nLab F-Yang-Mills equation

Idea

The $F$-Yang-Mills equation (or $F$-YM equation) arises from a generalization of the Yang-Mills action functional with a function, which scales the norm of the curvature form. Notable special cases include exponential Yang-Mills connections, $p$-Yang-Mills connections and Yang-Mills-Born-Infeld connections with positive and negative sign.

$F$-Yang-Mills action functional

Let $F$ be a strictly increasing $C^2$-function with $F(0)=0$. Let:

$$d_F \coloneqq\sup_{t\geq 0}\frac{tF'(t)}{F(t)}.$$

(Wei 22)

Since $F$ is a $C^2$ function, one can also consider the following constant:

$$d_{F'} =\sup_{t\geq 0}\frac{tF''(t)}{F'(t)}.$$

(Baba & Shintani 23, Definition 4.8, Baba 23)

Let $G$ be a compact Lie group with Lie algebra $\mathfrak{g}$ and $E\twoheadrightarrow B$ be a principal $G$-bundle with a orientable Riemannian manifold $B$. Let $\operatorname{Ad}(E)\coloneqq E\times_G\mathfrak{g}$ be its adjoint bundle. The $F$-Yang-Mills action functional (or $F$-YM action functional) is given by:

$$\operatorname{YM}_F\colon \Omega^1(B,\operatorname{Ad}(E))\rightarrow\mathbb{R}, \operatorname{YM}_F(A) \coloneqq\int_B F\left( \frac{1}{2}\|F_A\|^2 \right)\mathrm{d}\vol_g.$$

(Baba & Shintani 23, Definition 3.1, Baba 23)

For a flat connection $A\in\Omega^1(B,\operatorname{Ad}(E))$ (with $F_A=0$), one has $\operatorname{YM}_F(A)=F(0)\operatorname{vol}(M)$. Hence $F(0)=0$ is required to avert divergence for a non-compact manifold $B$, although this condition can also be left out as only the derivative $F'$ is of further importance.

$F$-Yang-Mills connections and equation

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

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

for all smooth families $A\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname{Ad}(E))$ with $A(0)=A$. This is the case iff the $F$-Yang–Mills equation (or $F$-YM equation) is fulfilled:

$$\mathrm{d}_A\star\left( F'\left( \frac{1}{2}\|F_A\|^2 \right)F_A \right) =0.$$

(Baba & Shintani 23, Corollary 3.4, Baba 23)

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

A $F$-Yang–Mills connection/field with:

• $F(t)=\exp(t)$ (or $F(t)=\exp(t)-1$ for normalization) is called (normed) exponential Yang–Mills connection/field. In this case, one has $d_{F'}=\infty$. The exponential and normed exponential Yang-Mills action functional are denoted with $\operatorname{YM}_\mathrm{e}$ and $\operatorname{YM}_\mathrm{e}^0$ respectively. (Matsura & Urakawa 95)

• $F(t)=\frac{1}{p}(2t)^{\frac{p}{2}}$ is called $p$-Yang–Mills connection/field. Usual Yang–Mills connections/fields are exactly the $2$-Yang–Mills connections/fields. In this case, one has $d_{F'}=\frac{p}{2}-1$. The $p$-Yang-Mills action functional is denoted with $\operatorname{YM}_p$.

• $F(t)=\sqrt{1-2t}-1$ or $F(t)=\sqrt{1+2t}-1$ is called Yang–Mills–Born–Infeld connection/field (or YMBI connection/field) with negative or positive sign respectively. In these cases, one has $d_{F'}=\infty$ and $d_{F'}=0$ respectively. The Yang-Mills-Born-Infeld action functionals with negative and positive sign are denoted with $\operatorname{YMBI}^-$ and $\operatorname{YMBI}^+$ respectively.

(Baba & Shintani 23, Example 3.2, Wei 22, Baba 23)

Stable $F$-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 $F$-Yang-Mills theory. $A$ is called a stable $F$-Yang-Mills connection (or stable $F$-YM connection), if:

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

for all smooth families $A\colon(-\varepsilon,\varepsilon)\rightarrow\Omega^1(B,\operatorname{Ad}(E))$ with $A(t)=A$. It is called weakly stable if only $\geq 0$ holds. A $F$-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 $F$-Yang-Mills connection, its curvature $F_A\in\Omega^2(B,\operatorname{Ad}(E))$ is also called (weakly) stable or unstable $F$-Yang-Mills field.

(Baba & Shintani 23, Definition 3.6, Baba 23)

Properties

Lemma

For a Yang-Mills connection with constant curvature, its stability as Yang-Mills connection implies its stability as exponential Yang-Mills connection.

(Matsura & Urakawa 95, Corollary 6.2)

Lemma

Every non-flat exponential Yang-Mills connection over $S^n$ with $n\geq 5$ and:

$$\|F_A\| \leq\sqrt{\frac{n-4}{2}}$$

is flat.

(Baba & Shintani 23, Proposition 4.14, Baba 23)

Lemma

Every non-flat Yang-Mills-Born-Infeld connection over $S^n$ with $n\geq 5$ and:

$$\|F_A\| \leq\sqrt{\frac{n-4}{n-2}}$$

is flat.

(Baba & Shintani 23, Proposition 4.13)

Theorem

For $n\gt 4(d_{F'}+1)$, every non-flat $F$-Yang-Mills connection over $S^n$ is unstable.

(Baba & Shintani 23, Theorem 1.2 and Corollary 4.12, Baba 23, Baba 23)

Lemma

For $n\gt 2p$, every non-flat $p$-Yang-Mills connection over $S^n$ is unstable.

Lemma

For $n\gt 4$, every non-flat Yang-Mills-Born-Infeld connection with positive sign over $S^n$ is unstable.

Lemma

For $0\leq d_{F'}\leq\frac{1}{6}$, every non-flat $F$-Yang-Mills connection over the Cayley plane $F_4/\operatorname{Spin}(9)$ is instable.

(Baba 23)

Related concepts

• Bi-Yang-Mills equation

References

• Fumiaki Matsura, Hajime Urakawa, On exponential Yang-Mills connections (1995), DOI:10.1016/0393-0440(94)00041-200041-2)

• Shihshu Walter Wei, On exponential Yang-Mills fields and p-Yang-Mills fields (2022), arXiv:2205.03016

• Kurando Baba, Kazuto Shintani, A Simons type condition for instability of F-Yang-Mills connections (2023), arXiv:2301.04291

• Kurando Baba, On instability of F-Yang-Mills connections (2023), Slides

See also:

• Wikipedia, F-Yang-Mills equations