Substituting $a=W^Ta'$ into the Chern--Simons term gives
$$
\boxed{K'=WKW^T}.
$$
This matrix is integral and symmetric. It is invertible because
$$
\det K'=(\det W)^2\det K=\det K\ne0.
$$
For $p'=Wp$ and $q'=Wq$,
$$
p'^TK'^{-1}q'
=p^TW^TW^{-T}K^{-1}W^{-1}Wq
=p^TK^{-1}q.
$$
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:
$$
\boxed{|\det K'|=|\det K|}.
$$

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-342/1/e/solution