nLab linearized Seiberg-Witten equations

Idea

The linearized Seiberg-Witten equations are a linearization of the non-linear Seiberg-Witten equations, hence the best approximation by a linear equation at a given solution, called Seiberg-Witten monopoles. They are obtained as the differential of the Seiberg-Witten map, which maps the space of principal connections and self-dual spinors to their evaluation of the Seiberg-Witten equations term, so that its solutions are exactly those mapped to the origin. (Due to the affinity of the space of principal connections and the non-linearity, this is not the kernel.)

Since the tangent space has the same dimension as its underlying manifold, the local virtual dimension of the Seiberg-Witten moduli space can be obtained from the linearization using the Atiyah-Singer index theorem.

Seiberg-Witten map

Consider the setup described in detail in Seiberg-Witten equations. The Seiberg-Witten map is given by:

$$f\colon \mathcal{A}\oplus\Gamma^\infty(S^+)\rightarrow\Omega_+^2(M,\mathfrak{u}(1))\oplus\Gamma^\infty(S^-), (A,\psi)\mapsto(F_A^+-\sigma(\psi),\partial_A\psi).$$

(Naber 11, Eq. (A.4.23))

$(A,\psi)\in\mathcal{A}$ is a solution of the Seiberg-Witten equations if and only if $f(A,\psi)=0$. the differential of the Seiberg-Witten map in it is given by:

$$D_{(A,\psi)}f =\begin{pmatrix} d_+ & -D_\psi \\ \frac{1}{2}\psi & -\partial_A \end{pmatrix}\colon \Omega^1(M,\mathfrak{u}(1))\oplus\Gamma^\infty(S^+)\rightarrow\Omega_+^2(M,\mathfrak{u}(1))\oplus\Gamma^\infty(S^-).$$

(Nicolaescu 00, Eq. (2.2.18), Naber 11, Eq. (A.4.24))

Linearized Seiberg-Witten equations

Let $(B,\phi)\in\Omega^1(M,\mathfrak{u}(1))\oplus\Gamma^\infty(S^+)$, then the linearized Seiberg-Witten equations at a solution $(A,\psi)\in\mathcal{A}\oplus\Gamma^\infty(S^+)$ are given by $D_{A,\psi}f(B,\phi)=0$ or expanded by:

$$d_+B -D_\psi\phi =0;$$

$$\frac{1}{2}B\cdot\psi -\partial_A\phi =0.$$

Elliptic complex

Let $act\colon\mathcal{G}\rightarrow\mathcal{A}\oplus\Gamma^\infty(S^+)$ be the action of the gauge group $\mathcal{G}=C^\infty(M,U(1))$ on a solution $(A,\psi)$. Since the Seiberg-Witten equations are gauge invariant, one has $f\circ act=0$ for the composition and therefore $D_{(A,\psi)}f\circ D_1 act=0$ for their differential, which yields an elliptic complex $\mathcal{E}(A,\phi)$ given by:

$$1 \rightarrow C^\infty(M,\mathfrak{u}(1)) \xrightarrow{D_1 act}\Omega^1(M,\mathfrak{u}(1))\oplus\Gamma^\infty(S^+) \xrightarrow{D_{(A,\psi)}f}\Omega_+^2(M,\mathfrak{u}(1))\oplus\Gamma^\infty(S^-) \rightarrow 1.$$

It has the cohomology vector spaces:

$$H^0(\mathcal{E}(A,\phi)) =ker(D_1 act);$$

$$H^1(\mathcal{E}(A,\phi)) =ker(D_{(A,\psi)}f)/img(D_1 act);$$

$$H^2(\mathcal{E}(A,\phi)) =(\Omega_+^2(M,\mathfrak{u}(1))\oplus\Gamma^\infty(S^-))/img(D_{(A,\psi)}f).$$

$H^0(\mathcal{E}(A,\phi))$ can be identified with the tangent space $T_1 Stab_{(A,\phi)}(\mathcal{G})$, so that $H^0(\mathcal{E}(A,\psi))=\{0\}$ is equivalent to $(A,\psi)$ being irreducible or $\psi\neq 0$. (Nicolaescu 00, Crl. 2.2.12, Naber 11, p. 393) $H^1(\mathcal{E}(A,\psi))$ can be identified with the tangent space $T_{[A,\psi]}\mathcal{M}$ of the Seiberg-Witten moduli space.

$$ind\mathcal{E}(A,\psi) =-\dim H^0(\mathcal{E}(A,\psi)) +\dim H^1(\mathcal{E}(A,\psi)) -\dim H^2(\mathcal{E}(A,\psi)) =\dim H^1(\mathcal{E}(A,\psi)) =dim T_{[A,\psi]}\mathcal{M}$$

(Naber 11, p. 394)

(For finite-dimensional vector spaces in the complex, these can replace the cohomology vector spaces in the index due to the homomorphism theorem?.) The Atiyah-Singer index theorem assures this analytic index to be equal to the topological index, which depends only on topological invariants.

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]

• Gregory L. Naber, Topology, Geometry and Gauge fields – Interactions, Applied Mathematical Sciences 141 (2011) [doi:10.1007/978-1-4419-7895-0]