CAPS AND PANTS: TOPOLOGICAL QUANTUM FIELDgeillan/research/thesis.pdfCAPS AND PANTS: TOPOLOGICAL QUANTUM FIELD THEORIES IN DIMENSION 2 1 Abstract. A two-dimensional Topological Quantum

1

CAPS AND PANTS: TOPOLOGICAL QUANTUM FIELDTHEORIES IN DIMENSION 2

GEILLAN ALY

Date: Dec. 31, 2009.

CAPS AND PANTS: TOPOLOGICAL QUANTUM FIELD THEORIES IN DIMENSION 2 1

Abstract. A two-dimensional Topological Quantum Field Theory is in-troduced and defined in an intuitive manner. The foundational conceptsincluding Frobenius Algebras, cobordisms, open and closedstrings aredefined in detail thus allowing a student with no prior knowledge of thesubject the ability to gain a strong grasp of basic aspects ofa TQFT. TheDijkgraaf-Witten Toy Model is worked out as an example.

1. Introduction

The notion of a Topological Quantum Field Theory (TQFT) can be intro-duced in a concise, rigorous and mathematically appealing way, withoutany reference to quantum field theory, the form this subject normally takesin Physics. For example, in two dimensions, a TQFT is an equivariant func-tor from the category of oriented compact surfaces to the category of linearalgebra where disjoint unions correspond to tensor products.

By considering a pair of pants, one immediately sees that a 2-d TQFT givesrise to an algebra, more precisely a commutative symmetric Frobenius al-gebra. In an analogous way, when the boundaries are allowed to includeclosed intervals, a 2-d TQFT gives rise to a symmetric Frobenius algebra,which is not necessarily commutative. This geometric pointof view turnsout to greatly enlighten the study of these algebras in the same way, for ex-ample, that the realization of a group as automorphisms of a covering spacecan shed light on the structure of a group. This will be the main theme ofthis general study.

The purpose of this paper is to provide a gentle introductionto the basics ofTQFT in a simple, but non-trivial manner. The attempt is to introduce thesubject to any interested party with a basic knowledge of algebra, topologyand category theory. Segal’s lecture notes [14] are a beautiful and stream-lined introduction to the subject. However, to ensure the general picture iseffectively delivered to the audience, Segal refrains from including manyimportant but elementary details and connections. At times, every sentencein Segal’s lecture notes as well as in the introduction of [13] read as a the-orem. Our purpose here is to work through the material and provide theintermediate steps that help draw the given conclusions. Much of the for-mal content is incorporated from Abrams [1] and Lauda and Pfeiffer [9].These sources should be consulted if the reader is interested in the formalcontent and structure of the presented material.

This paper is an attempt at a thorough but conceptual introduction to 2-dimensional TQFTs and their properties. We begin by definingthe 2-category

2 GEILLAN ALY

dCOB in section 2. Algebras with a Frobenius structure will beexploredin section 3 along with examples that will resurface later inthe discussion.In the following section we define a TQFT, a functor fromdCOB to linearalgebra. Properties of TQFTs are discussed and proven. Fromhere, we canfocus on the 2-category 2COB in section 5 and prove our first main theo-rem - a 2-dimensional TQFT over closed strings is essentially equivalentto a commutative finite-dimensional symmetric Frobenius algebra. Proper-ties of 2-dimensional TQFTs are given. In a parallel development, we ex-plore 2-dimensional TQFTs over open strings, an expansion of the closedstring case. We conclude the section by proving our second main theo-rem, namely that a 2-dimensional TQFT over open strings is equivalent toa not necessarily commutative Frobenius algebra. Finally,in section 6, theDijkgraaf-Witten Toy Model will be the illustrative example of a TQFT.Many illustrations and examples are provided to ensure the reader has a fullunderstanding of the material.

