# Ground State Degeneracy of 2+1D U(1) Chern Simons Theory?

I am a physics graduate student trying to understand more mathematical aspects of gauge theories.

How can I understand ground state degeneracy of a simple Chern Simons Theory: 2+1D U(1) $S= \int_M kAdA$ for different k- values (integers, rationals, irrationals?) on torus or other manifolds in a more mathematics/differential geometry oriented way? I have seen the calculation in physics literature, but I can't figure out how to translate it in differential geometry language.

I know it must be there somewhere in literature, but I am not able to find it.

• If you want to ask questions to mathematicians, it is important that you learn which words they can understand. For example: "ground state" is better understood by mathematicians as "lowest energy eigenspace" (in which case you should also specify which Hamiltonian you are talking the eigenvalues of, otherwise mathematicians will again complain that they don't know what you're talking about). Also "degeneracy" in math is called "of dimension greater than one"... – André Henriques Oct 31 '14 at 0:00
• Have you read the paper by Axelrod-DellaPietra-Witten on the subject? – André Henriques Oct 31 '14 at 0:04

The Abeian Chern-Simons theory you write down, can be written in a very generic form with a symmetric bilinear integer matrix $K_{IJ}$ with a path integral (or partition function): $$Z=\int DA \exp[i S]=\int DA \exp[i \int_{M^3} \frac{K_{IJ}}{4\pi} A_I dA_J]$$ The $\int DA$ integrates all possible $A$ configuration, while $\int_{M^3}$ integrates over the spacetime manifold. Please make sure that the level quantization $K_{IJ}$ and $4 \pi$ are in the correct form.

The ground state degeneracy ($\equiv Gsd$) on a spatial manifold $M^d$ and a time $S^1$ can be computed from the partition function as $$Z(M^d \times S^1)=\dim(\text{Hilbert space on } M^d)\equiv Gsd_{M^d}.$$ Below I give examples for various symmetric bilinear integer matrix $K_{IJ}$, and provide their ground state degeneracy on $T^2$ ($\equiv Gsd_{T^2}$),

$$Z(T^2 \times S^1)=\dim(\text{Hilbert space on } T^2)\equiv Gsd_{T^2}.$$

1. For $S=\int\frac{k}{4\pi} A dA$, with $k$ is an odd integer, it is a fermionic Chern-Simons TQFT which requires the manifold with a spin structure (i.e. spin manifold) to define the theory. It is fermionic in the sense that this TQFT can be realized in a Fermionic many-body quantum system (such as electrons, like filling-fraction 1/3 Langhlin quantum Hall state).

$$Gsd_{T^2}=k.$$

1. For $S=\int\frac{2k}{4\pi} A dA=\int\frac{k}{2\pi} A dA$, with $2k$ is an even integer, it is a bosonic Chern-Simons TQFT where the manifold with or without a spin structure can both define the theory. It is bosonic in the sense that this TQFT can be realized in a Bosonic many-body quantum system. For example the $Z_2$ topological order of Wen and $Z_2$ toric code of Kitaev, they are basically the same (bosonic) $Z_2$ gauge theory.

$$Gsd_{T^2}=2k.$$

1. For the most generic $S=\int\frac{K_{IJ}}{4\pi} A_I dA_J$, their ground state degeneracy on $T^2$ ($\equiv Gsd_{T^2}$), $$Gsd_{T^2}=|\det(K)|.$$ is determined by the determinant of $K$.

(1) For a more mathematical physics calculation, you may look for Belov-Moore: Classification of abelian spin Chern-Simons theories and Refs therein (those of Witten's). I believe that they provide the mathematics/differential geometry method and cite the Refs.

(2) For physics, you can check those Refs involving counting the ground states via line operators or via the distinct classes of quantum wavefunction in the Hilbert space: such as Classification of Abelian quantum Hall states and matrix formulation of topological fluids: PhysRevB.46.2290, and Degeneracy of Topological Order: PhysRevB.91.125124

See also the more involved various non-Abelian Cherm-Simons theory Gsd and dimensions of Hilbert space.