Seiberg–Witten invariant

In mathematics, Seiberg–Witten invariants are invariants of compact smooth 4-manifolds introduced by Witten (1994), using the Seiberg–Witten theory studied by Seiberg and Witten (1994a, 1994b) during their investigations of Seiberg–Witten gauge theory.

Seiberg–Witten invariants are similar to Donaldson invariants and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the moduli spaces of solutions of the Seiberg–Witten equations tend to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.

For detailed descriptions of Seiberg–Witten invariants see (Donaldson 1996), (Moore 2001), (Morgan 1996), (Nicolaescu 2000), (Scorpan 2005, Chapter 10). For the relation to symplectic manifolds and Gromov–Witten invariants see (Taubes 2000). For the early history see (Jackson 1995).

Spin^c-structures

The Seiberg-Witten equations depend on the choice of a complex spin structure, Spin^c, on a 4-manifold M. In 4 dimensions the group Spin^c is

(U(1)×Spin(4))/(Z/2Z),

and there is a homomorphism from it to SO(4). A Spin^c-structure on M is a lift of the natural SO(4) structure on the tangent bundle (given by the Riemannian metric and orientation) to the group Spin^c. Every smooth compact 4-manifold M has Spin^c-structures (though most do not have spin structures).

Seiberg–Witten equations

Fix a smooth compact 4-manifold M, choose a spin^c-structure s on M, and write W⁺, W⁻ for the associated spinor bundles, and L for the determinant line bundle. Write φ for a self-dual spinor field (a section of W⁺) and A for a U(1) connection on L. The Seiberg–Witten equations for (φ,A) are

D^A\phi=0

F^+_A=\sigma(\phi) + i\omega

where D^A is the Dirac operator of A, F_A is the curvature 2-form of A, and F_A⁺ is its self-dual part, and σ is the squaring map from W⁺ to imaginary self-dual 2-forms and $\omega$ is a real selfdual two form, often taken to be zero or harmonic.

The solutions (φ,A) to the Seiberg–Witten equations are called monopoles, as these equations are the field equations of massless magnetic monopoles on the manifold M.

The moduli space of solutions

The space of solutions is acted on by the gauge group, and the quotient by this action is called the moduli space of monopoles.

The moduli space is usually a manifold. A solution is called reducible if it is fixed by some non-trivial element of the gauge group which is equivalent to $\phi = 0$ . A necessary and sufficient condition for reducible solutions for a metric on M and self dual 2 forms $\omega$ is that the self-dual part of the harmonic representative of the cohomology class of the determinant line bundle is equal to the harmonic part of $\omega/2\pi$ . The moduli space is a manifold except at reducible monopoles. So if b₂⁺(M)≥1 then the moduli space is a (possibly empty) manifold for generic metrics. Moreover all components have dimension

(c_1(s)^2-2\chi(M)-3sign(M))/4.

The moduli space is empty for all but a finite number of spin^c structures s, and is always compact.

A manifold M is said to be of simple type if the moduli space is finite for all s. The simple type conjecture states that if M is simply connected and b₂⁺(M)≥2 then the moduli space is finite. It is true for symplectic manifolds. If b₂⁺(M)=1 then there are examples of manifolds with moduli spaces of arbitrarily high dimension.

Seiberg–Witten invariants

The Seiberg–Witten invariants are easiest to define for manifolds M of simple type. In this case the invariant is the map from spin^c structures s to Z taking s to the number of elements of the moduli space counted with signs.

If the manifold M has a metric of positive scalar curvature and b₂⁺(M)≥2 then all Seiberg–Witten invariants of M vanish.

If the manifold M is the connected sum of two manifolds both of which have b₂⁺≥1 then all Seiberg–Witten invariants of M vanish.

If the manifold M is simply connected and symplectic and b₂⁺(M)≥2 then it has a spin^c structure s on which the Seiberg–Witten invariant is 1. In particular it cannot be split as a connected sum of manifolds with b₂⁺≥1.

References

Donaldson, S. K. (1996), "The Seiberg-Witten equations and 4-manifold topology.", Bull. Amer. Math. Soc. (N.S.) 33 (1): 45–70, doi:10.1090/S0273-0979-96-00625-8, MR 1339810
Jackson, Allyn (1995), A revolution in mathematics
Morgan, John W. (1996), The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes 44, Princeton, NJ: Princeton University Press, pp. viii+128, ISBN 0-691-02597-5, MR 1367507
Moore, John Douglas (2001), Lectures on Seiberg-Witten invariants, Lecture Notes in Mathematics 1629 (2nd ed.), Berlin: Springer-Verlag, pp. viii+121, doi:10.1007/BFb0092948, ISBN 3-540-41221-2, MR 1830497
Nash, Ch. (2001), "Seiberg-Witten equations", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Nicolaescu, Liviu I. (2000), Notes on Seiberg-Witten theory (PDF), Graduate Studies in Mathematics 28, Providence, RI: American Mathematical Society, pp. xviii+484, ISBN 0-8218-2145-8, MR 1787219
Scorpan, Alexandru (2005), The wild world of 4-manifolds, American Mathematical Society, ISBN 978-0-8218-3749-8, MR 2136212 .
Seiberg, N.; Witten, E. (1994a), "Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory", Nuclear Phys. B 426 (1): 19–52, doi:10.1016/0550-3213(94)90124-4, MR 1293681 "Erratum", Nuclear Phys. B 430 (2), 1994: 485–486, doi:10.1016/0550-3213(94)00449-8, MR 1303306
Seiberg, N.; Witten, E. (1994b), "Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD", Nuclear Phys. B 431 (3): 484–550, doi:10.1016/0550-3213(94)90214-3, MR 1306869
Taubes, Clifford Henry (2000), Wentworth, Richard, ed., Seiberg Witten and Gromov invariants for symplectic 4-manifolds, First International Press Lecture Series 2, Somerville, MA: International Press, pp. vi+401, ISBN 1-57146-061-6, MR 1798809
Witten, Edward (1994), "Monopoles and four-manifolds.", Mathematical Research Letters 1 (6): 769–796, MR 1306021

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