Atiyah’s work on holomorphic vector bundles and gauge theories
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- by Simon Donaldson;
- Bull. Amer. Math. Soc. 58 (2021), 567-610
- DOI: https://doi.org/10.1090/bull/1748
- Published electronically: August 5, 2021
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Abstract:
The first part of the article surveys Atiyah’s work in algebraic geometry during the 1950s, mainly on holomorphic vector bundles over curves. In the second part we discuss his work from the late 1970s on mathematical aspects of gauge theories, involving differential geometry, algebraic geometry, and topology.References
- V. I. Arnol′d, The asymptotic Hopf invariant and its applications, Selecta Math. Soviet. 5 (1986), no. 4, 327–345. Selected translations. MR 891881
- Aravind Asok, Brent Doran, and Frances Kirwan, Yang-Mills theory and Tamagawa numbers: the fascination of unexpected links in mathematics, Bull. Lond. Math. Soc. 40 (2008), no. 4, 533–567. MR 2438072, DOI 10.1112/blms/bdn036
- M. F. Atiyah, Complex fibre bundles and ruled surfaces, Proc. London Math. Soc. (3) 5 (1955), 407–434. MR 76409, DOI 10.1112/plms/s3-5.4.407
- M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207. MR 86359, DOI 10.1090/S0002-9947-1957-0086359-5
- M. F. Atiyah, Vector bundles over an elliptic curve, Proc. London Math. Soc. (3) 7 (1957), 414–452. MR 131423, DOI 10.1112/plms/s3-7.1.414
- M. F. Atiyah, On analytic surfaces with double points, Proc. Roy. Soc. London Ser. A 247 (1958), 237–244. MR 95974, DOI 10.1098/rspa.1958.0181
- M. F. Atiyah, Geometry on Yang-Mills fields, Scuola Normale Superiore, Pisa, 1979. MR 554924
- M. F. Atiyah, Green’s functions for self-dual four-manifolds, Mathematical analysis and applications, Part A, Adv. Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 129–158. MR 634238
- M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15. MR 642416, DOI 10.1112/blms/14.1.1
- M. F. Atiyah, Instantons in two and four dimensions, Comm. Math. Phys. 93 (1984), no. 4, 437–451. MR 763752
- M. F. Atiyah, Magnetic monopoles in hyperbolic spaces, Vector bundles on algebraic varieties (Bombay, 1984) Tata Inst. Fund. Res. Stud. Math., vol. 11, Tata Inst. Fund. Res., Bombay, 1987, pp. 1–33. MR 893593
- M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
- M. F. Atiyah and R. Bott, The moment map and equivariant cohomology, Topology 23 (1984), no. 1, 1–28. MR 721448, DOI 10.1016/0040-9383(84)90021-1
- M. F. Atiyah, N. J. Hitchin, V. G. Drinfel′d, and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187. MR 598562, DOI 10.1016/0375-9601(78)90141-X
- Michael Atiyah and Nigel Hitchin, The geometry and dynamics of magnetic monopoles, M. B. Porter Lectures, Princeton University Press, Princeton, NJ, 1988. MR 934202, DOI 10.1515/9781400859306
- M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Deformations of instantons, Proc. Nat. Acad. Sci. U.S.A. 74 (1977), no. 7, 2662–2663. MR 458424, DOI 10.1073/pnas.74.7.2662
- M. F. Atiyah, N. J. Hitchin, and I. M. Singer, Self-duality in four-dimensional Riemannian geometry, Proc. Roy. Soc. London Ser. A 362 (1978), no. 1711, 425–461. MR 506229, DOI 10.1098/rspa.1978.0143
- M. F. Atiyah and J. D. S. Jones, Topological aspects of Yang-Mills theory, Comm. Math. Phys. 61 (1978), no. 2, 97–118. MR 503187
- M. F. Atiyah and R. S. Ward, Instantons and algebraic geometry, Comm. Math. Phys. 55 (1977), no. 2, 117–124. MR 494098
- H. Bateman, The Solution of Partial Differential Equations by Means of Definite Integrals, Proc. London Math. Soc. (2) 1 (1904), 451–458. MR 1576794, DOI 10.1112/plms/s2-1.1.451
- Wolf Barth and Klaus Hulek, Monads and moduli of vector bundles, Manuscripta Math. 25 (1978), no. 4, 323–347. MR 509589, DOI 10.1007/BF01168047
- C. P. Boyer, J. C. Hurtubise, B. M. Mann, and R. J. Milgram, The topology of instanton moduli spaces. I. The Atiyah-Jones conjecture, Ann. of Math. (2) 137 (1993), no. 3, 561–609. MR 1217348, DOI 10.2307/2946532
