nLab Yang-Mills 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 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.

All spaces $\Omega^k(B,Ad(E))$ are vector spaces, which from $B$ together with the choice of an $Ad$-invariant pairing on $\mathfrak{g}$ (which for semisimple $\mathfrak{g}$ must be proportional to its Killing form) inherits a local pairing $\langle-,-\rangle\colon\Omega^k(B,Ad(E))\times\Omega^k(B,Ad(E))\rightarrow\C^\infty(B)$. It defined the Hodge star operator by $\langle\omega,\eta\rangle vol_g=\omega\wedge\star\eta$ for all $\omega,\eta\in\Omega^k(B,Ad(E))$. Through postcomposition with integration, there is furthermore a scalar product $\langle-,-\rangle\colon\Omega^k(B,Ad(E))\times\Omega^k(B,Ad(E))\rightarrow\mathbb{R}_0^+$. Its induced norm is exactly the $L^2$ norm.

Definition

The Yang-Mills action functional is given by:

$$YM\colon \mathcal{A} \coloneqq\Omega^1(B,Ad(E))\rightarrow\mathbb{R}_0^+, YM(A) \coloneqq\int_B\|F_A\|^2\mathrm{d}vol_g \geq 0.$$

$\mathcal{A}$ is called configuration space.

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 $\alpha\colon I\rightarrow\mathcal{A}=\Omega^1(B,Ad(E))$ (hence continuously differentiable) fulfilling:

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

is a Yang-Mills flow.

Properties

Corollary

For a Yang-Mills connection $A\in\mathcal{A}$, the constant path on it is a Yang-Mills flow.

Lemma

For a Yang-Mills flow $\alpha\colon\mathbb{R}\supseteq I\rightarrow\mathcal{A}$, the composition $YM\circ\alpha\colon\mathbb{R}\supseteq I\rightarrow\mathbb{R}_0^+$ with the Yang-Mills action functional is monotonically descreasing and in particular fulfills:

$$(YM\circ\alpha)'(t) =-2\int_B\|\alpha'(t)\|^2\mathrm{d}\vol_g =-2(BYM\circ\alpha)(t) \leq 0.$$

$BYM\colon\mathcal{A}\rightarrow\mathbb{R}_0^+$ is the Bi-Yang-Mills action functional.

Proof

Using the flow equation yields:

$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|F_{\alpha(t)}\|^2 =\left\langle\frac{\mathrm{d}}{\mathrm{d}t}F_{\alpha(t)},F_{\alpha(t)}\right\rangle =\left\langle\mathrm{d}_{\alpha(t)}\alpha'(t),F_{\alpha(t)}\right\rangle =\left\langle\alpha'(t),\delta_{\alpha(t)}F_{\alpha(t)}\right\rangle =-\|\alpha'(t)\|^2.$$

Lemma

For an infinitely long Yang-Mills flow $\alpha\colon[0,\infty)\rightarrow\mathcal{A}$, the limit $\lim_{t\rightarrow\infty}\alpha(t)\in\mathcal{A}$ exists and is a Yang-Mills connection.

Proof

According to the previous lemma, the limit $\lim_{t\rightarrow\infty}(YM\circ\alpha)(t)\in\mathbb{R}_0^+$ exists, since the function $YM\circ\alpha\colon[0,\infty)\rightarrow\mathbb{R} _0^+$ is monotonically descreasing and bounded from below. The inequality in the previous lemma then yields $\lim_{t\rightarrow\infty}\alpha'(t)=0$, which implies that the limit $A=\lim_{t\rightarrow\infty}\alpha(t)$ exists. Using the flow equation then shows, that it is a Yang-Mills connection:

$$\delta_A F_A =\lim_{t\rightarrow\infty}\delta_{\alpha(t)}F_{\alpha(t)} =\lim_{t\rightarrow\infty}\alpha'(t) =0.$$

Related entries

On ordinary Yang-Mills theory (YM):

• D=2 YM, D=3 YM, D=4 YM, D=5 YM, D=6 YM, D=7 YM, D=8 YM

• Maxwell theory/electromagnetism (U(1) YM), Donaldson theory (SU(2) YM), quantum chromodynamics (SU(3) YM)

• Yang-Mills equation, linearized Yang-Mills equation, Yang-Mills instanton, Yang-Mills field, stable Yang-Mills connection, Yang-Mills moduli space, Yang-Mills flow, F-Yang-Mills equation, Bi-Yang-Mills equation

• Uhlenbeck's singularity theorem, Uhlenbeck's compactness theorem

On variants of Yang-Mills theory and on super Yang-Mills theory (SYM):

• Yang-Mills-Higgs equations, stable Yang-Mills-Higgs pair, Yang-Mills-Higgs flow

• Einstein-Yang-Mills theory, Einstein-Yang-Mills-Dirac theory, Einstein-Yang-Mills-Dirac-Higgs theory

• 3D Yang-Mills-Chern-Simons theory

• 3d superconformal gauge field theory: D=3 N=1 SYM, D=3 N=2 SYM, D=3 N=4 SYM

• 4d superconformal gauge field theory: D=4 N=1 SYM, D=4 N=2 SYM, D=4 N=4 SYM

• D=5 SYM, D=6 SYM, D=7 SYM, D=10 SYM

• topological Yang-Mills theory, topologically twisted D=4 super Yang-Mills theory

• Seiberg-Witten flow

References

• Casey Lynn Kelleher, Jeffrey Streets, Singularity formation of the Yang-Mills flow (2016), [arXiv:1602.03125]
• Alex Waldron, Long-time existence for Yang-Mills flow (2019), [arXiv:1610.03424]
• Pan Zhang, Gradient Flows of Higher Order Yang-Mills-Higgs Functionals (2020), [arXiv:2004.00420]

See also:

• Wikipedia, Yang-Mills flow