nLab Seiberg-Witten invariant

Idea

The Seiberg-Witten invariants are invariants for certain 4-manifolds (with $b_2^+-b_1$ odd and $b_2^+\geq 2$ for the Betti numbers) and spinᶜ structures on them. Their definition is based on the moduli space of the Seiberg-Witten equations, which is usually compact unlike the moduli space of the Yang-Mills equations, which first requires compactification. Seiberg-Witten invariants have therefore turned out to often be both easier in calculations and yielding stronger results than Donaldson invariants.

Basics

Let $M$ be a compact orientable Riemannian 4-manifold with a Riemannian metric $g$ and spinᶜ structure $\mathfrak{s}$. Both always exist. In particular the latter is a lift of the classifying map $\tau\colon M\rightarrow BSO(4)$ of the tangent bundle $TM\cong\tau^*\gamma_\mathbb{R}^4$ to a map $\mathfrak{s}\colon M\rightarrow BSpin^\mathrm{c}(4)$. Because of the exceptional isomorphism:

$$Spin^\mathrm{c}(4) \cong U(2)\times_{U(1)}U(2) =\{A^\pm\in U(2)|det(A^-)=det(A^+)\}$$

the spinᶜ structure $\mathfrak{s}$ consists of two complex plane bundles $W^\pm\twoheadrightarrow M$, called associated spinor bundles, with same determinant line bundle $L=det(W^\pm)$. Since it preserves the first Chern class one has $c_1(L)=c_1(W^\pm)\in H^2(M,\mathbb{Z})$. Furthermore let $W=W^-\oplus W^+$ be the Whitney sum of the spinor bundles.

Seiberg-Witten invariants

Let $\mathcal{M}_{M,\mathfrak{s},g,\eta}^\mathrm{SW}$ (or just $\mathcal{M}$ in the following) be the moduli space of the perturbed Seiberg-Witten equations, hence the space of its solution up to gauge, depending on the manifold $M$, the spinᶜ structure $\mathfrak{s}$, the metric $g$ and the disturbation $\eta$. Using the Atiyah-Singer index theorem, its dimension is given by the Euler characteristic and the signature of the underlying manifold as:

$$\dim\mathcal{M} =\frac{1}{4}\left( c_1(L)^2 -2\chi(M) -3\sigma(M) \right)$$

If the right expression is positive. Otherwise the moduli space is empty.

The group $\mathcal{G}=C^\infty(M,U(1))$ and its subgroup $\mathcal{G}_0=\{u\in\mathcal{G}|u(x_0)=1\}$ with a basepoint $x_0\in M$ act on the moduli space. The canonical projection $\mathcal{M}/\mathcal{G}_0\rightarrow\mathcal{M}/\mathcal{G}$ is a principal U(1)-bundles with Euler class $e\in H^2(M,\mathbb{Z})$.

If $b_2^+(M)-b_1(M)$ is odd, then $dim\mathcal{M}=2d$ is even and the Seiberg-Witten invariant can be defined as:

$$SW(M,\mathfrak{s},g,\eta) \coloneqq\int_{\mathcal{M}}e^d \in\mathbb{Z}.$$

If $b_2^+(M)\geq 2$, then $SW(M,\mathfrak{s})\coloneqq SW(M,\mathfrak{s},g,\eta)$ is independent of the metric $g$ and the disturbation $\eta$. With the space $Spin^\mathrm{c}(M)\cong H^2(M,\mathbb{Z})$ of spinᶜ structures, for which the isomorphism is given by the first Chern class, one can express the Seiberg-Witten invariant as a map:

$$SW\colon Spin^\mathrm{c}(M)\cong H^2(M,\mathbb{Z})\rightarrow\mathbb{Z}.$$

A similar map with the choice of a fundamental class $[M]\in H_4(M,\mathbb{Z})\cong\mathbb{Z}$, which is a generator, is the intersection form $H^2(M,\mathbb{Z})\rightarrow\mathbb{Z},c\mapsto\langle c^2,[M]\rangle$, which is essential for the classification of 4-manifolds.

A spinᶜ structure $\mathfrak{s}\in Spin^\mathrm{c}(M)$, or alternatively its corresponding cohomology class $c_1(L)\in H^2(M,\mathbb{Z})$, with $SW(M,\mathfrak{s})\neq 0$ is called basic class. Hence the basic classes are the support of the Seiberg-Witten invariant and there always exist only finitely many. Every basic class fulfills:

$$c^2 \geq 2\chi(M) +3\sigma(M).$$

Properties

Proposition

On a smooth 4-manifold $M$ with $b_2^+(M)=1$, which admits a Riemannian metric of positive scalar curvature, all Seiberg-Witten invariants vanish.

