Quantum groups and nonabelian braiding in quantum Hall systems

Abstract

Wave functions describing quasiholes and electrons in nonabelian quantum Hall states are well known to correspond to conformal blocks of certain coset conformal field theories. In this paper we explicitly analyse the algebraic structure underlying the braiding properties of these conformal blocks. We treat the electrons and the quasihole excitations as localised particles carrying charges related to a quantum group that is determined explicitly for the cases of interest. The quantum group description naturally allows one to analyse the braid group representations carried by the multi-particle wave functions. As an application, we construct the nonabelian braid group representations which govern the exchange of quasiholes in the fractional quantum Hall effect states that have been proposed by N. Read and E. Rezayi, recovering the results of C. Nayak and F. Wilczek for the Pfaffian state as a special case.

Figures (7)

• Figure 1: fusion diagram for the field $\sigma$. The diagram must be thought extended indefinitely in the $\lambda$-direction and up to $\Lambda=k$ in the $\Lambda$-direction (the case $k=3$ is as drawn here). On each line, we have drawn the Young diagram of the $sl(2)$ representation that resides on that line.
• Figure 2: The same diagram as in figure \ref{['sl2fusdiag']}, but this time each site in the diagram is labelled by the Young diagram for the $\widehat{sl(k=3)}_{2}$ weight of the field that resides there. The dot represents the empty diagram. Again, generalisation to arbitrary $k$ is straightforward. Note that in this picture, the weights label the fields unambiguously, whereas in figure \ref{['sl2fusdiag']}, one still has to take the field identifications (\ref{['fieldident']}) into account
• Figure 3: Diagram of an indecomposable representation as defined by (\ref{['repfrms']}). The dots represent the basis states $|\,j,m\,\rangle,$ in particular, we have written $|\,h\,\rangle$ for the highest weight state and $|\,l\,\rangle$ for the lowest weight state. The arrows $\rightarrow$ and $\leftarrow$ indicate the action of $L^{+}$ and $L^{-}$ resp.
• Figure 4: Diagram of an indecomposable representation as it would occur in the tensor product of two $U_{q}(sl(2))$-irreps at $q=e^{2\pi i/(k+2)}$. The dots represent the basis states in the module, the arrows $\rightarrow$ and $\leftarrow$ indicate the action of $L^{+}$ and $L^{-}$ resp. The split arrows are meant to indicate that the descendants of the state $|\,\psi\,\rangle$ are mapped onto linear combinations of descendants of $|\,\psi\,\rangle$ and $(L^{+})^{k+1}|\,l\,\rangle$
• Figure 5: This diagram shows two ways of going from one bracketing of a fourfold tensor product to another. The arrows denote the canonical isomorphisms given by the coassociator (or the truncated $6j$-symbols). The diagram will commute if the condition (\ref{['assoccond']}) on the coassociator is satisfied