THE GEOMETRY OF VECTOR BUNDLES AND

AN INTRODUCTION TO GAUGE THEORY

LECTURE 22

Professor Steven Bradlow Class Notes From Math 433

University of Illinois at Urbana-Champaign

March 13, 1998

Induced Connections II

(3)\underline{Induced Connections on E^{∗}}:

Given a connection D on E → B, we can define a connection D^{∗} on E^{∗} such that if s^{∗} ∈ Ω^{0}(E^{∗}) and t ∈ Ω^{0}(E), then [s^{∗}(t)]_{b} = s_{b}^{∗}(t_{b}), ∀b ∈ B, is a smooth funtion on the base B. So we can take the differential

ds^{∗}(t) = (D^{∗}s^{∗})(t) + s^{∗}(D(t)).

With respect to the local frame {e_{i}} for E, D = d + A and with respect to the dual frame {e_{i}^{∗}} for E^{∗}, D^{∗} = d + A^{∗}. Then we have

e_{i}^{∗}(b)(e_{j}(b)) = δ_{ij},

such that (i) de_{i}^{∗}(e_{j}) = 0, (ii) De_{i} = A_{ji}e_{k}, and (iii) D^{∗}e_{j}^{∗} = A_{kj}^{∗}e_{k}^{∗}. So we need

0 = de_{i}^{∗}(e_{j})
= (A_{ki}^{∗} ⊗ e_{k}^{∗})(e_{j}) + e_{i}^{∗} ⊗ (A_{kj}e_{k})
= A_{ji}^{∗} + A_{ij}.

Therefore we require

A^{∗} = −A^{t}.

Exercise 1. Check that if {A^{α}} is a collection of connection 1-forms for D, then {−(A^{α})^{t}} defines a connection on E^{∗}! (If so, then the connection clearly is the one we need!).

Exercise 2. Describe this in terms of horizontal lifting of curves.

Note. We can combine (2) and (3) to get D on (⨂^{r}E_{1}) ⊗ (⨂^{s}E_{2}) etc.

Remark. From (1), (2), and (3), we can obtain the connection on E_{1} ⊗ E_{2}^{∗} ≅ Hom(E_{2}, E_{1}).

(4)\underline{Connections on Hom(E_{1}, E_{2}) ≅ E_{2} ⊗ E_{1}^{∗}}:

Since Hom(E_{1}, E_{2}) ≅ E_{2} ⊗ E_{1}^{∗}, so we have the connection

D = D_{2} ⊗ I_{1} + I_{2} ⊗ D_{1}^{∗}

defined on Hom(E_{1}, E_{2}).

Direct description on Hom(E_{1}, E_{2}): Given h : E_{1} → E_{2} and fix bases {e_{i}^{(1)}} and {e_{i}^{(2)}} for E_{1} and E_{2}, respectively. Then h = [h_{ij}].

Typeset by AMS-TEX

Question. What is the D(h) ?

Let e_{ij} = e_{i}^{(2)} ⊗ e_{j}^{(1)^{∗}}. Then

\begin{aligned}D(e_{ij}) & = A_{ki}^{(2)}e_{k}^{(2)} ⊗ e_{j}^{(1)^{∗}} + e_{i}^{(1)} ⊗ A_{kj}^{(1)^{∗}}e_{k}^{(1)^{∗}} \\ & = A_{ki}^{(2)}e_{k}^{(2)} ⊗ e_{j}^{(1)^{∗}} − A_{jk}^{(1)}e_{i}^{(2)} ⊗ e_{k}^{(1)^{∗}}. \end{aligned}

But h = ∑ h_{ij}(e_{i}^{(2)} ⊗ e_{j}^{(1)^{∗}}), where we think of {e_{ij}} as a basis for Hom(E_{1}, E_{2}) via (e_{i}^{(2)} ⊗ e_{j}^{(1)^{∗}})(s) = e_{j}^{(1)^{∗}}(s)e_{i}^{(1)}. Exercise 3. Show that

D(h) = ∑(dh + A^{(2)}h − hA^{(1)})_{ij} e_{i}^{(2)} ⊗ e_{j}^{(1)^{∗}}.

I.e. if h = [h_{ij}] with respect to {e_{ij}}, then

D(h) = dh + A^{(2)}h − hA^{(1)}.

Remark. Special case: if E_{1} = E_{2} = E, then Hom(E_{1}, E_{2}) = End(E) and D on E induces D on End(E) such that with respect to local frames, if D = d + A on E and u ∈ Ω^{0}(End(E)), then

\begin{aligned}D(u) & = du + Au − uA \\ & = du + [A, u] \end{aligned}

on End(E). Note. We can use the extension of D on End(E) to define the Covariant Derivative

D : Ω^{p}(End(E)) → Ω^{p+1}(End(E))

Apply this covariant derivative to F_{D} ∈ Ω^{2}(End(E)) to compute D(F_{D}). With respect to local frames, D(F_{A}) = dF_{A} + [A, F_{A}], where D = d + A. Thus the Bianchi identity says D(F_{D}) = 0 ! (5)\underline{Connections on Pull-Back Bundles}:

Connection on f^{∗}(E) via

E
π
X  \overline{}_{f} B

Say D is a connection on E and {e_{i}^{α}} is local frames for E over U_{α}. Let {f^{∗}(e_{i}^{α})} be the “pull-back” frame for f^{∗}(E) over f^{−1}(U_{α}) with f^{∗}(e_{i}^{α})(x) = e_{i}^{α}(f(x)). Say D = d + A^{α} with respect to {e_{i}^{α}}. Claim. f^{∗}(A^{α}) defines connection 1-form on f^{−1}(U_{α}). (w.r.t. {f^{∗}(e_{i}^{α})}).

proof: Given a vector field Y on f^{−1}(U_{α})(∈ X), f^{∗}(A_{x}^{α})(Y) = A_{f(x)}^{α}(f_{∗}Y). That is, we define f^{∗}(D) on f^{∗}(E) such that

f^{∗}(D_{Y})(f^{∗}s) = D_{f∘Y} (s).

□ Note. Given an isomorphic bundle map (h is an isomorphism between two bundles)

E \overline{}^{h} F
[figure]
B

Let D be a connection on F. Then we can define h^{∗}(D) on E by h^{∗}(D)(s) = h^{−1}(D)(h(s)). Therefore h^{∗}(D) = h^{−1} ∘ D ∘ h.

273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801

E-mail address: bradlow@math.uiuc.edu