The Current State of the Property F Conjecture
Eric Rowell
April 2016
AMS Western Sectional
(2 + 1)Topological Quantum Field Theory
Surfaces with marked points:
(2 + 1)Topological Quantum Field Theory
Surfaces with marked points:
→
H
Reshetikhin-Turaev TQFTs
Modular Category C → TQFT. Roughly:
Reshetikhin-Turaev TQFTs
Modular Category C → TQFT. Roughly:
I
Marked points labelled by objects X ∈ C
Reshetikhin-Turaev TQFTs
Modular Category C → TQFT. Roughly:
I
Marked points labelled by objects X ∈ C
I
Gluing+Disjoint union axioms reduce to labelled disks, annuli
and trinions
Z
X
Y
Reshetikhin-Turaev TQFTs
Modular Category C → TQFT. Roughly:
I
Marked points labelled by objects X ∈ C
I
Gluing+Disjoint union axioms reduce to labelled disks, annuli
and trinions
Z
→
X
Y
H = Hom(X ⊗ Y , Z )
Mapping Class group
Definition
Let Σ be an oriented surface, with boundary ∂Σ, and n marked
points. The mapping class group MCG (Σ) is the group of
isotopy classes of orientation-preserving self-homeomorphisms of Σ.
Mapping Class group
Definition
Let Σ be an oriented surface, with boundary ∂Σ, and n marked
points. The mapping class group MCG (Σ) is the group of
isotopy classes of orientation-preserving self-homeomorphisms of Σ.
Remark
I
∂Σ is fixed pointwise
Mapping Class group
Definition
Let Σ be an oriented surface, with boundary ∂Σ, and n marked
points. The mapping class group MCG (Σ) is the group of
isotopy classes of orientation-preserving self-homeomorphisms of Σ.
Remark
I
∂Σ is fixed pointwise
I
the n points are fixed set-wise
Mapping Class group
Definition
Let Σ be an oriented surface, with boundary ∂Σ, and n marked
points. The mapping class group MCG (Σ) is the group of
isotopy classes of orientation-preserving self-homeomorphisms of Σ.
Remark
I
∂Σ is fixed pointwise
I
the n points are fixed set-wise
I
generated by Dehn twists and braid moves
The Braid Group
The Braid Group
A key example is the braid group Bn generated by σi :
The Braid Group
A key example is the braid group Bn generated by σi :
1
i
...
i+1
n
...
The Braid Group
A key example is the braid group Bn generated by σi :
1
i
...
i+1
n
...
Definition (Artin)
Bn is generated by σi , i = 1, . . . , n − 1 satisfying:
The Braid Group
A key example is the braid group Bn generated by σi :
1
i
...
i+1
n
...
Definition (Artin)
Bn is generated by σi , i = 1, . . . , n − 1 satisfying:
(R1) σi σi+1 σi = σi+1 σi σi+1
The Braid Group
A key example is the braid group Bn generated by σi :
1
i
...
i+1
n
...
Definition (Artin)
Bn is generated by σi , i = 1, . . . , n − 1 satisfying:
(R1) σi σi+1 σi = σi+1 σi σi+1
(R2) σi σj = σj σi if |i − j| > 1
Example (punctured Disk D 2 )
MCG
Example (punctured Disk D 2 )
MCG
=
B6
Example (punctured Disk D 2 )
MCG
=
B6
note: ∂D 2 = S 1
Example (punctured Disk D 2 )
MCG
Example (Torus T 2 )
MCG
=
B6
note: ∂D 2 = S 1
Example (punctured Disk D 2 )
MCG
=
B6
note: ∂D 2 = S 1
Example (Torus T 2 )
MCG
= SL(2, Z)
Mapping Class Group Action
Mapping Class Group Action
MCG (Σ) acts on H(Σ) (projectively ).
Mapping Class Group Action
MCG (Σ) acts on H(Σ) (projectively ).
