The Structure of Fusion Categories via Topological Quantum Field
Transcript of The Structure of Fusion Categories via Topological Quantum Field
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The Structure of Fusion Categoriesvia Topological Quantum Field Theories
Chris Schommer-Pries
Department of Mathematics, MIT
April 27th, 2011
Joint with Christopher Douglas and Noah Snyder
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Duality: Adjoint Functors
D. Kan
Definition
An adjunction is a pair of functors
F : C � D : G
and a natural bijection
HomD(Fx , y) ∼= HomC (x ,Gy).
F is left adjoint to G .
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Equivalent Formulation
HomD(Fx ,Fx) ∼= HomC (x ,GFx) HomC (Gy ,Gy) ∼= HomD(FGy , y)
idFx 7→ (ηx : x → GFx) idGy 7→ (εy : FGy → y)
Natural transformations...
unit η : idC → GF counit ε : FG → idD
Satisfying equations...
F1∗η→ FGF
ε∗1→ F = Fid→ F
Gη∗1→ GFG
1∗ε→ G = Gid→ G
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Equivalent Formulation
HomD(Fx ,Fx) ∼= HomC (x ,GFx) HomC (Gy ,Gy) ∼= HomD(FGy , y)
idFx 7→ (ηx : x → GFx) idGy 7→ (εy : FGy → y)
Natural transformations...
unit η : idC → GF counit ε : FG → idD
Satisfying equations...
F1∗η→ FGF
ε∗1→ F = Fid→ F
Gη∗1→ GFG
1∗ε→ G = Gid→ G
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Duality in any bicategory
Definition
An adjunction is a pair of 1-morphisms
F : C � D : G
and 2-morphismsη : idC → GF ε : FG → idD
satisfying ‘Zig-Zag’ equations:
F1∗η→ FGF
ε∗1→ F = Fid→ F
Gη∗1→ GFG
1∗ε→ G = Gid→ G
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Higher Category Theory
Use the theory of (∞, n)-categories.
Generalizes both topological spacesand categories.
Hueristically:C. Barwick C. Rezk
Objects: a, b, c , . . . ,
1-morphisms f , g , h, . . . ,2-morphisms, 3-morphisms, etc. (invertible above n)compositions...
a b
f
g
⇓ c
f ′
g ′
⇓
h
⇓
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Higher Category Theory
Use the theory of (∞, n)-categories.
Generalizes both topological spacesand categories.
Hueristically:C. Barwick C. Rezk
Objects: a, b, c , . . . ,
1-morphisms f , g , h, . . . ,
2-morphisms, 3-morphisms, etc. (invertible above n)compositions...
a b
f
g
⇓ c
f ′
g ′
⇓
h
⇓
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Higher Category Theory
Use the theory of (∞, n)-categories.
Generalizes both topological spacesand categories.
Hueristically:C. Barwick C. Rezk
Objects: a, b, c , . . . ,1-morphisms f , g , h, . . . ,
2-morphisms, 3-morphisms, etc. (invertible above n)compositions...
a b
f
g
⇓ c
f ′
g ′
⇓
h
⇓
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Higher Category Theory
Use the theory of (∞, n)-categories.
Generalizes both topological spacesand categories.
Hueristically:C. Barwick C. Rezk
Objects: a, b, c , . . . ,1-morphisms f , g , h, . . . ,2-morphisms, 3-morphisms, etc. (invertible above n)compositions...
a b
f
g
⇓
c
f ′
g ′
⇓
h
⇓
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Higher Category Theory
Use the theory of (∞, n)-categories.
Generalizes both topological spacesand categories.
Hueristically:C. Barwick C. Rezk
Objects: a, b, c , . . . ,1-morphisms f , g , h, . . . ,2-morphisms, 3-morphisms, etc. (invertible above n)compositions...
a b
f
g
⇓
c
f ′
g ′
⇓
h
⇓
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Higher Category Theory
Use the theory of (∞, n)-categories.
