nLab stable Yang-Mills-Higgs pair

Idea

A (weakly) stable Yang-Mills-Higgs pair (or (weakly) stable YMH connection) is a Yang-Mills-Higgs pair, around which the Yang-Mills-Higgs action functional is positive or even strictly positively curved. Yang-Mills-Higgs pairs are critical points of the Yang-Mills-Higgs action functional, where the first variational derivative vanishes. For (weakly) stable Yang-Mills connections, the second derivative is additionally required to be positive or even strictly positive.

Basics

Consider

• $G$ a Lie group and $\mathfrak{g}$ its Lie algebra,

• $X$ an orientable Riemannian manifold with Riemannian metric $g$ and volume form $vol_g$,

• $P\twoheadrightarrow X$ a principal $G$-bundle and $Ad(P)\coloneqq P\times_G\mathfrak{g}\twoheadrightarrow B$ its adjoint bundle,

• $\Phi\in\Omega^0(X,Ad(P))\cong\Gamma^\infty(X,Ad(P))$ a smooth section,

• $A\in\Omega_{\operatorname{Ad}}^1(P,\mathfrak{g})$ (affine space over $\Omega^1(X,Ad(P))$) a principal connection,

• $F_A\coloneqq\mathrm{d}A+\frac{1}{2}[A\wedge A]\in\Omega^2(X,Ad(P))$ its curvature. If $A$ is a Yang-Mills connection, $F_A$ is also called Yang-Mills field.

Definition

The Yang-Mills-Higgs action functional (or YMH action functional) is given by:

$$YMH\colon \Omega^1(X,Ad(P))\times\Gamma^\infty(X,Ad(P))\rightarrow\mathbb{R}, YMH(A,\Phi) \coloneqq\int_X\|F_A\|^2+\|\mathrm{d}_A\Phi\|^2\mathrm{d}vol_g.$$

$A$ and $\Phi$ are called a stable Yang-Mills-Higgs pair (or stable YMH pair) iff:

$$\frac{\mathrm{d}^2}{\mathrm{d}t^2}\operatorname{YMH}(\alpha(t),\varphi(t))\vert_{t=0} \gt 0$$

for all smooth families $\alpha\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname{Ad}(E))$ and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow\Gamma^\infty(X,Ad(P))$ with $\alpha(0)=A$ and $\varphi(0)=\Phi$. It is called weakly stable if only $\geq 0$ holds. For comparison, the condition for a Yang-Mills-Higgs pair (or YMH pair) is:

$$\frac{\mathrm{d}}{\mathrm{d}t}YMH(\alpha(t),\varphi(t))\vert_{t=0} =0.$$

(Hu & Hu 15, Cheng 21, Definition 3.1, Han, Jin & Wen 23)

Theorem

(Formula for Yang–Mills–Higgs stability) Let $\alpha\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(X,Ad(P))$ be a path with $A=\alpha(0)$, $B=\alpha'(0)$ and $C=\alpha''(0)$ and let $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(X,Ad(P))$ be a path with $\Phi=\varphi(0)$, $\Psi=\varphi'(0)$ and $\Omega=\varphi''(0)$, then:

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}t^2}YMH(\alpha(t),\varphi(t))\vert_{t=0} &=\int_X\|\mathrm{d}_A B\|^2 +\|\mathrm{d}_A \Psi+[B,\Phi]\|^2 +\langle\underbrace{\delta_A F_A-[\mathrm{d}_A\Phi,\Phi]}_{YMH},C\rangle \\ &+\langle\underbrace{\delta_A \mathrm{d}_A\Phi}_{YMH},\Omega\rangle +\langle F_A,[B\wedge B]\rangle +2\langle\mathrm{d}_A\Phi,[B,\Psi]\rangle. \end{aligned}$$

The Yang-Mills-Higgs equations are $\delta_A F_A-[\mathrm{d}_A \Phi,\Phi]=0$ as well as $\delta_A \mathrm{d}_A \Phi=0$ and therefore the formula simplifies for a Yang-Mills-Higgs pair $(A,\Phi)$. In this case it also becomes independent of $C$ and $\Omega$.