Examples
H(T 2 ) ∼
= Crank(C) , MCG (T 2 ) = SL(2, Z) action is by
0 −1
1 1
→ S,
→T
1 0
0 1
Mapping Class Group Action
MCG (Σ) acts on H(Σ) (projectively ).
Examples
H(T 2 ) ∼
= Crank(C) , MCG (T 2 ) = SL(2, Z) action is by
0 −1
1 1
→ S,
→T
1 0
0 1
H([D 2 , X , . . . , X ; Y ]) ∼
= Hom(Y , X ⊗n ), Bn action via
σi → IX⊗i−1 ⊗ cX ,X ⊗ IXn−i−1 ∈ End(X ⊗n ).
Mapping Class Group Action
MCG (Σ) acts on H(Σ) (projectively ).
Examples
H(T 2 ) ∼
= Crank(C) , MCG (T 2 ) = SL(2, Z) action is by
0 −1
1 1
→ S,
→T
1 0
0 1
H([D 2 , X , . . . , X ; Y ]) ∼
= Hom(Y , X ⊗n ), Bn action via
σi → IX⊗i−1 ⊗ cX ,X ⊗ IXn−i−1 ∈ End(X ⊗n ).
Note: For genus 0, modular not needed, braided is sufficient.
Common Constructions
Example
G a finite group, ω a 3-cocycle on G : the Drinfeld double D ω G
has Rep(D ω G ) modular.
Common Constructions
Example
G a finite group, ω a 3-cocycle on G : the Drinfeld double D ω G
has Rep(D ω G ) modular.
Example
g semisimple Lie algebra, q = e πi/`
Common Constructions
Example
G a finite group, ω a 3-cocycle on G : the Drinfeld double D ω G
has Rep(D ω G ) modular.
Example
g semisimple Lie algebra, q = e πi/`
g
Common Constructions
Example
G a finite group, ω a 3-cocycle on G : the Drinfeld double D ω G
has Rep(D ω G ) modular.
Example
g semisimple Lie algebra, q = e πi/`
g
Uq g
Common Constructions
Example
G a finite group, ω a 3-cocycle on G : the Drinfeld double D ω G
has Rep(D ω G ) modular.
Example
g semisimple Lie algebra, q = e πi/`
g
Uq g
Rep(Uq g)
Common Constructions
Example
G a finite group, ω a 3-cocycle on G : the Drinfeld double D ω G
has Rep(D ω G ) modular.
Example
g semisimple Lie algebra, q = e πi/`
g
usually modular.
Uq g
Rep(Uq g)
C(g, `)
Common Constructions
Example
G a finite group, ω a 3-cocycle on G : the Drinfeld double D ω G
has Rep(D ω G ) modular.
Example
g semisimple Lie algebra, q = e πi/`
g
Uq g
Rep(Uq g)
C(g, `)
usually modular.
Remark
C(g, `) are often denoted SU(N)k , SO(N)k , Sp(N)k etc.
Motivating Question
Question
When is the image of MCG (Σ) on H(Σ) a finite group?
Motivating Question
Question
When is the image of MCG (Σ) on H(Σ) a finite group?
Theorem (Ng-Schauenburg)
For Σ = T 2
Motivating Question
Question
When is the image of MCG (Σ) on H(Σ) a finite group?
Theorem (Ng-Schauenburg)
For Σ = T 2 always finite.
Motivating Question
Question
When is the image of MCG (Σ) on H(Σ) a finite group?
Theorem (Ng-Schauenburg)
For Σ = T 2 always finite.
Remark
For certain C, known before
e.g. Rep(D ω G ), Coste-Gannon-Ruelle.
Quantum Dimensions
Let X ∈ C be simple.
