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ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Caps and Pants: Topological Quantum FieldTheories In Dimension 2
Geillan Aly
University of Arizona
November 24, 2009
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Outline
Conventions
The Category 2COB
Frobenius Algebra
Defining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field TheoriesTwo dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Relevance
Area Field Theory
))RRRRRRRRRRRRRR
Topological QuantumField Theory
44jjjjjjjjjjjjjjj
**TTTTTTTTTTTTTTT
QuantumField Theory
Conformal Field Theory
55llllllllllllll
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Applications
A basic model for Quantum Field Theory
Knot Theory
String Theory
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Conventions
All manifolds are real, smooth, oriented and compact.
Space-time Σ is a 2 dimensional manifold.
Spaces X and Y are 1 dimensional manifolds.
Morphisms between manifolds are smooth maps.
For M a manifold, M is the orientation reversal of M.
View the empty set as a closed manifold of a givendimension.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Definition
A cobordism is a triple (Σ,X ,Y ) such that Σ is a manifold withan orientation preserving isomorphism
∂Σ → X ∐ Y .
Orientation determines the direction in which the cobordism
propagates:
(Σ,X ,Y ) propagates from X to Y .
(Σ,Y ,X ) propagates from Y to X .
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Example
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Example
Left-capRight-cap
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Example
Left-pair of pants Right-pair of pants
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Definition
For Σ′, Σ′′, 2 manifolds with common boundary X , form amanifold Σ = Σ′
⋃
X Σ′′ = Σ′′ ◦ Σ′ by sewing Σ′ and Σ′′ alongX .
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
The 2-category pre-2COB
Objects: 1 dimensional manifolds.
1-morphisms (Σ := X → Y ): 2 dim’l cobordisms (Σ,X ,Y ).
2-morphisms: orientation preserving isomorphismsα : Σ → Σ′ for cobordisms (Σ,X ,Y ) and (Σ′,X ,Y ) suchthat the following diagram commutes:
∂Σ
α|∂Σ
��
∼= // X ∐ Y
∂Σ′
∼=
;;vvvvvvvvv
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Definition
2COB, the category of 2 dimensional isomorphic cobordisms:
Objects: 1 dimensional manifolds up to isomorphism.
Morphisms: equivalence classes of 1-morphisms definedin terms of pre-2COB. Two 1-morphisms are equivalent ifthere exists a 2-morphism between them.
Composition: the sewing of cobordisms.
The identity morphism: X × I, X ∈ |2COB|, I a compactinterval.
A (Σ,X ,Y ) cobordism is a representative of an equivalenceclass.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Theorem
Given A, a finite dimensional C algebra, TFAE:
1 There exists an A module isomorphism
λ := A → (A)∗.
where A∗ = Hom C(A,C) has diagonal A-action.2 There exists a non-degenerate linear form η : A⊗C A → C
which is “associative”:
η(ab ⊗ c) = η(a ⊗ bc) for a, b, c ∈ A.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Definition
An algebra that satisfies either condition in the previoustheorem is a Frobenius algebra.
A symmetric Frobenius algebra is a Frobenius algebrasuch that η is induced by a linear map θ : A → C, called atrace map, such that θ(ab) = θ(ba) and
η(x ⊗ y) = θ(xy).
Thus η(a ⊗ b) = η(b ⊗ a).
TheoremIf A is a Frobenius algebra, then A∗ is a Frobenius algebra.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Given a Frobenius algebra A, define a multiplication map β onA⊗A:
β := A⊗A −→ A∑
ai ⊗ bi 7−→∑
aibi .
This consequently defines a comultiplication map β∗ on A∗:
β∗ A∗ −→ A∗ ⊗A∗ ∼= (A⊗A)∗
f 7−→ f ◦ β.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
As A and A∗ are isomorphic, A is prescribed a coalgebrastructure by defining the comultiplication map
α := (λ−1 ⊗ λ−1) ◦ β∗ ◦ λ
in terms of the following commutative diagram:
Aα //
λ
��
A⊗A
A∗β∗
// A∗ ⊗A∗
λ−1⊗λ−1
OO
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
General Idea
A TQFT Z assigns complex vector spaces Z (X ) and Z (Y ) to Xand Y . Based on the cobordism from X to Y , an elementZ (Σ) ∈ Hom C(Z (X ),Z (Y )) is assigned to Σ.
Z (X )Z (Σ) // Z (Y )
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Definition
A Topological Quantum Field Theory in dimension d is afunctor Z dCOB → Vect/C such that disjoint unions in dCOBmap to tensor products.
A TQFT is defined for a fixed dimension.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Property
Z (Σ) ∈ Z (∂Σ)
Proof.
