Abelian Chern-Simons theory with toral gauge group, modular tensor categories, and group categories

TL;DR

This work connects abelian toral Chern–Simons theory with modular tensor categories by constructing group-category MT categories from finite abelian data $(\mathcal{D},q,c)$ and showing that the induced projective representations of the mapping class group coincide with those from toral Chern–Simons theory. The authors formalize a lattice-quantization framework, invoke Nikulin lifting results to relate discriminant data to lattices, and use a detailed Heegaard-to-surgery translation to compare the MCG actions. A central contribution is the demonstration that the toral CS TQFTs admit equivalent MT-category descriptions via braided group categories parameterized by the third cohomology class $[h,s]$ in $H^3(A^{1}(\mathcal{D});\mathbb{Q}/\mathbb{Z})$. The results illuminate how extended, non-spin toral CS theories can be captured by group-category MTCs, guiding future work on spin theories and extended TQFTs. Overall, the paper provides a concrete algebraic realization of abelian toral CS data within the modular tensor category framework, with explicit constructions and equivalences grounded in lattice theory and group cohomology.

Abstract

Classical and quantum Chern-Simons with gauge group $\text{U}(1)^N$ were classified by Belov and Moore in \cite{belov_moore}. They studied both ordinary topological quantum field theories as well as spin theories. On the other hand a correspondence is well known between ordinary $(2+1)$-dimensional TQFTs and modular tensor categories. We study group categories and extend them slightly to produce modular tensor categories that correspond to toral Chern-Simons. Group categories have been widely studied in other contexts in the literature \cite{frolich_kerler},\cite{quinn},\cite{joyal_street},\cite{eno},\cite{dgno}. The main result is a proof that the associated projective representation of the mapping class group is isomorphic to the one from toral Chern-Simons. We also remark on an algebraic theorem of Nikulin that is used in this paper.

Figures (26)

• Figure 1: A counterclockwise twist of a boundary circle with respect to a second boundary circle. Instead of using parameterizations we depict distinguished basepoints. We provide visual markings to show the diffeomorphism.
• Figure 2: A counterclockwise braiding of two boundary circles with respect to the distinguished outer boundary circle. Instead of using parameterizations we depict distinguished basepoints. We provide visual markings to show the diffeomorphism.
• Figure 3: A $(k=5,l=2)$-ribbon graph. The diagrammatic presentation is depicted on the right.
• Figure 4: On the top is depicted a right twist (a $(1,1)$-ribbon graph). On the bottom is depicted a left twist (a $(1,1)$-ribbon graph). The diagrammatic presentation is depicted on the right for each.
• Figure 5: On the top is depicted a right braid (a $(2,2)$-ribbon graph). On the bottom is depicted a left braid (a $(2,2)$-ribbon graph). The diagrammatic presentation is depicted on the right for each.

Theorems & Definitions (79)

• Example 2.14
• Example 2.28
• Definition 2.43
• Definition 2.44
• Proposition 2.47: Segal
• Example 2.49
• Example 2.53
• Example 2.54
• Example 2.55
• Example 2.62