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.

Definition

In the following, consider

• $G$ be a compact Lie group

• with Lie algebra $\mathfrak{g}$,

• $P \twoheadrightarrow B$ a principal $G$-bundle over

• a compact orientable Riemannian manifold $(B,g)$,

and write

• $Ad_P \,\coloneqq\, P \times_G \mathfrak{g}$ for its adjoint bundle,

• $\Gamma^\infty(Ad_P)$ for the space of smooth sections,

• $\Omega^1_{conn}(P;\mathfrak{g}) \subset \Omega^1(P;\mathfrak{g})$ for the Lie algebra valued differential 1-forms which are (Ehresmann) connection forms

  (and recall that the gauge-invariants of the curvature 2-form $F_A$ of $A \in \Omega^1_{conn}(P;\mathfrak{g})$ descend to, hence are pulled back from, plain differential forms on $B$).

If the base space is not compact, then in the following the gauge field $A \in \Omega^1_{conn}(P;\mathfrak{g})$ and the Higgs field $\Phi \in \Gamma^\infty(Ad_P)$ are required to vanish at infinity (cf. Taubes 1982a, Equation (2.3)).

Action functional

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

(1)$$\begin{array}{ccc} \mathllap{ S_{YMH} \,\colon\, } \Omega^1_{conn}(P;\mathfrak{g}) \times \Gamma^\infty(Ad_P) &\longrightarrow& \mathbb{R} \\ \big(A, \Phi\big) &\mapsto& S_{YMH}(A,\Phi) \mathrlap{ \coloneqq \textstyle{\int_B} \big( \|F_A\|^2 + \|\mathrm{d}_A\Phi\|^2 \big) \mathrm{d}\vol_g \,. } \end{array}$$

(cf. Taubes 1982a, Equation (2.1))

Yang-Mills-Higgs equations and pairs

A pair consisting of a connection $A\in\Omega^1_{conn}(P;\mathfrak{g})$ and a section $\Phi\in\Gamma^\infty(B,Ad_P)$ is called a Yang-Mills-Higgs pair (or YMH pair) if it is a critical point of the Yang-Mills-Higgs action functional (1), hence if:

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

for all pairs of smooth families

$$\begin{array}{ccc} \mathllap{ \alpha\colon } (-\varepsilon, \varepsilon) & \longrightarrow & \Omega^1_{conn}(P;\mathfrak{g}) \\ \mathllap{ \varphi\colon } (-\varepsilon,\varepsilon) & \longrightarrow & \Gamma^\infty(Ad_P) \mathrlap{\,,} \end{array}$$

with $\alpha(0)=A and \varphi(0)=\Phi$,

(where $(-\epsilon, \epsilon) \subset \mathbb{R}$ denotes an open ball around the origin of the real line).

This is the case iff the Euler-Lagrange equations of motion are satisfies, here called the Yang-Mills-Higgs equations (or YMH equations):

(2)$$\begin{array}{ccc} \mathrm{d}_A \star F_A + \star[\Phi,\mathrm{d}_A\Phi] &=& 0 \\ \mathrm{d}_A\star\mathrm{d}_A\Phi &=& 0 \mathrlap{\,.} \end{array}$$

(cf. Taubes 1982a, Equations (2.2a) and (2.2b), Taubes 1984, Equation (1), Taubes 1985, Equations (A.1.1a) and (A.1.1b), but beware that Taubes 1984, Equation (1) is missing the second Hodge star operator in the first Yang-Mills-Higgs equation.)

Furthermore, the following Bianchi identities hold:

$$\begin{array}{ccc} \mathrm{d}_A F_A &=& 0 \\ \mathrm{d}_A\mathrm{d}_A\Phi +[\Phi,F_A] &=& 0 \end{array}$$

(cf. Taubes 1982a, Equations (2.2c) and (2.2d))

Using $\star^2=(-1)^{k(n-k)}$ (see there) and $\delta_A=(-1)^{n(k+1)+1}\star\mathrm{d}_A\star$ when applied to $k$-forms, the first Yang-Mills-Higgs equation (2) is equivalent to:

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

and the second one to:

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

Remark

For an abelian Lie group as structure group, its Lie algebra is also abelian and hence all Lie brackets vanish and the YMH equations (2) reduce to:

$$\begin{array}{ccc} \mathrm{d}\star\mathrm{d}A &=& 0 \\ \mathrm{d}\star\mathrm{d}\Phi &=& 0 \mathrlap{\,.} \end{array}$$

Properties

General

Essentially by definition:

Lemma

$A\in\Omega^1_{conn}(P; \mathfrak{g})$ is a Yang-Mills connection (a solution of the plain Yang-Mills equations) iff $(A,0)\in\Omega^1_{conn}(P)\times\Gamma^\infty(Ad_P)$ is a Yang-Mills-Higgs pair.

Relation to generalized Laplace equation

Let:

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

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:

$$\Delta_A\Phi \;=\; 0 \mathrlap{\,.}$$

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