Z (∂Σ) = Z (X ∐ Y )
= Z (X) ⊗ Z (Y )= Z (X )∗ ⊗ Z (Y )∼= Hom C(Z (X ),Z (Y ))∋ Z (Σ) (by functoriality)
It is essential that Z (·) be finite dimensional.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Property
Z (∅) ∼= C if the empty set is considered a closed 1 manifold.
Proof.
Z (X ) = Z (X ∐ ∅) = Z (X ) ⊗C Z (∅).
Thus
dim(Z (X )) = dim(Z (X ) ⊗C Z (∅)) = dim(Z (X )) · dim(Z (∅)).
Therefore Z (∅) must be a 1 dim’l vector space over C.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Property
Z (∅ × I) ∼= 1Z (∅) ∈ C for I a compact interval.
Proof.
Z (∅ × I) ∼= 1Z (∅) ∈ Hom C(Z (∅),Z (∅)) ∼= Hom C(C,C) ∼= C
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Property
Z (Σ) ∈ Z (∅) = C if Σ is a closed manifold.
Proof.Z (Σ) ∈ Z (∂Σ) = Z (∅) = C.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Definition
For M an object in 2COB, a connected component of M is aclosed (open) string if it is isomorphic to a circle (compact linesegment).
RemarkObjects in 2COB are composed of open and closed strings.
The sub-category 2COB|C is restricted to objects composedexclusively of closed strings.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Classifying 2COB|C
Let Σ be a two dimensional manifold whose boundary is adisjoint union of closed strings. Σ can be characterizedaccording to its genus and the number of closed strings in itsboundary.
We say Σ is a (Σ,g, k) manifold with genus g and number ofclosed strings k .
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Generating 2COB|C
A (Σ,g, k) manifold can be formed by sewing copies of cupsand pairs of pants.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof.
Utilizing a cup and pairs of pants, construct a (Σ,g, k) manifoldby induction on g and k . This construction is not unique.
If g = 0, sew k − 2 pants.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.If k < 3 sew the necessary cups.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.
If g = 1, sew pants together in the following manner
then sew a g = 0 piece with the appropriate number of k ’s.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.
If g > 1 sew the appropriate number of g = 1 pieces then sewa g = 0 piece with the necessary k .
If (Σ,g, k) = ∐ni=1(Σi ,gi , ki ) where
∑ni=1 gi = g,
∑ni=1 ki = k
then construct (Σ,g, k) by constructing each (Σi ,gi , ki).
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Theorem
In 2COB|C, a 2 dimensional TQFT is equivalent to a symmetric,commutative Frobenius algebra A .
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof.TQFT =⇒ symmetric, commutative Frobenius algebra A
S1 is the only non-empty closed connected oriented 1 manifold.
Define the vector space A := Z (S1).
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.
A right-cap R is a cobordism from S1 to ∅.
Define the trace map as Z (R) θ A → C.
Aθ // C
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.
A left-cap L is a cobordism from ∅ to S1.
Associate the unit map C → A to the elementZ (L) ∈ Hom C(C,A) such that 1C 7→ 1A.
∅∐ S1 ∼= S
1, and the following objects are equivalent in 2COB|C:
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.A left-pair of pants is a cobordism from (S1 ∐ S
1) to S1.
Associate to it the product A⊗A → A.
A⊗A // A
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.RemarkA right-pair of pants is a cobordism from S
1 to (S1 ∐ S1).
Associate to it comultiplication α ~v 7→ ~v ⊗ ~v, defined earlier.
A // A⊗A
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.Commutativity
Consider the cobordism[
S1 ∐ S
1 → S1]
≡[
S1 ∐ S
1 → S1]
.
There exists a homeomorphism from a pair of pants onto itselfgiving the necessary relations.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.Frobenius Property
Consider the following cobordism where the topology of themanifold does not change under the given diffeomorphism.
[
(S1 ∐ S1) ∐ S
1 → S1]
≡[
S1 ∐ (S1 ∐ S
1) → S1]
.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof con’t.Symmetric Frobenius algebra =⇒ TQFT
The above correspondences are bijections, thus the propertiesof a symmetric Frobenius algebra manifest themselves in aTQFT.
Left and right-pairs of pants, left and right-caps and cylindersgenerate an element in 2COB|C. Equivalent relations manifestisomorphic algebraic structures.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Definition
A cobordism is a 4-tuple (Σ,X ,Y , ∂cstr Σ). ∂Σ = X ∐ Y .
∂cstr Σ is the constrained boundary, a cobordism from ∂Xto ∂Y .
When ∂X = ∂Y = ∅, as before, refer to the 3-tuple (Σ,X ,Y ).
Again, the orientation of Σ determines the direction in which thecobordism propagates.
