Properties of Modular Categories and their Computation Consequences

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Properties of Modular Categoriesand
theirComputation Consequences
Eric C. Rowell, Texas AM U. UT Tyler, 21 Sept.
2007
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A Few Collaborators
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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

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Motivation
(Turaev)
Modular Categories
3-D TQFT
(Freedman)
definition
Top. Quantum Computer
(Kitaev)
Top. States (anyons)
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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
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Topological States FQHE
1011 electrons/cm2
particle exchange
9 mK
fusion
defectsquasi-particles
10 Tesla
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Topological Computation
Computation
Physics
output
measure
apply operators
braid
create particles
initialize
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MC Toy Model Rep(G)

• Irreps V1C, V2,,Vk
• Sums V?W, tens. prod. V?W, duals W
• Semisimple each W?imiVi
• Rep Sn EndG(V ?n)

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Modular Categories
deform axioms
group G
Rep(G)
Modular Category
Sn action (Schur-Weyl)
Bn action (braiding)
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Braid Group Bn Quantum Sn
Generated by
bi
Multiplication is by concatenation

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Modular Category

• Simple objects X0C,X1,XM-1
• Rep(G) properties
• Rep. Bn End(X?n) (braid group action)
• Non-degeneracy S-matrix invertible

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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)

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Partial Dictionary
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In Pictures
Simple objects Xi
Quasi-particles
Unit object X0
Vacuum
Particle exchange
Braiding
Create
X0 Xi ?Xi
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Two Hopf Algebra Constructions
quantum group
semisimplify
g Uqg Rep(Uqg)
F(g,q,L)
Lie algebra
qL-1
twisted quantum double
G D?G Rep(D?G)
finite group
Finite dimensional quasi-Hopf algebra
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Other Constructions

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

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Groethendieck Semiring

• Assume self-dual XX. For a MC D
• Xi ?Xj ?k Nijk Xk (fusion rules)
• Semiring Gr(D)(Ob(D),?,?)
• Encoded in matrices (Ni)jk Nijk

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Generalized Ocneanu Rigidity

• Theorem (see Etingof, Nikshych, Ostrik)
• For fixed fusion rules Nijk there are
  finitely many inequivalent modular categories
  with these fusion rules.

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Graphs of Fusion Rules

• Simple Xi multigraph Gi
• Vertices labeled by 0,,M-1

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Example F(g2,q,10)

• Rank 4 MC with fusion rules
• N111N113N123N222N233N3331
• N112N122 N2230

G1
Tensor Decomposable, 2 copies of Fibbonaci!
G2
0
2
1
3
G3
2
0
3
1
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More Graphs
Lie type B2 q9-1
D(S3)
Lie type B3 q12-1
Extra colors for different objects
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Classify Modular Categories
Rank of an MC of simple objects
Conjecture (Z. Wang 2003) The set MCs of rank
M is finite.
Verified for M1, 2 Ostrik, 3 and 4 ER,
Stong, Wang
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Analogy

• Theorem (E. Landau 1903)
• The set G Rep(G)N is finite.
• Proof Exercise (Hint Use class equation)

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Classification by Graphs

• Theorem (ER, Stong, Wang)
• Indecomposable, self-dual MCs of ranklt5 are
  determined and classified by

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Physical Feasibility
Realizable TQC
Bn action Unitary
i.e. Unitary Modular Category
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Two Examples
Unitary, for some q
Never Unitary, for any q
Lie type G2 q21-1
even part for Lie type B2 q9-1
For quantum group categories, can be complicated
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General Problem

• G discrete,
• ?(G)? U(N) unitary irrep.
• What is the closure of ?(G)? (modulo center)
• SU(N)
• Finite group
• SO(N), E7, other compact groups
• Key example ?i(Bn) ? U(Hom(X?n,Xi))

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Braid Group Reps.

• Let X be any object in a unitary MC
• Bn acts on Hilbert spaces End(X ?n)
• as unitary operators F(b), b a braid.
• The gate set F(bi), bi braid generators.

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Computational Power
Ui universal if promotions of Ui ?
U(kn)
qubits k2
Topological Quantum Computer universal
Fi(Bn) dense in SU(Ni)
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Dense Image Paradigm
Class P-hard Link invariant
F(Bn) dense
Universal Top. Quantum Computer

Eg. FQHE at ?12/5?
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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.

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Example 1

• Theorem
• F(sl2, q , L) has property F if and only if
  L2,3,4 or 6.
• (Jones 86, Freedman-Larsen-Wang 02)

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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.

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

• For modular category D define
• dim(X) TrD(IdX) ? R
• dim(D)?i(dim(Xi))2

IdX
dimEnd(X?n) ? dim(X)n
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Examples

• In Rep(D?G) all dim(V)??
• In F(sl2, q , L),
• dim(Xi)
• For L4 or 6, dim(Xi) ???L/2,
• for L2 or 3, dim(Xi) ??

sin((i1)?/L) sin(?/L)
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Property F Conjecture

• Conjecture (ER)
• Let D be a modular category. Then D has property
  F ?dim(C)??.
• Equivalent to dim(Xi)2 ?? for all simple Xi.

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Observations

• Wangs Conjecture is true for modular categories
  with dim(D)?? (Etingof,Nikshych,Ostrik)
• My Conjecture would imply Wangs for modular
  categories with property F.

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Current Problems

• Construct more modular categories (explicitly!)
• Prove Wangs Conj. for more cases
• Explore Density Paradigm
• Explore Finite Image Paradigm
• Prove Property F Conjecture

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• Thanks!

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SO(2k1) at L2(2k1)
k1
k2
k3
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Sketch of Proof (Mlt5)

• Show 1 Gal(K/Q) SM
• Use Gal(K/Q) constraints to determine
  (S, Ni)
  
• For each S find T rep. of SL(2,Z)
• 4. Find realizations.

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Braid Group Bn
Generated by
i1,,n-1
bi
Multiplication is by concatenation

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Some Number Theory

• Let pi(x) det(Ni - xI)
• and K
  Split(pi,Q).
• Study Gal(K/Q) always abelian!
• Nijk integers, Sij algebraic, constraints
  polynomials.

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Our Approach

• Study Gr(D) and reps. of SL(2,Z)
• Ocneanu Rigidity
• MCs Ni
• Verlinde Formula
• Ni determined by S-matrix

Finite-to-one
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Type G2 at L3s
L18
equivalent to
L21
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Half of SO(5) at odd L
L9
L11