A. Zee, Quantum Field Theory in a Nutshell · Problem IV.4.6

Field Theory Redux

Physics — QFT

Problem

Let

g(x)

be the element of a group

G

. The 1-form

v = gdg^\dagger

is known as the Cartan-Maurer form. Then

\text{tr} v^N

is trivially closed on an

N

-dimensional manifold since it is already an

N

-form. Consider

Q = \oint_{S^N} \text{tr} v^N

with

S^N

the

N

-dimensional sphere. Discuss the topological meaning of

Q

. These considerations will become important later when we discuss topology in field theory in chapter V.7. [Hint: Study the case

N = 3

and

G = SU(2)