Modular group representation

In mathematics, the modular group representation (or simply modular representation) of a modular tensor category ${\displaystyle {\mathcal {C}}}$ is a representation of the modular group ${\displaystyle {\text{SL}}_{2}(\mathbb {Z} )}$ associated to ${\displaystyle {\mathcal {C}}}$. It is from the existence of the modular representation that modular tensor categories get their name.¹

From the perspective of topological quantum field theory, the modular representation of ${\displaystyle {\mathcal {C}}}$ arrises naturally as the representation of the mapping class group of the torus associated to the Reshetikhin–Turaev topological quantum field theory associated to ${\displaystyle {\mathcal {C}}}$.² As such, modular tensor categories can be used to define projective representations of the mapping class groups of all closed surfaces.

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

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

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