a finiteness property for braided fusion categoriesrowell/rowellcordobanopause.pdf · the...
TRANSCRIPT
![Page 1: A Finiteness Property for Braided Fusion Categoriesrowell/RowellCordobanopause.pdf · The Conjecture Empirical Evidence Speculations and Connections A Finiteness Property for Braided](https://reader035.vdocuments.site/reader035/viewer/2022071212/602451d8a5c4683ec7342026/html5/thumbnails/1.jpg)
The ConjectureEmpirical Evidence
Speculations and Connections
A Finiteness Propertyfor
Braided Fusion Categories
Eric Rowell
Texas A&M University
La Falda, Argentina 2009
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Outline
1 The ConjectureBraided Fusion CategoriesDimensions and Braid Representations
2 Empirical EvidenceQuantum GroupsGroup Theoretical Categories
3 Speculations and ConnectionsWeakly Group Theoretical CategoriesRelated Questions
Eric Rowell A Finiteness Property for Braided Fusion Categories
![Page 3: A Finiteness Property for Braided Fusion Categoriesrowell/RowellCordobanopause.pdf · The Conjecture Empirical Evidence Speculations and Connections A Finiteness Property for Braided](https://reader035.vdocuments.site/reader035/viewer/2022071212/602451d8a5c4683ec7342026/html5/thumbnails/3.jpg)
The ConjectureEmpirical Evidence
Speculations and Connections
Outline
1 The ConjectureBraided Fusion CategoriesDimensions and Braid Representations
2 Empirical EvidenceQuantum GroupsGroup Theoretical Categories
3 Speculations and ConnectionsWeakly Group Theoretical CategoriesRelated Questions
Eric Rowell A Finiteness Property for Braided Fusion Categories
![Page 4: A Finiteness Property for Braided Fusion Categoriesrowell/RowellCordobanopause.pdf · The Conjecture Empirical Evidence Speculations and Connections A Finiteness Property for Braided](https://reader035.vdocuments.site/reader035/viewer/2022071212/602451d8a5c4683ec7342026/html5/thumbnails/4.jpg)
The ConjectureEmpirical Evidence
Speculations and Connections
Outline
1 The ConjectureBraided Fusion CategoriesDimensions and Braid Representations
2 Empirical EvidenceQuantum GroupsGroup Theoretical Categories
3 Speculations and ConnectionsWeakly Group Theoretical CategoriesRelated Questions
Eric Rowell A Finiteness Property for Braided Fusion Categories
![Page 5: A Finiteness Property for Braided Fusion Categoriesrowell/RowellCordobanopause.pdf · The Conjecture Empirical Evidence Speculations and Connections A Finiteness Property for Braided](https://reader035.vdocuments.site/reader035/viewer/2022071212/602451d8a5c4683ec7342026/html5/thumbnails/5.jpg)
The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
Some Axioms
Definition
A fusion category C is a monoidal category that is:
C-linear, abelian
finite rank: simple classes {X0 := 1,X1, . . . ,Xm−1}semisimple
rigid: duals X ∗, bX : 1→ X ⊗ X ∗, dX : X ∗ ⊗ X → 1
compatibility...
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
Braiding
Definition
A braided fusion (BF) category has (a natural family of)isomorphisms:
cX ,Y : X ⊗ Y → Y ⊗ X
satisfying, e.g.
cX ,Y⊗Z = (IdY ⊗cX ,Z )(cX ,Y ⊗ IdZ )
Further structure:
ribbon fusion categories: braiding and ∗ compatible
modular categories: Muger center trivial.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
Some (familiar) Sources of Braided Fusion Categories
Example
Quantum group U = Uqg with q` = −1.
subcategory of tilting modules T ⊂ Rep(U)
quotient C(g, `) of T by negligible morphisms is a BF category(ribbon).
Example
G a finite group, ω a 3-cocyle
semisimple quasi-triangular quasi-Hopf algebra DωG
Rep(DωG ) is a BF category (modular).
Generally, Drinfeld center Z(C) is BF if C is a fusion category.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
Grothendieck Semiring
Definition
Gr(C) := (Obj(C),⊕,⊗, 1) a unital based ring.
Define matrices(Ni )k,j := dim Hom(Xi ⊗ Xj ,Xk)
Rep. ϕ : Gr(C)→ End(Zm)
ϕ(Xi ) = Ni
Respects duals: ϕ(X ∗) = ϕ(X )T (self-dual ⇒ symmetric)
If C is braided, Gr(C) is commutative
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
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).
