nLab Yang-Mills-Higgs 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-Higgs action functional, then the 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 fuethermore a scalar product $\langle-,-\rangle\colon\Omega^k(B,Ad(E))\times\Omega^k(B,Ad(E))\rightarrow\mathbb{R}$. It’s induced norm is exactly the $L^2$ norm.

Definition

The Yang-Mills-Higgs action functional is given by:

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

Its first term is also called Yang-Mills action.

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

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

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

For an open interval $I\subseteq\mathbb{R}$, two $C^1$ maps $\alpha\colon I\rightarrow\Omega^1(B,Ad(E))$ and $\varphi\colon I\rightarrow\Gamma^\infty(B,Ad(E))$ (hence continuously differentiable) fulfilling:

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

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

are a Yang-Mills-Higgs flow.

Related entries

• Yang-Mills flow
• Seiberg-Witten flow

References

See also:

• Wikipedia, Yang-Mills-Higgs flow