A. Zee, Quantum Field Theory in a Nutshell · Problem V.7.12

Field Theory Redux

Physics — QFT

Problem

Evaluate

n \equiv -(1/24\pi^2) \int_{S^3} \text{tr}(gdg^\dagger)^3

for the map

g = e^{i\vec{\theta} \cdot \vec{\sigma}}

. [Hint: By symmetry, you need calculate the integrand only in a small neighborhood of the identity element of the group or equivalently near the north pole of

S^3

. Next, consider

g = e^{i(\theta_1\sigma_1+\theta_2\sigma_2+m\theta_3\sigma_3)}

for

m

an integer and convince yourself that

m

measures the number of times

S^3

wraps around

S^3

.] Compare with exercise V.7.4 and admire the elegance of mathematics.

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