×

zbMATH — the first resource for mathematics

The Seiberg-Witten equations and 4-manifold topology. (English) Zbl 0872.57023
Since 1982 the use of gauge theory, in the shape of the Yang-Mills instanton equations, has permeated research in 4-manifold theory. In the last part of the year 1994 this research area was turned on its head by the introduction of a new kind of differential-geometric equation by Seiberg and Witten. In the space of a few weeks long-standing problems were solved, new and unexpected results were found, along with simpler new proofs of extending ones, and new vistas for research opened up.
The paper under review is a report on some of these developments specially due to P. B. Kronheimer and T. S. Mrowka [Bull. Am. Math. Soc., New Ser. 30, 215-221 (1994; Zbl 0815.57010); Math. Res. Lett. 1, 797-808 (1994; Zbl 0851.57023)] and C. H. Taubes [Math. Res. Lett. 1, 809-822 (1994; Zbl 0853.57019); 2, 9-13 (1995; Zbl 0854.57019)] building on the seminal work of N. Seiberg [Phys. Lett. B 318, 469 (1993)] and N. Seiberg and E. Witten [Nucl. Phys. B 431, 581-640 (1994)]. It is written as an attempt to take stock of the progress stemming from this initial period of intense activity.
In conclusion a wonderful work which will be useful to all those interested in 4-manifold theory.