Generalizes both topological spacesand categories.
Hueristically:C. Barwick C. Rezk
Objects: a, b, c , . . . ,1-morphisms f , g , h, . . . ,2-morphisms, 3-morphisms, etc. (invertible above n)compositions...
a b
f
g
⇓ c
f ′
g ′
⇓
h
⇓
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Exmaples of (∞, n)-categories
Example
Cat the 2-category of small categories.More generally, any bicategory.
Example (Spaces = (∞, 0)-categories)
X a space
objects = points of X
1-morphisms = paths in X
2-morphisms = paths between paths
etc.
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Duality in any bicategory
Definition
An adjunction is a pair of 1-morphisms
F : C � D : G
and 2-morphismsη : idC → GF ε : FG → idD
satisfying ‘Zig-Zag’ equations:
F1∗η→ FGF
ε∗1→ F = Fid→ F
Gη∗1→ GFG
1∗ε→ G = Gid→ G
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Example
Monoidal Category (M,⊗) one object bicategory BM.
1-morphisms = objects of Mcomposition given by ⊗2-morphism = morphisms of M
Dual objects in M ↔ dual 1-morphisms in BM: x , x∗, and...
coevaluation η : 1→ x ⊗ x∗ evaluation ε : x∗ ⊗ x → 1
satisfying ‘Zig-Zag’ equations.
Example
M = Vect, a vector space x is dualizable ⇔ x is finite dimensional
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Example
Monoidal Category (M,⊗) one object bicategory BM.
1-morphisms = objects of Mcomposition given by ⊗2-morphism = morphisms of M
Dual objects in M ↔ dual 1-morphisms in BM: x , x∗, and...
coevaluation η : 1→ x ⊗ x∗ evaluation ε : x∗ ⊗ x → 1
satisfying ‘Zig-Zag’ equations.
Example
M = Vect, a vector space x is dualizable ⇔ x is finite dimensional
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Example
Monoidal Category (M,⊗) one object bicategory BM.
1-morphisms = objects of Mcomposition given by ⊗2-morphism = morphisms of M
Dual objects in M ↔ dual 1-morphisms in BM: x , x∗, and...
coevaluation η : 1→ x ⊗ x∗ evaluation ε : x∗ ⊗ x → 1
satisfying ‘Zig-Zag’ equations.
Example
M = Vect, a vector space x is dualizable ⇔ x is finite dimensional
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Fusion Categories
Definition
A Fusion Category is a monoidal semi-simple k-linear category, with
finitely many isom. classes of simples,
End(1) ∼= k,
left and right duals for all objects.
For simplicity, k = C.Sources of Fusion Categories:
Quantum Groups
Operator Algebras
Conformal Field Theory
Representations of Loop Groups
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P. Etingof D. Nikshych
V. Ostrik
Theorem (Etingof-Nikshych-Ostrik)
In any Fusion category, the functor
X 7→ X ∗∗∗∗
is canonically monoidally equivalent to id.
Why?
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P. Etingof D. Nikshych
V. Ostrik
Theorem (Etingof-Nikshych-Ostrik)
In any Fusion category, the functor
X 7→ X ∗∗∗∗
is canonically monoidally equivalent to id.
Why?
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Definition
A fusion category is pivotal if it admits a pivotal structure, i.e. a naturalmonoidal isomorphism X ∼= X ∗∗.
Definition
A fusion category is spherical if it admits a pivotal structure compatiblewith canonical X ∼= X ∗∗∗∗.
Conjecture (ENO)
All fusion categories are pivotal.
Conjecture
All pivotal categories are spherical.
Still Open
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Definition
A fusion category is pivotal if it admits a pivotal structure, i.e. a naturalmonoidal isomorphism X ∼= X ∗∗.
Definition
A fusion category is spherical if it admits a pivotal structure compatiblewith canonical X ∼= X ∗∗∗∗.
Conjecture (ENO)
All fusion categories are pivotal.
Conjecture
All pivotal categories are spherical.
