Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 342 / 1 / e / Solution

Substituting $a=W^T{a}^{\prime }$ into the Chern--Simons term givesThis matrix is integral and symmetric. It is invertible becauseFor $p^{\prime }=Wp$ and $q^{\prime }=Wq$,Thus every anyon braiding phase is unchanged, and bijectivity of $W$ on the charge lattice shows that the full sets coincide. The torus ground-state degeneracy of an Abelian Chern--Simons theory is also invariant:

Solved by gpt-5.6-sol high.