Monopoles and four-manifolds.

*(English)*Zbl 0867.57029Motivated by his joint work with N. Seiberg on \(N=2\) supersymmetric Yang-Mills theory in 4 dimensions, the author gives here a set of monopole equations dual to the antiself-duality equations which have since become known as the Seiberg-Witten equations. He discusses how they arise physically, formulates the mathematical properties of their solutions and defines new invariants using them, relates these invariants to Donaldson invariants, and proves a number of results concerning them. These new invariants have since revolutionized 4-manifold topology as they have led to simpler approaches to results previously proved using Donaldson invariants as well as providing a tool to prove many new results. One of the major breakthroughs in Donaldson theory due to P. B. Kronheimer and T. S. Mrowka [J. Differ. Geom. 41, No. 3, 573-734 (1995; Zbl 0842.57022)] shortly before these new invariants were discovered was the reformulation of the Donaldson invariants in terms of a finite number of basic classes that occur in a formula for the Donaldson series for a 4-manifold of simple type. The author gives the sketch of a physical argument that these basic classes as well as their coefficients are given through the new Seiberg-Witten invariants. Although his conjecture has been verified in a large number of examples, a full proof still remains a major topic of current work. Independent of this conjecture, these new invariants have provided an extremely powerful tool for studying 4-manifolds that has proved easier to use than the Donaldson invariant.

We describe here briefly the Seiberg-Witten equations and corresponding invariants. Let \(X\) be an oriented, closed 4-manifold with a Riemannian metric \(g\). One may always choose a \(\text{Spin}^c\) structure \(\xi\) on \((X,g)\), which may be described as a principal \(\text{Spin}^c(4)\) bundle \(\widetilde P\) covering the orthogonal frame bundle \(P\) over \(X\). Here \(\text{Spin}^c (4)= \text{Spin} (4)\times S^1/ \{\pm 1\}\) covers \(\text{SO}(4)= \text{Spin} (4)/ \{\pm 1\}\) on the fiber. Associated to this \(\text{Spin}^c\) structure are \(U(2)\) bundles of spinors \(S^\pm\) and their common determinant line bundle \(L\). The data for the Seiberg-Witten equations is a pair \((A,\psi)\), where \(A\) is a connection on \(L\) and \(\psi\) is a section of \(S^+\). There is an identification of the traceless endomorphisms of \(S^+\) with the self-dual 2-forms, through which the section \(\psi\) determines a 2-form \(q(\psi)\) using the trace-free part of the endomorphism \(\psi\otimes \psi^*\). The Seiberg-Witten equations are \[ F^+_A= q(\psi),\;D_A \psi=0. \] Here \(D_A\) is the Dirac operator formed using the connection on \(\widetilde P\) covering the Levi-Civita connection on \(P\) and the connection \(A\) on \(L\). There is a gauge group given by the automorphisms of \(\widetilde P\) which cover the identity automorphism of \(P\), which can be identified with maps from \(X\) to \(S^1\), and one can form the moduli space \({\mathcal M}\) of the gauge equivalence classes of solutions to these equations. As usual, the proper analytic formulation requires the use of appropriate Sobolev spaces. The virtual dimension of this moduli space is given by the index theorem to be \(d={1\over 4} (c_1(L)^2- (2\chi+ 3\sigma))\). To get a manifold of this dimension requires adding a small perturbation term \(ih\) to the right hand side of the first equation as well as a condition \(b^+_2\geq 1\) needed to generically avoid reducible solutions (those where \(\psi=0)\). In particular, there will generically be no solutions when \(d<0\).