Finally, to put the subject into perspective, the reader is encouraged to re-fer to [15] to explore the properties of a Quantum Field Theory (QFT) andhow they inspired Segal to form the mathematical axioms of a conformalfield theory. Segal’s ideas consequently paved the way to define a Topo-logical Quantum Field Theory (it was Witten who initially coined the term[17]). Segal used the properties of a QFT to establish the simple axiomswe will explore. The main difference between a QFT and a TQFT lies inwhat is emphasized. A QFT is a functor to an infinite-dimensional spaceand depends on the structure of the manifold and the definition of a metric,whereas a TQFT, a functor to a finite-dimensional space, onlydepends onthe structure of the manifold. TQFTs are very useful since they are a simpli-fication of QFTs; they provide an arena to test conjectures and calculationsthat are more difficult or possibly impractical to compute for a general QFT.

The author would like to extend their warmest acknowledgments to the fol-lowing for their guidance and support. First, I would like tothank my thesiscommittee, Dr. Pan Peng, and Dr. Linda Simonsen, with a special thanksto my advisor Dr. Doug Pickrell who introduced me to the subject. I wouldalso like to thank my friends and family in New York for their unwaveringsupport and encouragement. Finally, I would like to thank mydedicatedemotional and professional support network at the University of Arizonaincluding, not not limited to Dr. Bruce Bayly, Dr. ChristianCollberg, Vic-toria Frost, David Herzog, Dr. Klaus Lux, Victor Piercey, Dr. David Savitt,and Jordan Schettler.


1.1. Conventions. All manifolds are smooth, oriented and compact.Σ,representing space-time, isd-dimensional;X andY, representing space, are(d− 1)-dimensional. At this time,X andY will not have a boundary. Whenboundaries are introduced, it will be evident whether or notX or Y are man-ifolds with boundary. All maps from one manifold to another are smooth.If M is a manifold with a fixed orientation,M will denote the orientationreversal ofM. The empty set is a manifold of arbitrary dimension.

2. The Category dCOB

We begin with a definition that will be generalized as we expand our initialresults.

Definition 1. A cobordismis a triple (Σ,X,Y) whereΣ is a manifold with∂Σ = X∐ Y, the disjoint union ofX andY.

In a (Σ,X,Y) cobordism, orientation must be fixed to determine the directionin which the cobordism propagates. Therefore, let (Σ,X,Y) be a cobordismsuch thatΣ propagates fromX to Y; thus (Σ,Y,X) is a cobordism such thatΣ propagates fromY to X.

Definition 2. A d − 1 manifoldX is null-cobordantif there is a cobordismthat propagates fromX to the empty manifold. Observe that in this case,Xis the boundary of ad manifold.

The following are examples of cobordisms and null-cobordant manifolds:

Example 1. A cylinder C = S1 × [0, 1] is a cobordism (C, S1, S1) from S1

to S1 where theS1’s are the antipodal boundaries of the cylinder.

Example 2. Sn−1 is null-cobordant sinceSn−1 is the boundary ofDn, then-dimensional disk. Thus,D2 is a cobordism from the empty manifold toS

1. Refer to this cobordism as acap where (D2, S1, ∅) is a right-cap and(D2, ∅, S1) is a left-cap.

Left-capRight-cap

Example 3. Let Σ be a genus 0, 2-dimensional manifold such that∂Σ =

(S1 ∐ S1) ∐ S1, as intended in the diagram below. ThenΣ is a cobordism

4 GEILLAN ALY

betweenS1 ∐ S1 and S1. We will call Σ the pair of pants. Again, wedistinguish between a left and right pair of pants. (Σ, S1 ∐ S1, S1) is theleft-pair of pants. (Σ, S1, S1 ∐ S1) is the right-pair of pants.

Left-pair of pants Right-pair of pants

Definition 3. Let Σ′, Σ′′ bed manifolds where∂Σ′ = X1 ∐ X2 and∂Σ′′ =X2 ∐ X3. Then bysewingΣ′ andΣ′′ along X2 we form a new manifoldΣ = Σ′

⋃

X2Σ′′ = Σ′′ ◦ Σ′ with boundaryX1 ∐ X3.

