theories. Boston: Birkhauser, 1980 MR0614447(82m:81051)

7. Schenk, H.: On A Generalized Fourier Transform of Instan-tons over Flat Tori. Commun. Math. Phys. 116, 177–183 (1988)MR0939044 (89h:53076)

8. Seiberg, N. and Witten, E.: Gauge Dynamics and Compactifica-tion to Three Dimensions. [hep-th/9607163]

9. Seiberg, N. and Witten, E.: Electric-Magnetic Duality, MonopoleCondensation, and Confinement in N = 2 Supersymmetric Yang-Mills Theory. Nucl. Phys. B 426, 19 (1994) [hep-th/9407087]MR1293681 (95m:81202a)

10. Argyres, P.C. and Faraggi, A.E.: The Vacuum Structure andSpectrum of N = 2 Supersymmetric SU(n) Gauge Theory. Phys.Rev. Lett. 74, 3931 (1995) [hep-th/9411057]

11. Braam, P.J. and van Baal, P.: Nahm’s Transformation for In-stantons. Commun. Math. Phys. 122, 267 (1989) MR0994505(91a:58041)

12. Jardim, M.: Nahm Transform for Doubly-Periodic Instantons.math.dg/9910120; Spectral Curves and Nahm Transform for Dou-bly Periodic Instantons. math.ag/9909146

13. Donaldson, S.K. and Kronheimer, P.B.: The Geometry Of Four-Manifolds. Oxford Mathematical Monographs, New York: Claren-don Press, Oxford University Press, 1990 MR1079726 (92a:57036)

14. Hitchin, N.J.: The Self-duality Equations on a Riemann Sur-


face. Proc. Lond. Math. Soc. 55, 59 (1987); Stable Bundles andIntegrable Systems. Duke. Math. J. 54, 91 (1987) MR0887284(89a:32021) MR0885778 (88i:58068)

15. Callias, C.: Index Theorems On Open Spaces. Commun. Math.Phys. 62, 213 (1978) MR0507780 (80h:58045a)

16. Luty, M.A. and Taylor, W.I.: Varieties Of Vacua in ClassicalSupersymmetric Gauge Theories. Phys. Rev. D 53, 3399 (1996)[hep-th/9506098] MR1380943 (97d:81193)

17. Simpson, C.T.: Higgs Bundles And Local Systems. IHES Publ.Math. 75, 5 (1992); The Hodge Filtration On Nonabelian Coho-mology. In: Algebraic Geometry - Santa Cruz 1995, Proc. Symp.Pure Math. 62, Part 2, Providence, RI: Am. Math. Soc., 1997, p.217 MR1179076 (94d:32027) MR1492538 (99g:14028)

18. Konno, H.: Construction of The Moduli Space of Stable ParabolicHiggs Bundles on a Riemann Surface. J. Math. Soc. Japan 45,253 (1993) MR1206652 (94g:32027)

19. Donagi, R. and Witten, E.: Supersymmetric Yang-Mills TheoryAnd Integrable Systems. Nucl. Phys. B 460, 299 (1996) [hep-th/9510101] MR1377167 (97a:58076)

20. Hitchin, N.J., Karlhede, A., Lindstrom, U. and Rocek, M.: Hy-perkahler Metrics and Supersymmetry. Commun. Math. Phys.108, 535 (1987) MR0877637 (88g:53048)

2001m:53043 53C07 81T13

Ford, C. [Ford, Christopher]; Pawlowski, J. M. (IRL-DIAS) ;Tok, T. (D-TBNG-P) ; Wipf, A. (D-FSU-TP)

ADHM construction of instantons on the torus. (English.English summary)Nuclear Phys. B 596 (2001), no. 1-2, 387–414.The complete classification of instantons on R4 has been knownfor over 20 years, thanks to the celebrated ADHM construction. Inthe meantime, physicists and mathematicians have been studyinginstantons on R4 that are invariant under subgroups of translations.In particular, periodic instantons (that is, instantons on Td ×R4−d)have interesting physical interpretations both in QCD and stringtheory. The main goal is to understand why there are no charge oneinstantons on T4, while these exist over Td×R4−d for d = 1, 2, 3.

In this paper, the authors try to “render the ADHM construction bybrute force” and thus obtain periodic instantons explicitly. Althoughthey provide an interesting reformulation of the Nahm transform,technical difficulties stop them from fully realizing their program.


The case d = 2 (i.e. doubly periodic instantons) is then considered infurther detail. Marcos Jardim (1-MA)

2001k:53037 53C07 58J90 81T13 81T20

Etesi, Gabor (J-KYOTU) ; Hausel, Tamas (1-CA)

Geometric interpretation of Schwarzschild instantons.(English. English summary)J. Geom. Phys. 37 (2001), no. 1-2, 126–136.The authors consider the problem of finding finite energy abelianinstantons on the Euclidean Schwarzschild manifold. In mathematicalterms, this means finding a self-dual L2 harmonic 2-form with integercohomology class.

A non-topological L2 self-dual 2-form is explicitly given, which givesrise to an abelian instanton. It is then shown that these instantonscoincide with the SU(2)-instantons found by Charap-Duff.

The paper concludes with a complete characterization of the spaceof L2 harmonic forms on the Euclidean Schwarzschild space.


2001k:53038 53C07 53C28

Pauly, Marc [Pauly, Marc1] (F-PARIS11)

Spherical monopoles and holomorphic functions. (English.English summary)Bull. London Math. Soc. 33 (2001), no. 1, 83–88.The author considers (singular) monopoles over the round 3-sphere. Itis shown that they induce canonically a holomorphic function definedon an open subset of the twistor space (i.e., the space of geodesics onS3). No indication of the inverse construction is given.{See [M. Pauly, Math. Ann. 311 (1998), no. 1, 125–146; MR

99h:58026] for related work.} Marcos Jardim (1-MA)


2002a:81112 81R12 37K10 81T60

Morozov, A. [Morozov, A. Yu.] (RS-ITEP)

Integrability in Seiberg-Witten theory. (English. Englishsummary)Integrability : the Seiberg-Witten and Whitham equations (Edinburgh,1998), 93–102, Gordon and Breach, Amsterdam, 2000.The author speculates on the origins of integrability in Seiberg-Wittentheory, advocating the hypothesis that “integrability is the generalproperty of effective actions”.{For the entire collection see 2001i:37003} Marcos Jardim (1-MA)

