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.

Construction

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).

 

Definition of S-matrix entries.

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.

 

Formula for the Gauss sums of a modular tensor category.

 

Formula for the quantum dimension of a simple object.

References

1. Moore, G; Seiberg, N (1989-09-01). Lectures on RCFT (Rational Conformal Field Theory) (Report). doi:10.2172/7038633. OSTI 7038633.
2. Bakalov, Bojko; Kirillov, Alexander (2000-11-20). Lectures on Tensor Categories and Modular Functors. University Lecture Series. Vol. 21. Providence, Rhode Island: American Mathematical Society. doi:10.1090/ulect/021. ISBN 978-0-8218-2686-7. S2CID 52201867.