We begin our study with the 2-category pre-dCOB. Generally speaking,a 2-category is a categoryC such that for two objectsA, B ∈ |C—, theset Hom (A, B) is a category. See the appendix for a detailed exposition on2-categories. Objects in pre-dCOB are comprised of the class of (d − 1)-dimensional compact manifolds and 1-morphisms are cobordismsΣ in di-mensiond. 2-morphisms are orientation preserving homeomorphismsα :Σ → Σ′ for cobordisms (Σ,X,Y) and (Σ′,X,Y) such that the following dia-gram commutes:

∂Σ

α|∂Σ��

� // X∐ Y

∂Σ′�

;;wwwwwwww

From this, one can state that in pre-dCOB, two 1-morphismsΣ andΣ′ areequivalent if there is a 2-morphismα : Σ→ Σ′.

Definition 4. dCOBis the category ofd-dimensional homeomorphic cobor-disms. The objects are (d − 1)-dimensional compact manifolds up to iso-morphism. Morphisms are the equivalence classes ofd-dimensional 1-morphisms defined in the 2-category framework of pre-dCOB where two1-morphisms are equivalent if there exists a 2-morphismα : Σ → Σ′. Thecomposition of morphisms is identified with the sewing of cobordisms. Theidentity morphism for an objectX is the manifoldX× I whereI is a compactinterval.

The reader should note that this notation is used for convenience. Whenthe two-dimensional case is considered and boundaries are introduced in


section 5, this category will be denoted 2COB|C, a subcategory of 2COB.2COB will therefore be an expansion of the category we are currently fo-cusing on.

The perspective that two elements in pre-dCOB that are diffeomorphic un-der certain conditions are considered equivalent will be helpful in our devel-opment. We can see that a (Σ,X,Y) cobordism is actually a representativeof an equivalence class, where two manifolds are equivalentif and only ifthere exists a diffeomorphism between the surfaces (observe∂Σ � ∂Σ′).

We have now found the “Topology” in Topological Quantum Field Theory.

3. Frobenius Algebra

Definition 5. If K is a field, then analgebraA overK is a ring and aK-vector space overK such that (ka)b = k(ab) = a(kb) for all k ∈ K, a, b ∈ A.We will assume that all algebrasA have an identity element 1A.

The following are examples of algebras. It is clear that these examplessatisfy the axioms of an algebra.

• Examples of commutative algebras:

Example 4. A field L such thatK ⊂ L.

Example 5.Polynomial algebraK[X1, . . . ,Xn]. Observe that a poly-nomial algebra satisfies the requirements of an algebra, butis aninfinite-dimensional vector space overK. Thus, there is no require-ment that the algebraA be finite-dimensional.

Example 6.K valued functions on a nonempty set S.

• Examples of non-commutative algebras:

Example 7. Matrix algebraMn(K) where matrix multiplication isthe product operation.

Example 8. HomK(V,V) for any vector spaceV with compositionbeing the product operation. Observe that this example and example7 are equivalent with respect to a choice of basis whenV is finitedimensional.

Definition 6. Let A be aK algebra, andM a vector space overK. ThenM is a leftA-moduleif eacha, a′ ∈ A, m, m′ ∈ M andk ∈ K a productam∈ M is defined such that

• a(m+m′) = am+ am′

• (a+ a′)m= am+ a′m

6 GEILLAN ALY

• (aa′)m= a(a′m)• 1Am= m where 1A is the identity element inA• (ka)m= k(am) = a(km)

A rightA-moduleis defined analogously.

Remark 1. A is itself a leftA-module, denotedAA, with canonical leftmultiplication. This is called theleft regularA-module. Again, we can de-fine theright regularA-module,AA as a rightA-module with the canonicalright multiplication.

Definition 7. LetA be aK algebra andM a leftA-module. LetM∗ be thedual of M. ThenM∗ becomes a rightA-module if forψ ∈ M∗, a ∈ A,m ∈M,

(ψa)(m) = ψ(am).

The rightA moduleM∗ is thedual of M. Similarly, the dual of a rightA-module is a leftA-module.

Theorem 1. [4] LetA be a finite-dimensional algebra over a fieldK, thenthe following are equivalent:

(1) For AA and (AA)∗, two leftA-modules, there exists anA algebraisomorphismλ : AA→ (AA)∗.

