A. Zee, Quantum Field Theory in a Nutshell · Problem IV.5.5

Field Theory Redux

Physics — QFT

Problem

For a challenge show that tr FnF^n, which appears in higher dimensional theories such as string theory, are all total divergences. In other words, there exists a (2n−1)(2n - 1)-form ω2n−1(A)\omega_{2n-1}(A) such that tr Fn=dω2n−1(A)F^n = d\omega_{2n-1}(A). [Hint: A compact representation of the form ω2n−1(A)=∫01dtf2n−1(t,A)\omega_{2n-1}(A) = \int_0^1 dt f_{2n-1}(t, A) exists.] Work out ω5(A)\omega_5(A) explicitly and try to generalize knowing ω3\omega_3 and ω5\omega_5. Determine the (2n−1)(2n - 1)-form f2n−1(t,A)f_{2n-1}(t, A). For help, see B. Zumino et al., Nucl. Phys. B239:477, 1984.

Browse the problem library