MSC:
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
81T13 Yang-Mills and other gauge theories in quantum field theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] M. F. Atiyah and L. Jeffrey, Topological Lagrangians and cohomology, J. Geom. Phys. 7 (1990), no. 1, 119 – 136. · Zbl 0721.58056
[2] D. Austin, Equivariant Floer Theory for binary polyhedral spaces, Preprint. · Zbl 0821.57022
[3] D. Austin and P. Braam, Equivariant Floer cohomology, Topology (To appear). · Zbl 0834.57017
[4] S. Bradlow, G. Daskalopoulos, O. Garcia-Prada, R. Wentworth, Stable augmented bundles over Riemann surfaces, Vector bundles in algebraic geometry (Ed. Hitchin, Newstead, Oxbury), Cambridge UP, 1995. · Zbl 0827.14010
[5] S. K. Donaldson, Connections, cohomology and the intersection forms of 4-manifolds, J. Differential Geom. 24 (1986), no. 3, 275 – 341. · Zbl 0635.57007
[6] S. K. Donaldson, Irrationality and the \?-cobordism conjecture, J. Differential Geom. 26 (1987), no. 1, 141 – 168. · Zbl 0631.57010
[7] S. K. Donaldson, Polynomial invariants for smooth four-manifolds, Topology 29 (1990), no. 3, 257 – 315. · Zbl 0715.57007
[8] S. Donaldson, Yang-Mills invariants of four-manifolds, Geometry of low-dimensional manifolds, Vol I (Ed. Donaldson and Thomas), Cambridge UP, 1990. · Zbl 0836.57012
[9] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990. Oxford Science Publications. · Zbl 0820.57002
[10] S. K. Donaldson and D. P. Sullivan, Quasiconformal 4-manifolds, Acta Math. 163 (1989), no. 3-4, 181 – 252. · Zbl 0704.57008
[11] N. Elkies, A characterisation of the \(\mathbf{Z}^{n}\)-lattice, Math. Res. Letters (To appear).
[12] R. Fintushel and R. Stern, The blowup formula for Donaldson invariants, Preprint (1994). · Zbl 0869.57019
[13] Robert Friedman and John W. Morgan, Algebraic surfaces and 4-manifolds: some conjectures and speculations, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 1, 1 – 19. · Zbl 0662.57016
[14] R. Friedman and Z. Qin, The smooth invariance of the Kodaira dimension of a complex surface, Math. Res. Letters 1 (1994), 369-376, CMP 95:04.
[15] R. Gompf, A new construction of symplectic manifolds, Preprint (1994). · Zbl 0863.53025
[16] L. Gottsche, Modular forms and Donaldson invariants for \(4\)-manifolds with \(b_{+}=1\), Preprint (1995).
[17] M. Gromov, Pseudo holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307 – 347. · Zbl 0592.53025
[18] Mikhael Gromov and H. Blaine Lawson Jr., The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), no. 3, 423 – 434. · Zbl 0463.53025
[19] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59 – 126. · Zbl 0634.53045
[20] Arthur Jaffe and Clifford Taubes, Vortices and monopoles, Progress in Physics, vol. 2, Birkhäuser, Boston, Mass., 1980. Structure of static gauge theories. · Zbl 0457.53034
[21] Peter B. Kronheimer, Embedded surfaces in 4-manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 529 – 539. · Zbl 0746.53041
[22] D. Kotschick and P.Lisca, Instanton invariants of \(\mathbf{C}\mathbf{P}^{2}\) via topology, preprint. · Zbl 0844.57034
[23] P. B. Kronheimer and T. S. Mrowka, Gauge theory for embedded surfaces. I, Topology 32 (1993), no. 4, 773 – 826. · Zbl 0799.57007
[24] P. B. Kronheimer and T. S. Mrowka, Recurrence relations and asymptotics for four-manifold invariants, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 215 – 221. · Zbl 0815.57010
[25] P. Kronheimer and T. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Letters 1 (1994), 797-808, CMP 95:05. · Zbl 0851.57023
[26] C. LeBrun, Einstein metrics and Mostow rigidity, Math. Res. Letters 2 (1995), 1-8, CMP 95:07. · Zbl 0974.53035
[27] M. Manetti, On some components of moduli spaces of surfaces of general type, Compositio Math. 92 (1994), 285-297, CMP 94:15.
[28] J. Morgan, Z. Szabo and C. Taubes, The generalised Thom conjecture, In preparation.
[29] V. Pidstragach and A. Tyurin, Invariants of the smooth structure of an algebraic surface arising from the Dirac operator (English translation), Russian Math. Surveys (1994).
[30] V. Ya. Pidstrigach and A. N. Tyurin, The smooth structure invariants of an algebraic surface defined by the Dirac operator, Izv. Ross. Akad. Nauk Ser. Mat. 56 (1992), no. 2, 279 – 371 (Russian, with Russian summary); English transl., Russian Acad. Sci. Izv. Math. 40 (1993), no. 2, 267 – 351. · Zbl 0796.14024
[31] N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in \(N=2\) supersymmetric QCD, Nucl. Phys. B 431 (1994), 581-640, CMP 95:05. · Zbl 1020.81911
[32] A. Stipsicz, Donaldson invariants of certain symplectic \(4\)-manifolds, Crelles Journal (To appear). · Zbl 0828.53027
[33] D. Sullivan, Exterior \(d\), the local index and smoothability, IHES preprint (1995). · Zbl 0926.57027
[34] C. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Letters 1 (1994), 809-822, CMP 95:05. · Zbl 0853.57019
[35] C. Taubes, More constraints on symplectic forms from Seiberg-Witten equations, Math. Res. Letters 2 (1995), 9-14, CMP 95:07. · Zbl 0854.57019
[36] C. Taubes, The Seiberg-Witten and the Gromov invariants, Harvard Preprint (1995). · Zbl 0854.57020
[37] Michael Thaddeus, Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), no. 2, 317 – 353. · Zbl 0882.14003
[38] C. Vafa and E. Witten, A strong coupling test of S-duality, Nucl. Phys. B (To appear), CMP 95:04. · Zbl 0964.81522
[39] Edward Witten, Topological quantum field theory, Comm. Math. Phys. 117 (1988), no. 3, 353 – 386. · Zbl 0656.53078
[40] E. Witten, Supersymmetric Yang-Mills theory on a four-manifold, J. Math. Phys. 35 (1994), CMP 95:01. · Zbl 0822.58067
[41] E. Witten, Monopoles and 4-manifolds, Math. Res. Letters 1 (1994), 769-796, CMP 95:05.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.