This is a follow-up question on the topic that I opened a few days ago, Wilson Loops as raising operators.

The paper

Topological Degeneracy of Quantum Hall Fluids. X.G. Wen, A. Zee. Phys. Rev. B 58 no. 23 (1998), pp. 15717-15728. arXiv:cond-mat/9711223.

gives a nice derivation of the explicit ground states of the $U(1)$ Chern-Simons Theory on a torus in Section 2 on Abelian Quantum Hall States.

In particular Eq. (12) gives the generic form of a ground state $\psi(y) = \sum_{n=-\infty}^{\infty} c_{n} \ e^{i\ 2\pi ny}$. Due to the fact that the theory lives on a torus the ground state manifold is found to be $k$-fold degenerate.

My question: Is it possible (by direct calculation) to obtain the relations \begin{align} W(b)|n \rangle &= |n + 1 \text{ mod } |k| \rangle, \nonumber \\ W(a) |n \rangle &= e^{2\pi i n /k} |n \rangle. \end{align} from the previous question?

I don't have a particularly strong background in field theory so I am feeling somewhat uneasy when it comes to the explicit evaluation of the Wilson Loop (with its exponentiated gauge field and the path ordering) acting on the constructed state.

I am looking forward to your responses.

This is a follow-up question on the topic that I opened a few days ago, Wilson Loops as raising operators.

The paper

Topological Degeneracy of Quantum Hall Fluids. X.G. Wen, A. Zee. Phys. Rev. B 58 no. 23 (1998), pp. 15717-15728. arXiv:cond-mat/9711223.

gives a nice derivation of the explicit ground states of the $U(1)$ Chern-Simons Theory on a torus in Section 2 on Abelian Quantum Hall States.

In particular Eq. (12) gives the generic form of a ground state $\psi(y) = \sum_{n=-\infty}^{\infty} c_{n} \ e^{i\ 2\pi ny}$. Due to the fact that the theory lives on a torus the ground state manifold is found to be $k$-fold degenerate.

My question: Is it possible (by direct calculation) to obtain the relations \begin{align} W(b)|n \rangle &= |n + 1 \text{ mod } |k| \rangle, \nonumber \\ W(a) |n \rangle &= e^{2\pi i n /k} |n \rangle. \end{align} from the previous question?

I don't have a particularly strong background in field theory so I am feeling somewhat uneasy when it comes to the explicit evaluation of the Wilson Loop (with its exponentiated gauge field and the path ordering) acting on the constructed state.

I am looking forward to your responses.

This is a follow-up question on the topic that I opened a few days ago: Wilson Loops as raising operators

The paper http://arxiv.org/pdf/cond-mat/9711223v2.pdf gives a nice derivation of the explicit ground states of the U(1) Chern Simons Theory on a torus in Section 2 on Abelian Quantum Hall States.

In particular Eq. (12) gives the generic form of a ground state $\psi(y) = \sum_{n=-\infty}^{\infty} c_{n} \ e^{i\ 2\pi ny}$. Due to the fact that the theory lives on a torus the ground state manifold is found to be k-fold degenerate.

My question: Is it possible (by direct calculation) to obtain the relations \begin{align} W(b)|n \rangle &= |n + 1 \text{ mod } |k| \rangle, \nonumber \\ W(a) |n \rangle &= e^{2\pi i n /k} |n \rangle. \end{align} from the previous question?

I don't have a particularly strong background in field theory so I am feeling somewhat uneasy when it comes to the explicit evaluation of the Wilson Loop (with its exponentiated gauge field and the path ordering) acting on the constructed state.

I am looking forward to your responses.

This is a follow-up question on the topic that I opened a few days ago, Wilson Loops as raising operators.

The paper

Topological Degeneracy of Quantum Hall Fluids. X.G. Wen, A. Zee. Phys. Rev. B 58 no. 23 (1998), pp. 15717-15728. arXiv:cond-mat/9711223.

gives a nice derivation of the explicit ground states of the $U(1)$ Chern-Simons Theory on a torus in Section 2 on Abelian Quantum Hall States.

In particular Eq. (12) gives the generic form of a ground state $\psi(y) = \sum_{n=-\infty}^{\infty} c_{n} \ e^{i\ 2\pi ny}$. Due to the fact that the theory lives on a torus the ground state manifold is found to be $k$-fold degenerate.

My question: Is it possible (by direct calculation) to obtain the relations \begin{align} W(b)|n \rangle &= |n + 1 \text{ mod } |k| \rangle, \nonumber \\ W(a) |n \rangle &= e^{2\pi i n /k} |n \rangle. \end{align} from the previous question?

I don't have a particularly strong background in field theory so I am feeling somewhat uneasy when it comes to the explicit evaluation of the Wilson Loop (with its exponentiated gauge field and the path ordering) acting on the constructed state.

I am looking forward to your responses.