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 be a Hermitian connection on a Hermitian vector bundle over a Kähler manifold of dimension . Then the Hermitian Yang-Mills equations are
for some constant . Here we have
Notice that since is assumed to be a Hermitian connection, the curvature is skew-Hermitian, and so implies . When the underlying Kähler manifold is compact, may be computed using Chern-Weil theory. Namely, we have
Since and the identity endomorphism has trace given by the rank of , we obtain
where is the slope of the vector bundle , given by
and the volume of is taken with respect to the volume form .
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 , that are Hermitian, but for which the Levi-Civita connection is not Hermite-Einstein.)
When the Hermitian vector bundle has a holomorphic structure, there is a natural choice of Hermitian connection, the Chern connection. For the Chern connection, the condition that is automatically satisfied. The Hitchin-Kobayashi correspondence asserts that a holomorphic vector bundle admits a Hermitian metric 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 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 is , there is a splitting of the forms into self-dual and anti-self-dual forms. The complex structure interacts with this as follows:
When the degree of the vector bundle vanishes, then the Hermitian Yang-Mills equations become . By the above representation, this is precisely the condition that . That is, 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]
References
- Kobayashi, Shoshichi (1980), "First Chern class and holomorphic tensor fields", Nagoya Mathematical Journal, 77: 5–11, ISSN 0027-7630, MR 0556302
- Kobayashi, Shoshichi (1987), Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, ISBN 978-0-691-08467-1, MR 0909698
- Donaldson, S. K., Donaldson, S. K., & Kronheimer, P. B. (1990). The geometry of four-manifolds. Oxford University Press.