Nicolas Al Choueiry, Andrei Teleman: Reductive homogeneous spaces associated with real forms. A gauge-theoretical generalisation, 235-252

Abstract:

Let $G$ be a connected complex Lie group. A real form of $G$ is a closed subgroup $H\subset G$ whose Lie algebra ${\mathfrak{h}}$ is a real form of the Lie algebra ${\mathfrak{g}}$ of $G$. A pair $(G,H)$ of this type is reductive, and the corresponding quotient $G/H$ is a reductive homogeneous space whose canonical connection is torsion free.

Regarded as a principal $H$-bundle over $G/H$, $G$ comes with tensorial 1-form $\alpha$ of type $\mathrm{Ad}$ and a natural left invariant connection $A$. This remark suggests the following natural gauge theoretical generalisation of the class of reductive pairs of the form $(G,H)$ as above:

Let $H$ be an arbitrary Lie group. A triple , where is a principal $H$-bundle, $\alpha$ a tensorial 1-form of type $\mathrm{Ad}$ on $P$ and $A$ a connection on $P$ will be called admissible if the induced linear maps $A_y\to {\mathfrak{h}}$, $y\in P$, are all isomorphisms. If this is the case one obtains a canonical linear connection $\nabla^\alpha_A$ on $M$ and a canonical almost complex structure $J^\alpha_A$ on $P$ which, by a result of R. Zentner, is integrable if an only if the pair $(\alpha,A)$ satisfies a gauge invariant system of two first order differential equations. A triple as above will be called integrable, or Zentner triple, if this integrability condition for $J^\alpha_A$ is satisfied.

The main result of [1] states that any integrable triple with $M$ simply connected and $\nabla^\alpha_A$ complete can be identified with the triple associated with a real form of a complex Lie group; in particular there exists a complex Lie group $G$, a real form $H'\subset G$ of $G$ with $H'\simeq H$ such that the pair $(M,\nabla^\alpha_A)$ can be identified with the reductive homogeneous space $G/H'$ endowed with its canonical connection. In this article we explain the strategy of the proof of this classification result and we prove in detail a theorem which plays an important role in this strategy and is of independent interest. In the last section we introduce the moduli spaces of integrable pairs on a principal bundle, and we give explicit examples.

Key Words: Homogeneous space, principal bundles, connections, Lie group, real form.

2020 Mathematics Subject Classification: Primary 53C30; Secondary 22F30, 53C07.

Download the paper in pdf format here.