modular categories and applications irowell/rowellusa.pdfvthe category of f.d. c-vector spaces. 1 =...

ConnectionsWhat is a Modular Category?

Problem IProblem II

Modular Categoriesand Applications I

Eric Rowell

Texas A&M University

U. South Alabama, November 2009

Eric Rowell Modular Categories and Applications I


Problem IProblem II

Outline

1 ConnectionsTopological Quantum Computation

2 What is a Modular Category?Fusion CategoriesRibbon and Modular CategoriesFusion Rules and Dimensions

3 Problem IEnumerative QuestionsResults

4 Problem IIStructural QuestionsEmpirical Evidence



Problem IProblem II

Topological Quantum Computation

Main Collaborators

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.



Problem IProblem II


Mathematical Connections

Modular categories are related to:

Quantum invariants of links and 3-manifolds

Hopf algebras

subfactor inclusions (type II1 finite depth)

Kac-Moody algebras



Problem IProblem II


Quantum Computing: Overview

Top. Quantum ComputerTop. Phases

(i.e. anyons)

(2+1)D TQFT

Link Invariants

(Turaev)

Modular Categories(unitary)

(definition)

(Kitaev)

(Freedman)



Problem IProblem II


Topological States

Definition (Das Sarma, et al)

a system is in a topological phase if its low-energy effective fieldtheory is a topological quantum field theory.

Algebraic part: modular category.



Problem IProblem II


Fractional Quantum Hall Effect

1011 electrons/cm2

10 Tesla

defects=quasi-particles

9 mK

GaAs



Problem IProblem II


Topological Quantum Computation: Schematic

initialize create

particles

apply gates particle

exchange

output measure

Computation Physics

vacuum



Problem IProblem II

Fusion CategoriesRibbon and Modular CategoriesFusion Rules and Dimensions

Some Axioms

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 {X0 := 1,X1, . . . ,Xm−1}semisimple: Y ∼=

⊕i ciXi

rigid: duals X ∗, bX : 1→ X ⊗ X ∗, dX : X ∗ ⊗ X → 1

compatibility...



Problem IProblem II


First Example

Example

V the category of f.d. C-vector spaces.

1 = C1 is the only simple object: rank 1

V ∗ ⊗ VdV→ 1: dV (f ⊗ v) = f (v)

1bV→ V ⊗ V ∗: bV (x) = x

∑j vj ⊗ v j



Problem IProblem II


Braiding and Twists

Definition

A braided fusion (BF) category has isomorphisms:

cX ,Y : X ⊗ Y → Y ⊗ X

satisfying, e.g.

cX ,Y⊗Z = (IdY ⊗cX ,Z )(cX ,Y ⊗ IdZ )

Definition

A ribbon category has compatible ∗ and cX ,Y . Encoded in “twists”θX : X → X inducing V ∼= V ∗∗.



Problem IProblem II


The Braid Group

Definition

Bn has generators σi , i = 1, . . . , n − 1 satisfying:

σiσi+1σi = σi+1σiσi+1

σiσj = σjσi if |i − j | > 1



Problem IProblem II


Braid Group Representations

Fact

Braiding on C induces:

ΨX : CBn → End(X⊗n)

σi → Id⊗i−1X ⊗cX ,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 Vk = Hom(X⊗n,Xk) become Bn reps.



Problem IProblem II


Modular Categories

Ribbon categories have:

Consistent graphical calculus: braiding, twists, duality mapsrepresented by pieces of knot projections

canonical trace: trC : End(X )→ C = End(1)

trC(IdX ) := dim(X ) ∈ R× (generally not in Z≥0).

Invariants of links: given L with each component labelled byan object X find a braid β ∈ Bn s.t. β̂ = L thenKX (L) := trC(ΨX (β)).

Definition

Let Si ,j = trC(cXj ,XicXi ,Xj

), 0 ≤ i , j ≤ m − 1. C is modular ifdet(S) 6= 0.



Problem IProblem II


Grothendieck Semiring

Definition

Gr(C) := (Obj(C),⊕,⊗, 1) a unital based ring, with basis {Xi}i .

Define matrices(Ni )k,j := dim Hom(Xi ⊗ Xj ,Xk)

So: Xi ⊗ Xj =⊕m−1

k=0 Nki ,jXk

Rep. ϕ : Gr(C)→ Matm(Z)

ϕ(Xi ) = Ni

Respects duals: ϕ(X ∗) = ϕ(X )T (self-dual ⇒ symmetric)

If C is braided, Gr(C) is commutative



Problem IProblem II


Frobenius-Perron Dimensions

Definition

FPdim(X ) is the largest eigenvalue of ϕ(X )

FPdim(C) :=∑m−1

i=0 FPdim(Xi )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.



Problem IProblem II


Integrality

Definition

C is

integral if FPdim(X ) ∈ Z for all X

weakly integral if FPdim(C) ∈ Z



Problem IProblem II


Not Quite an Example

Example

Let G be a finite group. Rep(G ) category of f.d. C reps. of G isribbon (not modular).

FPdim(V ) = dimC(V ).

Gr(Rep(G )) is the representation ring.

Let {Vi} be the irreps. xi := dim(Vi ), V0 = C trivial rep.

S =

1 x1 · · · xm

x1. . . · · · x1xm

...... xixj

...xm xmx1 · · · x2

m

det(S) = 0 (rank 1 in fact).

twists: θi = 1 for all i .



