{{Short description|Representation of a modular tensor category}} In mathematics, the '''modular group representation''' (or simply '''modular representation''') of a modular tensor category <math>\mathcal{C}</math> is a representation of the modular group <math>\text{SL}_2(\mathbb{Z})</math> associated to <math>\mathcal{C}</math>. It is from the existence of the modular representation that modular tensor categories get their name.<ref>{{cite report |title=Lectures on RCFT (Rational Conformal Field Theory) |last1=Moore |first1=G |last2=Seiberg |first2=N |date=1989-09-01 |doi=10.2172/7038633 |page= |osti=7038633 |doi-access=free|url=https://www.osti.gov/biblio/7038633 }}</ref>
From the perspective of topological quantum field theory, the modular representation of <math>\mathcal{C}</math> arrises naturally as the representation of the mapping class group of the torus associated to the Reshetikhin–Turaev topological quantum field theory associated to <math>\mathcal{C}</math>.<ref name="Bakalov-2000">{{Cite book |last1=Bakalov |first1=Bojko |title=Lectures on Tensor Categories and Modular Functors |last2=Kirillov |first2=Alexander |date=2000-11-20 |publisher=American Mathematical Society |isbn=978-0-8218-2686-7 |series=University Lecture Series |volume=21 |location=Providence, Rhode Island |language=en |doi=10.1090/ulect/021 |s2cid=52201867}}</ref> 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 <math>\mathcal{C}</math>, it is a theorem that there is a finite-dimensional unitary representation <math>\rho_{\mathcal{C}}: \text{SL}_2(\mathbb{Z}) \to U(\mathbb{C}[\mathcal{L}])</math> where <math>\text{SL}_2(\mathbb{Z})</math> is the group of 2-by-2 invertible integer matrices, <math>\mathbb{C}[\mathcal{L}]</math> is a vector space with a formal basis given by elements of the set <math>\mathcal{L}</math> of isomorphism classes of simple objects, and <math>U(\mathbb{C}[\mathcal{L}])</math> denotes the space of unitary operators <math>\mathbb{C}[\mathcal{L}]</math> relative to Hilbert space structure induced by the canonical basis.<ref name="Bakalov-2000"/> Seeing as <math>\text{SL}_2(\mathbb{Z})</math> is sometimes referred to as the modular group, this representation is referred to as the modular representation of <math>\mathcal{C}</math>. It is for this reason that modular tensor categories are called 'modular'.
There is a standard presentation of <math>\text{SL}_2(\mathbb{Z})</math>, given by <math>\text{SL}_2( \mathbb{Z} ) = <\left. s , t \right| s^4 = 1 , \, \, (st)^3 = s^2></math>.<ref name="Bakalov-2000" /> Thus, to define a representation of <math>\text{SL}_2(\mathbb{Z})</math> it is sufficient to define the action of the matrices <math>s,t</math> and to show that these actions are invertible and satisfy the relations in the presentation. To this end, it is customary to define matrices <math>S,T</math> called the modular <math>S</math> and <math>T</math> matrices. The entries of the matrices are labeled by pairs <math>([A],[B])\in \mathcal{L}^2</math>. The modular <math>T</math>-matrix is defined to be a diagonal matrix whose <math>([A],[A])</math>-entry is the <math>\theta</math>-symbol <math>\theta_A</math>. The <math>([A],[B])</math> entry of the modular <math>S</math>-matrix is defined in terms of the braiding, as shown below (note that naively this formula defines <math>S_{A,B}</math> as a morphism <math>{\bf 1} \to {\bf 1}</math>, which can then be identified with a complex number since <math>\bf 1</math> is a simple object). center|thumb|Definition of S-matrix entries. The modular <math>S</math> and <math>T</math> matrices do not immediately give a representation of <math>\text{SL}_2(\mathbb{Z})</math> - they only give a projective representation. This can be fixed by shifting <math>S</math> and <math>T</math> by certain scalars. Namely, defining <math>\rho_{\mathcal{C}}(s) = (1/\mathcal{D}) \cdot S</math> and <math>\rho_{\mathcal{C}}(t)= (p_{\mathcal{C}}^-/p_{\mathcal{C}}^+)^{1/6} \cdot T</math> defines a proper modular representation,<ref name="Bakalov-2000"/> where <math display="inline">\mathcal{D}^2=\sum_{[A]\in\mathcal{L}}d_{A}^2</math> is the global quantum dimension of <math>\mathcal{C}</math> and <math>p_{\mathcal{C}}^-, \, \, p_{\mathcal{C}}^+</math> are the Gauss sums associated to <math>\mathcal{C}</math>, where in both these formulas <math>d_{A}</math> are the quantum dimensions of the simple objects. center|thumb|Formula for the Gauss sums of a modular tensor category. center|thumb|Formula for the quantum dimension of a simple object.
== References == <references /> Category:Topological quantum mechanics Category:Category theory Category:Representation theory Category:Representation theory of groups