Hermitian Yang–Mills connection

In mathematics, and in particular gauge theory and complex geometry, a Hermitian Yang–Mills connection (or Hermite-Einstein connection) is a Chern connection associated to an inner product on a holomorphic vector bundle over a Kähler manifold that satisfies an analogue of Einstein's equations: namely, the contraction of the curvature 2-form of the connection with the Kähler form is required to be a constant times the identity transformation. Hermitian Yang–Mills connections are special examples of Yang–Mills connections, 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.

Hermitian Yang–Mills equations

Hermite-Einstein connections arise as solutions of the Hermitian Yang-Mills equations. These are a system of partial differential equations on a vector bundle over a Kähler manifold, which imply the Yang-Mills equations. 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 skew-Hermitian, 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 Chern-Weil theory. 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 Einstein metric, solutions of the Hermitian Yang-Mills equations are often called Hermite-Einstein connections, as well as Hermitian Yang-Mills connections.

Examples

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 holomorphic structure, there is a natural choice of Hermitian connection, the Chern connection. For the Chern connection, the condition that ${\displaystyle F_{A}^{0,2}=0}$ is automatically satisfied. The Hitchin-Kobayashi correspondence 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 Kobayashi (1980, section 6). These equation imply the Yang-Mills equations in any dimension, and in real dimension four are closely related to the self-dual Yang-Mills equations that define instantons. 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]

See also

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

References

• Kobayashi, Shoshichi (1980), "First Chern class and holomorphic tensor fields", Nagoya Mathematical Journal, 77: 5–11, doi:10.1017/S0027763000018602, ISSN 0027-7630, MR 0556302, S2CID 118228189
• Kobayashi, Shoshichi (1987), Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, vol. 15, Princeton University Press, ISBN 978-0-691-08467-1, MR 0909698