Still Open
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Definition
A fusion category is pivotal if it admits a pivotal structure, i.e. a naturalmonoidal isomorphism X ∼= X ∗∗.
Definition
A fusion category is spherical if it admits a pivotal structure compatiblewith canonical X ∼= X ∗∗∗∗.
Conjecture (ENO)
All fusion categories are pivotal.
Conjecture
All pivotal categories are spherical.
Still Open
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Manifold Invariants
ZW
ZW1 ZW2
Locality of manifold invariants:
Reconstruct ZW from ZW1 and ZW2?
ZW = 〈ZW1 ,ZW2〉
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Manifold Invariants
ZW
ZW1 ZW2
Locality of manifold invariants:
Reconstruct ZW from ZW1 and ZW2?
ZW = 〈ZW1 ,ZW2〉
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Manifold Invariants
ZW
ZW1 ZW2
Locality of manifold invariants:
Reconstruct ZW from ZW1 and ZW2?
ZW = 〈ZW1 ,ZW2〉
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The Cobordism Category
Objects are closed compact(d − 1)-manifolds Ywith germ of d-manifold
Morphisms are compact d-manifolds W ,with ∂W = Y1 t Y2
up to equivalence.
variants:
extra structures: orientations, spin structures, etc
higher categories of cobordisms
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The Cobordism Category
Objects are closed compact(d − 1)-manifolds Ywith germ of d-manifold
Morphisms are compact d-manifolds W ,with ∂W = Y1 t Y2
up to equivalence.
variants:
extra structures: orientations, spin structures, etc
higher categories of cobordisms
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The Cobordism Category
Objects are closed compact(d − 1)-manifolds Ywith germ of d-manifold
Morphisms are compact d-manifolds W ,with ∂W = Y1 t Y2
up to equivalence.
variants:
extra structures: orientations, spin structures, etc
higher categories of cobordisms
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Topological Quantum Field Theories
Definition
A TQFT is a symmetricmonoidal functor:
Bord︸︷︷︸Cobordism Category
→ C︸︷︷︸Target Category
∅ ∈ Bord 7→ 1 ∈ C
M closed 7→ (1ZM→ 1)
M. Atiyah
G. Segal
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Topological Quantum Field Theories
Definition
A TQFT is a symmetricmonoidal functor:
Bord︸︷︷︸Cobordism Category
→ C︸︷︷︸Target Category
∅ ∈ Bord 7→ 1 ∈ C
M closed 7→ (1ZM→ 1)
M. Atiyah
G. Segal
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Distinguishing Manifolds?
0D, 1D, and 2D TFTs distinguish manifolds.
4D (unitary) TFTs cannot detect smooth structures.[Freedman-Kitaev-Nayak-Slingerland-Walker-Wang]
5D (unitary) TFTs can detect, if π1 = 0. [Kreck-Teichner]
≥ 6D (unitary) TFTs cannot detect homotopy type. [Kreck-Teichner]
Open Problem: Can 3D TFTs distinguish 3-manifolds?Evidence suggest “yes?”. [Calegari-Freedman-Walker]
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Distinguishing Manifolds?
0D, 1D, and 2D TFTs distinguish manifolds.
4D (unitary) TFTs cannot detect smooth structures.[Freedman-Kitaev-Nayak-Slingerland-Walker-Wang]
5D (unitary) TFTs can detect, if π1 = 0. [Kreck-Teichner]
≥ 6D (unitary) TFTs cannot detect homotopy type. [Kreck-Teichner]
Open Problem: Can 3D TFTs distinguish 3-manifolds?Evidence suggest “yes?”. [Calegari-Freedman-Walker]
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Distinguishing Manifolds?
0D, 1D, and 2D TFTs distinguish manifolds.
