Spin Geometry: Lecture 9

Shuhan Jiang

Example 3.6. Let S = ΛT ∗ M . Let h·, ·i be the Riemannian metric. Let ∇ be the Levi-Civita
connection. S is then a Dirac bundle. This follows from

g(γ(e)α, γ(e)β) = g(e ∧ α − ιe α, e ∧ β − ιe β)

= g(α, ιe (e ∧ β − ιe β) − e ∧ (e ∧ β − ιe β))
= g(α, ιe (e ∧ β) + e ∧ ιe β)
= g(α, β),

and Lemma 3.1.

/ on S is d + d∗ .
Exercise 3.3. Show that the Dirac operator D

Hint 3.2. In normal coordinates {xµ } around p ∈ M , we have

d = dxµ ∧ ∇µ , d∗ = −δ µν ι∂µ ∇ν .

at the point p.

Let S be a Dirac bundle. Let R denote the curvature of the metric connection ∇ on S. R
induces a section R of End(S) via the formula
1
R(σ) = γ µν (Rµν (σ)), (3.6)
2
where γ µν = γ(dxµ )γ(dxν ) and Rµν = R(∂µ , ∂ν ) are local sections of End(S).

Lemma 3.3. R is a self-adjoint operator, i.e., hσ1 , R(σ2 )i = hR(σ1 ), σ2 i.

Proof. Since ∇ is a metric connection, we have

hσ1 , Rµν (σ2 )i = −hRµν (σ2 ), σ2 i.

On the other hand, by (3.1),

hσ1 , γ µν (σ2 )i = (−1)2 hγ ν γ µ (σ1 ), σ2 i = −hγ µν (σ1 ), σ2 i.

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Proposition 3.5 (General Weitzenböck formula). Let (S, ∇, h·, ·i) be a Dirac bundle with
/
Dirac operator D.
/ 2 = ∆B + R,
D
where ∆B = ∇∗ ∇ is the Bochner Laplacian on S.

Proof. The proof follows from a direct computation. Let {xµ } be local normal coordinates.
1 µν
/ 2 = γ µ ∇µ (γ ν ∇ν ) = γ µν ∇µ ∇ν =
D (γ ∇µ ∇ν + γ νµ ∇ν ∇µ ) .
2
Noting that γ νµ = −2g µν − γ µν , we have
1
/ 2 = −g µν ∇µ ∇ν − γ µν ([∇µ , ∇ν ]) = ∆B + R.
D
2

/ 2 = ∆H , the
Example 3.7. Let’s take S to be the Dirac bundle ΛT ∗ M . In this case, D
Hodge Laplacian on ΛT ∗ M .

Lemma 3.4. We have

∆H = ∆B + Ric, (3.7)

where Ric(·) is the Weitzenböck operator. Using a local orthonormal frame {eµ }, we can
write
∑
n ∑
p
Ric(σ)(ek1 , · · · , ekp ) = (R(eµ , ekj )σ)(ek1 , · · · , eµ , · · · , ekp ),
µ=1 j=1

where R is the Riemannian curvature of M .

Proof. Let σ be a p-form. By (3.1),

1
R(σ)(ek1 , · · · , ekp ) = hγ(eµ )γ(eν )(R(eµ , eν )(σ)), ek1 ∧ · · · ∧ ekp i
2
1
= − hR(eµ , eν )(σ), γ(eµ )γ(eν )(ek1 ∧ · · · ∧ ekp )i
2
1
= hR(eµ , eν )(σ), eµ ∧ ιeν (ek1 ∧ · · · ∧ ekp ) + ιeµ (eν ∧ ek1 ∧ · · · ∧ ekp )i
2
1( )
= hR(eµ , eν )(σ), eµ ∧ ιeν (ek1 ∧ · · · ∧ ekp )i + hR(eν , eµ )(σ), ιeν (eµ ∧ ek1 ∧ · · · ∧ ekp )i
2
= hR(eµ , eν )(σ), eµ ∧ ιeν (ek1 ∧ · · · ∧ ekp )i
= Ric(σ)(ek1 , · · · , ekp ).

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Definition 3.11. Let A be an algebra bundle over M . An A-module F is said to be twisted
if there exists another A-module E such that

F ∼
= E ⊗ W,

where W is a vector bundle over M that is not isomorphic to the trivial line bundle K.

If W is a trivial vector bundle, then F is just a sum of copies of E. If W is non-trivial, then

twisting E with W makes the global topological properties of F ∼ = E ⊗ W differ from the
global topological properties of E.
Exercise 3.4. Let S be a Dirac bundle. Let E be a metric bundle equipped with a metric
connection.