- Sergey A. Cherkis and Anton Kapustin, $D_k$ gravitational instantons and Nahm equations, Adv. Theor. Math. Phys. 2 (1998), no. 6, 1287–1306 (1999). MR 1693628, DOI 10.4310/ATMP.1998.v2.n6.a3
- S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), no. 4, 453–460. MR 763753
- S. K. Donaldson, Nahm’s equations and the classification of monopoles, Comm. Math. Phys. 96 (1984), no. 3, 387–407. MR 769355
- S. K. Donaldson, Topological field theories and formulae of Casson and Meng-Taubes, Proceedings of the Kirbyfest (Berkeley, CA, 1998) Geom. Topol. Monogr., vol. 2, Geom. Topol. Publ., Coventry, 1999, pp. 87–102. MR 1734402, DOI 10.2140/gtm.1999.2.87
- S. K. Donaldson, Moment maps and diffeomorphisms, Asian J. Math. 3 (1999), no. 1, 1–15. Sir Michael Atiyah: a great mathematician of the twentieth century. MR 1701920, DOI 10.4310/AJM.1999.v3.n1.a1
- S. K. Donaldson, Moment maps in differential geometry, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 171–189. MR 2039989, DOI 10.4310/SDG.2003.v8.n1.a6
- J. J. Duistermaat and G. J. Heckman, On the variation in the cohomology of the symplectic form of the reduced phase space, Invent. Math. 69 (1982), no. 2, 259–268. MR 674406, DOI 10.1007/BF01399506
- Richard Earl and Frances Kirwan, Complete sets of relations in the cohomology rings of moduli spaces of holomorphic bundles and parabolic bundles over a Riemann surface, Proc. London Math. Soc. (3) 89 (2004), no. 3, 570–622. MR 2107008, DOI 10.1112/S0024611504014832
- Dan Freed, The Atiyah–Singer index theorem, Bull. Amer. Math. Soc. 58 (2021), no. 4,
- Dennis Gaitsgory and Jacob Lurie, Weil’s conjecture for function fields. Vol. 1, Annals of Mathematics Studies, vol. 199, Princeton University Press, Princeton, NJ, 2019. MR 3887650
- William M. Goldman, The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54 (1984), no. 2, 200–225. MR 762512, DOI 10.1016/0001-8708(84)90040-9
- Jens Gravesen, On the topology of spaces of holomorphic maps, Acta Math. 162 (1989), no. 3-4, 247–286. MR 989398, DOI 10.1007/BF02392839
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- G. Harder and M. S. Narasimhan, On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1974/75), 215–248. MR 364254, DOI 10.1007/BF01357141
- N. J. Hitchin, Monopoles and geodesics, Comm. Math. Phys. 83 (1982), no. 4, 579–602. MR 649818
- N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126. MR 887284, DOI 10.1112/plms/s3-55.1.59
- N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589. MR 877637
- G. Horrocks, Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964), 689–713. MR 169877, DOI 10.1112/plms/s3-14.4.689
- Lisa C. Jeffrey and Frances C. Kirwan, Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface, Ann. of Math. (2) 148 (1998), no. 1, 109–196. MR 1652987, DOI 10.2307/120993
- A. D. King and P. E. Newstead, On the cohomology ring of the moduli space of rank $2$ vector bundles on a curve, Topology 37 (1998), no. 2, 407–418. MR 1489212, DOI 10.1016/S0040-9383(97)00041-4
- Frances Clare Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984. MR 766741, DOI 10.2307/j.ctv10vm2m8
- Frances Kirwan, The cohomology rings of moduli spaces of bundles over Riemann surfaces, J. Amer. Math. Soc. 5 (1992), no. 4, 853–906. MR 1145826, DOI 10.1090/S0894-0347-1992-1145826-8
- Frances Kirwan, Geometric invariant theory and the Atiyah-Jones conjecture, The Sophus Lie Memorial Conference (Oslo, 1992) Scand. Univ. Press, Oslo, 1994, pp. 161–186. MR 1456466, DOI 10.1007/978-3-642-57916-5
- P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683. MR 992334
- P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I, Topology 32 (1993), no. 4, 773–826. MR 1241873, DOI 10.1016/0040-9383(93)90051-V
- David Mumford, Projective invariants of projective structures and applications, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 526–530. MR 175899
- Hiraku Nakajima and K\B{o}ta Yoshioka, Instanton counting on blowup. I. 4-dimensional pure gauge theory, Invent. Math. 162 (2005), no. 2, 313–355. MR 2199008, DOI 10.1007/s00222-005-0444-1