(2) There exists a non-degenerate linear formη : A⊗A → K which is“associative” in the following manner

η(ab⊗ c) = η(a⊗ bc) for a, b, c ∈ A.

(3) There exists a linear form f∈ A∗ whose kernel contains no non-trivial left or right ideals.

Definition 8. An algebra that satisfies any of the conditions in Theorem 1is aFrobenius algebra.

It is important to note that a Frobenius algebra is not a “type” of algebra.Rather, it is an algebra endowed with a given structure. It isan abuse oflanguage to state that “A is a Frobenius algebra”. One should rather statethat “A is an algebra endowed with a Frobenius structureη or f .” In general,we will use the former semi-correct language for the sake of brevity.

Proof. Part 1 (1)⇒ (2):

Forλ : AA→ (AA)∗, λ(ab) = aλ(b) for all a, b ∈ A. Thus for allx ∈ A,

λ(ab)(x) = (aλ(b))(x) = λ(b)(xa).

Thus define the linear formη : A⊗A → K where

η(x⊗ y) ≔ λ(y)(x).


Observe thatη is non-degenerate sinceλ is a K isomorphism. In otherwords, ifη(·⊗y) = 0 thenλ(y) = 0 and thusy = 0. Analogously,η(x⊗·) = 0impliesx = 0. Associativityη(xy⊗ z) = η(x ⊗ yz) follows fromλ(ab)(x) =(aλ(b))(x) = λ(b)(xa):

η(xy⊗ z) ≔ λ(z)(xy)= yλ(z)(x)= λ(yz)(x)= η(x⊗ yz)

Part 2 (2)⇒ (1):

Let η be a non-degenerate linear form onA. Defineλ : AA→ AA as

λ(y)x≔ η(x⊗ y).

λ is aK isomorphism sinceη is non-degenerate and is anA algebra isomor-phism by the associativity ofη.

Part 3 (2)⇒ (3):

Given a linear formη(x⊗ y), define a linear functionf ∈ A∗ as

f (x) ≔ η(x⊗ 1A).

Then f (xA) = 0 impliesη(xA⊗ 1A) = η(x⊗A) = 0. Thusx = 0 sinceη isnon-degenerate. Likewise,f (Ax) = 0 impliesx = 0. We conclude that thekernel of f contains no non-trivial left or right ideals.

Part 4 (3)⇒ (2):Given a linear functionf ∈ A∗ with no non-trivial left or right ideals, define

η(x⊗ y) = η(x, y) ≔ f (xy).

It is clear thatη satisfies the conditions of (2).

The hypothesis thatf does not have a non-trivial left or right ideal is neces-sary. Suppose thaty is in an idealI contained in the kernel off , thenxy ∈ Iimplies f (xy) = 0 = η(x⊗ y) andη is degenerate. �

Definition 9. A symmetric Frobenius algebrais a Frobenius algebra suchthat the non-degenerate linear formη defined in theorem 1 is induced by atrace mapθ : A→ C whereη(x, y) = θ(xy) thus implyingη(a, b) = η(b, a).

Remark 2. The term symmetric Frobenius algebra is not the universal termused in defining this structure. Moore, Segal[13], [14], Atiyah [2] andother authors refer to this structure simply as a Frobenius algebra whendiscussing TQFT. Yet it is clear that a Frobenius algebra differs from a

8 GEILLAN ALY

symmetric Frobenius algebra precisely based on the trace condition whichwe will see is essential in defining a TQFT. Although symmetric Frobeniusalgebras are contained in Frobenius algebras, it would be a disservice tothe reader if the exact terms used were not clear and if there was a failure toestablish the containment of the two types of structures. Itshould be notedfurther that Curtis and Reiner[4] among other sources refer to a symmetricFrobenius algebra simply as a symmetric algebra. So as to prevent anypossibility of confusion, we will remain abundantly clear by referring tothis structure as a symmetric Frobenius algebra, as found in[8].

Clearly ifA is a commutative algebra, i.e.ab= ba for all a, b ∈ A then acommutative Frobenius algebra is a symmetric Frobenius algebra.

