Hermitian Yang–Mills connection

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In

, and are often called instantons.

The Kobayashi–Hitchin correspondence proved by Donaldson, Uhlenbeck and Yau asserts that a holomorphic vector bundle over a compact Kähler manifold admits a Hermitian Yang–Mills connection if and only if it is slope polystable.

Hermite–Einstein connections arise as solutions of the Hermitian Yang–Mills equations. These are a system of

. Let ${\displaystyle A}$ be a Hermitian connection on a Hermitian vector bundle ${\displaystyle E}$ over a Kähler manifold ${\displaystyle X}$ of dimension ${\displaystyle n}$. Then the Hermitian Yang–Mills equations are:

${\displaystyle {\begin{aligned}&F_{A}^{0,2}=0\\&F_{A}\cdot \omega =\lambda (E)\operatorname {Id} ,\end{aligned}}}$

for some constant ${\displaystyle \lambda (E)\in \mathbb {C} }$. Here we have:

${\displaystyle F_{A}\wedge \omega ^{n-1}=(F_{A}\cdot \omega )\omega ^{n}.}$

Notice that since ${\displaystyle A}$ is assumed to be a Hermitian connection, the curvature ${\displaystyle F_{A}}$ is

, and so ${\displaystyle F_{A}^{0,2}=0}$ implies ${\displaystyle F_{A}^{2,0}=0}$. When the underlying Kähler manifold ${\displaystyle X}$ is compact, ${\displaystyle \lambda (E)}$ may be computed using . Namely, we have

${\displaystyle {\begin{aligned}\deg(E)&:=\int _{X}c_{1}(E)\wedge \omega ^{n-1}\\&={\frac {i}{2\pi }}\int _{X}\operatorname {Tr} (F_{A})\wedge \omega ^{n-1}\\&={\frac {i}{2\pi }}\int _{X}\operatorname {Tr} (F_{A}\cdot \omega )\omega ^{n}.\end{aligned}}}$

Since ${\displaystyle F_{A}\cdot \omega =\lambda (E)\operatorname {Id} _{E}}$ and the identity endomorphism has trace given by the rank of ${\displaystyle E}$, we obtain

${\displaystyle \lambda (E)=-{\frac {2\pi i}{n!\operatorname {Vol} (X)}}\mu (E),}$

where ${\displaystyle \mu (E)}$ is the slope of the vector bundle ${\displaystyle E}$, given by

${\displaystyle \mu (E)={\frac {\deg(E)}{\operatorname {rank} (E)}},}$

and the volume of ${\displaystyle X}$ is taken with respect to the volume form ${\displaystyle \omega ^{n}/n!}$.

Due to the similarity of the second condition in the Hermitian Yang–Mills equations with the equations for an

, solutions of the Hermitian Yang–Mills equations are often called Hermite–Einstein connections, as well as Hermitian Yang–Mills connections.

The Levi-Civita connection of a Kähler–Einstein metric is Hermite–Einstein with respect to the Kähler–Einstein metric. (These examples are however dangerously misleading, because there are compact Einstein manifolds, such as the Page metric on ${\displaystyle {\mathbb {C} P}^{2}\#{\overline {\mathbb {C} P^{2}}}}$, that are Hermitian, but for which the Levi-Civita connection is not Hermite–Einstein.)

When the Hermitian vector bundle ${\displaystyle E}$ has a

. For the Chern connection, the condition that ${\displaystyle F_{A}^{0,2}=0}$ is automatically satisfied. The asserts that a holomorphic vector bundle ${\displaystyle E}$ admits a Hermitian metric ${\displaystyle h}$ such that the associated Chern connection satisfies the Hermitian Yang–Mills equations if and only if the vector bundle is polystable. From this perspective, the Hermitian Yang–Mills equations can be seen as a system of equations for the metric ${\displaystyle h}$ rather than the associated Chern connection, and such metrics solving the equations are called Hermite–Einstein metrics.

The Hermite–Einstein condition on Chern connections was first introduced by

. In particular, when the complex dimension of the Kähler manifold ${\displaystyle X}$ is ${\displaystyle 2}$, there is a splitting of the forms into self-dual and anti-self-dual forms. The complex structure interacts with this as follows:

${\displaystyle \Lambda _{+}^{2}=\Lambda ^{2,0}\oplus \Lambda ^{0,2}\oplus \langle \omega \rangle ,\qquad \Lambda _{-}^{2}=\langle \omega \rangle ^{\perp }\subset \Lambda ^{1,1}}$

When the degree of the vector bundle ${\displaystyle E}$ vanishes, then the Hermitian Yang–Mills equations become ${\displaystyle F_{A}^{0,2}=F_{A}^{2,0}=F_{A}\cdot \omega =0}$. By the above representation, this is precisely the condition that ${\displaystyle F_{A}^{+}=0}$. That is, ${\displaystyle A}$ is an ASD instanton. Notice that when the degree does not vanish, solutions of the Hermitian Yang–Mills equations cannot be anti-self-dual, and in fact there are no solutions to the ASD equations in this case.^[1]

• Einstein manifold
• Deformed Hermitian Yang–Mills equation
• Gauge theory (mathematics)

• Kobayashi, Shoshichi (1980), "First Chern class and holomorphic tensor fields", Nagoya Mathematical Journal, 77: 5–11,
  S2CID 118228189
• Kobayashi, Shoshichi (1987), Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15,
  MR 0909698