nLab Yang-Mills flow

Idea

For a scalar function, a curve whose derivative is opposite to its gradient is called a gradient flow. It always points down the way 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 Yang-Mills action functional, then the gradient flow is called Yang-Mills flow. It is described by the Yang-Mills equation and can be used to find Yang-Mills connections, the critical points, as well as study stable Yang-Mills connections, the local minima.

The Yang-Mills flow is named after Chen Ning Yang and Robert Mills, who introduced Yang-Mills theory in 1954. But it was first studied by Michael Atiyah and Raoul Bott in 1982. It was also studied by Simon Donaldson in the context of the Kobayashi-Hitchin correspondence (or Donaldson-Uhlenbeck-Yau theorem).

Basics

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $E\twoheadrightarrow B$ be a principal $G$-bundle with a compact orientable Riemannian manifold $B$ having a metric $g$ and a volume form $vol_g$. Let $Ad(B)\coloneqq E\times_G\mathfrak{g}$ be its adjoint bundle. One has $\Omega_{Ad}^k(E,\mathfrak{g})\cong\Omega^k(B,Ad(E))$, which are either under the adjoint representation $Ad$ invariant Lie algbera valued or vector bundle valued differential forms. Since the Hodge star operator $\star$ is defined on the base manifold $B$ as it requires the metric $g$ and the volume form $vol_g$, the second spaces are usually used.

Definition

The Yang-Mills action functional is given by:

$$YM\colon \Omega^1(B,Ad(E))\rightarrow\mathbb{R}, YM(A) \coloneqq\int_B\|F_A\|^2\mathrm{d}vol_g \geq 0.$$

Hence the gradient of the Yang-Mills action functional gives exactly the Yang-Mills equations:

$$grad(YM)(A) =-\delta_A F_A.$$

For an open interval $I\subseteq\mathbb{R}$, a $C^1$ map $\alpha\colon I\rightarrow\Omega^1(B,Ad(E))$ (hence continuously differentiable) fulfilling:

$$\alpha'(t) =-grad(YM)(\alpha(t)) =-\delta_{\alpha(t)}F_{\alpha(t)}$$

is a Yang-Mills flow.

Related entries

• Yang-Mills-Higgs flow
• Seiberg-Witten flow

References

See also:

• Wikipedia, Yang-Mills flow