In contrast to Donaldson theory where the lack of compactness causes many technical difficulties, here \({\mathcal M}\) is shown to be compact through use of the Weitzenböck formula for the Dirac operator. This formula plays a special role in a number of other results, including the fact that for a fixed metric there are only a finite number of \(\text{Spin}^c\) structures where the moduli space is nonempty, and for a positive scalar curvature metric there are only reducible solutions (and none of those generically if \(b^+_2 \geq 1)\). As in Donaldson theory, one uses a homology orientation of \(H^1(X;\mathbb{R}) \oplus H^2_+ (X;\mathbb{R})\) to orient the moduli space when it is a manifold. When \(d=0\) (which occurs for example if the \(\text{Spin}^c\) structure comes from an almost complex structure on the tangent bundle) the oriented moduli space will consist of a finite number of signed points. In this case the Seiberg-Witten invariant \(\text{SW} (\xi)\) is defined to be the sum of the signs at these points. When \(d>0\) and is even, the first Chern class \(\mu\) of the restriction of the base point fibration to \({\mathcal M}\) is used to define \(\text{SW} (\xi)= \int_{\mathcal M} \mu^{d/2}\). The invariant is defined to be 0 when \(d\) is odd. As long as \(b^+_2>1\), reducible connections can be avoided in paths \((g_t,h_t)\) of metric-perturbation pairs, and a cobordism argument gives that \(\text{SW} (\xi)\) is a diffeomorphism invariant independent of generic \((g,h)\). In particular, all Seiberg-Witten invariants must vanish if \(X\) has a metric of positive scalar curcature. When \(b^+_2=1\), the paper gives a wall crossing formula which shows how the invariant changes when a path goes through a wall corresponding to a reducible connection when \(b_1=0\). More general wall crossing formulas have since been given by T. J. Li and A. Liu [General wall crossing formula, Math. Res. Lett. 2, No. 6, 797-810 (1995; Zbl 0871.57017)] and play an important role in the analysis of the case \(b^+_2 = 1\).

A class \(L\) with non-vanishing Seiberg-Witten invariant is called a basic class – there is a physically based argument that the basic classes with \(d=0\) are the same as in Donaldson theory. For \(b^+_2>1\), the basic classes occur in pairs as the conjugate \(\text{Spin}^c\) structure \(\overline\xi\) has \(\text{SW} (\overline\xi) =\pm \text{SW} (\xi)\). Besides establishing the main properties of the invariant, a method is given for computing it for Kähler manifolds where the invariant is given a holomorphic reformulation. There it is shown that the invariant can only be nonzero when \(d=0\), and for the \(\text{Spin}^c\) structures determined by \(\pm K\) with \(K\) the canonical class one has \(\text{SW} (\pm K)= \pm 1\).

It is also shown that for minimal surfaces of general type the only basic classes are \(\pm K\), proving that the canonical class is a diffeomorphism invariant of these surfaces. Another major result in the paper is that the Seiberg-Witten invariant vanishes whenever \(X=X_1 \# X_2\) with \(b^+_2 (X_i)\geq 1\). There has been a lot of important work that has been based on the Seiberg-Witten invariants introduced in this paper, and we want to inform the reader of a few of the more important papers that have appeared. P. B. Kronheimer and T. S. Mrowka [Math. Res. Lett. 1, No. 6, 797-808 (1994; Zbl 0851.57023)] gave a proof of the Thom conjecture which states that a holomorphic curve in \(\mathbb{C} P^2\) minimizes the genus of any surface which represents the same homology class. They had earlier proved forms of the generalized Thom conjecture and given a generalized adjunction inequality for other manifolds using Donaldson theory, but these proofs did not work for \(\mathbb{C} P^2\). Their earlier results can be reproven using Seiberg-Witten theory. J. W. Morgan, M. G. Szabo and C. H. Taubes [J. Differ. Geom. (to appear)] have proven that a smooth symplectic curve of nonnegative self-intersection in a symplectic four-manifold is genus minimizing, and have given a product formula for computing Seiberg-Witten invariants which allows them to give a non-vanishing result for the Seiberg-Witten invariants of certain generalized connected sum manifolds. C. H. Taubes has published a number of papers [Math. Res. Lett. 1, No. 6, 809-822 (1994; Zbl 0853.57019); ibid. 2, No. 1, 9-13 (1995; Zbl 0854.57019); ibid. 2, No. 2, 221-238 (1995; Zbl 0854.57020); SW \(\Rightarrow\) Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves, J. Am. Math. Soc. 9, No. 3, 845-918 (1996; Zbl 0867.53025)] which have generalized results for Kähler surfaces to symplectic four-manifolds, and identified Seiberg-Witten invariants with Gromov invariants. C. LeBrun has found a number of differential geometric applications of the Seiberg-Witten invariants, including proving uniqueness results for Einstein metrics for smooth compact quotients of complex hyperbolic 2-space [Einstein metrics and Mostow rigidity, Math. Res. Lett. 2, No. 1, 1-8 (1995; Zbl 0974.53035)] and finding infinitely many compact simply connected smooth four-manifolds which do not admit Einstein metrics but still satisfy the strict Hitchin-Thorpe inequality [ibid. 3, No. 2, 133-147 (1996; Zbl 0856.53035)]. Furuta has used the Seiberg-Witten moduli space to prove a slightly weaker version of the 11/8 conjecture. There are also now available expository treatments of the Seiberg-Witten invariants, including a paper by S. K. Donaldson [The Seiberg-Witten equations and 4-manifold topology, Bull. Am. Math. Soc. 33, No. 1, 45-70 (1996; Zbl 0872.57023)] and a book by J. W. Morgan [The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Princeton Math. Notes 44 (1996; Zbl 0846.57001)].