Quantum Dimensions
Let X ∈ C be simple. NXZ ,Y := dim Hom(X ⊗ Y , Z ) defines a
fusion matrix [NX ]Z ,Y = NXZ ,Y
Quantum Dimensions
Let X ∈ C be simple. NXZ ,Y := dim Hom(X ⊗ Y , Z ) defines a
fusion matrix [NX ]Z ,Y = NXZ ,Y
Definition
Define dim(X ) to be the maximal eigenvalue of NX .
Quantum Dimensions
Let X ∈ C be simple. NXZ ,Y := dim Hom(X ⊗ Y , Z ) defines a
fusion matrix [NX ]Z ,Y = NXZ ,Y
Definition
Define dim(X ) to be the maximal eigenvalue of NX .
I
X is a pointed object if dim(X ) = 1
Quantum Dimensions
Let X ∈ C be simple. NXZ ,Y := dim Hom(X ⊗ Y , Z ) defines a
fusion matrix [NX ]Z ,Y = NXZ ,Y
Definition
Define dim(X ) to be the maximal eigenvalue of NX .
I
X is a pointed object if dim(X ) = 1
I
X is an integral object if dim(X ) ∈ Z
Quantum Dimensions
Let X ∈ C be simple. NXZ ,Y := dim Hom(X ⊗ Y , Z ) defines a
fusion matrix [NX ]Z ,Y = NXZ ,Y
Definition
Define dim(X ) to be the maximal eigenvalue of NX .
I
X is a pointed object if dim(X ) = 1
I
X is an integral object if dim(X ) ∈ Z
I
X is a weakly integral object if dim(X )2 ∈ Z.
Quantum Dimensions
Let X ∈ C be simple. NXZ ,Y := dim Hom(X ⊗ Y , Z ) defines a
fusion matrix [NX ]Z ,Y = NXZ ,Y
Definition
Define dim(X ) to be the maximal eigenvalue of NX .
I
X is a pointed object if dim(X ) = 1
I
X is an integral object if dim(X ) ∈ Z
I
X is a weakly integral object if dim(X )2 ∈ Z.
I
C is pointed, integral or weakly integral if all simple X ∈ C are.
Examples
Example
Fibonacci: C(g2 , 15)
Examples
Example
Fibonacci: C(g2 , 15)
f × f = 1 + f . dim(f ) =
√
1+ 5
2
Examples
Example
Fibonacci: C(g2 , 15)
√
f × f = 1 + f . dim(f ) = 1+2 5
" −4iπ/5
#
e
e −4iπ/5
σ1 7→
,
0
e 3iπ/5
#
" 3iπ/5
e
0
σ2 7→
−e 3iπ/5 e −4iπ/5
Examples
Example
Fibonacci: C(g2 , 15)
√
f × f = 1 + f . dim(f ) = 1+2 5
" −4iπ/5
#
e
e −4iπ/5
σ1 7→
,
0
e 3iπ/5
#
" 3iπ/5
e
0
σ2 7→
−e 3iπ/5 e −4iπ/5
|ρ(Bn )| = ∞
Examples
Example
Fibonacci: C(g2 , 15)
√
f × f = 1 + f . dim(f ) = 1+2 5
" −4iπ/5
#
e
e −4iπ/5
σ1 7→
,
0
e 3iπ/5
#
" 3iπ/5
e
0
σ2 7→
−e 3iπ/5 e −4iπ/5
|ρ(Bn )| = ∞
Example
Ising: SO(8)1
Examples
Example
Fibonacci: C(g2 , 15)
√
f × f = 1 + f . dim(f ) = 1+2 5
" −4iπ/5
#
e
e −4iπ/5
σ1 7→
,
0
e 3iπ/5
#
" 3iπ/5
e
0
σ2 7→
−e 3iπ/5 e −4iπ/5
|ρ(Bn )| = ∞
Example
Ising: SO(8)1
σ × σ = 1 + ψ, σ × ψ √
= σ,
ψ × ψ = 1. dim(σ) = 2
Examples
Example
Fibonacci: C(g2 , 15)
