Fractional Quantum Hall Effect- degeneracy of ground state (Tong's notes)

[binbagsss]

Hi , I'm looking at the argument in David Tongs notes (http://www.damtp.cam.ac.uk/user/tong/qhe/three.pdf) for ground state degeneracy on depending on the topology of the manifold (page 97, section 3.2.4).

I follow up to getting equation 3.31 but I'm stuck on the comment after : ' But such an algebra of operators can’t be realized on a single vacuum state.'

I have no idea where this comment comes from- (also I then don't understand the smallest representation comment, dimension m, and where the action comes from).

Any links to relevant background knowledge assumed to follow these comments of this section, or any help, would be greatly appreciated. many thanks.

 

[king vitamin]

In the end this is purely a math problem: we know that there is the following identity on the ground state manifold,
$$
T_1 T_2 T_1^{-1} T_2^{-1} | 0 \rangle = e^{2 \pi i/m} | 0 \rangle,
$$
where $m$ is a positive integer. Given that, what is the dimension of the manifold $| 0 \rangle$? It turns out to just be equivalent to asking what dimension of matrices do you need to have the two matrices $T_{1,2}$ satisfy
$$
T_1 T_2 = e^{2 \pi i/m} T_2 T_1.
$$
Here I will admit that I don't remember my abstract algebra well enough to know how to solve this problem in complete generality, but you should be able to convince yourself pretty quickly that there are no one-dimensional solutions, so you need more than one ground state. With a little bit of trial-and-error for small values of $m$ I quickly found the $m$-dimensional solution
$$
T_1 =
\begin{pmatrix}
1 & 0 & 0 & \cdots & 0 \\
0 & e^{2 \pi i/m} & 0& \cdots & 0 \\
0 & 0 & e^{4 \pi i/m} & \cdots & 0 \\
\vdots & \vdots & \vdots & \ddots & \vdots \\
0 & 0 & 0 & \cdots & e^{2 \pi i (m-1)/m}
\end{pmatrix},
\qquad
T_2 =
\begin{pmatrix}
0 & 0 & 0 & \cdots & 0 & 1 \\
1 & 0 & 0 & \cdots & 0 & 0 \\
0 & 1 & 0 & \cdots & 0 & 0 \\
\vdots &\vdots &\vdots &\ddots & \vdots & \vdots \\
0 & 0 & 0 & \cdots & 1 & 0
\end{pmatrix}
$$
Wen's textbook claims that there is only a single $m$-dimensional irreducible representation of the above algebra, so this construction must be it. But I'd go over to the Algebra forum if you want a full proof of that statement. In any case, this proves that representations of the above algebra must be direct sums of the above representation, so the ground state must have degeneracy $m b$ for some positive integer $b$. Unfortunately, I don't think this argument tells you that $b=1$ for genus 1, but if you're planning on completing Tong's notes I think you will eventually be able to show this from Chern-Simons theory.