Classification of abelian spin Chern-Simons theories

TL;DR

This work provides a complete quantum classification of abelian spin Chern-Simons theories with gauge group U(1)^N by translating the problem from classical lattice data to robust quantum invariants. The authors identify three key invariants—the signature modulo 24, the discriminant group with its bilinear form, and the equivalence class of a quadratic refinement—constrained by the Gauss-Milgram formula, which together determine the modular representation of the theory. The construction of the ground-state wavefunctions uses a detailed Kaehler structure on the space of connections and Siegel-Narain theta functions to realize conformal blocks, with a magnetic translation group governing large gauge transformations. They also prove a converse: every allowed quartet of invariants arises from some lattice, and they discuss applications to fractional quantum Hall fluids, including explicit relations for Hall conductivities and vortex charges.

Abstract

We derive a simple classification of quantum spin Chern-Simons theories with gauge group T=U(1)^N. While the classical Chern-Simons theories are classified by an integral lattice the quantum theories are classified differently. Two quantum theories are equivalent if they have the same invariants on 3-manifolds with spin structure, or equivalently if they lead to equivalent projective representations of the modular group. We prove the quantum theory is completely determined by three invariants which can be constructed from the data in the classical action. We comment on implications for the classification of fractional quantum Hall fluids.

Figures (5)

• Figure 1: The Picard group is a disjoint union of the torii. The spin line bundles are represented by $2^{2g}$ points in the $g-1$ component of the Picard group. There is no natural origin of $\mathrm{Pic}_{g-1}$.
• Figure 2: Figure a) shows that the line bundle $\mathcal{L}_{\sigma}$ has non-zero curvature, and thus the parallel transport does not define the lift of the gauge group. In figure b) we present the gauge invariant configuration space $\mathcal{C}$. The parallel transport along straight line path on figure a) corresponds to the holonomy around the closed loop in $\mathcal{C}/\mathcal{C}_{\Lambda}$.
• Figure 3: In figure a) we illustrate the construction of the twisted vector bundle $E_{\omega}$ over $\Sigma_g\times S^1_-$. The twist is defined by identification of $E|_{t=1}$ with $E|_{t=0}$ by the gauge transformation $g$ corresponding to $\omega$. In figure b) we present the surface $\Delta$ (a pair of pants) which is used to prove the cocycle relation.
• Figure 4: In Figure a) we present genus two Riemann surface with marking. In Figure b) we use the $a$ and $b$ cycles to cut the Riemann surface $\Sigma_g$ to obtain a connected $4g$-gon. The straight white lines represent the canonical paths which were used to trivialize the line bundle $\mathcal{L}_{\sigma}.$
• Figure 5: The fundamental domain of the dual lattice $\Lambda^*$ for the Example 5. The basis vectors of the dual lattice are $\omega^1$ and $\omega^2$. The lattice $\Lambda$ is spanned by the vectors $\bar{e}_1,\bar{e}_2$. The red dots together with the origin represent the discriminant group $A$. The circle on the top represents the values of the quadratic refinement $q_{\omega^1}$.