√
f × f = 1 + f . dim(f ) = 1+2 5
" −4iπ/5
#
e
e −4iπ/5
σ1 7→
,
0
e 3iπ/5
#
" 3iπ/5
e
0
σ2 7→
−e 3iπ/5 e −4iπ/5
|ρ(Bn )| = ∞
Example
Ising: SO(8)1
σ × σ = 1 + ψ, σ × ψ √
= σ,
ψ × ψ = 1. dim(σ) = 2
Braiding
via:

1
0 0 1
0
1 1 0

R = √12 
 0 −1 1 0
−1 0 0 1
Examples
Example
Fibonacci: C(g2 , 15)
√
f × f = 1 + f . dim(f ) = 1+2 5
" −4iπ/5
#
e
e −4iπ/5
σ1 7→
,
0
e 3iπ/5
#
" 3iπ/5
e
0
σ2 7→
−e 3iπ/5 e −4iπ/5
Example
Ising: SO(8)1
σ × σ = 1 + ψ, σ × ψ √
= σ,
ψ × ψ = 1. dim(σ) = 2
Braiding
via:

1
0 0 1
0
1 1 0

R = √12 
 0 −1 1 0
−1 0 0 1
|ρ(Bn )| = ∞
|ρσ (Bn )| < ∞
Property F Conjectures
Conjecture (Naidu-R, Etingof-R-Witherspoon, R-Stong-Wang)
I
Classic: Let C be a braided fusion category. Then
|ρX (Bn )| < ∞ for all X , n ⇔ C is weakly integral.
Property F Conjectures
Conjecture (Naidu-R, Etingof-R-Witherspoon, R-Stong-Wang)
I
Classic: Let C be a braided fusion category. Then
|ρX (Bn )| < ∞ for all X , n ⇔ C is weakly integral.
I
Genus g : Let C be a modular category. Then
|ρ(MCG (Σ))| < ∞ for all Σ with any labels X ∈ C ⇔ C
weakly integral.
Property F Conjectures
Conjecture (Naidu-R, Etingof-R-Witherspoon, R-Stong-Wang)
I
Classic: Let C be a braided fusion category. Then
|ρX (Bn )| < ∞ for all X , n ⇔ C is weakly integral.
I
Genus g : Let C be a modular category. Then
|ρ(MCG (Σ))| < ∞ for all Σ with any labels X ∈ C ⇔ C
weakly integral.
I
Object-wise: Let C be a braided fusion category, and X ∈ C
simple. Then |ρX (Bn )| < ∞ for all n ⇔ dim(X )2 ∈ Z.
Property F Conjectures
Conjecture (Naidu-R, Etingof-R-Witherspoon, R-Stong-Wang)
I
Classic: Let C be a braided fusion category. Then
|ρX (Bn )| < ∞ for all X , n ⇔ C is weakly integral.
I
Genus g : Let C be a modular category. Then
|ρ(MCG (Σ))| < ∞ for all Σ with any labels X ∈ C ⇔ C
weakly integral.
I
Object-wise: Let C be a braided fusion category, and X ∈ C
simple. Then |ρX (Bn )| < ∞ for all n ⇔ dim(X )2 ∈ Z.
We will say X has property F if |ρX (Bn )| < ∞ for all n.
Theorem (Etingof,R,Witherspoon)
Let G be a finite group, ω any 3-cocycle.
Theorem (Etingof,R,Witherspoon)
Let G be a finite group, ω any 3-cocycle. For all X ∈ Rep(D ω G ),
ρX : Bn → End(X ⊗n ) has |ρX (Bn )| < ∞.
Theorem (Etingof,R,Witherspoon)
Let G be a finite group, ω any 3-cocycle. For all X ∈ Rep(D ω G ),
ρX : Bn → End(X ⊗n ) has |ρX (Bn )| < ∞.
Proof.
