Properties of Modular Categories and their Computation Consequences Eric C. Rowell, Texas A&M U. UT Tyler, 21 Sept. 2007

A Few Collaborators Z. Wang (Microsoft) M. Larsen (Indiana) S. Witherspoon (TAMU) P. Etingof (MIT) Y. Zhang (Utah, Physics)

Publications/Preprints [Franko,ER,Wang] JKTR 15, no. 4, 2006 [Larsen,ER,Wang] IMRN 2005, no. 64 [ER] Contemp. Math. 413, 2006 [Larsen, ER] MP Camb. Phil. Soc. [ER] Math. Z 250, no. 4, 2005 [Etingof,ER,Witherspoon] preprint [Zhang,ER,et al] preprint

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

What is a Topological Phase? [Das Sarma, Freedman, Nayak, Simon, Stern] “…a system is in a topological phase if its low-energy effective field theory is a topological quantum field theory…” Working definition…

Topological States: FQHE electrons/cm 2 10 Tesla defects=quasi-particles particle exchange fusion 9 mK

Topological Computation initialize create particles apply operators braid outputmeasure ComputationPhysics

MC Toy Model: Rep(G) CIrreps: {V 1 =C, V 2,…,V k } Sums V  W, tens. prod. V  W, duals W* Semisimple: each W=  i m i V i Rep: S n End G (V  n )

Modular Categories group GRep(G) Modular Category deform axioms S n action (Schur-Weyl) B n action (braiding) 9

Braid Group B n “Quantum S n ” Generated by: 1 i i+1 n Multiplication is by concatenation: = bi =bi =

Modular Category CSimple objects {X 0 =C,X 1,…X M-1 } + Rep(G) properties Rep. B n End(X  n ) (braid group action) Non-degeneracy: S-matrix invertible

Uses of Modular Categories Link, knot and 3-manifold invariants Representations of mapping class groups Study of (special) Hopf algebras “Symmetries” of topological states of matter. (analogy: 3D crystals and space groups)

Partial Dictionary Simple objects X i Indecomposable particle types B n -actionParticle exchange CX0 =CCX0 =CVacuum state X i *Antiparticle X 0 X i  X i * Creation

In Pictures Simple objects X i Quasi-particles Braiding Particle exchange Unit object X 0 Vacuum X 0 X i  X i * Create

Two Hopf Algebra Constructions g U q g Rep(U q g) F(g,q,L) Lie algebra quantum group q L =-1 semisimplify G G D  G Rep(D  G) finite group twisted quantum double Finite dimensional quasi-Hopf algebra

Other Constructions Direct Products of Modular Categories Doubles of Spherical Categories Minimal Models, RCFT, VOAs, affine Kac- Moody, Temperley-Lieb, and von Neumann algebras…

Groethendieck Semiring D Assume self-dual: X=X*. For a MC D: X i  X j =  k N ijk X k (fusion rules) DDSemiring Gr(D):=(Ob(D), ,  ) Encoded in matrices (N i ) jk = N ijk 17

Generalized Ocneanu Rigidity Theorem: (see [Etingof, Nikshych, Ostrik]) For fixed fusion rules { N ijk } there are finitely many inequivalent modular categories with these fusion rules.

Simple X i multigraph G i : Vertices labeled by 0,…,M-1 Graphs of Fusion Rules N ijk edges j k

Example: F(g 2,q,10) Rank 4 MC with fusion rules: N 111 =N 113 =N 123 =N 222 =N 233 =N 333 =1; N 112 =N 122 = N 223 =0 G1:G1: 0123 G2:G2: 0213 G3:G3: Tensor Decomposable, 2 copies of Fibbonaci!