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
(Weak) Integrality
Definition
C is
integral if FPdim(X ) ∈ Z for all X
weakly integral if FPdim(C) ∈ Z
[Etingof,Nikshych,Ostrik ’05]: C integral iff C ∼= Rep(H), H f.d.s.s. quasi-Hopf alg.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
A Consequence
Lemma
C weakly integral iff FPdim(X )2 ∈ Z for all simple X .
Proof.
Exercise. Use Galois argument.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
The Braid Group
Definition
Bn has generators σi , i = 1, . . . , n − 1 satisfying:
σiσi+1σi = σi+1σiσi+1
σiσj = σjσi if |i − j | > 1
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
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.
Vk irred. as Bn reps. if ΨX is surjective.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
Braid Group Images
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.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
Property F
Definition
Say C has property F if |ΨX (Bn)| <∞ for all X and n.
· · · ⊂ ΨX (Bn) ⊂ ΨX (Bn+1) ⊂ · · ·so if no property F, |ΨX (Bn)| =∞ for all n >> 0
If Y ⊂ X⊗k then ΨX (Bkn) � ΨY (Bn)so to verify prop. F, check for generating X .
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
First Examples
Examples
C(sl2, 4) C(g2, 15) Rep(DS3) Z(12E6)
rank 3 2 8 10
FPdim(Xi )√
2 1+√
52 2, 3
√3 + {1, 2, 3}
Prop. F? Yes No Yes No
12E6 is a non-braided rank 3 fusion categorywith X⊗2 = 1⊕ 2X ⊕ Y , Y⊗2 = 1.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Braided Fusion CategoriesDimensions and Braid Representations
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.?
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Lie Types A and C
Proposition (Jones ’86, Freedman,Larsen,Wang ’02)
C(slk , `) has property F if and only if ` ∈ {k , k + 1, 4, 6}.
Proposition (Jones ’89, Larsen,R,Wang ’05)
C(sp2k , `) has property F if and only if ` = 10 and k = 2.
Approach:
Take V generating “vector rep.” and q = eπi/`
ΨV (CBn) is quotient of Hecken(q2) or BMWn(−q2k+1, q)
only weakly integral in these cases
(FPdim(V ) ∈ {1,√
2,√
3,√
5, 3}).
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Lie types B and D
Conjecture
C(so2k+1, 4k + 2) has property F
C(so2m, 2m) has property F
Difficulty: spin objects Vε. Description of ΨVε(CBn)?
FPdim(Vε) ∈ {√
2k + 1,√
m}Verified for k ≤ 4, m ≤ 5
Property F fails otherwise [Larsen,R,Wang ’05].
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Some Details
P = C(sop, 2p), p primeset X := Vεsimples:{1,Z ,X ,X ′,Y1, . . . ,Yk}FPdim(X ) = FPdim(X ′) =
√p
FPdim(Yi ) = 2, FPdim(Z ) = 1
dim Hom(X⊗n,X ) = pn−1
2 +12
dim Hom(X⊗n,X ′) = pn−1
2 −12
Bratteli Diagram
…
… …
X
XX
1
1
Y1 Yk
Y1 Yk Z
… …
X X
… …
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Guesses?
Look for a series of finite (simple) groups with irreps of dimensions:
pn−1
2 +12 and p
n−12 −12
Any guesses?
Conjecture
PSp(2n, p) (Weil representation.)
This has been verified for p = 3, 5 and 7
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Exceptional Type Example
Proposition
Property F conjecture is true for C(g2, `).
Proof.
(outline) Let X be “7-dimensional” object, assume 3 | `.1 For ` >> 0, dim Hom(X 3,X ) = 4 and B3 acts irreducibly.
2 Spec(ΨX (σ1)): {q−12, q2,−q−6,−1}.3 |ΨX (B3)| =∞ for 0 << ` (use [R,Tuba ’09?])
4 Check FPdim(X )2 6∈ Z. Verify for small `.
For 3 - `, use [R ’08] for FPdim.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Main Tool
C is group-theoretical if
Z(C) ∼= Rep(DωG ) [Natale ’03], or
Z(C) ∼= Z(P), P a pointed category.
Proposition (Etingof,R.,Witherspoon ’08)
Braided group-theoretical categories C have property F.
Proof.