2001i:53039 53C07 14D21 14H70 53C28

Small, A. J. (IRL-MNTH)

Osculation and singularity of charge 2 (complexified) BPSmonopoles. (English. English summary)Internat. J. Math. 11 (2000), no. 7, 943–948.Complexified BPS monopoles can be described via the singularitylocus J ⊂C3 of the complex Higgs field. J intersects the degeneracylocus of the complex Higgs field on a null curve C. It is shown inthis paper that the dual curve C∗ ⊂ T ′P1 does not coincide with themonopole’s spectral curve for a generic rank-two monopole of chargetwo. Marcos Jardim (1-MA)

[References]

1. C. Anand, Uniton Bundles, Comm. Anal. Geom. 3(3-4) (1995),371–419. MR1371204 (96k:58052)

2. M. Atiyah and N. J. Hitchin, The Geometry and Dynamics ofMagnetic Monopoles, Princeton Univ. Press, Princeton, 1988.MR0934202 (89k:53067)

3. A. Cayley, Phil. Trans. Royal Soc. London, Vol. CLI (1861),225–239.

4. G. Darboux, Lecons sur la Theorie Generale des Surfaces III,Grauthier-Villars, Paris, 1894.

5. R. Hartshorne, Algebraic Geometry, Springer Verlag, New York,1977. MR0463157 (57 #3116)

6. N. J. Hitchin, Monopoles and Geodesics, Comm. Math. Phys. 83(1982), 579–602. MR0649818 (84i:53071)

7. J. Hurtubise, Analysis on Manifolds, Some Problems LinkingAlgebraic and Differential Geometry, D. Phil. Thesis, OxfordUniversity, 1982.

8. J. Hurtubise, SU(2) Monopoles of Charge 2, Comm. Math. Phys.


92 (1983), 195–202. MR0728865 (85j:32050)9. J. Hurtubise, The Asymptotic Higgs Field of a Monopole, Comm.

Math. Phys. 97 (1985), 381–389. MR0778622 (86f:58036)10. J. Hurtubise, Twistors and the Geometry of Bundles over P2(C),

Proc. London Math. Soc. 55(3) (1987), 450–464. MR0907228(88j:14023)

11. A. J. Small, Minimal Surfaces in R3 and Algebraic Curves, Diff.Geom. Appl. 2 (1992), 369–384. MR1243536 (94h:53014)

12. A. J. Small, Singularity Criteria for (Complexified) BPS Mono-poles, Math. Proc. Camb. Phil. Soc. Part 1, 129 (2000), 59–71.MR1757778 (2001e:53028)

13. A. J. Small, Duality for a Class of Minimal Surfaces in Rn+1,Tohoku Math. J. 51 (1999), 585–601. MR1725627 (2000h:53012)

14. E. T. Whittaker and G. N. Watson, A Course of Modern Anal-ysis, Cambridge Univ. Press, Cambridge, 1902. MR1424469(97k:01072)

15. R. S. Ward and R. O. Wells, Twistor Geometry and Field Theory,Cambridge Univ. Press, 1990. MR1054377 (91b:32034)

2001g:53057 53C07 53C26 53C29 53C80 58D27

Kraan, Thomas C. (NL-LEID-N)

Instantons, monopoles and toric hyperKahler manifolds.(English. English summary)Comm. Math. Phys. 212 (2000), no. 3, 503–533.The paper is concerned with SU(n) calorons, i.e. instantons on R3×S1. In particular, the metric in the moduli space of charge one caloronsis explicitly computed and shown to be toric hyper-Kahler.

Moreover, the limits as the radius of the circle S1 tends to zeroand to infinity are discussed. It is shown that the metric on themoduli space of charge one calorons approaches the metric of themoduli space of charge one instantons on R4 as S1 gets infinitelylarge. Similarly, the metric on the moduli space of charge one caloronsapproaches the metric of the moduli space of charge one monopoleson R3 as the S1 is contracted to a point.

Such phenomena were previously conjectured from considerationsin string theory and QCD. It would also be rather interesting to havea precise mathematical description at the level of the anti-self-dualityequations. Marcos Jardim (1-MA)

[References]

1. Atiyah, M.F. and Hitchin, N.J.: The Geometry and Dynamics


of Magnetic Monopoles. Princeton: Princeton Univ. Press, 1988MR0934202 (89k:53067)

2. Atiyah, M.F., Hitchin, N.J., Drinfeld, V.G., Manin, Yu.I.: Phys.Lett. 65 A, 185 (1978); Atiyah, M.F.: Geometry of Yang-Millsfields. Fermi lectures, Scuola Normale Superiore, Pisa, 1979MR0598562 (82g:81049)

3. Bogomol’nyi, E.B.: Yad. Fiz. 24, 861 (1976); Sov. J. Nucl. 24, 449(1976); Prasad, M.K. and Sommerfield, C.M.: Phys. Rev. Lett.35, 760 (1975) MR0443719 (56 #2082)

4. Braam, P.J. and van Baal, P.: Commun. Math. Phys. 122, 267(1989) MR0994505 (91a:58041)

5. Connell, S.A.: The dynamics of the SU(3) charge (1, 1) magneticmonopole. (1991), ftp://maths.adelaide.edu.au/pure/murray/oneone.tex,unpublished preprint

6. Corrigan, E.: Unpublished, quoted in [41]7. Corrigan, E. and Goddard, P.: Ann. Phys. (N.Y.) 154, 253 (1984)

MR0741204 (85k:81104)8. Donaldson, S.K.: Commun. Math. Phys. 93, 453 (1984)

MR0763753 (86m:32043)9. Donaldson, S.K.: Commun. Math. Phys. 96, 387 (1984)

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27. Lee, K., Weinberg, E.J., Yi, P.: Phys. Rev. D 54, 1633 (1996)MR1404609 (97e:58034)

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eds. N. Craigie, e.a. Singapore: World Scientific, 1982, p. 87MR0766754 (86e:53058)

38. Nahm, W.: Self-dual monopoles and calorons. In: Lect. Notes inPhysics. 201, eds. G. Denardo, e.a. 1984, p. 189 MR0782145

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2001g:53056 53C07 58D27

Jarvis, Stuart (4-OXMT)

A rational map for Euclidean monopoles via radial scattering.J. Reine Angew. Math. 524 (2000), 17–41.As is well-known, there is a correspondence between the moduli spaceof SU(2) monopoles and the space of rational maps from P1 to itself.This result of S. K. Donaldson [Comm. Math. Phys. 96 (1984), no. 3,387–407; MR 86c:58039] has been generalised for all compact groupsby various authors.

