Caps and Pants: Topological Quantum Field Theories In …geillan/research/capsnpants.pdf · Conventions The Category 2COB Frobenius Algebra Deﬁning a Topological Quantum Field Theory

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




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





Relevance

Area Field Theory

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Topological QuantumField Theory

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**TTTTTTTTTTTTTTT

QuantumField Theory

Conformal Field Theory

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Applications

A basic model for Quantum Field Theory

Knot Theory

String Theory





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.





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 .





Example





Example

Left-capRight-cap





Example

Left-pair of pants Right-pair of pants





Definition

For Σ′, Σ′′, 2 manifolds with common boundary X , form amanifold Σ = Σ′

⋃

X Σ′′ = Σ′′ ◦ Σ′ by sewing Σ′ and Σ′′ alongX .





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





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.





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.





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.





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.





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





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 )





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.





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.





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.





Property

Z (∅ × I) ∼= 1Z (∅) ∈ C for I a compact interval.

Proof.

Z (∅ × I) ∼= 1Z (∅) ∈ Hom C(Z (∅),Z (∅)) ∼= Hom C(C,C) ∼= C





Property

Z (Σ) ∈ Z (∅) = C if Σ is a closed manifold.

Proof.Z (Σ) ∈ Z (∂Σ) = Z (∅) = C.





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.






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 .






Generating 2COB|C

A (Σ,g, k) manifold can be formed by sewing copies of cupsand pairs of pants.






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.






Proof con’t.If k < 3 sew the necessary cups.






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.






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).


Relations





Theorem

In 2COB|C, a 2 dimensional TQFT is equivalent to a symmetric,commutative Frobenius algebra A .






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).






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






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:






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






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






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.






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]

.






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.






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.






Example

(Σ,1 1 ∐ 1 1,1 1,1 ∐ 1) is a cobordism from two line segmentsto one.






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.






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






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 .






Proof

Construct a (Σ,g, k + l) manifold as before, then add l copiesof the following piece.






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






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).






Property

There exists linear maps

ıa : Oaa → A

a

a

andıa : A → Oaa.

a

a






Property

ıa is an algebra homomorphism since for φi ∈ A,

ıa(φ1φ2) = ıa(φ1)ıa(φ2).

a

a

a

a

a

a

a

a






Property

ıa preserves the identity in the sense that ıa(1A) = 1a.

a

a

a

a






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






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






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.






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).






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.






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.


Documents

Caps and Pants: Topological Quantum Field Theories In …geillan/research/capsnpants.pdf · Conventions The Category 2COB Frobenius Algebra Deﬁning a Topological Quantum Field Theory