Braided functor C ↪→ Z(C) ∼= Rep(DωG ).Reduces to Rep(DωG ).Bn acts on (DωG )⊗n as monomial group.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Useful Criterion
Proposition (Drinfeld,Gelaki,Nikshych,Ostrik)
An integral modular category C is group-theoretical if and only ifthere exists a D ⊂ C such that
D is symmetric and
(D′)ad ⊂ D
Here D′ is the Muger center:
{X : cX ,Y cY ,X = IdX⊗Y allY ∈ D}
Lad is “spanned” by subobjects of all X ⊗ X ∗.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Some Applications
Results (Naidu,R)
If√
2k + 1 ∈ Z, C(so2k+1, 4k + 2) has property F.
If√
m ∈ Z, C(so2m, 2m) has property F.
If C a BF category with FPdim(Xi ) ∈ {1, 2} and X ∗ ∼= X forall X , C has property F.
If C is an integral modular category with FPdim(C) < 36, thenC has property F. cf. [Natale ’09?]
Approach: show certain subcategories are group-theoretical.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
More Examples
Example
Let A be an abelian group, χ nondeg. sym. bilinear form on A andτ = ±1/
√|A|.
Tambara-Yamagami categories T Y(A, χ, τ) have simple objectsA ∪ {m}with fusion rules:
m ⊗ a = m, m⊗2 =∑a∈A
a
and associativity defined via χ.T Y(A, χ, τ) is a (spherical) fusion category, soZ(T Y(A, χ, τ)) is a modular category.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
Properties of Z(T Y(A, χ, τ))
Remarks
Z(T Y(A, χ, τ))
has simple objects of dimensions 1, 2 and√|A|,
is weakly integral,
is not always group-theoretical when integral (i.e. when√|A| ∈ Z),
has rank |A|(|A|+7)2 ,
Z2-graded:Z(T Y(A, χ, τ)) = ZT Y(A, χ, τ)+ ⊕ZT Y(A, χ, τ)−
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Quantum GroupsGroup Theoretical Categories
First Results
Property F for Z(T Y(A, χ, τ)) is mostly open.
Results
ZT Y(A, χ, τ)+ is group-theoretical (so has prop. F)[Naidu,R]
Z(T Y(A, χ, τ)) is group-theoretical iff L = L⊥ for somesubgroup L ⊂ A.
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Weakly Group Theoretical CategoriesRelated Questions
Weakly Group Theoretical Categories
Definition
D is nilpotent if Dad ⊃ (Dad)ad ⊃ · · · converges to Vec.
C is weakly group theoretical if Z(C) ∼= ZD for D nilpotent.
C weakly group theoretical ⇒ C weakly integral
Conjecturally, ⇐, so
Do weakly group theoretical categories have property F?
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Weakly Group Theoretical CategoriesRelated Questions
Related Problems
Question
If C has property F, does Z(C) also?
Do braided nilpotent categories have property F?(known if C is integral)
Description of braiding?
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Weakly Group Theoretical CategoriesRelated Questions
Braided Vector Spaces
Let R ∈ Mm2(C) be a unitary solution to:R1R2R1 = R2R1R2 where R1 = (R ⊗ I ) and R2 = (I ⊗ R) and Rhas finite order.
Question
Image of Bn → U(Cmn) finite?
Results
If R comes from DωG : Yes.
For m = 2: Yes [Franko,R,Wang ’05], [Franko, Thesis].
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Weakly Group Theoretical CategoriesRelated Questions
Conversely...
ΨX : CBn → End(X⊗n) “non-local” while for X ∈ Rep(DωG ) Bn
acts locally on X⊗n.
Definition
Say ΨX can be unitarily localized if there is a unitary R-matrix Rand a v.s. V so that ΨX (Bn) is realized as Bn acting on V⊗n viaR.
Fact
Reps. from C(sl2, 4) [Franko,R,Wang ’05] and C(sl2, 6) can beunitarily localized.and are weakly integral with property F.
Question (Wang)
Unitarily localized iff property F?
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Weakly Group Theoretical CategoriesRelated Questions
Link Invariants
If C is a ribbon fusion category, X ∈ C, L a link:
IX (L) := trC(ΨX (β))
is a C-valued link invariant, where β = L.
Question
Is computing (i.e. approximating, probabilistically) IX (L) easy(polynomial-time) or hard (NP, assuming P 6= NP!)?
Appears to coincide with: Is ΨX (Bn) finite or infinite?Related to topological quantum computers: weak or powerful?(original motivation of Freedman, et al).
Eric Rowell A Finiteness Property for Braided Fusion Categories
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The ConjectureEmpirical Evidence
Speculations and Connections
Weakly Group Theoretical CategoriesRelated Questions
Thank You!
Eric Rowell A Finiteness Property for Braided Fusion Categories