nLab Bi-Yang-Mills equation

Idea

The Bi-Yang-Mills equations (or Bi-YM equations) arise from a generalization of the Yang-Mills action functional, so that the curvature is replaced by its adjoint derivative. While a vanishing curvature gives a flat connection, its vanishing adjoint derivative gives a Yang-Mills connection. Hence by only requiring local extrema, Bi-Yang-Mills connections are to Yang-Mills connections what these are to flat connections. While Yang-Mills connections can be seen as a non-linear generalization of harmonic maps, Bi-Yang-Mills connections can be seen as a non-linear generalization of biharmonic maps.

Bi-Yang-Mills 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 Bi-Yang-Mills action functional (or Bi-YM action functional) is given by:

$$\operatorname{BiYM}\colon \Omega^1(B,\operatorname{Ad}(E))\rightarrow\mathbb{R}, \operatorname{BiYM}(A) \coloneqq\int_B\|\delta_A F_A\|^2\mathrm{d}\vol_g.$$

(Chiang 13, Eq. (9))

Bi-Yang-Mills connections and equation

A connection $A\in\Omega^1(B,\operatorname{Ad}(E))$ is called Bi-Yang-Mills connection (or Bi-YM connection) it it is a critical point of the Bi-Yang-Mills action functional, hence if:

$$\frac{\mathrm{d}}{\mathrm{d}t}\operatorname{BiYM}(A(t))\vert_{t=0} =0$$

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

(Chiang 13, Eq. (5.1) and (6.1))

This is the case iff the Bi-Yang-Mills equation (or Bi-YM equation) is fulfilled:

$$(\delta_A\mathrm{d}_A+\mathcal{R}_A)(\delta_A F_A) =0.$$

(Chiang 13, Eq. (10), (5.2) and (6.3))

For a Bi-Yang-Mills connection $A\in\Omega^1(B,\operatorname{Ad}(E))$, its curvature $F_A\in\Omega^2(B,\operatorname{Ad}(E))$ is called Bi-Yang-Mills field (or Bi-YM field).

Stable Bi-Yang-Mills connections

Analogous to (weakly) stable Yang-Mills connections and (weakly) stable Yang-Mills-Higgs connections, one can also consider the positivity of the second derivative in Bi-Yang-Mills theory. $A$ is called a stable Bi-Yang-Mills connection (or stable Bi-YM connection), if:

$$\frac{\mathrm{d}^2}{\mathrm{d}t^2}\operatorname{BiYM}(A(t))\vert_{t=0} \gt 0$$

for all smooth families $A\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname{Ad}(E))$ with $A(0)=A$. It is called weakly stable if only $\geq 0$ holds. A Bi-Yang–Mills connection, which is not weakly stable, is called unstable. If $A\in\Omega^1(B,\operatorname{Ad}(E))$ is a (weakly) stable or unstable Bi-Yang-Mills connection, its curvature $F_A\in\Omega^2(B,\operatorname{Ad}(E))$ is also called (weakly) stable or unstable Bi-Yang-Mills field.

(Chiang 13, Definition 6.3.2)

Property

Lemma

Yang-Mills connections are weakly Bi-Yang-Mills connections.

(Chiang 13, Proposition 6.3.3)

Related concepts

• F-Yang-Mills equation

References

• Yuan-Jen Chiang, Developments of Harmonic Maps, Wave Maps and Yang-Mills Fields into Biharmonic Maps, Biwave Maps and Bi-Yang-Mills Fields, ISBN 978-3034805339

See also:

• Wikipedia, Bi-Yang-Mills equations