In this paper, the author proposes a new correspondence betweenmonopoles with semisimple, compact, gauge group G and certainspaces of rational maps that are different from those in Donaldson’stheorem and its generalisations.

More precisely, let G be a compact semisimple Lie group, and letR be a fixed conjugacy class in its Lie algebra. Then R is isomorphicto the flag manifold GC/P . It is shown that there is a one-to-onecorrespondence between the framed moduli space of G-monopoles ofcharge k whose Higgs field takes values in R at infinity, and the spaceof degree k rational maps from P1 to GC/P . Marcos Jardim (1-MA)

[References]

1. M. F. Atiyah, Magnetic monopoles in hyperbolic space, in:


Vector bundles on Algebraic Varieties, OUP, 1987. MR0893593(88i:32045)

2. M. F. Atiyah and N. J. Hitchin, The geometry and dynamics ofmonopoles, Princeton 1988. MR0934202 (89k:53067)

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6. S. K. Donaldson, Anti-self-dual Yang-Mills connections over com-plex algebraic surfaces and stable vector bundles, Proc. Lond.Math. Soc. 30 (1985), 1–26. MR0765366 (86h:58038)

7. S. K. Donaldson, Boundary-value problems for Yang-Mills fields,J. Geom. Phys. 8 (1992), 89–122. MR1165874 (93d:53033)

8. S. K. Donaldson, The approximation of instantons, Geom. Funct.Anal. 3 (1993), 179–200. MR1209301 (94k:58030)

9. G.-Y. Guo, Differential Geometry of holomorphic bundles, D.Phil. thesis, Oxford 1993.

10. R. S. Hamilton, Harmonic maps of manifolds with boundary,Springer Lect. Notes Math. 471 (1975). MR0482822 (58 #2872)

11. N. J. Hitchin, Monopoles and geodesics, Comm. Math. Phys. 83(1982), 579–602. MR0649818 (84i:53071)

12. N. J. Hitchin, On the construction of monopoles, Comm. Math.Phys. 89 (1983), 145–190. MR0709461 (84m:53076)

13. N. J. Hitchin, N. S. Manton and M. K. Murray, Symmetricmonopoles, Nonlin. 8 (1995), 661–692. MR1355037 (96j:53026)

14. C. J. Houghton, N. S. Manton and P. M. Sutcliffe, Rational maps,monopoles, and skyrmions, DAMPT preprint 1997. MR1607196(98m:58016)

15. J. Hurtubise, Monopoles and rational maps: a note on a theorem ofDonaldson, Comm. Math. Phys. 100 (1985), 191–196. MR0804459(87b:53113)

16. J. Hurtubise, The classification of monopoles for the classicalgroups, Comm. Math. Phys. 120 (1989), 613–641. MR0987771(90c:53182)

17. J. Hurtubise and M. K. Murray, On the construction of monopolesfor the classical groups, Comm. Math. Phys. 122 (1989), 35–89.MR0994495 (91d:58037)

18. J. Hurtubise and M. K. Murray, Monopoles and their spectraldata, Comm. Math. Phys. 133 (1990), 487–508. MR1079792


(92a:32041)19. A. Jaffe and C. H. Taubes, Vortices and Monopoles, Birkhauser,

Boston 1980. MR0614447 (82m:81051)20. S. Jarvis, Euclidean Monopoles and rational maps, Proc. London

Math. Soc. 77 (1998), 170–192. MR1625475 (99h:58024)21. S. Jarvis and P. Norbury, Compactification of hyperbolic mono-

poles, Nonlin. 10 (1997), 1073–1092. MR1473375 (98k:58035)22. S. Jarvis and P. Norbury, Zero and infinite curvature limits of

hyperbolic monopoles, Bull. London Math. Soc. 29 (1997), 737–744. MR1468062 (98i:53034)

23. A. Munari, D. Phil. Thesis, Oxford 1993.24. M. K. Murray, Non-abelian magnetic monopoles, Comm. Math.

Phys. 96 (1984), 539–565. MR0775045 (86e:53057)25. M. K. Murray and M. Singer, Spectral curves of non-integral

hyperbolic monopoles, Nonlin. 9 (1996), 973–997. MR1399482(97h:53026)

26. C. T. Simpson, Constructing Variations of Hodge structure usingYang-Mills theory and applications to uniformization, J. Amer.Math. Soc. 1 (1988), 867–918. MR0944577 (90e:58026)

27. D. Stuart, The Geodesic Approximation for the Yang-Mills-HiggsEquations, Comm. Math. Phys. 166 (1994), 149–190. MR1309545(96e:58037)

2001d:53056 53C27 58J50 58J60

Branson, Thomas (1-IA) ; Hijazi, Oussama (F-NANC-IE)

Improved forms of some vanishing theorems in Riemannianspin geometry. (English. English summary)Internat. J. Math. 11 (2000), no. 3, 291–304.From the summary: “We improve the hypotheses on some vanishingtheorems over a Riemannian spin manifold [see T. P. Branson andO. Hijazi, Internat. J. Math. 8 (1997), no. 7, 921–934; MR 98k:58229].The improved hypotheses are uniform, in the sense that they arethe same for each of an infinite sequence of bundles in each evendimension. They are also elementary, in the sense that they involveonly the bottom eigenvalue of the Yamabe operator on scalars, andthe pointwise action of the Weyl conformal curvature tensor on two-forms. In particular, they do not make reference to the higher spinbundles on which the conclusion holds.” Marcos Jardim (1-MA)

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and Killing Spinors on Riemannian Manifolds, Seminarbericht,Vol. 108, Humboldt-Universitat zu Berlin, 1990. MR1084369(92h:53055)

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Soc. London Ser. A, 370 (1980), 173–191. MR0563832 (81i:81057)17. E. Stein and G. Weiss, Generalization of the Cauchy-Riemann

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2001f:53052 53C07 53C28 81T13

Houghton, C. J. [Houghton, Conor J.] (4-CAMB-A) ;Manton, N. S. (4-CAMB-A) ; Romao, N. M. (4-CAMB-A)

On the constraints defining BPS monopoles. (English.English summary)Comm. Math. Phys. 212 (2000), no. 1, 219–243.N. J. Hitchin [Comm. Math. Phys. 89 (1983), no. 2, 145–190; MR84m:53076] described the twistor approach to SU(2) monopoles (i.e.solutions of the Bogomol′nyı equations on R3), showing that these areequivalent to the so-called spectral curves on the mini-twistor spaceTP1, the total space of the holomorphic tangent bundle of P1.