1. Enough to consider Bn acting on End((D ω G )⊗n )
Theorem (Etingof,R,Witherspoon)
Let G be a finite group, ω any 3-cocycle. For all X ∈ Rep(D ω G ),
ρX : Bn → End(X ⊗n ) has |ρX (Bn )| < ∞.
Proof.
1. Enough to consider Bn acting on End((D ω G )⊗n )
2. σi acts monomially on the basis of tensor products of
δg h ∈ D ω G .
Theorem
I
Object-wise Property F true for pointed categories.
Theorem
I
Object-wise Property F true for pointed categories.
I
Genus g Property F conj. true for Rep(DG ) for Σ with
∂Σ ∈ {∅, S 1 }. (Fjelstad-Fuchs)
Theorem
I
Object-wise Property F true for pointed categories.
I
Genus g Property F conj. true for Rep(DG ) for Σ with
∂Σ ∈ {∅, S 1 }. (Fjelstad-Fuchs)
I
Genus g Property F conj. true for SU(2)2 and SU(2)4 for Σ
with ∂Σ = ∅. (G. Wright)
Theorem
I
Object-wise Property F true for pointed categories.
I
Genus g Property F conj. true for Rep(DG ) for Σ with
∂Σ ∈ {∅, S 1 }. (Fjelstad-Fuchs)
I
Genus g Property F conj. true for SU(2)2 and SU(2)4 for Σ
with ∂Σ = ∅. (G. Wright)
I
Object-wise Property F true for braided group-theoretical
categories C ⊂ Rep(D ω G ).
Quantum Groups
A complete list of weakly integral categories from (simple)
quantum groups:
I
SU(2)2 , SU(2)4 , SU(N)1 , SU(4)2 , SU(3)3 , Sp(4)2 = SO(5)2
Quantum Groups
A complete list of weakly integral categories from (simple)
quantum groups:
I
SU(2)2 , SU(2)4 , SU(N)1 , SU(4)2 , SU(3)3 , Sp(4)2 = SO(5)2
Property F: various authors (see Freedman-Larsen-Wang,
CMP 228)
Quantum Groups
A complete list of weakly integral categories from (simple)
quantum groups:
I
SU(2)2 , SU(2)4 , SU(N)1 , SU(4)2 , SU(3)3 , Sp(4)2 = SO(5)2
Property F: various authors (see Freedman-Larsen-Wang,
CMP 228)
I
SO(N)1 , SO(N)2
Quantum Groups
A complete list of weakly integral categories from (simple)
quantum groups:
I
SU(2)2 , SU(2)4 , SU(N)1 , SU(4)2 , SU(3)3 , Sp(4)2 = SO(5)2
Property F: various authors (see Freedman-Larsen-Wang,
CMP 228)
I
SO(N)1 , SO(N)2 Property F: R-Wenzl
Quantum Groups
A complete list of weakly integral categories from (simple)
quantum groups:
I
SU(2)2 , SU(2)4 , SU(N)1 , SU(4)2 , SU(3)3 , Sp(4)2 = SO(5)2
Property F: various authors (see Freedman-Larsen-Wang,
CMP 228)
I
SO(N)1 , SO(N)2 Property F: R-Wenzl
I
C(g, `) for some small ` for eN , g2 (mostly pointed).
Quantum Groups
A complete list of weakly integral categories from (simple)
quantum groups:
I
SU(2)2 , SU(2)4 , SU(N)1 , SU(4)2 , SU(3)3 , Sp(4)2 = SO(5)2
Property F: various authors (see Freedman-Larsen-Wang,
CMP 228)
I
SO(N)1 , SO(N)2 Property F: R-Wenzl
I
C(g, `) for some small ` for eN , g2 (mostly pointed). Property
F: trivial.
Quantum Groups
A complete list of weakly integral categories from (simple)
quantum groups:
I
SU(2)2 , SU(2)4 , SU(N)1 , SU(4)2 , SU(3)3 , Sp(4)2 = SO(5)2
Property F: various authors (see Freedman-Larsen-Wang,
CMP 228)
I
SO(N)1 , SO(N)2 Property F: R-Wenzl
I
C(g, `) for some small ` for eN , g2 (mostly pointed). Property
F: trivial.