More Graphs D(S 3 ) Lie type B 2 q 9 =-1 Lie type B 3 q 12 =-1 Extra colors for different objects…

Classify Modular Categories Verified for: M=1, 2 [Ostrik], 3 and 4 [ER, Stong, Wang] Conjecture (Z. Wang 2003): The set { MCs of rank M } is finite. Rank of an MC: # of simple objects

Analogy Theorem (E. Landau 1903): The set { G : |Rep(G)|=N } is finite. Proof: Exercise (Hint: Use class equation)

Classification by Graphs Theorem: (ER, Stong, Wang) Indecomposable, self-dual MCs of rank<5 are determined and classified by:

Physical Feasibility Realizable TQC B n action Unitary i.e. Unitary Modular Category 25

Two Examples Unitary, for some q Never Unitary, for any q Lie type G 2 q 21 =-1 “even part” for Lie type B 2 q 9 =-1 For quantum group categories, can be complicated…

General Problem G discrete,  (G)  U(N) unitary irrep. What is the closure of  (G)? (modulo center) SU(N) Finite group SO(N), E 7, other compact groups… Key example:  i ( B n )  U(Hom(X  n,X i ))

Braid Group Reps. Let X be any object in a unitary MC B n acts on Hilbert spaces End(X  n ) as unitary operators:  a braid. The gate set: {  b i )}, b i braid generators.

Computational Power {U i } universal if {promotions of U i }  U(k n ) Topological Quantum Computer universal  i (B n ) dense in SU(N i ) qubits: k=2

Dense Image Paradigm Universal Top. Quantum Computer Class #P-hard Link invariant  (B n ) dense Eg. FQHE at =12/5?

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

Example 1 Theorem: F(sl 2, q, L) has property F if and only if L=2,3,4 or 6. (Jones ‘86, Freedman-Larsen-Wang ‘02)

Example 2 Theorem: [Etingof,ER,Witherspoon] Rep(D  G) has property F for any finite group G and 3-cocycle . More generally, true for braided group- theoretical fusion categories. 33

Finite Group Paradigm Non-Universal Top. Quantum Computer Modular Cat. with prop. F Abelian anyons, FQHE at =5/2? Poly-time Link invariant quantum error correction?

Categorical Dimensions D For modular category D define D R dim(X) = Tr D (Id X ) =  R D dim(D)=  i (dim(X i )) 2 Id X dim[End(X  n )]  [dim(X)] n

Examples In Rep(D  G) all dim(V)  In F(sl 2, q, L), dim(X i ) =  For L=4 or 6, dim(X i )  [  L/2],  for L=2 or 3, dim(X i )  sin((i+1)  /L) sin(  /L)

Property F Conjecture Conjecture: (ER) DD C  Let D be a modular category. Then D has property F  dim(C) .  Equivalent to: dim(X i ) 2  for all simple X i.

Observations D Wang’s Conjecture is true for modular categories with dim(D)  (Etingof,Nikshych,Ostrik) My Conjecture would imply Wang’s for modular categories with property F.

Current Problems Construct more modular categories (explicitly!) Prove Wang’s Conj. for more cases Explore Density Paradigm Explore Finite Image Paradigm Prove Property F Conjecture

Thanks!

SO(2k+1) at L=2(2k+1) k=2 k=3 k=1

Sketch of Proof (M<5) Q 1.Show: 1 Gal(K/Q) S M Q 2.Use Gal(K/Q) + constraints to determine (S, {N i }) Z 3.For each S find T rep. of SL(2,Z) 4. Find realizations.

Braid Group B n Generated by: 1 i i+1 n Multiplication is by concatenation: = bi =bi = i=1,…,n-1

Some Number Theory Let p i (x) = det(N i - xI) Q and K = Split({p i },Q). QStudy Gal(K/Q): always abelian! N ijk integers, S ij algebraic, constraints polynomials.

Our Approach DStudy Gr(D) and reps. of SL(2,Z) Ocneanu Rigidity: MCs {N i } Verlinde Formula: {N i } determined by S-matrix Finite-to-one

Type G 2 at L=3s L=21 L=18 equivalent to

Half of SO(5) at odd L L=11 L=9