On the Classification of Modular Tensor Categories

On the Classification of Modular Tensor Categories Eric C. Rowell, Texas A&M U. UT Dallas, 19 Dec. 2007

A Few Collaborators See www.arxiv.org

Connections (Turaev) Modular Categories 3-D TQFT (Freedman) definition Top. Quantum Computer (Kitaev) Top. States (anyons)

Topological States: FQHE 1011 electrons/cm2 particle exchange 9 mK fusion defects=quasi-particles 10 Tesla

Topological Computation Computation Physics output measure apply operators braid create particles initialize

Conceptual MTC deform axioms group G Rep(G) Modular Category Snaction (Schur-Weyl) Bnaction (braiding)

Algebraic Constructions quantum group semisimplify gUqg Rep(Uqg) C(g,q,L) Lie algebra qL=-1 twisted quantum double GDGRep(DG) finite group Finite dimensional quasi-Hopf algebra

New from Old CandD MTCs • Direct products: CD • Sub-MTCs: CD( D=CC Müger) • D(S), S spherical

Invariants of MTCs • Rank: # of simple objects Xi • Dimensions: dim(X) = • S-matrix: Sij= • Twists: i= IdX j i i

-1 0 1 1 1 0 0 1 Structure • Fusion rules: Xi Xj= kNijkXk • Bn action on End(Xn): : iIdX cXX IdX • PSL(2,Z) representation:  T = iji S,

Properties of MTCs

Analogy/Counting N-dim’l space groups Rank N UMTCs

Wang’s Conjecture Conjecture (Z. Wang 2003): #{CMTC: rank(C)=M }   True for M4!

Another Analogy Theorem (E. Landau 1903): #{ G : |Rep(G)|=N }   Proof: Exercise (Hint: Use class equation)

Graphs of Self-dual Fusion Rules Simple XimultigraphGi Vertices labeled by 0,…,M-1 Nijk edges Gi= j k Note: for non-self-dual, need arrows…

0 1 2 3 Example: C(g2,q,10) Rank 4 MTC with graphs: G1: Not prime, product of 2 copies of Fibbonaci! G2: 0 2 1 3 G3: 0 3 2 1

Prime Self-dual UMTCs, rank4 Theorem: (ER, Stong, Wang) Fusion graphs are: 1 3 4 2

Property F A modular category D has property F if the subgroup: F(Bn)  GL(End(Vn)) is finite for all objects V in D.

Property F Conjecture Conjecture: (ER) D a UMTC has property Fdim(Xi)2  for all simple Xi D

Property F Hierarchy

Prop. F vs. Invariant Complexity

Exotic MTCs • Observation: All MTCs rank4 have quantum group realizations. • Conjecture (Moore & others): All MTCs “come from” quantum groups. • Probably false: counterexamples called EXOTIC

Two Potentially Exotic MTCs • D(E) a rank 10 category: “doubled ½E6” • D(H) a rank 12 category: “doubled ½Haagerup” related to subfactor of index ½(5+ 13) • Analogous to finite simple sporadic groups • Difficult to make precise & prove…

Physical Feasibility Realizable TQC Bn action Unitary i.e. Unitary Modular Category

Two Examples Unitary, for some q Never Unitary Lie type G2 q21=-1 Half ofLie type B2 q9=-1 Known for quantum group categories

Thanks!

On the Classification of Modular Tensor Categories