We describe here briefly the Seiberg-Witten equations and corresponding invariants. Let \(X\) be an oriented, closed 4-manifold with a Riemannian metric \(g\). One may always choose a \(\text{Spin}^c\) structure \(\xi\) on \((X,g)\), which may be described as a principal \(\text{Spin}^c(4)\) bundle \(\widetilde P\) covering the orthogonal frame bundle \(P\) over \(X\). Here \(\text{Spin}^c (4)= \text{Spin} (4)\times S^1/ \{\pm 1\}\) covers \(\text{SO}(4)= \text{Spin} (4)/ \{\pm 1\}\) on the fiber. Associated to this \(\text{Spin}^c\) structure are \(U(2)\) bundles of spinors \(S^\pm\) and their common determinant line bundle \(L\). The data for the Seiberg-Witten equations is a pair \((A,\psi)\), where \(A\) is a connection on \(L\) and \(\psi\) is a section of \(S^+\). There is an identification of the traceless endomorphisms of \(S^+\) with the self-dual 2-forms, through which the section \(\psi\) determines a 2-form \(q(\psi)\) using the trace-free part of the endomorphism \(\psi\otimes \psi^*\). The Seiberg-Witten equations are \[ F^+_A= q(\psi),\;D_A \psi=0. \] Here \(D_A\) is the Dirac operator formed using the connection on \(\widetilde P\) covering the Levi-Civita connection on \(P\) and the connection \(A\) on \(L\). There is a gauge group given by the automorphisms of \(\widetilde P\) which cover the identity automorphism of \(P\), which can be identified with maps from \(X\) to \(S^1\), and one can form the moduli space \({\mathcal M}\) of the gauge equivalence classes of solutions to these equations. As usual, the proper analytic formulation requires the use of appropriate Sobolev spaces. The virtual dimension of this moduli space is given by the index theorem to be \(d={1\over 4} (c_1(L)^2- (2\chi+ 3\sigma))\). To get a manifold of this dimension requires adding a small perturbation term \(ih\) to the right hand side of the first equation as well as a condition \(b^+_2\geq 1\) needed to generically avoid reducible solutions (those where \(\psi=0)\). In particular, there will generically be no solutions when \(d<0\).

In contrast to Donaldson theory where the lack of compactness causes many technical difficulties, here \({\mathcal M}\) is shown to be compact through use of the Weitzenböck formula for the Dirac operator. This formula plays a special role in a number of other results, including the fact that for a fixed metric there are only a finite number of \(\text{Spin}^c\) structures where the moduli space is nonempty, and for a positive scalar curvature metric there are only reducible solutions (and none of those generically if \(b^+_2 \geq 1)\). As in Donaldson theory, one uses a homology orientation of \(H^1(X;\mathbb{R}) \oplus H^2_+ (X;\mathbb{R})\) to orient the moduli space when it is a manifold. When \(d=0\) (which occurs for example if the \(\text{Spin}^c\) structure comes from an almost complex structure on the tangent bundle) the oriented moduli space will consist of a finite number of signed points. In this case the Seiberg-Witten invariant \(\text{SW} (\xi)\) is defined to be the sum of the signs at these points. When \(d>0\) and is even, the first Chern class \(\mu\) of the restriction of the base point fibration to \({\mathcal M}\) is used to define \(\text{SW} (\xi)= \int_{\mathcal M} \mu^{d/2}\). The invariant is defined to be 0 when \(d\) is odd. As long as \(b^+_2>1\), reducible connections can be avoided in paths \((g_t,h_t)\) of metric-perturbation pairs, and a cobordism argument gives that \(\text{SW} (\xi)\) is a diffeomorphism invariant independent of generic \((g,h)\). In particular, all Seiberg-Witten invariants must vanish if \(X\) has a metric of positive scalar curcature. When \(b^+_2=1\), the paper gives a wall crossing formula which shows how the invariant changes when a path goes through a wall corresponding to a reducible connection when \(b_1=0\). More general wall crossing formulas have since been given by T. J. Li and A. Liu [General wall crossing formula, Math. Res. Lett. 2, No. 6, 797-810 (1995; Zbl 0871.57017)] and play an important role in the analysis of the case \(b^+_2 = 1\).