There are several examples of a symmetric Frobenius algebraessential toour needs:

Example 9. Consider the algebraMn(K) whereθ(a) = tr(a), the trace func-tion. η(a · b, c) = tr((ab)c) = tr(a(bc)) = η(a, b · c)

Example 10. The fieldC is a Frobenius algebra overR with one possibleform θ(a + bi) ≔ a. Another form could be based on a different map. Forexample consider the formθ(2+ 3i) ≔ 7; θ(1− i) ≔ 4.[8]

Example 11. Let G be a group (for our purposes, a finite group),R a ring.By considering the set of allR linear combinations of elements ofG, wecan define thegroup ring R[G]. If a, b are elements ofR[G], thena+ b isdefined pointwise, anda · b is defined by the distributive law of the ring.

θ(a · b) is the coefficient of the identity element ofa · b. This clearly definesa Frobenius algebra sinceθ(a · b, c) is the coefficient of the identity elementof (a · b) · c = a · b · c = a · (b · c). The coefficient of the identity element ofa · (b · c) is θ(a, b · c).

Furthermore,θ is non-degenerate. Suppose fora ∈ R[G], θ(R[G]a) = 0.Thenθ(g−1a) = 0 for eachg ∈ G. Thus,θ(g−1a) is the coefficient ofg in a,implying thata = 0. Similarly, θ(aR[G]) = 0 impliesa = 0.

Example 12. Let R[G] be a group ring, define thecenter of a group ringasthe set of elements inR[G] which commute with all other elements inR[G].

Claim 1. p ∈ Z(R[G]) ⇔ p is constant over the conjugacy classes of G.


Proof. If p = λ1g1 + . . . + λngn then forh ∈ G,

h · p · h−1 = h(λ1g1 + . . . + λngn)h−1

= hλ1g1h−1 + . . . + hλngnh−1

= λ1hg1h−1 + . . . + λnhgnh−1

= p⇔ λi = λ j wheng j = hgih−1

�

Theorem 2. [1] If A is a Frobenius algebra over a fieldK with form f , thenevery other Frobenius form onA is given by u· f for u an invertible elementinA.

Proof. Let u ∈ A be a unit. Then for anya ∈ A whereu · f (ax) = f (uax) =0 for all a ∈ A, ua = 0 and thereforea = 0. Thus there are no non-trivialideals in the kernel andu · f is a Frobenius form.

Now considerg ∈ A∗ a Frobenius form not equal tof . Then by the proofin theorem 1,g = λ(u) = u · f for someu ∈ A. Sinceg is a Frobenius form,the mapλ′ ≔ gβ is an isomorphismA → A∗, whereβ : A → End (A),the map which takes an elementa ∈ A to the map “multiplication bya”.Thus, there is av ∈ A such thatf = λ′(v) = v · g = vu · f . Then we see thatλ(1A) = f = uv· f = λ(vu) implies that 1A = vu. Sinceλ is an isomorphism,u is a unit inA.

�

Theorem 3. If A is a Frobenius algebra, thenA∗ is a Frobenius algebra.

Proof. LetA be a Frobenius algebra with formf ∈ A∗. By theorem 1, allelements ofA∗ are of the forma· f for a ∈ A. AsA andA∗ are isomorphic,define mutiplication inA∗ by (a· f )(b· f ) ≔ (ab· f ). Now defineτ : A∗ → Kto be “evaluation at 1A.” Then the identityτ(ax · f ) = f (ax) letsA∗, withstructureτ, be a Frobenius algebra. �

Definition 10. A coalgebraA over a fieldK is a vector spaceA with twoK-linear maps:

α : A→ A⊗A andδ : A→ K

such that the following diagrams commute

Aα //

α

��

A⊗A

IA⊗α��

A⊗Aα⊗IA // A⊗A⊗A

10 GEILLAN ALY

AIA

**UUUUUUUUUUUUUUUUUUUUUα //

α

��

A⊗A

IA⊗δ��

A⊗Aδ⊗IA // K ⊗A � A � A⊗ K

The mapα is thecomultiplicationmap,δ is thecounit map with axiomscoassociativity[(a⊗ (b⊗c) = (a⊗b)⊗c] and thecounitcondition [A⊗K �K ⊗A is a natural isomorphism].