4D (unitary) TFTs cannot detect smooth structures.[Freedman-Kitaev-Nayak-Slingerland-Walker-Wang]
5D (unitary) TFTs can detect, if π1 = 0. [Kreck-Teichner]
≥ 6D (unitary) TFTs cannot detect homotopy type. [Kreck-Teichner]
Open Problem: Can 3D TFTs distinguish 3-manifolds?Evidence suggest “yes?”. [Calegari-Freedman-Walker]
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Distinguishing Manifolds?
0D, 1D, and 2D TFTs distinguish manifolds.
4D (unitary) TFTs cannot detect smooth structures.[Freedman-Kitaev-Nayak-Slingerland-Walker-Wang]
5D (unitary) TFTs can detect, if π1 = 0. [Kreck-Teichner]
≥ 6D (unitary) TFTs cannot detect homotopy type. [Kreck-Teichner]
Open Problem: Can 3D TFTs distinguish 3-manifolds?Evidence suggest “yes?”. [Calegari-Freedman-Walker]
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Turaev-Viro-Barrett-Westbury Construction: a 3D TQFT
Input:C a Spherical Category
triangulate your 3-manifold
Label using data from C
Weighted average over all labelings gives invariant.
In 2010...
Theorem (Turaev-Virelizier, Balsam-Kirillov)
This gives a tqft which is local down to 1-manifolds.
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Turaev-Viro-Barrett-Westbury Construction: a 3D TQFT
Input:C a Spherical Category
triangulate your 3-manifold
Label using data from C
Weighted average over all labelings gives invariant.
In 2010...
Theorem (Turaev-Virelizier, Balsam-Kirillov)
This gives a tqft which is local down to 1-manifolds.
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Theorem (Douglas-SP-Snyder)
Fusion, Pivotal, and Spherical Categories all give rise to fully localextended 3D TQFTs.
Moreover the structure of the TQFTs reflects the structure of fusioncategories.
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Tangential Structures on Manifolds
a manifold M has a tangent bundle τclassified by a map
G → O(n)
M BO(n)
BG
τ
G = SO(n) Orientation
G = Spin(n) (universal cover of SO(n)) Spin structure
G = 1 framing
etc
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Tangential Structures on Manifolds
a manifold M has a tangent bundle τclassified by a mapG → O(n)
M BO(n)
BG
τ
G = SO(n) Orientation
G = Spin(n) (universal cover of SO(n)) Spin structure
G = 1 framing
etc
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different sorts of fusion categories give different tqfts.
Theorem (Douglas-SP-Snyder)
G name of structure kind of category
SO(3)† Orientation Spherical
SO(2) Combing Pivotal
1 = SO(1) Framing Fusion
† This group might change slightly.
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2D (non-local) TQFTs
Theorem (Folklore)
The category of (non-local) oriented 2D tqfts in C is equivalent tocategory of commutative Frobenius algebras in C .
[R. Dijkgraaf, L. Abrams, S. Sawin, B. Dubrovin, Moore-Segal, . . . ]
unit multiplication comultiplication counit
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1D TQFTs
Theorem (1D Cobordism Hypothesis)
The category of 1D oriented tqfts in C is equivalent to the groupoid ofdualizable objects of C , denoted k(C fd)
coevaluation-
+ evaluation+
-
Zig-Zag equations:
= =
F1∗η→ FGF
ε∗1→ F = Fid→ F
Gη∗1→ GFG
1∗ε→ G = Gid→ G
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2D Local TQFTs
Like 1D tqfts, but with 2D bordisms too.
Objects (0-manifolds) have duals
1-morphisms (1-manifolds)also have duals
F ◦ G⇓ ε
id
id⇓ η
G ◦ F
Zig-Zag Equation: =
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2D Local TQFTs
Like 1D tqfts, but with 2D bordisms too.
