nLab Seiberg-Witten flow

Idea

For a scalar-valued function, a curve whose derivative is the negative of its gradient is called a gradient flow. It always points down the direction of steepest descent and hence is monotonically descreasing with respect to the scalar function. It is then possible to study its convergence to critical points, especially those that are local minima.

If the scalar function is the Seiberg-Witten action functional, then the gradient flow is called Seiberg-Witten flow. It is described by the Seiberg-Witten equation and can be used to find solutions of the Seiberg-Witten equations, which are the critical points.

The Seiberg-Witten flow was introduced by Lorenz Schabrun in his 2010 PhD thesis. Schabrun 10 It is named after Nathan Seiberg und Edward Witten, who introduced Seiberg-Witten theory in 1994.

Basics

Let $M$ be a compact orientable Riemannian 4-manifold with Riemannian metric $g\in\Gamma^\infty(S^2T^*M)$ and spinᶜ structure $\mathfrak{s}\colon M\rightarrow BSpin^\mathrm{c}(4)$. Because of the exceptional isomorphism: (Perutz 2002, p. 2)

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

the spinᶜ structure $\mathfrak{s}$ consists of two complex plane bundles $W^\pm\twoheadrightarrow M$, called associated spinor bundles (whose sections are called (anti) self-dual spinors), with same determinant line bundle $L=det(W^\pm)$. Since the determinant line bundle preserves the first Chern class, one has $c_1(\mathfrak{s})\coloneqq c_1(L)=c_1(W^\pm)$ with $c_1(\mathfrak{s})mod 2=w_2(M)$. (Perutz 2002, p. 2) Let $\mathcal{A}$ be the space of connections on $L$.

Definition

The Seiberg-Witten action functional is given by:

$$\begin{aligned} S_{SW}(A,\varphi) &=\int_M|F_A^+|^2 +|\nabla_A\varphi|^2 +\frac{1}{4}scal|\varphi|^2 +\frac{1}{8}|\varphi|^4\mathrm{d}vol_g \\ &=\int_M\frac{1}{2}|F_A|^2 +|\nabla_A\varphi|^2 +\frac{1}{4}scal|\varphi|^2 +\frac{1}{8}|\varphi|^4\mathrm{d}vol_g +\pi^2c_1(\mathfrak{s})^2[M] \end{aligned}$$

(Schabrun 10, Eq. (4) & (6))

The last expression uses Chern-Weil theory to express the Chern class, which as a constant has no effect on the flow equations and can therefore be obmitted, independent of the connection $A\in\mathcal{A}$ as:

$$c_1(\mathfrak{s})^2[M] =\frac{1}{4\pi^2}\int_M|F_A^+|^2-|F_A^-|^2\mathrm{d} vol_g.$$

(Schabrun 10, Eq. (5))

The gradients of the Seiberg-Witten action functional are given by:

$$grad(S_{SW})(A,\Phi)_1 =\mathrm{d}^*F_A +i Im\langle\nabla_A\Phi,\Phi\rangle,$$

$$grad(S_{SW})(A,\Phi)_2 =\nabla_A^*\nabla_A\Phi -\frac{1}{4}(scal+\|\Phi\|^2)\Phi.$$

(Schabrun 10, Eq. (7) & (8))

For an open interval $I\subseteq\mathbb{R}$, two $C^1$ maps $\alpha\colon I\rightarrow\mathcal{A}$ and $\varphi\colon I\rightarrow\Gamma^\infty(M,W^+)$ (hence continuously differentiable) fulfilling:

$$\alpha'(t) =-grad(S_{SW})(\alpha(t),\varphi(t))_1 =-\mathrm{d}^*F_{\alpha(t)} -i Im\langle\nabla_{\alpha(t)}\varphi(t),\varphi(t)\rangle,$$

$$\varphi'(t) =-grad(S_{SW})(\alpha(t),\varphi(t))_2 =-\nabla_{\alpha(t)}^*\nabla_{\alpha(t)}\varphi(t) -\frac{1}{4}(scal+\|\varphi(t)\|^2)\varphi(t).$$

are a Seiberg-Witten flow.

(Schabrun 10, Eq. (9) & (10))

Related entries

Articles about Seiberg-Witten theory:

• Ordinary theory:

  • Seiberg-Witten equations

  • linearized Seiberg-Witten equations

  • Seiberg-Witten invariant

  • simple type conjecture

  • 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

• Yang-Mills flow
• Yang-Mills-Higgs flow

References

• Liviu Nicolaescu, Notes on Seiberg-Witten theory, American Mathematical Society (2000) [ISBN:978-0-8218-2145-9, pdf]
• Tim Perutz, Basics of Seiberg-Witten theory (May 2002), [pdf]
• Min-Chun Hong, Lorenz Schabrun, Global Existence for the Seiberg-Witten Flow (2009), [arXiv:0909.1855]
• Lorenz Schabrun, Seiberg-Witten Flow in Higher Dimensions (2010), [arXiv:1003.1765]

See also:

• Wikipedia, Seiberg-Witten flow