Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 342 / 1 / c / Solution

Because $t^{T}M=n{t}^T$,Thus a fundamental quasiparticle has fractional constituent number

Let $q$ be the gauge-charge vector of a local electron. Trivial full braiding with every integer quasiparticle $q^{\prime }$ requiresEquivalently $K^{-1}q=\ell$ for some $\ell \in {\mathbb{Z}}^n$, so the allowed local charges lie in the local-particle latticeUnit constituent number requiresFinally its self-exchange angle isFor integer ${\ell }_I$, one has ${\ell }_I^2\equiv {\ell }_I(\mathrm{mod}2)$, and henceThe exchange phase is fermionic precisely when this integer is odd. Therefore