Theorem (Jones,Larsen-R-Wang,R)
Every C = C(g, `) not listed above has some X ∈ C with
|ρX (Bn )| = ∞ for n 0.
Quantum Groups
A complete list of weakly integral categories from (simple)
quantum groups:
I
SU(2)2 , SU(2)4 , SU(N)1 , SU(4)2 , SU(3)3 , Sp(4)2 = SO(5)2
Property F: various authors (see Freedman-Larsen-Wang,
CMP 228)
I
SO(N)1 , SO(N)2 Property F: R-Wenzl
I
C(g, `) for some small ` for eN , g2 (mostly pointed). Property
F: trivial.
Theorem (Jones,Larsen-R-Wang,R)
Every C = C(g, `) not listed above has some X ∈ C with
|ρX (Bn )| = ∞ for n 0.
So the Classic Property F conj. true for quantum groups.
Yang-Baxter eqn.
Definition
(R, V ) is a braided vector space if R ∈ Aut(V ⊗ V ) satisfies
(R ⊗ IV )(IV ⊗ R)(R ⊗ IV ) = (IV ⊗ R)(R ⊗ IV )(IV ⊗ R)
Yang-Baxter eqn.
Definition
(R, V ) is a braided vector space if R ∈ Aut(V ⊗ V ) satisfies
(R ⊗ IV )(IV ⊗ R)(R ⊗ IV ) = (IV ⊗ R)(R ⊗ IV )(IV ⊗ R)
Get local Bn -reps (ρR , V ⊗n ) by
Yang-Baxter eqn.
Definition
(R, V ) is a braided vector space if R ∈ Aut(V ⊗ V ) satisfies
(R ⊗ IV )(IV ⊗ R)(R ⊗ IV ) = (IV ⊗ R)(R ⊗ IV )(IV ⊗ R)
Get local Bn -reps (ρR , V ⊗n ) by
ρR (σi ) = IV⊗i−1 ⊗ R ⊗ IV⊗n−i−1
Yang-Baxter eqn.
Definition
(R, V ) is a braided vector space if R ∈ Aut(V ⊗ V ) satisfies
(R ⊗ IV )(IV ⊗ R)(R ⊗ IV ) = (IV ⊗ R)(R ⊗ IV )(IV ⊗ R)
Get local Bn -reps (ρR , V ⊗n ) by
ρR (σi ) = IV⊗i−1 ⊗ R ⊗ IV⊗n−i−1
ρR (σi )
v1 ⊗ · · · ⊗ vi ⊗ vi+1 ⊗ · · · ⊗ vn −→ v1 ⊗ · · · ⊗ R(vi ⊗ vi+1 ) ⊗ · · · ⊗ vn
Square Peg, Round Hole?
Definition (R,Wang)
Anyon X is localizable if there is a braided vector space (R, W )
and
Square Peg, Round Hole?
Definition (R,Wang)
Anyon X is localizable if there is a braided vector space (R, W )
and injective algebra maps τn : CρX (Bn ) → End(W ⊗n )
Square Peg, Round Hole?
Definition (R,Wang)
Anyon X is localizable if there is a braided vector space (R, W )
and injective algebra maps τn : CρX (Bn ) → End(W ⊗n ) such that
the following diagram commutes:
CBn
ρn
CρX (Bn )
ρR
τn
&
/ End(W ⊗n )
Conjecture (R-Wang,Galindo-Hong-R)
Let X ∈ C, C a braided fusion category.
