# Chiral Spin Liquid(CSL), Chern number, and the ground state degeneracy(GSD)

Consider a 2D gapped CSL with a nonzero Chern number $m$, then is the GSD of the system on a torus directly related to the Chern number $m$?

For example, see this article, in the last paragraph on page 7, the authors give the 4-fold GSD from the Chern number $m=\pm2$ for a CSL. I can not understand the explanation, can anybody present an intuitive illustration or a simple mathematical proof ? I will be very appreciated, thank you very much.

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Free arXiv version: arxiv.org/abs/1110.0116 – Qmechanic Jan 21 '14 at 18:54

Based on the paper, the answer is $|m|^2$. They suggest in their p.8, Eq.36, the effective theory is a Chern-Simons theory $$\frac{1}{4\pi}\int K_{IJ} a_I \wedge d a_J$$ with the $K_{IJ}$ bilinear K matrix as
$$K_{IJ}={\begin{pmatrix}m & 0\\ 0 & -m\end{pmatrix}}$$.
The up $m$ labels one sector and the lower $m$ labels the other sector. The degeneracy(GSD) is computed by a generalizing level-$k$ U(1) Chern-Simons theory(GSD=$k$) to a bilinear K matrix U(1)$^n$ Chern-Simons theory. GSD=$|\det(K)|=|m|^2$. This GSD result for GSD=$|\det(K)|$ is a well-known fact.
Yes, a good question. the 1-form $a$ field represents so-called Wilson-line operator or simply anyons. In the lattice model, this Wilson-line operator or anyons can be regarded as the string or cycle on the lattice(such as string-net or toric-code, etc), See for example, page 4 of this paper: Boundary Degeneracy of Topological Order - Sec: Mutual Chern-Simons, Zk gauge theory, Toric code and String-net model and this paper: Quantum codes on a lattice with boundary – Idear Jan 22 '14 at 18:59