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Complex structure on an associated bundle by realification

Physical scope

Pure mathematical well-definedness result; it has no physical assumptions or empirical claim.

Statement

Let $H$ be a finite-dimensional real Lie group, let $π : P \to M$ be a smooth right principal $H$ -bundle, let $W$ be a finite-dimensional complex vector space, and let $ρ : H \to {G L}_{C} (W)$ be a smooth complex representation. Regard $W$ as the underlying real vector space $W_{R}$ and $ρ$ as the real representation $ρ_{R}$ . The associated quotient $P \times_{H, ρ_{R}} W_{R}$ has the smooth real vector-bundle structure supplied by the associated-vector-bundle theorem. The map $J [p, w] = [p, i w]$ is well-defined and smooth and satisfies $J^{2} = - i d$ . It therefore gives this real bundle a complex vector-bundle structure of rank $\dim_{C} W$ , with transition maps $ρ (g_{j i})$ in any local principal trivializations $s_{i} = s_{j} g_{j i}$ . This assertion specifies the complex structure induced by $J$ and makes no uniqueness claim about other possible complex structures on the same real bundle.

Statement references

Smooth principal $G$ -bundle

Principal g bundle and associated fiber bundle

Associated vector bundles are well-defined

Facts & Assumptions

Given: A smooth right principal $H$ -bundle $P \to M$ , a finite-dimensional complex vector space $W$ , and a smooth complex representation $ρ$ .

[F1]

Principal g bundle and associated fiber bundle: the associated quotient uses the relation $(p h, w) \sim (p, ρ (h) w)$ .

[F2]

Associated vector bundles are well-defined: for a smooth right principal bundle and a smooth finite-dimensional real representation, this quotient has a smooth real vector-bundle structure whose transition maps are the representation matrices of the principal transition functions.

Proof

1.1F2givenalgebra

The underlying real space $W_{R}$ is finite-dimensional. In a complex basis, the inclusion ${G L}_{C} (W) ↪ {G L}_{R} (W_{R})$ sends a matrix $A$ to the real block matrix $(\begin{array}{cc} Re A & - Im A \\ Im A & Re A \end{array})$ ; its entries depend linearly on the real and imaginary parts of those of $A$ , so the inclusion is smooth. Therefore $ρ_{R}$ is a smooth real representation. Applying [F2] supplies the smooth real vector-bundle structure and the local charts induced by sections $s_{i} : U_{i} \to P$ , with transition maps $ρ_{R} (g_{j i})$ .

2.1F1F2givenstep 1.1algebra∎

If $(p h, w) \sim (p, ρ (h) w)$ , then $J [p h, w] = [p h, i w] = [p, ρ (h) i w] = [p, i ρ (h) w] = J [p, ρ (h) w],$ because $ρ (h)$ is complex-linear. Thus $J$ is well-defined on the quotient. In the local charts from step 1.1 it acts by $(x, w) \mapsto (x, i w)$ , so it is smooth and $J^{2} = - i d$ . Each fiber consequently has the complex scalar action $(a + i b) [p, w] = [p, a w + i b w]$ . The chart transitions $ρ (g_{j i})$ commute with multiplication by $i$ , so they are complex-linear; the resulting complex bundle has rank $\dim_{C} W$ .

Depends on

Dependency tree · 2 levels

Sources

John M. Lee, Introduction to Smooth Manifolds, 2nd ed.