Subsequently, consider surfaces whose boundaries are openand closed strings, i.e. objects and morphisms in 2COB.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Example
(Σ,1 1 ∐ 1 1,1 1,1 ∐ 1) is a cobordism from two line segmentsto one.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Definition
Let B0 be a set of elements of an R−module, R a commutativering with unity.
A B0 decorated string is an oriented compact 1-manifoldwith a labeling of the boundary components by elements ofB0.
A B0 decorated morphism is a cobordism with labeling ofthe connected components of the constrained boundary byelements of B0.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
The category B0-decorated 2COB
Objects: B0-decorated strings.
Morphisms: B0-decorated morphisms with labelingconsistent with the labeling of the unconstrained part of theboundary.
c
b
a
b c
a
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Theorem
Let Σ be a two dimensional manifold whose boundary is adisjoint union of closed and open strings. Σ can be categorizedaccording to its genus and the number of closed and openstrings in the boundary.
Σ is a (Σ,g, k , l) manifold with genus g, number of closedstrings k and number of open strings l .
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Proof
Construct a (Σ,g, k + l) manifold as before, then add l copiesof the following piece.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
In a TQFT, a cobordism with open strings gives a linear map:
c
b
a
b c
a
Oab ⊗Obc// Oac
Line segment ba is a cobordism from b to a. The set ofmorphisms from b to a is a vector space.
Z (ba) = Hom C(b,a) = Oab
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
TheoremIn the undecorated category 2COB, a two dimensional TQFTZ 2COB → Complex Vector Spaces is equivalent to thefollowing algebraic structure:
A finite dimensional, symmetric and commutativeFrobenius algebra over C with a non-degenerate trace mapθA A → C.
A finite dimensional, symmetric, not necessarilycommutative Frobenius algebra over C with anon-degenerate trace map θZ (I) Z (I) → C.
A homomorphism ıA → Z (I) such that ı(1A) = 1Z (I) andthe image of A is in the center of Z (I).
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Property
There exists linear maps
ıa : Oaa → A
a
a
andıa : A → Oaa.
a
a
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Property
ıa is an algebra homomorphism since for φi ∈ A,
ıa(φ1φ2) = ıa(φ1)ıa(φ2).
a
a
a
a
a
a
a
a
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Property
ıa preserves the identity in the sense that ıa(1A) = 1a.
a
a
a
a
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Property
ıa is central, it maps into the center of Oaa. ıa(φ)ψ = ψıa(φ) forall φ ∈ A, ψ ∈ Oaa.
a
aa
aa
a
a
a
a
a
a
a
a
a
a
a
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Property
ıa and ıa are adjoints θ(ıa(ψ)φ) = θa(ψıa(φ)) for all ψ ∈ Oaa,φ ∈ A.
a
a
a
a
a
a
a
a
a
a
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
PropertyThe Cardy Condition
For a space Oab with basis ψµ, Oba is its dual space with basisψν . Let πa
b Oaa → Obb be defined in the following manner.
πab(ψ) =
∑
p
ψµψψν .
This implies the Cardy condition
πab = ıb ◦ ıa.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
The Cardy condition can be understood using this diagram.
a
a
a
b
b
a b
b
ba
a
b
b
ba
ab
ba
a
The association is as follows:
Oaa // Oab ⊗Oba// Oba ⊗Oab
// Obb
An element ψ maps to:
ψ � // cµνψµ ⊗ ψν � // cµνψν ⊗ ψµ
� // cµνψνψµ
where cµ0ν0 = θb(ψµ0ψψν0).
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
Moving Forward
One can consider a TQFT over closed strings as a restriction ofa TQFT over both open and closed strings. What is unclear iswhether the converse is true.
Moore and Segal’s work explore this question using K-theory.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension
ConventionsThe Category 2COB
Frobenius AlgebraDefining a Topological Quantum Field Theory
Two dimensional Topological Quantum Field Theories
Two dimensional TQFTs with Closed StringsTwo dimensional TQFTs with Open Strings
References
Lowell Abrams, Two dimensional topological quantum field theories andfrobenius algebras, J. Knot Theory and its Ramifications 5 (1996),569–587.
Joachim Kock, Frobenius algebras and 2d topological quantum fieldtheories, Cambridge University Press, New York, 2004.
Aaron D. Lauda and Hendryk Pfeiffer, Open-closed strings:Two-dimensional extended tqfts and frobenius algebras, (2008).
Gregory W. Moore and Graeme Segal, D-branes and k-theory in 2dtopological field theory, arXiv:hep-th/0609042v1, 2006.
Graeme Segal, Lecture 1, topological field theories,http://www.cgtp.duke.edu/ITP99/, July-August 1999.
Geillan Aly Caps and Pants: Topological Quantum Field Theories In Dimension