1. Show that the twisted Clifford module S ⊗ E is also a Dirac bundle.

2. In particular, take S = ΛT ∗ M and prove Proposition 2.8.

If there exists a non-trivial line bundle L on M , then every A-module E is twisted because

E∼
= (E ⊗ L∨ ) ⊗ L,

where L∨ is the dual bundle of L. Therefore, a more interesting question is: Given two
non-isomorphic A-modules E and F with rankE ≤ rankF , can one find a vector bundle W
such that F ∼
= E ⊗ W?
Let A be an algebra bundle over M . Let E and F be two A-modules over M . Let
⨿
HomA (E, F ) := HomAx (Ex , Fx ),
x∈M

where HomAx (Ex , Fx ) = {f ∈ HomK (Ex , Fx )|f (a(e)) = a(f (e), ∀e ∈ Ex and a ∈ Ax }.
Exercise 3.5. Show that HomA (E, F ) is vector bundle over M .
Proposition 3.6. Let A be a complex algebra bundle over M . Let E be a complex A-module
such that Ex is the unique complex irreducible representation of Ax for all x ∈ M . Then
every A-module F can be obtained as the twisted A-module

F ∼
= E ⊗ HomA (E, F ).

Proof. The bundle morphism

HomA (E, F ) ⊗ E → F
(x, φx ⊗ ex ) 7→ (x, φx (ex )).

is a bundle isomorphism due to Schur’s lemma.

Let n = dim M be even. Suppose that there exists a complex Cl(M )-module Σ such that
Σx is the complex irreducible representation of Cl(n) for all x ∈ M .

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Corollary 3.1. Every complex Cl(M )-module S can be obtained by twisting S with the vector
bundle W = HomCl(M ) (Σ, S).

We will see later if M admits a “spin structure”, then Σ exists. Moreover, it can be equipped
with a Dirac bundle structure which is determined by solely the Riemannian metric and the
spin structure of M .

3 Principal Bundles and Connections

Let G be a Lie group. A left/right G-torsor is a manifold endowed with a free, transitive
left/right G-action. Every G-torsor is diffeomorphic to G as smooth manifolds.

Definition 3.12. A fiber bundle is a triple (E, M, π), where E and M are smooth manifolds,
π : P → M is a smooth map, such that for any x ∈ M ,

• Ex := π −1 (x) is diffeomorphic to a smooth manifold F ;

• there exist an open neighborhood U of x and a diffeomorphism

φ : E|U := π −1 (U ) → U × F

of the form φ = (π, ψ).

A principal G-bundle is a fiber bundle (P, M, π), where P is endowed with a right G-action,
such that for any x ∈ M ,

• Px := π −1 (x) is a right G-torsor;

• the local trivialization map around x

φ : P |U := π −1 (U ) → U × G

is G-equivariant.

By definition, the base manifold M of a principal G-bundle π : P → M is isomorphic to the

orbit space P/G.

Proposition 3.7. Let P be a manifold equipped with a right G-action. π : P → P/G is a

principal G-bundle if and only the G-action is free and proper.

Proof. This follows from the slice theorem for proper Lie group actions.

Proposition 3.8. Let π : P → M be a principal G-bundle. The following statements are

equivalent:

1. P is a trivial bundle;

4
 2. P has a Lie group bundle structure;

3. P has a global section, i.e., a smooth map s : M → P such that π ◦ s = idM .

Proof. If P is trivial, then P ∼

= M × G, which is a Lie group bundle.
If P has a Lie group bundle structure, then we can define a global section of P by sending
each x ∈ M to the identity element of Px .
If P has a global section s, then we can define a map

M ×G→P
(x, g) 7→ s(x)g,

which is a diffeomorphism between M × G and P because it is bijective, smooth, and locally

a diffeomorphism.
Example 3.8. Let E be a vector bundle over M . Let
⨿
Fr(E) := Fr(Ex ),
x∈M

where Fr(Ex ) is the set of frames of the vector space Ex . (Recall that a frame {ei }m
i=1 of a
vector space V of rank m is equivalent to a linear isomorphism

e : Km → V

where e(0, · · · , 1, · · · , 0) = ei . The general linear group GL(m; K) acts on Fr(Ex ) from right
via
eg = e ◦ g, e ∈ Fr(Ex ),
and makes Fr(Ex ) into a right GL(m; K)-torsor.
Let φE : E|U → U × Km be a local trivialization of E. We obtain a bijective map
⨿
φFr(E) : Fr(Ex ) → U × GL(m; K)
x∈U

(x, ex ) 7→ (x, φE
x ◦ ex ).

Therefore, Fr(E) is a principal GL(m; K)-bundle.

Example 3.9. The Lie group U(1) acts freely and properly on
{ }
S 2n+1 := z ∈ Cn+1 | |z0 |2 + · · · + |zn |2 = 1

in the diagonal manner. We obtain the following principal U(1)-bundle:

π : S 2n+1 → CPn
z 7→ [z],

which is sometimes called the Hopf fibration.