A class \(L\) with non-vanishing Seiberg-Witten invariant is called a basic class – there is a physically based argument that the basic classes with \(d=0\) are the same as in Donaldson theory. For \(b^+_2>1\), the basic classes occur in pairs as the conjugate \(\text{Spin}^c\) structure \(\overline\xi\) has \(\text{SW} (\overline\xi) =\pm \text{SW} (\xi)\). Besides establishing the main properties of the invariant, a method is given for computing it for Kähler manifolds where the invariant is given a holomorphic reformulation. There it is shown that the invariant can only be nonzero when \(d=0\), and for the \(\text{Spin}^c\) structures determined by \(\pm K\) with \(K\) the canonical class one has \(\text{SW} (\pm K)= \pm 1\).

It is also shown that for minimal surfaces of general type the only basic classes are \(\pm K\), proving that the canonical class is a diffeomorphism invariant of these surfaces. Another major result in the paper is that the Seiberg-Witten invariant vanishes whenever \(X=X_1 \# X_2\) with \(b^+_2 (X_i)\geq 1\). There has been a lot of important work that has been based on the Seiberg-Witten invariants introduced in this paper, and we want to inform the reader of a few of the more important papers that have appeared. P. B. Kronheimer and T. S. Mrowka [Math. Res. Lett. 1, No. 6, 797-808 (1994; Zbl 0851.57023)] gave a proof of the Thom conjecture which states that a holomorphic curve in \(\mathbb{C} P^2\) minimizes the genus of any surface which represents the same homology class. They had earlier proved forms of the generalized Thom conjecture and given a generalized adjunction inequality for other manifolds using Donaldson theory, but these proofs did not work for \(\mathbb{C} P^2\). Their earlier results can be reproven using Seiberg-Witten theory. J. W. Morgan, M. G. Szabo and C. H. Taubes [J. Differ. Geom. (to appear)] have proven that a smooth symplectic curve of nonnegative self-intersection in a symplectic four-manifold is genus minimizing, and have given a product formula for computing Seiberg-Witten invariants which allows them to give a non-vanishing result for the Seiberg-Witten invariants of certain generalized connected sum manifolds. C. H. Taubes has published a number of papers [Math. Res. Lett. 1, No. 6, 809-822 (1994; Zbl 0853.57019); ibid. 2, No. 1, 9-13 (1995; Zbl 0854.57019); ibid. 2, No. 2, 221-238 (1995; Zbl 0854.57020); SW \(\Rightarrow\) Gr: From the Seiberg-Witten equations to pseudo-holomorphic curves, J. Am. Math. Soc. 9, No. 3, 845-918 (1996; Zbl 0867.53025)] which have generalized results for Kähler surfaces to symplectic four-manifolds, and identified Seiberg-Witten invariants with Gromov invariants. C. LeBrun has found a number of differential geometric applications of the Seiberg-Witten invariants, including proving uniqueness results for Einstein metrics for smooth compact quotients of complex hyperbolic 2-space [Einstein metrics and Mostow rigidity, Math. Res. Lett. 2, No. 1, 1-8 (1995; Zbl 0974.53035)] and finding infinitely many compact simply connected smooth four-manifolds which do not admit Einstein metrics but still satisfy the strict Hitchin-Thorpe inequality [ibid. 3, No. 2, 133-147 (1996; Zbl 0856.53035)]. Furuta has used the Seiberg-Witten moduli space to prove a slightly weaker version of the 11/8 conjecture. There are also now available expository treatments of the Seiberg-Witten invariants, including a paper by S. K. Donaldson [The Seiberg-Witten equations and 4-manifold topology, Bull. Am. Math. Soc. 33, No. 1, 45-70 (1996; Zbl 0872.57023)] and a book by J. W. Morgan [The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Princeton Math. Notes 44 (1996; Zbl 0846.57001)].

Reviewer: T.Lawson (New Orleans)

##### MSC:

57R57 | Applications of global analysis to structures on manifolds |

57N13 | Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010) |

58D27 | Moduli problems for differential geometric structures |

81T13 | Yang-Mills and other gauge theories in quantum field theory |

81T60 | Supersymmetric field theories in quantum mechanics |