Modular group representation

In mathematics, the modular group representation (or simply modular representation) of a modular tensor category $𝒞$ is a representation of the modular group ${\text{SL}}_2(\mathbb{Z})$ associated to $𝒞$. It is from the existence of the modular representation that modular tensor categories get their name.

From the perspective of topological quantum field theory, the modular representation of $𝒞$ arrises naturally as the representation of the mapping class group of the torus associated to the Reshetikhin–Turaev topological quantum field theory associated to $𝒞$. As such, modular tensor categories can be used to define projective representations of the mapping class groups of all closed surfaces.

Construction

Associated to every modular tensor category $𝒞$, it is a theorem that there is a finite-dimensional unitary representation ${\rho }_{𝒞}:{\text{SL}}_2(\mathbb{Z})\to U(ℂ[ℒ])$ where ${\text{SL}}_2(\mathbb{Z})$ is the group of 2-by-2 invertible integer matrices, $ℂ[ℒ]$ is a vector space with a formal basis given by elements of the set $ℒ$ of isomorphism classes of simple objects, and $U(ℂ[ℒ])$ denotes the space of unitary operators $ℂ[ℒ]$ relative to Hilbert space structure induced by the canonical basis. Seeing as ${\text{SL}}_2(\mathbb{Z})$ is sometimes referred to as the modular group, this representation is referred to as the modular representation of $𝒞$. It is for this reason that modular tensor categories are called 'modular'.

There is a standard presentation of ${\text{SL}}_2(\mathbb{Z})$, given by ${\text{SL}}_2(\mathbb{Z})=<s,t|s^4=1,(st{)}^3=s^2>$. Thus, to define a representation of ${\text{SL}}_2(\mathbb{Z})$ it is sufficient to define the action of the matrices $s,t$ and to show that these actions are invertible and satisfy the relations in the presentation. To this end, it is customary to define matrices $S,T$ called the modular $S$ and $T$ matrices. The entries of the matrices are labeled by pairs $([A],[B])\in {ℒ}^2$. The modular $T$-matrix is defined to be a diagonal matrix whose $([A],[A])$-entry is the $\theta$-symbol ${\theta }_A$. The $([A],[B])$ entry of the modular $S$-matrix is defined in terms of the braiding, as shown below (note that naively this formula defines $S_{A,B}$ as a morphism $\mathrm{𝟏}\to \mathrm{𝟏}$, which can then be identified with a complex number since $\mathrm{𝟏}$ is a simple object).

Definition of S-matrix entries.

The modular $S$ and $T$ matrices do not immediately give a representation of ${\text{SL}}_2(\mathbb{Z})$ - they only give a projective representation. This can be fixed by shifting $S$ and $T$ by certain scalars. Namely, defining ${\rho }_{𝒞}(s)=(1/𝒟)\cdot S$ and ${\rho }_{𝒞}(t)=(p_{𝒞}^{-}/p_{𝒞}^{+}{)}^{1/6}\cdot T$ defines a proper modular representation, where ${𝒟}^2=\sum _{[A]\in ℒ}{d}_A^2$ is the global quantum dimension of $𝒞$ and $p_{𝒞}^{-},p_{𝒞}^{+}$ are the Gauss sums associated to $𝒞$, where in both these formulas $d_A$ are the quantum dimensions of the simple objects.

Formula for the Gauss sums of a modular tensor category.

Formula for the quantum dimension of a simple object.