Conjecture (R-Wang,Galindo-Hong-R)
Let X ∈ C, C a braided fusion category. The following are
equivalent:
Conjecture (R-Wang,Galindo-Hong-R)
Let X ∈ C, C a braided fusion category. The following are
equivalent:
I
X has property F ,
Conjecture (R-Wang,Galindo-Hong-R)
Let X ∈ C, C a braided fusion category. The following are
equivalent:
I
X has property F ,
I
X is (quasi-)localizable,
Conjecture (R-Wang,Galindo-Hong-R)
Let X ∈ C, C a braided fusion category. The following are
equivalent:
I
X has property F ,
I
X is (quasi-)localizable,
I
X is weakly-integral
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
1. We need a non-standard deformation: Uq0 soN gen’d by Bi with
I
[Bi , Bj ] = 0 for |i − j| > 1
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
1. We need a non-standard deformation: Uq0 soN gen’d by Bi with
I
I
[Bi , Bj ] = 0 for |i − j| > 1
Bi2 Bi±1 − (q + q −1 )Bi Bi±1 Bi + Bi±1 Bi2 = Bi±1 .
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
1. We need a non-standard deformation: Uq0 soN gen’d by Bi with
I
I
[Bi , Bj ] = 0 for |i − j| > 1
Bi2 Bi±1 − (q + q −1 )Bi Bi±1 Bi + Bi±1 Bi2 = Bi±1 .
2. Uq0 soN End(X ⊗n ).
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
1. We need a non-standard deformation: Uq0 soN gen’d by Bi with
I
I
[Bi , Bj ] = 0 for |i − j| > 1
Bi2 Bi±1 − (q + q −1 )Bi Bi±1 Bi + Bi±1 Bi2 = Bi±1 .
2. Uq0 soN End(X ⊗n ).
3. The quantum torus: Tq2N (n) gen’d by ui :
I
[ui , uj ] = 0 |i − j| > 1,
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
1. We need a non-standard deformation: Uq0 soN gen’d by Bi with
I
I
[Bi , Bj ] = 0 for |i − j| > 1
Bi2 Bi±1 − (q + q −1 )Bi Bi±1 Bi + Bi±1 Bi2 = Bi±1 .
2. Uq0 soN End(X ⊗n ).
3. The quantum torus: Tq2N (n) gen’d by ui :
I
I
[ui , uj ] = 0 |i − j| > 1,
ui ui+1 = qui+1 ui ,
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
1. We need a non-standard deformation: Uq0 soN gen’d by Bi with
I
I
[Bi , Bj ] = 0 for |i − j| > 1
Bi2 Bi±1 − (q + q −1 )Bi Bi±1 Bi + Bi±1 Bi2 = Bi±1 .
2. Uq0 soN End(X ⊗n ).
3. The quantum torus: Tq2N (n) gen’d by ui :
I
I
I
[ui , uj ] = 0 |i − j| > 1,
ui ui+1 = qui+1 ui ,
ui2N = 1.
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
1. We need a non-standard deformation: Uq0 soN gen’d by Bi with
I
I
[Bi , Bj ] = 0 for |i − j| > 1
Bi2 Bi±1 − (q + q −1 )Bi Bi±1 Bi + Bi±1 Bi2 = Bi±1 .
2. Uq0 soN End(X ⊗n ).
3. The quantum torus: Tq2N (n) gen’d by ui :
I
I
I
[ui , uj ] = 0 |i − j| > 1,
ui ui+1 = qui+1 ui ,
ui2N = 1.
4. Uq0 soN → Tq2N (n) via Bj → i
uj +uj−1
.
q−q −1
Theorem (R-Wenzl)
A fundamental spin object X ∈ SO(N)2 has property F .
Proof (for N odd, q = e πi/(2N) )
1. We need a non-standard deformation: Uq0 soN gen’d by Bi with
I
I
[Bi , Bj ] = 0 for |i − j| > 1
Bi2 Bi±1 − (q + q −1 )Bi Bi±1 Bi + Bi±1 Bi2 = Bi±1 .
