nLab Yang-Mills-Higgs equations

Idea

The Yang-Mills-Higgs equations (or YMH equations) arise from a generalization of the Yang-Mills action functional with a section, which in physics represents the Higgs field.

Yang-Mills-Higgs action functional

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 $B$. Let $\operatorname{Ad}(E)\coloneqq E\times_G\mathfrak{g}$ be its adjoint bundle. The Yang-Mills-Higgs action functional (or YMH action functional) is given by:

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

(Taubes 82a, Equation (2.1))

Yang-Mills-Higgs equations and pairs

A connection $A\in\Omega^1(B,\operatorname{Ad}(E))$ and a section $\Phi\in\Gamma^\infty(B,\operatorname{Ad}(E))$ are called Yang-Mills-Higgs pair (or YMH pair) if they are a critical point of the Yang-Mills-Higgs action functional, hence if:

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

for all smooth families $\alpha\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname{Ad}(E))$ with $\alpha(0)=A$ and $\varphi\colon(-\varepsilon,\varepsilon)\rightarrow\Gamma^\infty(B,\operatorname{Ad}(E))$ with $\varphi(0)=\Phi$.

This is the case iff the Yang-Mills-Higgs equations (or YMH equations) are fulfilled:

$$\mathrm{d}_A\star F_A +\star[\Phi,\mathrm{d}_A\Phi] =0;$$

$$\mathrm{d}_A\star\mathrm{d}_A\Phi =0.$$

(Taubes 82a, Equations (2.2a) and (2.2b), Taubes 84, Equation (1), Taubes 85, Equations (A.1.1a) and (A.1.1b))

(Taubes 84, Equation (1) is missing the second Hodge star operator in the first Yang-Mills-Higgs equation.)

Furthermore the following Bianchi identities hold:

$$\mathrm{d}_A F_A =0;$$

$$\mathrm{d}_A\mathrm{d}_A\Phi +[\Phi,F_A] =0.$$

(Taubes 82a, Equations (2.2c) and (2.2d))

Furthermore the Higgs field is required to vanish at infinity:

$$\lim_{|x|\rightarrow\infty}|\Phi|(x) =0.$$

(Taubes 82a, Equation (2.3))

A solution $(A,\Phi)$ of the Yang-Mills-Higgs equations is called Yang-Mills-Higgs pair.

Using $\star^2=(-1)^{k(n-k)}$ and $\delta_A=(-1)^{n(k+1)+1}\star\mathrm{d}_A\star$ when applied to $k$-forms, the first Yang-Mills-Higgs equation can also be rewritten as:

$$\delta_A F_A +[\Phi,\mathrm{d}_A\Phi] =0.$$

The second Yang-Mills-Higgs equation can also be rewritten as:

$$\delta_A\mathrm{d}_A\Phi =0.$$

Properties

Lemma

$A\in\Omega^1(B,\operatorname{Ad}(E))$ is a Yang-Mills connection iff $(A,0)\in\Omega^1(B,\operatorname{Ad}(E))\times\Gamma^\infty(B,\operatorname{Ad}(E))$ is a Yang-Mills-Higgs pair.

Abelian Yang-Mills-Higgs equation

For an abelian Lie group as structure group, its Lie algebra is also abelian and hence all Lie brackets vanish and makes the Yang-Mills-Higgs equations into:

$$\mathrm{d}\star\mathrm{d}A =0;$$

$$\mathrm{d}\star\mathrm{d}\Phi =0.$$

Connection with generalized Laplace equation

Let:

$$\Delta_A \coloneqq\delta_A\mathrm{d}_A +\mathrm{d}_A\delta_A\colon \Omega^k(B,\operatorname{Ad}(E))\rightarrow\Omega^k(B,\operatorname{Ad}(E))$$

be a generalized Laplace operator.

The Bianchi identity $\mathrm{d}_A F_A=0$ and the first Yang-Mills-Higgs equation $\delta_A F_A=-[\Phi,\mathrm{d}_A\Phi]$ combine to:

$$\Delta_A F_A =-\mathrm{d}_A[\Phi,\mathrm{d}_A\Phi].$$

The trivial identity $\delta_A\Phi=0$ (since $\Phi$ is a $0$-form whose degree cannot be lowered any further) and the second Yang-Mills-Higgs equation $\delta_A\mathrm{d}_A\Phi=0$ combine to:

$$\Delta_A\Phi =0.$$

Related entries

• stable Yang-Mills-Higgs pair

• Einstein-Yang-Mills-Dirac-Higgs theory

References

• Clifford Taubes, The existence of a non-minimal solution to the $\operatorname{SU}(2)$ Yang-Mills-Higgs equations on $\mathbb{R}^3$ Part I, Communications in Mathematical Physics 86 (1982) 257–298 [doi:10.1007/BF01206014]

• Clifford Taubes, The existence of a non-minimal solution to the $\operatorname{SU}(2)$ Yang-Mills-Higgs equations on $\mathbb{R}^3$ Part II, Communications in Mathematical Physics 86 (1982) 299–320 [doi:10.1007/BF01212170]

• Clifford Taubes, On the Yang–Mills–Higgs equations, Bulletin of the American Mathematical Society 10 (1984) 295–297 [doi:10.1090/s0273-0979-1984-15254-6]

• Clifford Taubes, Min-max theory for the Yang-Mills-Higgs equations, Communications in Mathematical Physics 97 (1985) 295–297 [doi:10.1007/BF01221215]

See also:

• Wikipedia, Yang-Mills-Higgs equations