In mathematics, the modular group representation (or simply modular representation) of a modular tensor category C {\displaystyle {\mathcal {C}}} is a representation of the modular group SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} associated to C {\displaystyle {\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 C {\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 C {\displaystyle {\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 C {\displaystyle {\mathcal {C}}} , it is a theorem that there is a finite-dimensional unitary representation ρ C : SL 2 ( Z ) → U ( C [ L ] ) {\displaystyle \rho _{\mathcal {C}}:{\text{SL}}_{2}(\mathbb {Z} )\to U(\mathbb {C} [{\mathcal {L}}])} where SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} is the group of 2-by-2 invertible integer matrices, C [ L ] {\displaystyle \mathbb {C} [{\mathcal {L}}]} is a vector space with a formal basis given by elements of the set L {\displaystyle {\mathcal {L}}} of isomorphism classes of simple objects, and U ( C [ L ] ) {\displaystyle U(\mathbb {C} [{\mathcal {L}}])} denotes the space of unitary operators C [ L ] {\displaystyle \mathbb {C} [{\mathcal {L}}]} relative to Hilbert space structure induced by the canonical basis.3 Seeing as SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} is sometimes referred to as the modular group, this representation is referred to as the modular representation of C {\displaystyle {\mathcal {C}}} . It is for this reason that modular tensor categories are called 'modular'.
There is a standard presentation of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} , given by SL 2 ( Z ) =< s , t | s 4 = 1 , ( s t ) 3 = s 2 > {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )=<\left.s,t\right|s^{4}=1,\,\,(st)^{3}=s^{2}>} .4 Thus, to define a representation of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} it is sufficient to define the action of the matrices s , t {\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 S , T {\displaystyle S,T} called the modular S {\displaystyle S} and T {\displaystyle T} matrices. The entries of the matrices are labeled by pairs ( [ A ] , [ B ] ) ∈ L 2 {\displaystyle ([A],[B])\in {\mathcal {L}}^{2}} . The modular T {\displaystyle T} -matrix is defined to be a diagonal matrix whose ( [ A ] , [ A ] ) {\displaystyle ([A],[A])} -entry is the θ {\displaystyle \theta } -symbol θ A {\displaystyle \theta _{A}} . The ( [ A ] , [ B ] ) {\displaystyle ([A],[B])} entry of the modular S {\displaystyle S} -matrix is defined in terms of the braiding, as shown below (note that naively this formula defines S A , B {\displaystyle S_{A,B}} as a morphism 1 → 1 {\displaystyle {\bf {1}}\to {\bf {1}}} , which can then be identified with a complex number since 1 {\displaystyle {\bf {1}}} is a simple object).
The modular S {\displaystyle S} and T {\displaystyle T} matrices do not immediately give a representation of SL 2 ( Z ) {\displaystyle {\text{SL}}_{2}(\mathbb {Z} )} - they only give a projective representation. This can be fixed by shifting S {\displaystyle S} and T {\displaystyle T} by certain scalars. Namely, defining ρ C ( s ) = ( 1 / D ) ⋅ S {\displaystyle \rho _{\mathcal {C}}(s)=(1/{\mathcal {D}})\cdot S} and ρ C ( t ) = ( p C − / p C + ) 1 / 6 ⋅ T {\displaystyle \rho _{\mathcal {C}}(t)=(p_{\mathcal {C}}^{-}/p_{\mathcal {C}}^{+})^{1/6}\cdot T} defines a proper modular representation,5 where D 2 = ∑ [ A ] ∈ L d A 2 {\textstyle {\mathcal {D}}^{2}=\sum _{[A]\in {\mathcal {L}}}d_{A}^{2}} is the global quantum dimension of C {\displaystyle {\mathcal {C}}} and p C − , p C + {\displaystyle p_{\mathcal {C}}^{-},\,\,p_{\mathcal {C}}^{+}} are the Gauss sums associated to C {\displaystyle {\mathcal {C}}} , where in both these formulas d A {\displaystyle d_{A}} are the quantum dimensions of the simple objects.
References
Moore, G; Seiberg, N (1989-09-01). Lectures on RCFT (Rational Conformal Field Theory) (Report). doi:10.2172/7038633. OSTI 7038633. /wiki/Doi_(identifier) ↩
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. 978-0-8218-2686-7 ↩
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. 978-0-8218-2686-7 ↩
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. 978-0-8218-2686-7 ↩
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. 978-0-8218-2686-7 ↩