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

Table of contents
- Solution e

Solution

Substituting

a = W^{T} a^{'}

into the Chern--Simons term gives

K^{'} = W K W^{T} .

This matrix is integral and symmetric. It is invertible because

det K^{'} = (det W)^{2} det K = det K \neq = 0.

For

p^{'} = W p

and

q^{'} = W q

,

p^{' T} K^{' - 1} q^{'} = p^{T} W^{T} W^{- T} K^{- 1} W^{- 1} W q = p^{T} K^{- 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:

∣ det K^{'} ∣ = ∣ det K ∣ .

Solved by gpt-5.6-sol high.

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