Next, we define a multiplication mapβ onA⊗A:

β : A⊗A → A∑

ai ⊗ bi 7→∑

aibi

which defines a comultiplication mapβ∗ onA∗:

β∗ : A∗ → A∗ ⊗ A∗ � (A⊗A)∗∑

f 7→ f ◦ β

As a Frobenius algebraA andA∗ are isomorphic,A is prescribed a coal-gebra structure by definingα, the comultiplication map (λ−1 ⊗ λ−1) ◦ β∗ ◦ λ:

Aα //

λ

��

A⊗

A

A∗β∗ // A∗

⊗

A∗

λ−1⊗λ−1

OO

A is coassociative and cocommutative by the definition ofα. α can be alsoused to define multiplication inA∗ since (a· f )(b· f ) = [a· f⊗b· f ]◦α = ab· f .Let g: K → A denote the unit map. The commutativity of

A∗

g∗

��>>>

>>>>

>>f ◦ β(a)

((PPPPPPPPPPPP

A

λ

OO

f // K a //

OO

f (a) = f ◦ β(a)(1A)

guarantees the commutativity of

A∗β∗ // A∗

⊗

A∗g∗⊗IA∗ // K

⊗

A∗

λ−1

��

A

λ

OO

α // A⊗

A

λ⊗λ

OO

f⊗IA // K⊗

A

Since the top row isIA∗ , the bottom row isIA. Thus f is the counit inA.

CAPS AND PANTS: TOPOLOGICAL QUANTUM FIELD THEORIES IN DIMENSION 211

4. Defining a Topological Quantum Field Theory

We are now in a position to define a Topological Quantum Feld Theory. Themodel to keep in mind is the following: consider a (Σ,X,Y) cobordism. ATopological Quantum Field TheoryZ assigns complex vector spacesZ(X)andZ(Y) to X andY respectively. Furthermore, based on the structure of thecobordism fromX to Y, an elementZ(Σ) ∈ Hom (Z(X),Z(Y)) is assignedto Σ. In its full generality, a TQFT is a functor fromdCOB toB, a subcat-egory ofR modules over a commutative ringR with unity where disjointunions correspond to tensor products. The following diagram illustrates theconcept.

Z(X)Z(Σ) // Z(Y)

ConsiderS, the surface ofΣ as a “function” of time, traveling from X toY. The elementZ(Σ) ∈ Hom (Z(X),Z(Y)) can be determined based on theinterpretation of information asS propagates fromX to Y.

Note that a TQFT is defined on a manifold with a fixed dimension.

Definition 11. A Topological Quantum Field Theoryin dimensiond is amonoidal functorZ : dCOB→ Vect/K.

In this case, the monoidal structure that is preserved is thecorrespondencebetween disjoint unions indCOB and tensor products in the category ofLinear Algebra. In its fullest generality, a TQFT is a functor to B, a subcat-egory ofR modules, whereR is a commutative ring with unity.

Definition 12. A TQFT is reducedif for some intervalI , Z(X × I ) = id|Z(X)

for each objectX.

Remark 3. For Σ = X× I for some interval I, Z(Σ)2 = Z(Σ) sinceΣ◦Σ � Σ.Thus we can suppose that a TQFT is reduced by denoting id|Z(X) as theimage of X× I. Henceforth, we will always assume that a TQFT is reduced.

12 GEILLAN ALY

We will elaborate on the meaning of the definition of a TQFT by showingthat this is equivalent to a more expanded definition given byAtiyah [2].

Atiyah [2] states that aTopological Quantum Field Theoryin dimensiondconsists of the following two components and three axioms:

• a finite, positive-dimensional complex vector spaceZ(X) associatedto each oriented (d− 1)-dimensional manifoldX. Under orientationpreserving diffeomorphisms ofX, the vector space behaves functo-rially.• an elementZ(Σ) ∈ Z(∂Σ) associated to eachd-dimensional manifoldΣwith boundary∂Σ. Again, under orientation preserving diffeomor-phisms ofΣ, Z(Σ) behaves functorially.

