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

Represent quasiparticle worldlines by conserved currents $j_I^{\alpha }$ and add the minimal-coupling termUp to the corresponding sign convention for charge, variation of the Abelian Chern--Simons theory givesFor a static quasiparticle of integer gauge-charge vector $q$, integration over a small surrounding disk yields the Chern--Simons flux attachment relationA quasiparticle of charge $q^{\prime }$ carried once around this flux acquires the Aharonov--Bohm phaseso

The constituent-particle current has charge vector $t=(1,\dots ,1{)}^T$. Its integral over the same disk isFor $K=I_n$ and $q=e_J$, this gives $N=1$. The self-exchange angle is ${\theta }_{qq}=\pi e_J^T{e}_J=\pi$, so the excitation is a fermion with one unit of microscopic particle number and may be identified with an electron.

Since $M^2=nM$, direct multiplication gives the matrix inverseprovided $1+np\ne 0$, as required by invertibility of the K-matrix. For $q=e_J$,The attached gauge-flux vector is thereforeRelative to the $K=I_n$ electron in part a, every one of the $n$ gauge fields consequently carries the additional flux

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

Let $x$ and $y$ denote the two oriented noncontractible cycles of the torus. Their transport operators intersect once, so the mutual braiding phase gives the Wilson-loop algebra of an Abelian Chern--Simons theoryThe integer charge vectors are identified modulo local electrons: $q$ and $q+K\ell$ have identical braiding with every quasiparticle. The distinct topological charges therefore form the finite quotient group

Choose a ground state that diagonalizes the mutually commuting $x$-cycle operators. Acting with $T_y(q)$ multiplies their eigenvalues by the character $q^{\prime }\mapsto e^{2\pi i{q}^{\prime T}{K}^{-1}q}$. This pairing is nondegenerate on $\mathcal{A}$: if the character is trivial for every $q^{\prime }$, then $K^{-1}q$ is integral and $q$ is the trivial class. Thus the $|\mathcal{A}|$ states obtained from inequivalent $q$ have distinct loop eigenvalues and are orthogonal. The torus ground-state degeneracy of an Abelian Chern--Simons theory is consequently at least $|\det K|$.

The all-ones matrix has eigenvalue $n$ along $t$ and eigenvalue zero on the $(n-1)$-dimensional perpendicular subspace. Hence $K=I_n+pM$ has eigenvalues $1+np,1,\dots ,1$, and