nLab Seiberg-Witten equations

Idea

The Seiberg-Witten equations are partial differential equations for connections of principal U(1)-bundles and smooth sections of their spinor bundles. Their moduli space is used to define the Seiberg-Witten invariants capable to study smooth structures on 4-manifolds. Since the gauge group of the Seiberg-Witten equations is abelian, many calculations could be simplified with new methods provided by them.

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 equations

Covariant derivatives$\nabla\colon\Gamma^\infty(L)\rightarrow\Gamma^\infty(T^*M\otimes L)$ on the determinant line bundle $L$, which are linear and fulfill the Leibniz rule, induce principal connections on the frame bundle $Fr_U(L)$, which is a principal U(1)-bundle, using the following isomorphisms:

$$\Hom(\Gamma^\infty(L),\Gamma^\infty(T^*M\otimes L)) \cong\Gamma^\infty\underline{Hom}(L,T^*M\otimes L) \cong\Gamma^\infty(T^*M\otimes L^*\otimes L) \cong\Gamma^\infty(T^*M\otimes\underline{End}(L)) \cong\Omega^1(M,\underline{End}(L)) \cong\Omega^1(M,Ad Fr_U(L)) \cong\Omega_{Ad}^1(Fr_U(L),\mathfrak{u}(1))$$

(In particular, since $L$ is a line bundle, one has $\underline{End}(L)\cong\underline{\mathfrak{u}(1)}$, which can be seen with the fact that the identity provides a global section.)

Since the first unitary group $U(1)$ is abelian, the Seiberg-Witten equations obtain a strong simplification compared to the Yang-Mills equations, which are usually considered with non-abelian gauge groups. In particular this can be seen for the curvature form simplifying to $F_A=d A$. Its self-dual part is then given by:

$$F_A^+ =\frac{1}{2}(F_A+\star F_A) =\frac{1}{2}(d A+\star d A).$$

Smooth sections of $W^-$, whose space is denoted $\Gamma^\infty(W^-)$ (or short $\Gamma(W^-)$), are called antiselfdual spinor fields (or short ASD spinor fields). Smooth sections of $W^+$, whose space is denoted $\Gamma^\infty(W^+)$ (or short $\Gamma(W^+)$), are called selfdual spinor fields (or short SD spinor fields). A connection like above induces a Dirac operator $D^A\colon\Gamma^\infty(W^+)\rightarrow\Gamma^\infty(W^-)$. Furthermore the Riemannian metric induces scalar products $\langle-,-\rangle\colon W^\pm\oplus W^\pm\rightarrow\underline{\mathbb{R}}$.

Now the Seiberg-Witten equations are partial differential equations for a connection $A\in\Omega^1(M,\mathfrak{u}(1))$ and a self-dual spinor field $\phi\in\Gamma^\infty(W^+)$, often given as undisturbed Seiberg-Witten equations without and disturbed Seiberg-Witten equations with a self-dual form $\eta\in\Omega_+^2(M,\mathfrak{u}(1))$ as:

$$D^A\phi =0;$$

$$F_A^+ +\tau(\phi) +\eta =0.$$

Considering the undisturbed Seiberg-Witten equations with a vanishing spinor field $\phi=0$ yields the anti self-dual Yang-Mills equations (ASDYM equations) $F_A^+=0$.

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 equations