TQFTs satisfy the axioms laid out by Atiyah[2]:

1. SinceX is X with opposite orientation, letZ(X) � Z(X)∗ whereZ(X)∗

is the dual ofZ(X). Again, this isomorphism behaves functorially underisomorphisms ofX.

2. Z(X1 ∐ X2) = Z(X1) ⊗C Z(X2).3. GivenΣ′ and Σ′′ with ∂Σ′ = U ∐ V ∂Σ′′ = V ∐ W so thatΣ′ andΣ′′ are sewn along the common boundaryV to form Σ, then Z(Σ) =〈Z(Σ′),Z(Σ′′)〉 where〈 , 〉 denotes the natural pairing

Z(U) ⊗ Z(V) ⊗ Z(V)∗ ⊗ Z(W) → Z(U) ⊗ Z(W).

Thus ifV = ∅, andΣ = Σ′ ∐ Σ′′ thenZ(Σ) = Z(Σ′) ⊗ Z(Σ′′).

The functoriality requirement is based on the orientation preserving diffeo-morphisms that act on the spaces. Iff : X1 → X2 is an orientation pre-serving diffeomorphism,f induces an isomorphismZ( f ) : Z(X1) → Z(X2).Under composition,Z is covariant: forg: X2 → X3 an orientation preserv-ing diffeomorphism,g ◦ f induces the composition

Z(g ◦ f ) = Z(g) ◦ Z( f ) : Z(X1)→ Z(X3).

If X1 andX2 are boundaries of twod manifoldsΣ1 andΣ2 respectively, andf can be extended to an orientation preserving diffeomorphism fromΣ1 toΣ2, thenZ( f ) can be extended such thatZ( f ) : Z(Σ1)→ Z(Σ2).

Several properties can be deduced from Atiyah’s definition that begin tosynchronize it with the given definition. These properties are immediateconsequences of Atiyah’s definition and illustrate the motivation behind re-duced theories. In this regard, remark 3 can actually be seenas a propertyin Atiyah’s definition as well.

Property 4.0.1. For a (Σ,X,Y) cobordism,Z(Σ) ∈ Hom (Z(X),Z(Y)).


Lemma 1. If V and W are finite-dimensional vector spaces over a fieldK and V∗ is the dual space of V, then V∗ ⊗K W � HomK(V,W). Theisomorphism is via the map

T : V∗ ⊗W // Hom(V,W)

φ ⊗ w � // Tφ⊗w

where

Tφ⊗w(v) = φ(v)w

Proof. (of Lemma 1): Let{vi} and {wj} be the bases forV andW respec-tively and let{v∗i } be the basis forV∗. ThenV∗ ⊗K W has basis{v∗i ⊗K wj}

with

dim (V∗ ⊗K W) = dim (V∗) dim (W).

Next, observe that

dim ( HomK(V,W) ) = dim (V) dim (W) = dim (V∗) dim (W).

As dim (V∗ ⊗K W) = dim ( HomK(V,W)) and both are vector spaces overK, these vector spaces are isomorphic since all vector spacesover a givenfield with the same dimension are isomorphic.

�

Proof. (of Property 4.0.1):

Z(∂Σ) = Z(X ∐ Y)= Z(X) ⊗ Z(Y)= Z(X)∗ ⊗ Z(Y)� Hom C(Z(X),Z(Y))

Observe, it is essential thatZ(·) be a finite-dimensional complex vectorspace. �

Property 4.0.2. If the empty set is considered as a closed (d−1)-dimensionalmanifold, thenZ(∅) = C.

Proof.

Z(X) = Z(X∐ ∅) = Z(X) ⊗C Z(∅).

Thus

dim(Z(X)) = dim(Z(X) ⊗C Z(∅)) = dim(Z(X)) · dim(Z(∅)).