(Nicolaescu 00, Crl. 2.3.8)

Proposition

Let $M$ be a closed oriented Riemannian 4-manifold with $b_2^+(M)\geq 2$.

• If $SW_M(\mathfrak{s})=0$ for all spinᶜ structures $\mathfrak{s}$, then $M$ doesn’t admit symplectic structures.

• If $|SW_M(\mathfrak{s})|\neq 1$ for all spinᶜ structures $\mathfrak{s}$, then $M$ doesn’t admit symplectic structures.

(Nicolaescu 00, Crl. 3.3.33)

A combination of both previous results shows that if $M$ has a Riemannian metric of positive scalar curvature, it doesn’t admit symplectic structures.

Proposition

For two compact oriented Riemannian 4-manifolds $M$ and $N$ with $b_2^+(M),b_2^+(N)\geq 1$, one has:

$$SW_{M#N}(\mathfrak{s}) =0$$

for all spinᶜ structures $\mathfrak{s}$.

(Nicolaescu 00, Thrm. 4.6.1)

A combination of both previous results shows that symplectic manifolds don’t decompose as a connected sum $M#N$ with $b_2^+(M),b_2^+(N)\geq 1$. (Nicolaescu 00, Crl. 4.6.2) Since the classification of 4-manifolds is extremely difficult and still unresolved, many attempts have therefore focussed on symplectic manifolds.

Let $\mathfrak{s}_n$ be the spinᶜ structure on the orientation-reveresed second complex projective space $\overline{\mathbb{C}P^2}$ with $c_1(\mathfrak{s}_n)=(2n+1)c_1(S^5)$ (with the latter being the first Chern class of the principal U(1)-bundle $S^5\twoheadrightarrow\mathbb{C}P^2$ and a generator $c_1(S^5)\in H^2(\mathbb{C}P^2,\mathbb{Z})\cong\mathbb{Z}$).

Proposition

(Blow-up formula) For a compact orientable Riemannian 4-manifold $M$ with a spinᶜ structure $\mathfrak{s}$, one has:

$$|SW_{M#\overline{\mathbb{C}P^2},\mathfrak{s}#\mathfrak{s}_n}| =\begin{cases} 0 & dim\mathcal{M}_{M,\mathfrak{s}}\lt\pm n(n+1) \\ |SW_M(\mathfrak{s})| & dim\mathcal{M}_{M,\mathfrak{s}}\geq n(n+1) \end{cases}$$

(Nicolaescu 00, Thrm. 4.6.7)

Proposition

For a Kähler surface? $M$ with canonical spinᶜ structure $\mathfrak{s}_0$, one has:

$$SW_M(\mathfrak{s}_0) =1;$$

$$SW_M^+(\mathfrak{s}_0) =1$$

for $b_2^+(M)\geq 2$ and $b_2^+(M)=1$, respectively. If $M$ is a K3 surface, then $\overline\mathfrak{s}_0=\mathfrak{s}_0$ and it is the only basic class.

(Nicolaescu 00, Thrm. 3.3.2 & Crl. 3.3.3)

Related concepts

Articles about Seiberg-Witten theory:

• Ordinary theory: Seiberg-Witten equations, linearized Seiberg-Witten equations, Seiberg-Witten invariant, Seiberg-Witten moduli space, Seiberg-Witten flow
• Adapted theory: D=6 Seiberg-Witten theory, D=7 Seiberg-Witten theory (over G₂ manifolds), D=8 Seiberg-Witten theory (over Spin(7) manifolds), quaternionic Seiberg-Witten theory, pseudo-Riemannian Seiberg-Witten theory

References

• Liviu Nicolaescu, Notes on Seiberg-Witten theory, American Mathematical Society (2000) [ISBN:978-0-8218-2145-9, pdf]

• Jürgen Einhorn?, Thomas Friedrich, Seiberg-Witten theory (pdf)

• Simon Donaldson, The Seiberg-Witten equations and 4-manifold topology (pdf)

• Matilde Marcolli, Seiberg-Witten gauge theory, pdf

See also:

• Wikipedia, Seiberg-Witten invariants