The moduli space of gauge equivalence classes of monopoles canthen be characterised as the space of complex curves in TP1 satisfyinga number of transcendental constraints.

Attempts to formulate such constraints explicitly have been madeby E. Corrigan and P. Goddard [Comm. Math. Phys. 80 (1981),no. 4, 575–587; MR 83b:81078] and by N. M. Ercolani and A. Sinha[Comm. Math. Phys. 125 (1989), no. 3, 385–416; MR 91a:58147]. Thepresent paper presents a new version of the Ercolani-Sinha constraints,clarifying their relation with the Corrigan-Goddard approach.

As an application, the authors apply the Ercolani-Sinha method tocompute the spectral curve of the tetrahedral 3-monopole discussedby Hitchin, N. S. Manton and M. K. Murray [Nonlinearity 8 (1995),no. 5, 661–692; MR 96j:53026]. Marcos Jardim (1-MA)

[References]

1. Abramowitz, M. and Stegun, I.A.: Handbook of MathematicalFunctions. National Bureau of Standards, 1965

2. Atiyah, M.F. and Hitchin, N.J.: The Geometry and Dynamics ofMagnetic Monopoles. Princeton, NJ: Princeton University Press,1988 MR0934202 (89k:53067)

3. Corrigan, E. and Goddard, P.: An n Monopole Solution with 4n−1 Degrees of Freedom. Commun. Math. Phys. 80, 575–587 (1981)MR0628513 (83b:81078)

4. Ercolani, N. and Sinha, A.: Monopoles and Baker Functions. Com-mun. Math. Phys. 125, 385–416 (1989) MR1022520 (91a:58147)


5. Forgacs, P., Horvath, Z. and Palla, L.: Finitely Separated Mul-timonopoles Generated as Solitons. Phys. Lett. B 109, 200–204(1982) MR0644081 (83b:81082)

6. Griffiths, P. and Harris, J.: Principles of Algebraic Geometry. NewYork: Wiley, 1978 MR0507725 (80b:14001)

7. Hitchin, N.J.: Monopoles and Geodesics. Commun. Math. Phys.83, 579–602 (1982) MR0649818 (84i:53071)

8. Hitchin, N.J.: On the Construction of Monopoles. Commun. Math.Phys. 89, 145–190 (1983) MR0709461 (84m:53076)

9. Hitchin, N.J., Manton, N.S. and Murray, M.K.: Symmetric Mono-poles. Nonlinearity 8, 661–692 (1995); dg-ga/9503016 MR1355037(96j:53026)

10. Houghton, C.J. and Sutcliffe, P.M.: Tetrahedral and Cu-bic Monopoles. Commun. Math. Phys. 180, 343–361 (1996);hep-th/9601146 MR1405955 (97g:53029)

11. Houghton, C.J. and Sutcliffe, P.M.: Octahedral and Dodecahe-dral Monopoles. Nonlinearity 9, 385–401 (1996); hep-th/9601147MR1384481 (97c:53039)

12. Hurtubise, J.: SU(2) Monopoles of Charge 2. Commun. Math.Phys. 92, 195–202 (1983) MR0728865 (85j:32050)

13. O’Raifeartaigh, L., Rouhani, S. and Singh, L.P.: Explicit Solutionof the Corrigan-Goddard Conditions for n Monopoles for SmallValues of the Parameters. Phys. Lett. B 112, 369–372 (1982)MR0658809 (84j:81083)

14. Sutcliffe, P.M.: BPS Monopoles. Int. J. Mod. Phys. A 12, 4663–4705 (1997); hep-th/9707009 MR1474144 (98j:53032)

15. Ward, R.S.: A Yang-Mills-Higgs Monopole of Charge 2. Commun.Math. Phys. 79, 317–325 (1981) MR0627055 (84i:81077)

2001e:53028 53C07 14D21 32L81 53C28

Small, A. J. (IRL-MNTH)

Singularity criteria for (complexified) BPS monopoles.Math. Proc. Cambridge Philos. Soc. 129 (2000), no. 1, 59–71.As was shown by N. J. Hitchin [Comm. Math. Phys. 83 (1982), no. 4,579–602; MR 84i:53071], an SU(2) monopole on R3 is equivalentto a holomorphic rank 2 bundle E over T (the total space of theholomorphic tangent bundle of P1). Generically, the restriction ofsuch a bundle to the image of a global section of T→ P1 is trivial.The so-called jumping lines are parametrised by an analytic set J ⊂C3 = H0(P1,T).

Here, C3 is also thought of as the complexification of R3, wherethe monopole lives. The analytic continuation of the monopole has


singularities over J . Conversely, an analytic set J also determinesthe monopole. One then asks whether it is possible to describe thesingularity locus J , as in the case of instantons [see M. F. Atiyahand R. S. Ward, Comm. Math. Phys. 55 (1977), no. 2, 117–124; MR 58#13029].

The paper under review reports on the author’s first step towardsthis goal, explicitly describing the singularity locus J associated tocharge 1 monopoles and charge 2 axially symmetric monopoles. Itis shown that in the first case J is given by the union of certainquadric surfaces in C3. In the second case, the singularity locus ismore complicated; it is described in Section 4.

REVISED (November, 2002)Marcos Jardim (1-MA)

2001c:53060 53C26 53C07 58J60

Hitchin, Nigel (4-OX)

L2-cohomology of hyperkahler quotients. (English. Englishsummary)Comm. Math. Phys. 211 (2000), no. 1, 153–165.Building on earlier results by Jost and Zuo, the author gives necessaryconditions which imply that the L2 harmonic forms on noncompactcomplete hyper-Kahler manifolds lie in the middle dimension and areinvariant under the isometry group.

