Modular Categories and Applications I

Transcription

1 I Modular Categories and Applications I Texas A&M University U. South Alabama, November 2009

2 Outline I 1 Topological Quantum Computation 2 Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions 3 Enumerative Questions Results 4 I Structural Questions Empirical Evidence

3 Main Collaborators I Topological Quantum Computation Zhenghan Wang, Microsoft Station Q Michael Larsen, Indiana U. Seung-moon Hong, U. Toledo (Ohio) Deepak Naidu, Texas A&M U. Sarah Witherspoon, Texas A&M U.

4 I Mathematical Topological Quantum Computation Modular categories are related to: Quantum invariants of links and 3-manifolds Hopf algebras subfactor inclusions (type II 1 finite depth) Kac-Moody algebras

5 I Quantum Computing: Overview Topological Quantum Computation Modular Categories (unitary) (Turaev) (2+1)D TQFT Link Invariants (definition) (Freedman) Top. Phases (i.e. anyons) (Kitaev) Top. Quantum Computer

6 Topological States I Topological Quantum Computation Definition (Das Sarma, et al) a system is in a topological phase if its low-energy effective field theory is a topological quantum field theory. Algebraic part: modular category.

7 I Fractional Quantum Hall Effect Topological Quantum Computation electrons/cm 2 GaAs 9 mk defects=quasi-particles 10 Tesla

8 I Topological Quantum Computation Topological Quantum Computation: Schematic Computation Physics output measure apply gates particle exchange initialize create particles vacuum

9 Some Axioms I Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Definition A fusion category is a monoidal category (C,, 1) that is: C-linear: Hom(X, Y ) f.d. vector space abelian: X Y finite rank: simple objects {X 0 := 1, X 1,..., X m 1 } semisimple: Y = i c ix i rigid: duals X, b X : 1 X X, d X : X X 1 compatibility...

10 First Example I Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Example V the category of f.d. C-vector spaces. 1 = C 1 is the only simple object: rank 1 V V d V 1: d V (f v) = f (v) 1 b V V V : b V (x) = x j v j v j

11 Braiding and Twists I Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Definition A braided fusion (BF) category has isomorphisms: c X,Y : X Y Y X satisfying, e.g. c X,Y Z = (Id Y c X,Z )(c X,Y Id Z ) Definition A ribbon category has compatible and c X,Y. Encoded in twists θ X : X X inducing V = V.

12 The Braid Group I Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Definition B n has generators σ i, i = 1,..., n 1 satisfying: σ i σ i+1 σ i = σ i+1 σ i σ i+1 σ i σ j = σ j σ i if i j > 1

13 I Braid Group Representations Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Fact Braiding on C induces: Ψ X : CB n End(X n ) σ i Id i 1 X c X,X Id n i 1 X X is not always a vector space End(X n ) semisimple algebra (multi-matrix). simple End(X n )-mods V k = Hom(X n, X k ) become B n reps.

14 Modular Categories I Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Ribbon categories have: Consistent graphical calculus: braiding, twists, duality maps represented by pieces of knot projections canonical trace: tr C : End(X ) C = End(1) tr C (Id X ) := dim(x ) R (generally not in Z 0 ). Invariants of links: given L with each component labelled by an object X find a braid β B n s.t. ˆβ = L then K X (L) := tr C (Ψ X (β)). Definition Let S i,j = tr C (c Xj,X i c Xi,X j ), 0 i, j m 1. C is modular if det(s) 0.

15 I Grothendieck Semiring Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Definition Gr(C) := (Obj(C),,, 1) a unital based ring, with basis {X i } i. Define matrices (N i ) k,j := dim Hom(X i X j, X k ) Rep. ϕ : Gr(C) Mat m (Z) So: X i X j = m 1 k=0 Nk i,j X k ϕ(x i ) = N i Respects duals: ϕ(x ) = ϕ(x ) T (self-dual symmetric) If C is braided, Gr(C) is commutative

16 I Frobenius-Perron Dimensions Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Definition FPdim(X ) is the largest eigenvalue of ϕ(x ) FPdim(C) := m 1 i=0 FPdim(X i) 2 (a) FPdim(X ) > 0 (b) FPdim : Gr(C) C is a unital homomorphism (c) FPdim is unique with (a) and (b). If FPdim(X ) = dim(x ) for all X, C is pseudo-unitary.

17 Integrality I Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Definition C is integral if FPdim(X ) Z for all X weakly integral if FPdim(C) Z

18 I Not Quite an Example Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Example Let G be a finite group. Rep(G) category of f.d. C reps. of G is ribbon (not modular). FPdim(V ) = dim C (V ). Gr(Rep(G)) is the representation ring. Let {V i } be the irreps. x i := dim(v i ), V 0 = C trivial rep. 1 x 1 x m. S = x.. 1 x1 x m det(s) = 0 (rank 1 in fact)... x i x j. x m x m x 1 xm 2 twists: θ i = 1 for all i.

19 I Some Sources of Modular Categories Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Example Quantum group U = U q g with q = e π i /l. Example subcategory of tilting modules T Rep(U) quotient C(g, l) of T by negligible morphisms is (often) modular. G a finite group, ω a 3-cocyle semisimple quasi-triangular quasi-hopf algebra D ω G Rep(D ω G) is modular, and integral. Conjecture (Folk) All modular categories come from these 2 families.