Problem IProblem II


Some Sources of Modular Categories

Example

Quantum group U = Uqg with q = eπ i /`.

subcategory of tilting modules T ⊂ Rep(U)

quotient C(g, `) of T by negligible morphisms is (often)modular.

Example

G a finite group, ω a 3-cocyle

semisimple quasi-triangular quasi-Hopf algebra DωG

Rep(DωG ) is modular, and integral.

Conjecture (Folk)

All modular categories come from these 2 families.



Problem IProblem II


Everyone’s Favorite Example

Example (Fibbonaci)

quantum group category C(g2, 15)

Rank 2: simple objects 1, X

FPdim(X ) = τ = 1+√

52

S =

(1 ττ −1

)θ0 = 1, θX = e4π i /5

X ⊗ X = 1⊕ X



Problem IProblem II


Dictionary

Categories Physics

Simple objects Xi Indecomposable particle types ti1 vacuum type

dual objects X ∗ Antiparticles

End(X ) State space

cX ,Y particle exchange

det(S) 6= 0 particle types distinguishable

Xi ⊗ Xj =⊕

k Nki ,jXk fusion channels ti ? tj → tk

Nki,j dim(Xk )

dim(Xi ) dim(Xj )Prob(ti ? tj → tk) ?



Problem IProblem II


Landscape of Modular Categories

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.



Problem IProblem II

Enumerative QuestionsResults

General Problem

Problem

Classify all modular categories, up to equivalence.

For physicists: a classification of algebraic models for FQHliquids

For topologists: a classification of (most? all?) quantum linkinvariants

For algebraists: a classification of (all?) factorizable Hopfalgebras.

would include a classification of finite groups...too ambitious!



Problem IProblem II


Wang’s Conjecture

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(Xi ) and Nki ,j) bounded.

m = 2: true for fusion cats., m = 3: true for ribbon cats.



Problem IProblem II


Reduction

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 somemodular category C.

Definition

C and D are Grothendieck equivalent if Gr(C) ∼= Gr(D).



Problem IProblem II


More Modest Goal

Problem

Classify modular categories:

that are pseudo-unitary (so dim(Xi ) ≥ 1)

up to Grothendieck equivalence

for small ranks m (say ≤ 12).



Problem IProblem II


Fusion Graphs

For each simple Xi ∈ C define a graph Gi :Vertices: simple objects Xj

(directed) Edges: Nki ,j edges from Xj to Xk .

undirected if Nki ,j = N j

i ,k (e.g. if self-dual)

Nijk

j kGi



Problem IProblem II


Rank ≤ 4 Classification

[R,Stong,Wang ’09]. Determined (up to Gr. semiring) by a singlefusion graph:

1

2

4

3



Problem IProblem II


Non-self-dual: Rank 5

Assume X 6∼= X ∗ for at least one X .Determined by:



Problem IProblem II


Integral: Rank ≤ 6

Proposition

Assume C is an integral modular category of rank ≤ 6. Then C iseither:

1 pointed (FPdim(Xi ) = 1 for simple Xi ) or

2 Gr(C) ∼= Gr(Rep(DG )) (FPdim(C) = 36).

Follows from [Naidu,R] and computer search.



Problem IProblem II

Structural QuestionsEmpirical Evidence

Braid Group Images

Let C be any braided fusion category.

Question

Given X and n, what is ΨX (Bn)?

(F) Is it finite or infinite?

(U) If unitary and infinite, what is ΨX (Bn)?

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 ΨX (Bn) ⊃∏

k SU(Vk), Vk irred. subreps.

(M): n ≥ 5 solvable ΨX (Bn) implies abelian.



Problem IProblem II


Property F

Definition

Say C has property F if |ΨX (Bn)| <∞ for all X and n.



Problem IProblem II


Examples

Examples

C(sl2, 4) C(g2, 15) Rep(DS3)

rank 3 2 8

FPdim(Xi )√

2 1+√

52 2, 3

Prop. F? Yes No Yes



Problem IProblem II


Property F Conjecture

Conjecture

A braided fusion category C has property F if and only if it isweakly integral (FPdim(C) ∈ Z).

Clear for pointed categories (FPdim(Xi ) = 1)

E.g.: does Rep(H) have prop. F for H f.d., s.s., quasi-4,quasi-Hopf alg.?



Problem IProblem II


Relationship to Link Invariants

Recall: C ribbon, X ∈ C, L a link: get invariant KX (L).

Question

Is computing (randomized approximation) KX (L) easy: FPRASableor hard: #P-hard, not FPRASable, assuming P 6= NP?

Appears to coincide with: Is ΨX (Bn) finite or infinite?Related to topological quantum computers: weak or powerful?



Problem IProblem II


Lie Types A and C

Proposition (Jones ’86, Freedman,Larsen,Wang ’02)

C(slk , `) has property F if and only if ` ∈ {2, 3, 4, 6}.

Proposition (Jones ’89, Larsen,R,Wang ’05)

C(sp2k , `) has property F if and only if ` = 10 and k = 2.

Only weakly integral in these cases

(FPdim(Xi ) ∈ {1,√

2,√

3, 2,√

5, 3}).



Problem IProblem II


Finite Group Doubles

Proposition (Etingof,R,Witherspoon)

Rep(DωG ) has property F for any ω,G .

Recall: Rep(DωG ) is integral.



Problem IProblem II


Integral, Low-dimension

Proposition (Naidu,R)

Suppose C is an integral modular category with FPdim(C) < 36.Then C has property F .



Problem IProblem II


Intermission...

Thank you!


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