More precisely, let M be a complete hyper-Kahler manifold of realdimension 4k such that one of the Kahler forms ω = bβ with β havinglinear growth (i.e. ‖β(x)‖ ≤ c1 · ρ(x0, x) + c2, where ρ(x0, x) is theRiemannian distance from a point x0 ∈M , and c1, c2 are constants).Then any L2 harmonic form is primitive and of type (k, k) withrespect to any complex strucuture on M .

The theorem is then applied to various examples, most notably theones in connection with gauge-theoretical hyper-Kahler moduli spaces(of Higgs bundles/monopoles/instantons). Marcos Jardim (1-MA)

[References]

1. Atiyah, M. F., Hitchin, N. J., Drinfel’d, V. G., Manin, Yu. I.:Construction of instantons. Phys. Lett. A 65, 185–187 (1978)MR0598562 (82g:81049)

2. Atiyah, M. F., Hitchin, N. J.: The geometry and dynamics of mag-netic monopoles. M. B. Porter Lectures, Princeton, NJ: PrincetonUniversity Press, 1988 MR0934202 (89k:53067)

3. Baldwin, P.: L2 solutions of Dirac equations. Ph.D. Thesis, Cam-


bridge, 19994. Calabi, E.: Metriques kahleriennes et fibres holomorphes.

Ann. Sci. Ecole Norm. Sup. 12, 269–294 (1979) MR0543218(83m:32033)

5. de Rham, G.: Differential manifolds. Berlin-Heidelberg-New York:Springer Verlag, 1988

6. Dodziuk, J.: Vanishing theorems for square-integrable harmonicforms. Proc. Indian Acad. Sci. Math. Sci. 90, 21–27 (1981).MR0653943 (83h:58006)

7. Gibbons, G. W.: The Sen conjecture for fundamental monopolesof distinct types. Phys. Lett. B 382, 53–59 (1996) MR1398847(97j:81255)

8. Gromov, M.: Kahler hyperbolicity and L2-Hodge theory, J. Dif-ferential Geom. 33, 263–292 (1991) MR1085144 (92a:58133)

9. Hausel, T.: Vanishing of intersection numbers on the moduli spaceof Higgs bundles. Adv. Theor. Math. Phys. 2, 1011–1040 (1998)MR1688480 (2000g:14017)

10. Hitchin, N. J., Karlhede, A., Lindstrom, U., Rocek, M.: Hy-perkahler metrics and supersymmetry. Commun. Math. Phys.108, 535–589 (1987) MR0877637 (88g:53048)

11. Hitchin, N. J.: The self-duality equations on a Riemann sur-face. Proc. London Math. Soc. 55, 59–126 (1987) MR0887284(89a:32021)

12. Hitchin, N. J.: Integrable systems in Riemannian geometry. InSurveys in Differential Geometry Vol. 4, C.-L. Terng and K.Uhlenbeck, (eds.), Cambridge, MA: International Press, 1999, pp.21–80 MR1726926 (2000j:53046)

13. Jost, J., Zuo,K.: Vanishing theorems for L2-cohomology on infinitecoverings of compact Kahler manifolds and applications in alge-braic geometry. Preprint No. 70, Max-Planck-Institut fur Math-ematik in den Naturwissenschaften, Leipzig (1998) MR1730897(2001f:32033)

14. Kimyeong Lee, Weinberg, E. J., Piljin Yi: Electromagnetic du-ality and SU(3) monopoles. Phys. Lett. B 376,, 97–102 (1996)MR1395564 (97c:81183)

15. Nakajima, H.: Monopoles and Nahm’s equations. In: Einstein met-rics and Yang-Mills connections. Proceedings of the 27th Taniguchiinternational symposium, held at Sanda, Japan, December 6–11,1990, eds T. Mabuchi et al., Lect. Notes Pure Appl. Math. 145,New York: Marcel Dekker, 1993 MR1215288 (94b:53055)

16. Nakajima, H.: Lectures on Hilbert Schemes of points on surfaces,AMS University Lecture Series 18, AMS (Providence) (1999)


MR1711344 (2001b:14007)17. Nekrasov, N., Schwarz, A.: Instantons on noncommutative R4,

and (2, 0) superconformal six dimensional theory, Commun. Math.Phys. 198, 689–703 (1998) MR1670037 (2000e:81198)

18. Nelson, E.: Analytic vectors. Ann. of Math. 70, 572–615 (1959)MR0107176 (21 #5901)

19. Segal, G. B., Selby, A.: The cohomology of the space of mag-netic monopoles, Commun. Math. Phys. 177, 775–787 (1996)MR1385085 (97a:58022)

20. Sen, A.: Dyon-monopole bound states, self-dual harmonic formson the multi-monopole moduli space, and SL(2,Z)-invariance ofstring theory. Phys. Lett. B 329, 217–221 (1994) MR1281578(95e:81187)

21. Sethi, S., Stern, M.,Zaslow, E.: Monopole and dyon bound statesin N = 2 supersymmetric Yang-Mills theories. Nuclear Phys. B457, 484–510 (1995) MR1367232 (97a:81234)

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2001b:53024 53C07 53C43 58D27

Mukai, Mariko [Mukai-Hidano, Mariko] (J-TOKYMGS)

Moduli spaces of solutions to the gauge-theoretic equationsfor harmonic maps.Harmonic morphisms, harmonic maps, and related topics (Brest,1997), 211–241, Chapman & Hall/CRC Res. Notes Math., 413,Chapman & Hall/CRC, Boca Raton, FL, 2000.In this paper, the author considers a modification of Hitchin’s self-duality equations over a Riemann surface Σ [see N. J. Hitchin, Proc.London Math. Soc. (3) 55 (1987), no. 1, 59–126; MR 89a:32021].

More precisely, let P be a principal G-bundle over Σ, and let µbe the slope (i.e. degree/rank) of the associated vector bundle; asusual, G is a compact Lie group. Consider the pair (A,ϕ) consistingof a connection A on P and an endomorphism-valued 1-form ϕ. Theso-called gauge-theoretic equations for harmonic maps are given by{

FA + 12 [ϕ, ϕ] =−2πiµ,

dAϕ = 0, d∗Aϕ = 0.

