Modular group representation

In mathematics, the modular group representation (or simply modular representation) of a modular tensor category ${\mathcal {C}}$ is a representation of the modular group ${\text{SL}}_{2}(\mathbb {Z} )$ associated to ${\mathcal {C}}$. 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 ${\mathcal {C}}$ arrises naturally as the representation of the mapping class group of the torus associated to the Reshetikhin–Turaev topological quantum field theory associated to ${\mathcal {C}}$. 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 ${\mathcal {C}}$, it is a theorem that there is a finite-dimensional unitary representation $\rho _{\mathcal {C}}:{\text{SL}}_{2}(\mathbb {Z} )\to U(\mathbb {C} [{\mathcal {L}}])$ where ${\text{SL}}_{2}(\mathbb {Z} )$ is the group of 2-by-2 invertible integer matrices, $\mathbb {C} [{\mathcal {L}}]$ is a vector space with a formal basis given by elements of the set ${\mathcal {L}}$ of isomorphism classes of simple objects, and $U(\mathbb {C} [{\mathcal {L}}])$ denotes the space of unitary operators $\mathbb {C} [{\mathcal {L}}]$ 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 ${\mathcal {C}}$. 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 ${\text{SL}}_{2}(\mathbb {Z} )=<\left.s,t\right|s^{4}=1,\,\,(st)^{3}=s^{2}>$. 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 $([A],[B])\in {\mathcal {L}}^{2}$. 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 ${\bf {1}}\to {\bf {1}}$, which can then be identified with a complex number since ${\bf {1}}$ 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 $\rho _{\mathcal {C}}(s)=(1/{\mathcal {D}})\cdot S$ and $\rho _{\mathcal {C}}(t)=(p_{\mathcal {C}}^{-}/p_{\mathcal {C}}^{+})^{1/6}\cdot T$ defines a proper modular representation, where ${\textstyle {\mathcal {D}}^{2}=\sum _{[A]\in {\mathcal {L}}}d_{A}^{2}}$ is the global quantum dimension of ${\mathcal {C}}$ and $p_{\mathcal {C}}^{-},\,\,p_{\mathcal {C}}^{+}$ are the Gauss sums associated to ${\mathcal {C}}$, 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.