nLab stable Yang-Mills-Higgs pair

Idea

A (weakly) stable Yang-Mills-Higgs connection (or (weakly) stable YMH connection) is a Yang-Mills-Higgs connection, around which the Yang-Mills-Higgs action functional is positive or even strictly positively curved:

Yang-Mills-Higgs connections 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,

• $B$ an orientable Riemannian manifold with Riemannian metric $g$ and volume form $\operatorname{vol}_g$,

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

• $\Phi\in\Omega_{\operatorname{Ad}}^0(E,\mathfrak{g})\cong\Omega^0(B,\operatorname{Ad}(E))\cong\Gamma^\infty(B,\Ad(E))$ a smooth section,

• $A\in\Omega_{\operatorname{Ad}}^1(E,\mathfrak{g})\cong\Omega^1(B,\operatorname{Ad}(E))$ a principal connection,

• $F_A\coloneqq\mathrm{d}A+\frac{1}{2}[A\wedge A]\in\Omega_{\operatorname{Ad}}^2(E,\mathfrak{g})\cong\Omega^2(B,\operatorname{Ad}(E))$ 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:

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

$A$ 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 $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow\Gamma^\infty(B,\operatorname{Ad}(E))$ 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:

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

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

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)

References

• Zhi Hu and 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 and Yang Wen; Stability and energy identity for Yang-Mills-Higgs pairs (2023), Journal of Mathematical Physics 64, arxiv:2303.00270

See also

• stable Yang-Mills connection
• Yang-Mills-Higgs equations

References

See also:

• Wikipedia, stable Yang-Mills-Higgs pair