If Σ has genus zero and µ = 0, each solution gives a harmonic mapfrom Σ to G. The case of G = SU(2) and genus one was stud-ied by Hitchin [J. Differential Geom. 31 (1990), no. 3, 627–710; MR91d:58050]. Finally, these equations were also considered in full gener-


ality by G. Valli [Ann. Inst. H. Poincare Anal. Non Lineaire 6 (1989),no. 3, 233–245; MR 90f:58035].

The purpose of this paper is to discuss the basic properties (e.g.,deformation theory, metric structures) of the moduli space of solutionsto the equations above. The moduli space of solutions correspondingto harmonic maps (case µ = 0) and the moduli space of solutions asthe conformal structure of Σ is varied are also considered.{For the entire collection see 2000j:53001}


2001a:53046 53C07 53C26 53D12 58D27

Ohnita, Yoshihiro (J-TOKYMGS)

Gauge-theoretic equations for symmetric spaces and certainminimal submanifolds in moduli spaces.Harmonic morphisms, harmonic maps, and related topics (Brest,1997), 193–209, Chapman & Hall/CRC Res. Notes Math., 413,Chapman & Hall/CRC, Boca Raton, FL, 2000.This paper contains the proofs of a set of results previously an-nounced by the author [Surikaisekikenkyusho Kokyuroku No. 1044(1998), 76–85; MR 99m:53046]. The goal is to use a generalisation ofHitchin’s equations for symmetric spaces (see the mentioned reviewfor a description) to construct totally geodesic, complex Lagrangiansubmanifolds of the hyper-Kahler moduli space of solutions to theusual Hitchin’s equations.

The constructions presented here are very interesting, and theresults are potentially generalisable for other contexts within gaugetheory.{For the entire collection see 2000j:53001}


2000k:58011 58D27 53C07 55R45 57M50

Bryan, Jim (1-TULN) ; Sanders, Marc (1-TULN)

Instantons on S4 and CP2, rank stabilization, and Bottperiodicity. (English. English summary)Topology 39 (2000), no. 2, 331–352.Let MGn

k (X) be the moduli space of based Gn-instantons on X ofcharge k, where Gn is SU(n), SO(n) or Sp(n/2). In [J. DifferentialGeom. 29 (1989), no. 1, 163–230; MR 90f:58023], C. H. Taubes hasshown that, in the direct limit topology, limk→∞ MGn

k (X) is homotopi-cally equivalent to the set of based maps X →BGn.

In this paper, the authors study the homotopy type of the direct


limit as n →∞ for X = S4 and X = CP2. Most of the results hereestablished were obtained earlier [see P. Norbury and M. Sanders,Proc. Amer. Math. Soc. 124 (1996), no. 7, 2193–2201; MR 96i:58021;M. Sanders, Trans. Amer. Math. Soc. 347 (1995), no. 10, 4037–4072;MR 96m:58030; J. Bryan and M. Sanders, Proc. Amer. Math. Soc. 125(1997), no. 12, 3763–3768; MR 98b:58028]. The main novelty is thatthe authors now provide a unified approach to all groups mentionedabove.

The proof is based upon Donaldson’s correspondence between in-stantons on S4 and CP2 and framed holomorphic vector bundles overCP2 and CP2, respectively [see S. K. Donaldson, Comm. Math. Phys.93 (1984), no. 4, 453–460; MR 86m:32043; N. Buchdahl, J. DifferentialGeom. 37 (1993), no. 3, 669–687; MR 94e:53025]. Such bundles canbe constructed via monads, and the authors analyze the direct limittopology of the space of ADHM-type data.

Their main result is the rank stabilization theorem: limn→∞ MGn

k (X)is homotopically equivalent to the classifying space for the automor-phism group of ADHM-type data.

As a by-product, this result is used along with Taubes’ result toobtain a novel proof of the homotopy equivalences in the eight-foldBott periodicity spectrum. Marcos Jardim (1-MA)

[References]

1. C.P. Boyer, J.C. Hurtubise, B.M. Mann, R.J. Milgram, The topol-ogy of instanton moduli spaces. I: The Atiyah-Jones conjecture,Ann. Math. 137 (1993) 561–609. MR1217348 (94h:55010)

2. C.P. Boyer, B.M. Mann, Homology operations on instantons, J.Differential Geom. 28 (1988) 423–465. MR0965223 (89m:58044)

3. J. Bryan, Symplectic geometry and the relative Donaldson in-variants of CP

2, Forum Math. 9 (1997) 325–365. MR1441925

(98b:57047)4. J. Bryan, M. Sanders, The rank stable topology of instantons

on CP2, Proc. Amer. Math. Soc. 125(12) (1997) 3763–3768.

MR1425114 (98b:58028)5. N. Buchdahl, Instantons on CP2, J. Differential Geom. 24 (1986).

MR0857374 (88b:32066)6. N. Buchdahl, Instantons on nCP2, J. Differential Geom. 37

(1993). MR1217165 (94e:53025)7. R.L. Cohen, J. Jones, Monopoles, braid groups, and the Dirac

operator, Comm. Math. Phys. 158 (1993) 241–266. MR1249594(95d:58126)


8. S.K. Donaldson, Instantons and geometric invariant the-ory, Comm. Math. Phys. 93 (1984) 453–460. MR0763753(86m:32043)

9. S.K. Donaldson, P.B. Kronheimer, The Geometry of Four-Manifolds, Oxford Mathematical Monographs, Oxford UniversityPress, Oxford, 1990. MR1079726 (92a:57036)

10. G. Horrocks, Vector bundles on the punctured spectrum of a localring, Proc. London Math. Soc. 14(3) (1964) 689–713. MR0169877(30 #120)

11. J.C. Hurtubise, Instantons and jumping lines, Comm. Math. Phys.105 (1986) 107–122. MR0847130 (87g:14009)

12. J.C. Hurtubise, R.J. Milgram, The Atiyah-Jones conjecturefor ruled surfaces, J. Math. 466 (1995) 111–143. MR1353316(96j:58025)