20 I Everyone s Favorite Example Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Example (Fibbonaci) quantum group category C(g 2, 15) Rank 2: simple objects 1, X FPdim(X ) = τ = ( ) 1 τ S = τ 1 θ 0 = 1, θ X = e4π i /5 X X = 1 X

21 Dictionary I Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Categories Physics Simple objects X i Indecomposable particle types t i 1 vacuum type dual objects X Antiparticles End(X ) State space c X,Y particle exchange det(s) 0 particle types distinguishable X i X j = k Nk i,j X k fusion channels t i t j t k Ni,j k dim(x k) dim(x i ) dim(x j ) Prob(t i t j t k )?

22 I Landscape of Modular Categories Fusion Categories Ribbon and Modular Categories Fusion Rules and Dimensions Question Are there exotic (not quantum group or Hopf algebra) modular categories? How many modular categories are there? Can we characterize the braid group images? We focus on 2 and 3.

23 General Problem I Enumerative Questions Results Problem Classify all modular categories, up to equivalence. For physicists: a classification of algebraic models for FQH liquids For topologists: a classification of (most? all?) quantum link invariants For algebraists: a classification of (all?) factorizable Hopf algebras. would include a classification of finite groups...too ambitious!

24 Wang s Conjecture I Enumerative Questions Results Conjecture Fix m 1. There are finitely many modular categories of rank m. Only verified in the following situations: m 4 C is weakly integral. FPdim(C) (hence FPdim(X i ) and Ni,j k ) bounded. m = 2: true for fusion cats., m = 3: true for ribbon cats.

25 Reduction I Enumerative Questions Results Theorem (Ocneanu Rigidity) Fix a unital based ring R. There are at most finitely many (fusion, ribbon) modular categories C with Gr(C) = R. Up to finite ambiguity, enough consider: Problem Classify all unital based rings R such that R = Gr(C) for some modular category C. Definition C and D are Grothendieck equivalent if Gr(C) = Gr(D).

26 More Modest Goal I Enumerative Questions Results Problem Classify modular categories: that are pseudo-unitary (so dim(x i ) 1) up to Grothendieck equivalence for small ranks m (say 12).

27 Fusion Graphs I Enumerative Questions Results For each simple X i C define a graph G i : Vertices: simple objects X j (directed) Edges: N k i,j edges from X j to X k. undirected if N k i,j = Nj i,k (e.g. if self-dual) N ij k G i j k

28 I Rank 4 Classification Enumerative Questions Results [R,Stong,Wang 09]. Determined (up to Gr. semiring) by a single fusion graph:

29 I Non-self-dual: Rank 5 Enumerative Questions Results Assume X = X for at least one X. Determined by:

30 Integral: Rank 6 I Enumerative Questions Results Proposition Assume C is an integral modular category of rank 6. Then C is either: 1 pointed (FPdim(X i ) = 1 for simple X i ) or 2 Gr(C) = Gr(Rep(DG)) (FPdim(C) = 36). Follows from [Naidu,R] and computer search.

31 Braid Group Images I Structural Questions Empirical Evidence Let C be any braided fusion category. Question Given X and n, what is Ψ X (B n )? (F) Is it finite or infinite? (U) If unitary and infinite, what is Ψ X (B n )? see [Freedman,Larsen,Wang 02], [Larsen,R,Wang 05] (M) If finite, what are minimal quotients? see [Larsen,R. 08 AGT] For example: (U): typically Ψ X (B n ) k SU(V k), V k irred. subreps. (M): n 5 solvable Ψ X (B n ) implies abelian.

32 Property F I Structural Questions Empirical Evidence Definition Say C has property F if Ψ X (B n ) < for all X and n.

33 Examples I Structural Questions Empirical Evidence Examples C(sl 2, 4) C(g 2, 15) Rep(DS 3 ) rank FPdim(X i ) , 3 Prop. F? Yes No Yes

34 I Property F Conjecture Structural Questions Empirical Evidence Conjecture A braided fusion category C has property F if and only if it is weakly integral (FPdim(C) Z). Clear for pointed categories (FPdim(X i ) = 1) E.g.: does Rep(H) have prop. F for H f.d., s.s., quasi-, quasi-hopf alg.?

35 I Relationship to Link Invariants Structural Questions Empirical Evidence Recall: C ribbon, X C, L a link: get invariant K X (L). Question Is computing (randomized approximation) K X (L) easy: FPRASable or hard: #P-hard, not FPRASable, assuming P NP? Appears to coincide with: Is Ψ X (B n ) finite or infinite? Related to topological quantum computers: weak or powerful?

36 Lie Types A and C I Structural Questions Empirical Evidence Proposition (Jones 86, Freedman,Larsen,Wang 02) C(sl k, l) has property F if and only if l {2, 3, 4, 6}. Proposition (Jones 89, Larsen,R,Wang 05) C(sp 2k, l) has property F if and only if l = 10 and k = 2. Only weakly integral in these cases (FPdim(X i ) {1, 2, 3, 2, 5, 3}).

37 I Finite Group Doubles Structural Questions Empirical Evidence Proposition (Etingof,R,Witherspoon) Rep(D ω G) has property F for any ω, G. Recall: Rep(D ω G) is integral.

38 I Integral, Low-dimension Structural Questions Empirical Evidence Proposition (Naidu,R) Suppose C is an integral modular category with FPdim(C) < 36. Then C has property F.

39 I Structural Questions Empirical Evidence Intermission... Thank you!