- M. S. Narasimhan, Geometry of moduli spaces of vector bundles, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars Éditeur, Paris, 1971, pp. 199–201. MR 429901
- M. S. Narasimhan and S. Ramanan, Moduli of vector bundles on a compact Riemann surface, Ann. of Math. (2) 89 (1969), 14–51. MR 242185, DOI 10.2307/1970807
- M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI 10.2307/1970710
- Nikita A. Nekrasov, Seiberg-Witten prepotential from instanton counting, Adv. Theor. Math. Phys. 7 (2003), no. 5, 831–864. MR 2045303
- Nikita A. Nekrasov and Andrei Okounkov, Seiberg-Witten theory and random partitions, The unity of mathematics, Progr. Math., vol. 244, Birkhäuser Boston, Boston, MA, 2006, pp. 525–596. MR 2181816, DOI 10.1007/0-8176-4467-9_{1}5
- P. E. Newstead, Topological properties of some spaces of stable bundles, Topology 6 (1967), 241–262. MR 232015, DOI 10.1016/0040-9383(67)90037-7
- P. E. Newstead, Stable bundles of rank $2$ and odd degree over a curve of genus $2$, Topology 7 (1968), 205–215. MR 237500, DOI 10.1016/0040-9383(68)90001-3
- Johan Råde, On the Yang-Mills heat equation in two and three dimensions, J. Reine Angew. Math. 431 (1992), 123–163. MR 1179335, DOI 10.1515/crll.1992.431.123
- Lorenzo Sadun and Jan Segert, Non-self-dual Yang-Mills connections with nonzero Chern number, Bull. Amer. Math. Soc. (N.S.) 24 (1991), no. 1, 163–170. MR 1067574, DOI 10.1090/S0273-0979-1991-15978-1
- Simon Salamon, Quaternionic Kähler manifolds, Invent. Math. 67 (1982), no. 1, 143–171. MR 664330, DOI 10.1007/BF01393378
- S. M. Salamon, Differential geometry of quaternionic manifolds, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 1, 31–55. MR 860810
- Graeme Segal, The topology of spaces of rational functions, Acta Math. 143 (1979), no. 1-2, 39–72. MR 533892, DOI 10.1007/BF02392088
- R. L. E. Schwarzenberger, Vector bundles on the projective plane, Proc. London Math. Soc. (3) 11 (1961), 623–640. MR 137712, DOI 10.1112/plms/s3-11.1.623
- Clifford Henry Taubes, Stability in Yang-Mills theories, Comm. Math. Phys. 91 (1983), no. 2, 235–263. MR 723549
- Clifford Henry Taubes, Min-max theory for the Yang-Mills-Higgs equations, Comm. Math. Phys. 97 (1985), no. 4, 473–540. MR 787116
- Clifford Henry Taubes, The stable topology of self-dual moduli spaces, J. Differential Geom. 29 (1989), no. 1, 163–230. MR 978084
- Michael Thaddeus, Conformal field theory and the cohomology of the moduli space of stable bundles, J. Differential Geom. 35 (1992), no. 1, 131–149. MR 1152228
- Michael Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), no. 2, 317–353. MR 1273268, DOI 10.1007/BF01232244
- L. M. Sibner, R. J. Sibner, and K. Uhlenbeck, Solutions to Yang-Mills equations that are not self-dual, Proc. Nat. Acad. Sci. U.S.A. 86 (1989), no. 22, 8610–8613. MR 1023811, DOI 10.1073/pnas.86.22.8610
- Alex Waldron, Long-time existence for Yang-Mills flow, Invent. Math. 217 (2019), no. 3, 1069–1147. MR 3989258, DOI 10.1007/s00222-019-00877-2
- R. S. Ward, On self-dual gauge fields, Phys. Lett. A 61 (1977), no. 2, 81–82. MR 443823, DOI 10.1016/0375-9601(77)90842-8
- A. Weil, Generalisation des fonctions abeliennes, J. Math. Pures Appl. 17 47-87 (1938)
- Edward Witten, On quantum gauge theories in two dimensions, Comm. Math. Phys. 141 (1991), no. 1, 153–209. MR 1133264
- Scott Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. (2) 117 (1983), no. 2, 207–234. MR 690844, DOI 10.2307/2007075
Bibliographic Information
- Simon Donaldson
- Affiliation: Simons Center for Geometry and Physics, Stony Brook University, New York; and Department of Mathematics, Imperial College, London, United Kingdom
- MR Author ID: 59010
- Received by editor(s): June 10, 2021
- Published electronically: August 5, 2021
- © Copyright 2021 American Mathematical Society
- Journal: Bull. Amer. Math. Soc. 58 (2021), 567-610
- MSC (2020): Primary 53C05; Secondary 14D21, 14H60, 58E15
- DOI: https://doi.org/10.1090/bull/1748
- MathSciNet review: 4311555