13. A. King, Instantons and holomorphic bundles on the blown upplane, Ph.D. thesis, Oxford, 1989.

14. F. Kirwan, Geometric invariant theory and the Atiyah-Jonesconjecture, in: O.A. Laudal, B. Jahren (Eds), Proceedings of theSophus Lie Memorial Conference, Scandinavian University Press,1994. MR1456466 (98e:58036)

15. E. Lupercio, Ph.D. thesis, Stanford, 1995.16. P. Norbury, M. Sanders, Real instantons, Dirac operators and

quaternionic classifying spaces, Proc. Amer. Math. Soc. 124(7)(1996). MR1327032 (96i:58021)

17. C. Okonek, M. Schneider, H. Spindler, Vector Bundles on Com-plex Projective Spaces, Birkhauser, Basel, 1980. MR0561910(81b:14001)

18. M. Sanders, Classifying spaces and Dirac operators coupled toinstantons, Trans. Amer. Math. Soc. 347(10) (1995). MR1311915(96m:58030)

19. C. Taubes, The stable topology of self-dual moduli spaces, J.Differential Geom. 29 (1989). MR0978084 (90f:58023)

20. Y. Tian, The Atiyah-Jones conjecture for classical groups andBott periodicity. J. Differential Geom., to appear. MR1420352(97j:58019)


2000j:53031 53C07 53C29 53C80

Miyagi, Sayuri (J-OSAKC-P)

Yang-Mills instantons on seven-dimensional manifold of G2holonomy. (English. English summary)Modern Phys. Lett. A 14 (1999), no. 37, 2595–2604.In this brief paper the author investigates Yang-Mills instantons on a7-manifold given by an R4 bundle over the 3-sphere with respect to aRiemannian metric with G2 holonomy constructed by G. W. Gibbons,D. N. Page and C. N. Pope [Comm. Math. Phys. 127 (1990), no. 3,529–553; MR 91f:53039]; see also R. L. Bryant and S. M. Salamon[Duke Math. J. 58 (1989), no. 3, 829–850; MR 90i:53055].

After a quick but useful review of the relevant physics literature, theinstanton equations are explicitly solved under a spherical symmetryansatz. Marcos Jardim (1-MA)

2000i:53076 53C29 58J90 81T13 81T60

Selivanov, K. (RS-ITEP)

Flat connections with non-diagonalizable holonomies.(English. English summary)Phys. Lett. B 471 (1999), no. 2-3, 171–173.Summary: “Recently the long-standing puzzle about counting theWitten index in N = 1 supersymmetric gauge theories was resolved [E.Witten, J. High Energy Phys. 1998, no. 2, Paper 6, 43 pp. (electronic);MR 99e:81214]. The resolution was based on the existence (for higherorthogonal SO(N), N ≥ 7, and exceptional gauge groups) of flatconnections on T 3 which have commuting holonomies but cannot begauged to a Cartan torus. A number of papers have been publishedwhich studied moduli spaces and some topological characteristics ofthose flat connections. In the present paper an explicit description ofsuch flat connections for the basic case of Spin(7) is given.”



2000h:53065 53C29 53C07 53C80 58J60

Bilge, Ayse Humeyra (TR-ISTNT) ; Dereli, Tekin (TR-MET-P) ;Kocak, Sahin (TR-ANA)

Monopole equations on 8-manifolds with Spin(7) holonomy.(English. English summary)Comm. Math. Phys. 203 (1999), no. 1, 21–30.The authors study a generalisation of the Seiberg-Witten monopoleequations from 4- to 8-dimensional manifolds with Spin(7) holonomy.It is shown that the pair of equations proposed is elliptic, thus openingup interesting mathematical possibilities.{See [J. Math. Phys. 38 (1997), no. 9, 4804–4814; MR 98h:53117;

Lett. Math. Phys. 36 (1996), no. 3, 301–309; MR 96m:53023] for earlierrelated work by the authors.} Marcos Jardim (1-MA)

[References]

1. Seiberg, N., Witten, E.: Nucl. Phys. B426, 19 (1994) MR1293681(95m:81202a)

2. Witten, E.: Math. Res. Lett. 1, 764 (1994) MR1306021(96d:57035)

3. Flume, R.: O’Raifeartaigh, L., Sachs, I.: Brief resume of theSeiberg-Witten theory. hep-th/9611118

4. Donaldson, S.K., Thomas, R.P.: Gauge theory in higher dimen-sions. Oxford University preprint, 1996 MR1634503 (2000a:57085)

5. Baulieu, L., Kanno, H., Singer, I.M.: Special quantum field theo-ries in eight and other dimensions. hep-th/9704167

6. Acharya, B.S., O’Loughlin, M., Spence, B.: Higher dimensionalanalogues of Donaldson-Witten Theory. hep-th/9705138

7. Hull, C.M.: Higher dimensional Yang-Mills theories and topolog-ical terms. hep-th/9710165

8. Salamon, D.: Spin Geometry and Seiberg-Witten Invariants. April1996 version, Book to appear

9. Bilge, A.H., Dereli, T., Kocak. S.: Seiberg-Witten equations on R8.In: The Proceedings of 5th Gokova Geometry-Topology Conference,Edited by S. Akbulut, T. Onder, R. Stern, TUBITAK, Ankara,1997, p. 87 MR1456162 (98d:57054)

10. Bilge, A.H., Dereli, T., Kocak, S.: J. Math. Phys. 38, 4804 (1997)MR1468670 (98h:53117)

11. Corrigan, E., Devchand, C., Fairlie, D., Nuyts, J.: Nucl. Phys.B214, 452 (1983) MR0698892 (84i:81058)

12. Ward, R.S.: Nucl. Phys. B236, 381 (1984) MR0739812(86b:53073)


13. Bilge, A.H., Dereli, T., Kocak, S.: Lett. Math. Phys. 36, 301(1996) MR1376941 (96m:53023)

14. Gursey, F., Tze, C.-H.: On the Role of Division, Jordan andRelated Algebras in Particle Physics. Singapore: World Scientific,1996 MR1626607 (99g:81061)

15. Fairlie, D., Nuyts, J.: J. Phys. A17, 2867 (1984) MR0771770(86e:53056)

16. Fubini, S., Nicolai, H.: Phys. Lett. B155, 369 (1985) MR0793375(86m:81101)

17. Grossman, B., Kephart, T.W., Stasheff, J.D.: Commun. Math.Phys. 96, 431 (1984) (Erratum:ibid, 100 311 (1985)) MR0775040(87b:53048) MR0804465 (87b:53049)

