THE GEOMETRY OF VECTOR BUNDLES AND

AN INTRODUCTION TO GAUGE THEORY

LECTURE 32

Professor Steven Bradlow Class Notes From Math 433

University of Illinois at Urbana-Champaign

April 17, 1998

Characteristic Classes for Real Bundles

Before we return to Characteristic Classes for real bundles, we go back to complex bundles : we had Chern Classes defined by the generating function

c(A) = det(I + \frac{i}{2π} A) = ∏_{i=1}^{n}(1 + x_{i}) (where x_{i} are the eigenvalues of \frac{i}{2π} A) .

Chern Characters defined by the generating function

ch(A) = Tr (exp(\frac{i}{2π} A)) = ∑_{i=1}^{n} exp(x_{i}) .

Another important characteristic class is the Todd class

T d(A) = ∏_{j=1}^{n} \frac{x_{j}}{1 − e^{−x_{j}}} .

Note.

• When applied to A = F = curvature, series terminates at terms of deg = \frac{dimB}{2} .

• For a complex bundle E, T d(E) = 1 + \frac{1}{2}c_{1}(E) + \frac{1}{12}(c_{1}^{2}(E) + c_{2}(E)) + · · · .

• Appears in Riemann-Roch Theorem.

For a holomorphic bundle (complex bundle such that E is a complex manifold and π is holomorphic) over a complex manifold E →^{π} X, we have an integer

χ(E) = bundle version of Euler characteristic of a manifold.

= index of a differential operator.

The Riemann-Roch Theorem relates this (global, topological) quantity χ(E) to characteristic classes:

χ(E) = ∫_{X} T d(T′X) ∧ ch(E) .

This is an example of an Index Theorem which relates the (integer valued) index of a differential operator Ω^{0}(E_{1}) → Ω^{0}(E_{2}) to ∫_{B}(characteristic classes of T B and bundles).

Typeset by AMS-TEX

If X is a Riemann surface (dim_{ℂ} X = 1), then Rank(T′X) = 1 and T d(T′X) = 1 + \frac{1}{2}c_{1}(T′X). So

T d(T′X) ∧ ch(E) = (1 + \frac{1}{2}c_{1}(T′X)) ∧ (r + c_{1}(E)) .

Thus

\begin{aligned}∫_{X} T d(T′X) ∧ ch(E) & = ∫_{X} (1 + \frac{1}{2}c_{1}(T′X)) ∧ (r + c_{1}(E)) \\ & = ∫_{X} c_{1}(E) + \frac{r}{2} ∫_{X} c_{1}(T′X) . \end{aligned}

Fact / Definition.

• ∫_{X} c_{1}(E) ≡ deg(E) .

• deg(T′X) = 2 − 2g, where g = genus of Riemann surface X.

So the Riemann-Roch Theorem:

χ(E) = deg(E) + Rank(E)(1 − g) .

Back to the Characteristic Classes for Real Bundles

Recall the \underline{First Approach} from last time:

Given a real bundle E_{ℝ} with fiber ℝ^{n} and GL(n, ℝ), we can define a complex bundle E_{ℂ} = E_{ℝ} ⊗ ℂ with fiber ℂ^{n} and GL(n, ℂ)(↩ GL(n, ℝ)).

Exercise 1. Show that E_{ℂ} = E_{ℝ} ⊕ E_{ℝ} as real bundle.

Hint: Look in detail at GL(n, ℝ) ↪ GL(n, ℂ) ↪ GL(2n, ℝ). □

Note. Given any complex bundle, we can look at underlying real bundle uning ℂ^{n} ≅ ℝ^{2n} to realize fibers as real and GL(n, ℂ) ↪ GL(2n, ℝ) to obtain real transition functions.

Therefore we can define

c_{k}(E_{ℝ}) ≡ c_{k}(E_{ℂ}) .

Note. For k odd, c_{k}(E_{ℝ}) = 0.

Proof: Given a complex bundle E_{ℂ}, we can define \overline{E_{ℂ}}. If g_{αβ} ∈ GL(n, ℂ) are transition functions for E_{ℂ}, then \overline{g_{αβ}} ∈ GL(n, ℂ) are transition functions for \overline{E_{ℂ}}. We now use the following key facts:

Key Facts.

(1) If E_{ℂ} = E_{ℝ} ⊗ ℂ, then \overline{E_{ℂ}} = E_{ℂ}.

(2) c_{k}(E_{ℂ}) = (−1)^{k}c_{k}(\overline{E_{ℂ}}), for any complex bundle E_{ℂ}.

Hint: Given a connection D on E_{ℂ}, we can get a connection \overline{D} on E_{ℂ} such that F\overline{D} = \overline{F_{D}}, then i F\overline{D} = −(\overline{i F_{D}}). □

Thus c_{k}(E_{ℂ}) = (−1)^{k}c_{k}(\overline{E_{ℂ}}) = (−1)^{k}c_{k}(E_{ℂ}). □
So we can get c_{2k}(E_{ℝ}) in H^{4k}(B, ℝ).

\underline{Second Approach}: Look at GL(n, ℝ) invariant polynomials on Mat_{n}(ℝ).

\underline{Easier}: Look at O(n)-invariant polynomials on Lie(O(n)) = \{ A| A^{t} + A = 0 \}.

\underline{Why} can we always do this: Fix a Riemann metric on E_{ℝ} and use orthonormal frames and orthogonal connections, then the curvature F is Lie(O(n))-valued and the transition functions are O(n)-valued. Hence give any O(n)-invariant P : Lie(O(n)) → ℝ, we can get [P (\frac{1}{2π} F)] ∈ H^{∗}(B, ℤ).

So what can we say about such P ?

Fact. A + A^{t} = 0, A cannot be diagonalized: Since

\{ \begin{matrix}A & + A^{t} = 0 \\ A & = [ \begin{matrix}λ_{1} \\ & . & . & . \\ & & & & λ_{n} \end{matrix} ] \end{matrix} ⇒ λ_{i} = 0 !

But A is conjugate (via T ∈ O(n)) to

\begin{aligned}[ & \begin{aligned}0 & λ_{1} \\ −λ_{1} & 0 \\ & & . & . & . \\ & & & & & 0 & λ_{k} \\ & & & & & −λ_{k} & 0 \end{aligned} & ] \\ & n = 2k \end{aligned} or \begin{aligned}[ & \begin{aligned}0 & λ_{1} \\ −λ_{1} & 0 \\ & & . & . & . \\ & & & & & 0 & λ_{k} \\ & & & & & −λ_{k} & 0 \\ & & & & & & & 0 \end{aligned} & ] \\ & n = 2k + 1 \end{aligned} .

Thus we only need to consider P ’s on such A. Write

P [ \begin{matrix}0 & λ_{1} \\ −λ_{1} & 0 \\ & & 0 & λ_{2} \\ & & −λ_{2} & 0 \\ & & & & . & . & . \\ & & & & & & & . & . & . \end{matrix} ] = P (λ_{1}, ⋯, λ_{n}) .

273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801 E-mail address: bradlow@math.uiuc.edu