Objects (0-manifolds) have duals
1-morphisms (1-manifolds)also have duals
F ◦ G⇓ ε
id
id⇓ η
G ◦ F
Zig-Zag Equation: =
Chris Schommer-Pries (MIT) April 27th, 2011 23 / 35
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Layers of dualizability
3-Cat
monoidal3-Cat
dual 2-mor
dual 1-mordual 1-mor
dual objects
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Layers of dualizability
3-Cat
monoidal3-Cat
dual 2-mor
dual 1-mordual 1-mor
dual objects
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Layers of dualizability
3-Cat
monoidal3-Cat
dual 2-mor
dual 1-mor
dual 1-mor
dual objects
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Layers of dualizability
3-Cat
monoidal3-Cat
dual 2-mor
dual 1-mor
dual 1-mor
dual objects
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Fully-dualizable
Fully-dualizable is dualizable on all levels:
Definition
If C is a symmetric monoidal n-category, there is a filtration
C fd = C0 ⊆ C1 ⊆ · · · ⊆ Cn−1 ⊆ C
where Ci = the maximal sub-n-category where j-morphisms have bothduals if i ≤ j ≤ n − 1.
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Baez-Dolan Cobordism Hypothesis
J. Baez
J. Dolan
“Bordn is the free symmetricmonoidal n-category withduality”
Theorem (Hopkins-Lurie)
Fun(Bordfrn ,C) ' k(Cfd)
M. Hopkins
J. Lurie
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Theorem (Douglas-SP-Snyder)
Fusion categories are fully-dualizable objects in the symmetric monoidal3-category TC. (Tensor Categories)
Corollary
Fusion categories give rise to fully-local extended 3D tqfts.
What is TC?
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Theorem (Douglas-SP-Snyder)
Fusion categories are fully-dualizable objects in the symmetric monoidal3-category TC. (Tensor Categories)
Corollary
Fusion categories give rise to fully-local extended 3D tqfts.
What is TC?
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The 3-category of Tensor Categories
Example
Algebras, Bimodules, Bimodule maps = a (monoidal) 2-category
Definition
TC =
objects: Tensor Categories (monoidal k-linear)
1-morphisms: Bimodule Categories
2-morphisms and 3-morphisms: Bimodule Functors and BimoduleNatural Transformations
Monoidal for Deligne tensor product.
Proposition (Douglas-SP-Snyder)
TC is a symmetric monoidal (∞, 3)-category.
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The 3-category of Tensor Categories
Example
Algebras, Bimodules, Bimodule maps = a (monoidal) 2-category
Definition
TC =
objects: Tensor Categories (monoidal k-linear)
1-morphisms: Bimodule Categories
2-morphisms and 3-morphisms: Bimodule Functors and BimoduleNatural Transformations
Monoidal for Deligne tensor product.
Proposition (Douglas-SP-Snyder)
TC is a symmetric monoidal (∞, 3)-category.
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A Basic Principle
and a Theorem
If G acts on B, then G acts on Map(B,C ).
O(3) acts on Bordfr3 by change of framing.
O(3)→ Aut(k(Cfd))
Theorem (Hopkins-Lurie)
Fun(BordGn ,C) ' [k(Cfd)]hG .
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A Basic Principle
and a Theorem
If G acts on B, then G acts on Map(B,C ).
O(3) acts on Bordfr3 by change of framing.
O(3)→ Aut(k(Cfd))
Theorem (Hopkins-Lurie)
Fun(BordGn ,C) ' [k(Cfd)]hG .
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A Basic Principle and a Theorem
If G acts on B, then G acts on Map(B,C ).
O(3) acts on Bordfr3 by change of framing.
O(3)→ Aut(k(Cfd))
Theorem (Hopkins-Lurie)
Fun(BordGn ,C) ' [k(Cfd)]hG .
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So O(3) acts on the “space” of fusion categories.What is the action?
points in O(3) self-equivalences k(Cfd)→ k(Cfd)
paths in O(3) natural isomorphisms
paths between paths in O(3) natural 2-isomorphism
etc
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So O(3) acts on the “space” of fusion categories.What is the action?
points in O(3) self-equivalences k(Cfd)→ k(Cfd)
paths in O(3) natural isomorphisms
paths between paths in O(3) natural 2-isomorphism
etc
Chris Schommer-Pries (MIT) April 27th, 2011 30 / 35
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So O(3) acts on the “space” of fusion categories.What is the action?
points in O(3) self-equivalences k(Cfd)→ k(Cfd)
paths in O(3) natural isomorphisms
paths between paths in O(3) natural 2-isomorphism
etc
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In more detail...