18. Joyce, D.D.: Invent. Math. 123, 507 (1996) MR1383960(97d:53052)

19. Lawson, H.B., Michelsohn, M-L.: Spin Geometry. Princeton, NJ:Princeton U.P., 1989 MR1031992 (91g:53001)

20. John, F.: Partial Differential Equations. Berlin-Heidelberg-NewYork: Springer-Verlag, 1982 MR0831655 (87g:35002)

2001f:53140 53C43 53C07 58E20

Mukai, Mariko [Mukai-Hidano, Mariko] (J-TOKYMGS) ;Ohnita, Yoshihiro (J-TOKYMGS)

Gauge-theoretic equations for harmonic maps intosymmetric spaces. (English. English summary)The Third Pacific Rim Geometry Conference (Seoul, 1996), 195–209,Monogr. Geom. Topology, 25, Internat. Press, Cambridge, MA, 1998.Given a Riemann surface M and a compact Lie group G, itis known that harmonic maps from M to G admit a gauge-theoretic formulation [see N. J. Hitchin, J. Differential Geom. 31(1990), no. 3, 627–710; MR 91d:58050]. Indeed, each such map pro-duces a solution (A,ϕ) to the equations FA + 1

2 [ϕ, ϕ] = 0, dAϕ =dA ∗ϕ = 0, where A is a connection on a principal G-bundle over Mand ϕ is a section of the adjoint bundle.

In this paper, the authors study the above equations from thegauge-theoretic point of view (deformation theory, construction of themoduli space, removability of point singularities). This is done in aslightly generalised setup, considering not only compact Lie groupsbut compact symmetric spaces as a target space.

(See also previous work by Mukai [in Harmonic morphisms, har-monic maps, and related topics (Brest, 1997), 211–241, Chapman &Hall/CRC, Boca Raton, FL, 2000; MR 2001b:53024].)


{For the entire collection see 2000m:53002}Marcos Jardim (1-MA)

2001b:14057 14H70 14H81 37J35 81R12 81T60

Donagi, Ron Y. (1-PA)

Seiberg-Witten integrable systems.Surveys in differential geometry : integral systems [integrable systems],83–129, Surv. Differ. Geom., IV, Int. Press, Boston, MA, 1998.Integrable systems have long been a strong focal point of research forboth mathematicians and physicists. In this excellent survey article,the author explores a new aspect of such interaction: the connectionbetween algebraic geometry and high energy physics found in Seiberg-Witten theory. The upshot of this interaction is that structures arisingin the physics of certain supersymmetric Yang-Mills theories can berealised as algebraically integrable systems.

The survey is divided into four sections. The first reviews the rel-evant physics in a way particularly accessible to mathematicians.Section 2 consists of a brief review of algebraically integrable systems.Section 3 revisits the paper by the author and E. Witten [NuclearPhys. B 460 (1996), no. 2, 299–334; MR 97a:58076] emphasising themathematical rather than the physical aspects. In the final section,periodic Toda lattices are also used to solve another class of super-symmetric Yang-Mills theories [see E. J. Martinec and N. P. Warner,Nuclear Phys. B 459 (1996), no. 1-2, 97–112; MR 97g:58080].

This is a very interesting and useful review article, especially forgraduate students and young research mathematicians and physiciststrying to get a broader view of how algebraic geometry has be-come increasingly important in modern non-perturbative high energyphysics.{For the entire collection see 2000g:00041}



2000j:53046 53C21 14H70 34M55 37J35 53C07 53C26

Hitchin, Nigel (4-CAMB)

Integrable systems in Riemannian geometry.Surveys in differential geometry : integral systems [integrable systems],21–81, Surv. Differ. Geom., IV, Int. Press, Boston, MA, 1998.In this extraordinary survey, the author tours various branches ofRiemannian geometry in which the theory of integrable systems hasbeen used in the past few years. More precisely, these are the existenceof immersed minimal tori inside a compact semisimple Lie group, thestudy of self-dual Einstein manifolds with a 3-dimensional isometrygroup, and the study of certain hyper-Kahler manifolds based onsolutions to Nahm’s equations.

The first topic is treated in Section 2, which contains an accountof two approaches to the problem. The first approach is based on thestudy of harmonic maps from a 2-dimensional torus into a compactsemisimple Lie group G, which are in turn equivalent to flat GC con-nections. The second method is christened the loop group approach,and involves the existence of a rather specific object: a polynomialKilling field over the torus. Both approaches are finally brought to-gether via the construction of the so-called spectral curve. Both theharmonic map and the polynomial Killing field mentioned above canbe reconstructed from the spectral curve, which explicitly displays theintegrability of the original problem.

Section 3 is dedicated to the second topic. A particular class ofEinstein metrics on four manifolds with self-dual Weyl tensor isgeometrically interpreted as the problem of finding an isomonodromicdeformation of a given meromorphic connection on a trivial bundleover P1. The general isomonodromic deformation problem for 4 pointsreduces to a Painleve equation, which, for some specific values of theparameters, can be explicitly solved in terms of theta functions. Sucha point of view again reveals the integrability of the original problem.

Before addressing the third topic, the author surveys “the role ofYang-Mills equations as an overarching structure yielding integrableequations in Riemannian geometry” (see also the book by L. J. Masonand N. M. J. Woodhouse [Integrability, self-duality, and twistor theory,Oxford Univ. Press, New York, 1996; MR 98f:58002]). In particular,the two examples previously discussed can be regarded as incarnationsof the Yang-Mills equations.

Finally, Section 5 reviews a set of results in which the methodsof integrable systems are used to provide concrete information abouthyper-Kahler metrics. The key ingredients are the hyper-Kahler quo-tient construction and Nahm’s equations. The main goal here is to


obtain an explicit expression for the Kahler potential associated to theL2 metric on the moduli spaces of monopoles (which are equivalentto solutions of Nahm’s equations via the so-called Nahm transform).A few examples are also closely analysed.

This survey is a beautiful piece of mathematics, and containsnumerous historical and inspirational remarks, as well as an extensivebibliography. Its reading is highly recommended.{For the entire collection see 2000g:00041}


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