Suppose we have two topological phases described with two modular tensor categories C1 and C2. 1. modular category. In the first part of this notes, we will introduce some basic ideas of category theory. ... integral modular categories by gauging nite group symmetries of pointed modular categories. There are a total of 70 UMTCs of rank $\leq 4$ (Note that some authors would have counted as 35 MTCs.) r is called the rank of the category. A pointed modular category is a modular category in which all simple objects are invertible, that is, X ⊗ X ⁎ ≅ 1. tegories (theprototypical example being the category of representations of an al-gebraic group). Such categories have the same fusion rules as Rep (A) where A is a finite abelian group. A cyclic modular category is a pointed modular category with the same fusion rules as Rep (Z n). In particular, Theo-rem 4.5.2 in [T] says that in any such category, Nk ij= D 2 X ‘ Sm i‘ S m j‘ S m k‘ Sm 0‘; where D2 = P i S m2 0i. Let C be a s.w.i. A modular tensor category (MTC) in the sense of V. Turaev determines uniquely a (2+1)-topological quantum field theory (TQFT) [Tu] (a seemingly different definition appeared in [MS1].) My question is: what kind of mathematics does a modular tensor category … Theorem 1.1. The full de nition of a modular tensor category is to long to give here. This theorywas simplified and furtherdevelopedby Deligne and Milne in their classical paper [DelM]. Modular tensor categories are the algebraic data that faithfully encode (2 + 1)-TQFTs [50], and are used to describe anyonic properties of certain quantum systems (see [29] [7] [56] and the references therein). Much of (V1)-(V3) is automatic in any modular tensor category. Download PDF Abstract: We classify all unitary modular tensor categories (UMTCs) of rank $\leq 4$. I jumped into this subject and almost have no background. The extensive and general theory underlying this construction also establishes the operator product expansion for intertwining operators, which correspond to … The exterior tensor product C1 ⊠ C opp 2, where \opp" means reversing the braiding, gives a new modular tensor category. Abstract: We review the construction of braided tensor categories and modular tensor categories from representations of vertex operator algebras, which correspond to chiral algebras in physics. A topological phase is described with a modular tensor category where each anyon corresponds to an irreducible objects. Title: On classification of modular tensor categories. Authors: Eric Rowell, Richard Stong, Zhenghan Wang. I have studied the definition of a modular tensor category. If FPDim(C) = 25 then C is (a) B I I where B is pointed with dimension 2 and I is an Ising modular category or We outline the most important details[Row1]: I The category contains nitely (r) many simple objects (up to isomorphism). This is a primal example of what is known as a fusion category. the modular category of highest weight integrable bg-modules of level ‘is a fusion category. If FPDim(C) = 4q2 then C is (a) The gauging of the particle-hole symmetry of a pointed modular cat-egory of order q2 or (b) A generalized Tambara-Yamagami category 2. So the remaining content of (2) is that S ˜= D 1Sm. Shortly afterwards, the theory of tensor categories(i.e.,monoidalabeliancategories)becameavibrantsubject,withspec- The classification of MTCs is motivated by the application of MTCs to topological quantum computing [F,Ki1,FKW,FLW1,FKLW,P], and by the In this paper, wewillusetheterms(2+1)-TQFT,orjust A modular tensor category is a category with some extra structures, and provides the mathematical foundation of some physical concepts, especially the topological orders. the notion of a modular tensor category (MTC) [50]. 4.If V 2C, then we have a natural structure of a G-representation on V = Hom k(V;k): given f 2V , the action is given by (gf)(v) = f(g 1v) (the inverse is just to make it a left action rather than a right action). (ii)
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