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

2. A Few Collaborators Z. Wang (Microsoft) R. Stong (Rice)
S.-M. Hong (Indiana)
See

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

4. Topological States: FQHE
1011 electrons/cm2
particle exchange
9 mK
fusion
defects=quasi-particles
Experimentally achieved for FQHE \nu=1/3, others predicted…. 2D critical: in 3D, particle exchange (more or less) trivial.
10 Tesla

5. Topological Computation
Physics
output
measure
apply operators
braid
Emphasize: topological states of matter, braiding unitary, look for sufficiently entangling braiding, measurement is hermitian: projection onto one
Of the eigenstates.
create
particles
initialize

6. Conceptual MTC group G Rep(G) Sn action (Schur-Weyl) Bn action
Modular Category
deform
axioms
Sn action
(Schur-Weyl)
Bn action
(braiding)
Replace a group by a category: move from classical to quantum

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

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

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

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

11. Property Description Properties of MTCs Self-dual Prime D≠ C×C
X  X* for all XD
Prime
D≠ C×C
Property F
(Bn) finite for all XD.
Unitary
End(Xn) Hilbert,
 unitary rep.

12. Analogy/Counting N 1 2 3 4 5 17 230 4783 finite 8 24 36 ?
N-dim’l space groups
Rank N
UMTCs

13. Wang’s Conjecture Conjecture (Z. Wang 2003): #{C MTC: rank(C)=M }  
Analogy: 3D crystals correspond to 230 space groups. Classifying MCs will classify topological states?
True for M4!

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

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

16. Example: C(g2,q,10) Rank 4 MTC with graphs: G1: G2: G3:1 2 3
Not prime, product of
2 copies of Fibbonaci!
G2:
G2 at a 10th root of unity 2 1 3
G3: 3 2 1

17. Prime Self-dual UMTCs, rank4
Theorem: (ER, Stong, Wang) Fusion graphs are: 1 3
Assume self-dual for simplicity. Only 1 non-self-dual of ranks 3 and 4. 2 4

18. 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.

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

20. Property F Hierarchy dim(Xi)2 dim(Xi) dim(Xi)=1 C(so2k+1,4k+2)
D(TY) ? DG
f.d. q-Hopf?
DA, A abelian
C(sln,q,n+1)

21. Prop. F vs. Invariant Complexity
Construct.
Prop. F?
Invariant
Complexity
C(sl2,q,L)
L=2,3,4,6
Arf, H1(M,Z/3Z) Jones
P if L=2,3,4,6
#P-hard else
C(sln,q,L)
L=n+1,4,6
classical
HOMFLYPT
P if L=n+1,4,6
C(so2k+1,q,L)
L=4k+2
H1(M,Z/NZ) Kauffman
P if L=4k+2
Rep(DG)
Yes
#{Hom(1(K),G)}
P if G solvable
? else?

22. 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

23. 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…

24. Physical Feasibility Realizable TQC Bn action Unitary
Positive hermitian form….
i.e. Unitary Modular Category

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

26. Thanks!