[Paper Review] Equivariant Modular Categories via Dijkgraaf-Witten Theory

Jennifer Maier, Thomas Nikolaus+1 more|arXiv (Cornell University)|Mar 15, 2011

Algebraic structures and combinatorial models23 references4 citations

TL;DR

This paper constructs $J$-equivariant modular tensor categories using Dijkgraaf-Witten theory with a weak action of a finite group $J$ on a finite group $G$. It introduces an equivariant Drinfel'd double as a $J$-ribbon algebra, generalizing the standard Drinfel'd double, and shows that the resulting category is modular and carries a natural $J$-action, completing a categorical orbifold square for equivariant topological field theories.

ABSTRACT

Based on a weak action of a finite group J on a finite group G, we present a geometric construction of J-equivariant Dijkgraaf-Witten theory as an extended topological field theory. The construction yields an explicitly accessible class of equivariant modular tensor categories. For the action of a group J on a group G, the category is described as the representation category of a J-ribbon algebra that generalizes the Drinfel'd double of the finite group G.

Motivation & Objective

• To construct a geometric realization of $J$-equivariant extended topological field theories using Dijkgraaf-Witten theory.
• To provide a concrete class of equivariant modular tensor categories arising from finite group actions.
• To generalize the Drinfel'd double construction to incorporate group actions via a $J$-ribbon algebra.
• To complete the categorical orbifold square by showing that the modularization of a $J$-equivariant category yields the standard Drinfel'd double.
• To establish a link between twisted bundles, action groupoids, and modular categories through character theory and 2-linearization.

Proposed method

• Uses a weak action of a finite group $J$ on a finite group $G$ to define a $J$-equivariant Dijkgraaf-Witten theory as an extended TFT.
• Constructs the equivariant Drinfel'd double as a $J$-ribbon algebra over the group algebra $\mathbb{K}[G]$, generalizing the standard Drinfel'd double $\mathcal{D}(G)$.
• Employs 2-linearization of spans over action groupoids $M//G$ to realize the category of representations as a modular category.
• Applies a cohomological description of twisted bundles via $G$-equivariant class functions on $M \times G$, with invariance under conjugation.
• Uses character theory for action groupoids, including orthogonality relations and a generalized Burnside theorem, to classify irreducible representations.
• Establishes the modularization of the category $\mathcal{B}(G\triangleleft H)$-mod as equivalent to $\mathcal{D}(G)$-mod via induction along the regular representation of $J=H/G$.

Experimental results

Research questions

• RQ1 How can a $J$-equivariant modular tensor category be geometrically constructed from a weak action of $J$ on a finite group $G$?
• RQ2 What is the structure of the equivariant Drinfel'd double as a $J$-ribbon algebra, and how does it generalize the standard Drinfel'd double?
• RQ3 Can the modularization of a premodular category arising from a normal subgroup inclusion $G \triangleleft H$ be realized as a $J$-equivariant modular category with $J=H/G$?
• RQ4 How do twisted sectors and fusion rules in the equivariant theory relate to the inertia groupoid $\Lambda(M//G)$ and $G$-orbits on $A = \{(m,g) \mid g.m = m\}$?
• RQ5 What is the role of the regular representation of $J$ in inducing the modular category from the equivariant one, and how does this relate to orbifold constructions?

Key findings

• The equivariant Dijkgraaf-Witten theory yields a $J$-equivariant modular tensor category that is equivalent to the representation category of the equivariant Drinfel'd double.
• The number of irreducible representations of the action groupoid $M//G$ equals the number of isomorphism classes of objects in the inertia groupoid $\Lambda(M//G)$, i.e., $|I| = |\mathrm{Iso}(\Lambda(M//G))|$.
• The character of the regular representation of $M//G$ is given by $\chi_H(m,g) = \delta(g,1)\cdot|G|$, and it contains each irreducible representation with multiplicity equal to its dimension.
• The category $\mathcal{B}(G\triangleleft H)$-mod is premodular and modularizable, with its modularization equivalent to $\mathcal{D}(G)$-mod via induction along the regular representation of $J=H/G$.
• The irreducible characters of $M//G$ form an orthogonal basis for the space of class functions under the bilinear form $\langle f,f'\rangle = \frac{1}{|G|}\sum_{g,m} f(m,g^{-1})f'(m,g)$.
• The generalized Burnside theorem holds: $\sum_{i\in I} |d_i|^2 = |M||G|$, where $d_i = \dim V_i$ for simple representations $V_i$ of $M//G$.