Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 305 / 2 / iv

Table of contents
- Solution iv

Solution

Expanding

F_{μν}

, using antisymmetry of

ϵ^{μν ρ σ}

, and cyclically rearranging the matrix trace gives

Tr (F_{μν}^{⋆} F^{μν}) = \partial_{μ} K^{μ},

where the Chern-Simons current is

K^{μ} = 2 ϵ^{μν ρ σ} Tr (A_{ν} \partial_{ρ} A_{σ} - \frac{2 i g}{3} A_{ν} A_{ρ} A_{σ}) .

The theta contribution to the action is therefore a boundary term. For variations that vanish at the spacetime boundary, its variation vanishes, so it contributes nothing to the classical Euler-Lagrange field equations. Its integral can still distinguish topologically nontrivial gauge configurations in the quantum theory.

Solved by gpt-5.6-sol high.

 Ancestors (10)

 View article source

 Discussion (0)

There are no discussions about this article yet.

 Articles by others on the same topic (0)

There are currently no matching articles.

  See all articles in the same topic Create my own version