nLab Yang-Mills-Higgs 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-Higgs action functional, then this gradient flow is called Yang-Mills-Higgs flow. It is described by the Yang-Mills-Higgs equations and can be used to find Yang-Mills-Higgs pairs, the critical points, as well as study stable Yang-Mills-Higgs pairs, the local minima.

The Yang-Mills flow is named after Chen Ning Yang and Robert Mills, who introduced Yang-Mills theory in 1954, and Peter Higgs, who proposed the Higgs field in 1964.

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 $Ad(E)\coloneqq E\times_G\mathfrak{g}$ 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-algebra 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 space is 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-Higgs action functional is given by:

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

Its first term is also called Yang-Mills action. $\mathcal{A}$ is called configuration space.

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

$$grad(YMH)(A,\Phi)_1 =\delta_A F_A +[\Phi,\mathrm{d}_A\Phi],$$

$$grad(YMH)(A,\Phi)_2 =\delta_A\mathrm{d}_A\Phi.$$

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

$$\alpha'(t) =-grad(YMH)(\alpha(t),\varphi(t))_1 =-\delta_{\alpha(t)}F_{\alpha(t)} -[\varphi(t),\mathrm{d}_{\alpha(t)}\varphi(t)]$$

$$\varphi'(t) =-grad(YMH)(\alpha(t),\varphi(t))_2 =-\delta_{\alpha(t)}\mathrm{d}_{\alpha(t)}\varphi(t)$$

is a Yang-Mills-Higgs flow.

Properties

Corollary

For a Yang-Mills-Higgs pair $(A,\Phi)\in\mathcal{A}$, the constant path on it is a Yang-Mills-Higgs flow.

Lemma

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

$$(YMH\circ(\alpha,\varphi))'(t) =-2\int_B\|\alpha'(t)\|^2+\|\varphi'(t)\|^2\mathrm{d}\vol_g \leq 0.$$

Proof

Using the flow equations yields:

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

$$\frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\|\mathrm{d}_{\alpha(t)}\varphi(t)\|^2 =\left\langle\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{d}_{\alpha(t)}\varphi(t),\mathrm{d}_{\alpha(t)}\varphi(t)\right\rangle =\left\langle\mathrm{d}_{\alpha(t)}\varphi'(t) +[\alpha'(t),\varphi(t)],\mathrm{d}_{\alpha(t)}\varphi(t)\right\rangle =\langle\alpha'(t),[\varphi(t),\mathrm{d}_{\alpha(t)}\varphi(t)]\rangle +\langle\varphi'(t),\delta_{\alpha(t)}\mathrm{d}_{\alpha(t)}\varphi(t)\rangle;$$

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

Lemma

For an infinitely long Yang-Mills-Higgs flow $(\alpha,\varphi)\colon[0,\infty)\rightarrow\mathcal{A}$, the limits $(\lim_{t\rightarrow\infty}\alpha(t),\lim_{t\rightarrow\infty}\varphi(t))\in\mathcal{A}$ exist and are a Yang-Mills-Higgs pair.

Proof

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

$$\delta_A F_A+[\Phi,\mathrm{d}_A\Phi] =\lim_{t\rightarrow\infty}\delta_{\alpha(t)}F_{\alpha(t)}+[\varphi(t),\mathrm{d}_{\alpha(t)}\varphi(t)] =\lim_{t\rightarrow\infty}\alpha'(t) =0;$$

$$\delta_A\mathrm{d}_A\Phi =\lim_{t\rightarrow\infty}\delta_{\alpha(t)}\mathrm{d}_{\alpha(t)}\varphi(t) =\lim_{t\rightarrow\infty}\varphi'(t) =0.$$

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

• Pan Zhang, Gradient Flows of Higher Order Yang-Mills-Higgs Functionals (2020), [arXiv:2004.00420]
• Changpeng Pan, Zhenghan Shen, Pan Zhang, The Limit of the Yang-Mills-Higgs Flow for twisted Higgs pairs (2023), [arXiv:2301.02552]

See also:

• Wikipedia, Yang-Mills-Higgs flow