2. Uq0 soN End(X ⊗n ).
3. The quantum torus: Tq2N (n) gen’d by ui :
I
I
I
[ui , uj ] = 0 |i − j| > 1,
ui ui+1 = qui+1 ui ,
ui2N = 1.
4. Uq0 soN → Tq2N (n) via Bj → i
uj +uj−1
.
q−q −1
Factors over End(X ⊗n ).
Proof (cont.)
Proof (cont.)
−2
2
5. From all this: End(S ⊗n ) ∼
+ un−1
i
= h1, u12 + u1−2 , . . . , un−1
Proof (cont.)
−2
2
5. From all this: End(S ⊗n ) ∼
+ un−1
i
= h1, u12 + u1−2 , . . . , un−1
PN−1 4j 2 2j
γ
6. ρX (σi ) → √N j=0 q ui where |γ| = 1.
Proof (cont.)
−2
2
5. From all this: End(S ⊗n ) ∼
+ un−1
i
= h1, u12 + u1−2 , . . . , un−1
PN−1 4j 2 2j
γ
6. ρX (σi ) → √N j=0 q ui where |γ| = 1. A Gaussian
representation.
Proof (cont.)
−2
2
5. From all this: End(S ⊗n ) ∼
+ un−1
i
= h1, u12 + u1−2 , . . . , un−1
PN−1 4j 2 2j
γ
6. ρX (σi ) → √N j=0 q ui where |γ| = 1. A Gaussian
representation.
7. ρR (σi ) generate a finite group (Jones,Galindo-R).
I
Drinfeld: does weakly integral imply weakly group-theoretical?
I
Drinfeld: does weakly integral imply weakly group-theoretical?
I
R-Wang: If (R, V ) is a braided vector space with R unitary
and finite order then |ρR (Bn )| < ∞
I
Drinfeld: does weakly integral imply weakly group-theoretical?
I
R-Wang: If (R, V ) is a braided vector space with R unitary
and finite order then |ρR (Bn )| < ∞
I
Cui, Galindo, Plavnik, Wang: Every weakly integral modular C
is obtained by gauging a pointed modular category.
I
Drinfeld: does weakly integral imply weakly group-theoretical?
I
R-Wang: If (R, V ) is a braided vector space with R unitary
and finite order then |ρR (Bn )| < ∞
Cui, Galindo, Plavnik, Wang: Every weakly integral modular C
is obtained by gauging a pointed modular category.
L
Gauging: (C, G ) → D = g ∈G Dg , C = D0 .
I
I
Drinfeld: does weakly integral imply weakly group-theoretical?
I
R-Wang: If (R, V ) is a braided vector space with R unitary
and finite order then |ρR (Bn )| < ∞
Cui, Galindo, Plavnik, Wang: Every weakly integral modular C
is obtained by gauging a pointed modular category.
L
Gauging: (C, G ) → D = g ∈G Dg , C = D0 . now equivariantize:
B = DG . dim(B) = |G |2 dim(C).
I
Current Efforts/Open Problems
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Next talk: P. Gustafson considering Rep(D ω G ) for Σ with
∂Σ = ∅ and ω 6= 1.
Current Efforts/Open Problems
I
Next talk: P. Gustafson considering Rep(D ω G ) for Σ with
∂Σ = ∅ and ω 6= 1.
I
Show that property F is invariant under gauging.
Current Efforts/Open Problems
I
Next talk: P. Gustafson considering Rep(D ω G ) for Σ with
∂Σ = ∅ and ω 6= 1.
I
Show that property F is invariant under gauging.
I
Verify property F for TY (A, χ, τ )Z2 .
Current Efforts/Open Problems
I
Next talk: P. Gustafson considering Rep(D ω G ) for Σ with
∂Σ = ∅ and ω 6= 1.
I
Show that property F is invariant under gauging.
I
Verify property F for TY (A, χ, τ )Z2 .
I
Verify property F for weakly group-theoretical braided fusion
categories.
Thank you!