Proof

For the calculations for the first term see stable Yang-Mills connection. One has the following derivatives for the covariant derivative:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\mathrm{d}_{\alpha(t)}\varphi(t) &=\frac{\mathrm{d}}{\mathrm{d}t}\left( \mathrm{d}\varphi(t) +[\alpha(t),\varphi(t)] \right) \\ &=\mathrm{d}\varphi'(t) +[\alpha(t),\varphi'(t)] +[\alpha'(t),\varphi(t)] =\mathrm{d}_{\alpha(t)}\varphi'(t) +[\alpha'(t),\varphi(t)]; \end{aligned}$$

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}t^2}d_{\alpha(t)}\varphi(t) &=\mathrm{d}\varphi''(t) +[\alpha(t),\varphi''(t)] +2[\alpha'(t),\varphi'(t)] +[\alpha''(t),\varphi(t)] \\ &=\mathrm{d}_{\alpha(t)}\varphi''(t) +2[\alpha'(t),\varphi'(t)] +[\alpha''(t),\varphi(t)], \end{aligned}$$

which combine together into the following second derivative for the additional part of the Yang-Mills-Higgs action functional:

$$\begin{aligned} \frac{\mathrm{d}^2}{\mathrm{d}t^2}(YMH-YM)(\alpha(t),\varphi(t))\vert_{t=0} &=\int_X\left\|\frac{\mathrm{d}}{\mathrm{d}t}\mathrm{d}_{\alpha(t)}\varphi(t)\right\|^2 +\left\langle\mathrm{d}_{\alpha(t)}\varphi(t),\frac{\mathrm{d}^2}{\mathrm{d}t^2}\mathrm{d}_{\alpha(t)}\varphi(t)\right\rangle\mathrm{d}vol_g\vert_{t=0} \\ &=\int_X\|\mathrm{d}_{\alpha(t)}\varphi'(t)+[\alpha'(t),\varphi(t)]\|^2 \\ &+\left\langle \mathrm{d}_{\alpha(t)}\varphi(t),\mathrm{d}_{\alpha(t)}\varphi''(t) +2[\alpha'(t),\varphi'(t)] +[\alpha''(t),\varphi(t)]\right\rangle\mathrm{d}vol_g \end{aligned}$$

Since the first and second term are always positive, leaving both out directly implies:

Corollary

If $A\in\Omega^1(X,Ad(P))$ and $\Phi\in\Gamma^\infty(X,Ad(P))$ are a Yang-Mills-Higgs pair with:

$$\int_X\langle F_A,[B\wedge B]\rangle +2\langle\mathrm{d}_A\Phi,[B,\Psi]\rangle\mathrm{d}vol_g \gt 0\;\text{(or}\geq 0\text{)}$$

for all $B\in\Omega^1(X,Ad(P))$ and $\Psi\in\Gamma^\infty(X,Ad(P))$, then it is stable (or weakly stable).

Properties

Theorem

Let $(A,\Phi)$ be a weakly stable YMH pair on $S^n$.

• If $n=4$, then $\mathrm{d}_A\star F_A=0$ (meaning $A$ is a YM connection), $\mathrm{d}_A\Phi=0$ and $\|\Phi\|=1$.
• If $n\geq 5$, then $F_A=0$ (meaning $A$ is flat), $\mathrm{d}_A\Phi=0$ and $\|\Phi\|=1$.

(Han, Jin & Wen 23)

See also

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

References

• Zhi Hu, Sen Hu; Degenerate and Stable Yang-Mills-Higgs Pair (2015) [arxiv:1502.01791]

• Da Rong Cheng, Stable Solutions to the Abelian Yang–Mills–Higgs Equations on $S^2$ and $T^2$ (2021), Journal of Geometric Analysis 31, pp. 9551–9572, [doi:10.1007/s12220-021-00619-y]

• Xiaoli Han, Xishen Jin, Yang Wen; Stability and energy identity for Yang-Mills-Higgs pairs (2023), Journal of Mathematical Physics 64, [arxiv:2303.00270 doi:10.1063/5.0130905]

See also:

• Wikipedia, stable Yang-Mills-Higgs pair