π0O(3) = Z/2, non-trivial element: (F ,⊗) 7→ (F ,⊗op).
π1O(3) gives the Serre automorphism (natural automorphism ofidentity functor)
in componentsSF : F → F
is an invertible F -F -bimodule category.
π2O(3) = 0
π3O(3) = Z gives the anomaly. aF ∈ C×
No other data since TC is just a 3-category.
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In more detail...
π0O(3) = Z/2, non-trivial element: (F ,⊗) 7→ (F ,⊗op).
π1O(3) gives the Serre automorphism (natural automorphism ofidentity functor)in components
SF : F → F
is an invertible F -F -bimodule category.
π2O(3) = 0
π3O(3) = Z gives the anomaly. aF ∈ C×
No other data since TC is just a 3-category.
Chris Schommer-Pries (MIT) April 27th, 2011 31 / 35
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In more detail...
π0O(3) = Z/2, non-trivial element: (F ,⊗) 7→ (F ,⊗op).
π1O(3) gives the Serre automorphism (natural automorphism ofidentity functor)in components
SF : F → F
is an invertible F -F -bimodule category.
π2O(3) = 0
π3O(3) = Z gives the anomaly. aF ∈ C×
No other data since TC is just a 3-category.
Chris Schommer-Pries (MIT) April 27th, 2011 31 / 35
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evL ev
Theorem (Douglas-SP-Snyder)
The Serre Automorphism of a fusion category F is the bimodulification of
(F ,⊗)→ (F ,⊗)
x 7→ x∗∗
π1O(3) ∼= Z/2⇒ square of the Serre is trivial!
Corollary
The bimodulification of x 7→ x∗∗∗∗ is trivial.
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Some 3D structure groups
1
Spin(3)
Spinc(3)
SO(2)
Orpo
SO(3)
O(3)
[framing]
[combing]
[Atiyah 2-framing]
kill w1
kill p1kill βw2
kill w2
fusion
pivotal
spherical
if ourTQFTagrees withTuraev-Viro
Conjecture:anomalyvanishes
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Some 3D structure groups
1
Spin(3)
Spinc(3)
SO(2)
Orpo
SO(3)
O(3)
[framing]
[combing]
[Atiyah 2-framing]
kill w1
kill p1kill βw2
kill w2
fusion
pivotal
spherical
if ourTQFTagrees withTuraev-Viro
Conjecture:anomalyvanishes
Chris Schommer-Pries (MIT) April 27th, 2011 33 / 35
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Some 3D structure groups
1
Spin(3)
Spinc(3)
SO(2)
Orpo
SO(3)
O(3)
[framing]
[combing]
[Atiyah 2-framing]
kill w1
kill p1kill βw2
kill w2
fusion
pivotal
spherical
if ourTQFTagrees withTuraev-Viro
Conjecture:anomalyvanishes
Chris Schommer-Pries (MIT) April 27th, 2011 33 / 35
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Some 3D structure groups
1
Spin(3)
Spinc(3)
SO(2)
Orpo
SO(3)
O(3)
[framing]
[combing]
[Atiyah 2-framing]
kill w1
kill p1kill βw2
kill w2
fusion
pivotal
spherical
if ourTQFTagrees withTuraev-Viro
Conjecture:anomalyvanishes
Chris Schommer-Pries (MIT) April 27th, 2011 33 / 35
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A new version of ENO conjecture
Conjecture
All framed extended 3D tqfts in TC can be extended to oriented tqfts.
Evidence one dimension lower...
Theorem
All framed extended 2D tqfts in Alg can be extended to oriented tqfts.
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The End
Chris Schommer-Pries (MIT) April 27th, 2011 35 / 35