ThereforeZ(∅) must be a 1-dimensional vector space overC. �

14 GEILLAN ALY

Remark 4. The two previous properties show that ifΣ is a closed manifolda TQFT calculates an invariantΨΣ associated to a closed manifoldΣ. Thisinvariant can be computed by decomposingΣ into a series of manifolds thatare easier to compute.

Property 4.0.3. If the empty set is considered as a closedd-dimensionalmanifold, then writing the empty set as∅ × I , where the empty set hasdimension (d− 1) andI is an interval, we getZ(∅ × I ) = 1Z(∅) = 1 ∈ C.

The notation, while at the outset may seem ambiguous, is standard in Atiyah’swork. There are attempts to clarify the notation but those endeavors lead toconfusion in other discussions on TQFTs.

Proof.Z(∅) ∈ Z(∂∅) = Hom C(C,C).

So letZ(∅) = c ∈ C. This yields the following relation:

c = Z(∅) = Z(∅ ◦ ∅) = Z(∅) · Z(∅) = c2

Since we have a reduced TQFT based on Remark 3, we exclude the trivialcase, and letc = 1.

�

Property 4.0.4. If ∂Σ = (∐ki=1Xi) ∐ (∐l

j=1X′j) then

Z(Σ) ∈ Hom C

(

⊗ki=1 Z(Xi),⊗

lj=1Z(X′j)

)

.

Proof. Follows inductively from the axioms. �

Property 4.0.5. If Σ is a closed manifold (a manifold with empty bound-ary), thenZ(Σ) ∈ Z(∅) = C.

Proof. If Σ is a closed manifold, thenZ(Σ) ∈ Z(∂Σ) = Z(∅) = C. �

4.1. Connecting to Atiyah. Aside from not using the power of categorytheory to prove the above properties, Atiyah’s definition isdistinct fromthe given one in one crucial manner. In terms of associating sewing withcomposition, a priori, Atiyah’s definition provides us withmore freedomthan the given definition. Resolving the differences allows us to bring thetwo definitions in line, and we can thus understand how they are equivalent.

Consider the following situation in which one would want to sew a right-pair of pants to one leg of a right-pair of pants. Atiyah’s definition does notrestrict this action whereas the given definition does. In order to resolve theforgoing impasse of our definitions, one would sew a cylinderto the other“leg” of the pants. This acts as an identity on the other leg, which does noteffect the calculation of the TQFT.


Atiyah’s Definition

Resolution

In the general case, given a (Σ,X,Y) cobordism, sewing a surface toY′, asubset ofY, can be achieved by attaching identity cobordisms toY\Y′.

5. Two-Dimensional Topological Quantum Field Theories

We now prove that a two-dimensional TQFT is equivalent to a commutativesymmetric Frobenius algebra.

Definition 13. For M an object in 2COB, a component ofM is a closedstring if it is homeomorphic to a circle and anopen stringif it is homeo-morphic to a compact line segment.

The objects in the category 2COB are therefore composed of both openand closed strings1. We can define a sub-category 2COB|C to be the sub-category whose objects are restricted exclusively to closed strings, earlierdefined as 2COB. It should be emphasized that what was earlierdefined tobe dCOB is nowdCOB|C, while the expansion of the category to includeopen strings is nowdCOB. All results from section 4 still hold fordCOB.

The reader should note that objects with open strings in 2COBare equiv-alence classes of manifolds with corners. The structure anddetail in usingthe term will be suppressed for the sake of brevity and simplicity withoutlosing much rigor in this work.

5.1. Two-Dimensional TQFTs with Closed Strings. At this time, we willconsider surfaces (two manifolds) whose boundary is a disjoint union ofclosed strings. By expanding the Classification Theorem found below, we

1The objects are isomorphism classes of 1-dimensional compact manifolds which re-duce to two equivalence classes represented by an open string (compact line segment) anda closed string (S1).

16 GEILLAN ALY

follow with a generalization. This paves the way for the maintheorem ofthe section.

Theorem 4. Classification Theorem. LetΣ be a compact two-dimensionalclosed manifold. ThenΣ can be classified, up to diffeomorphism, accordingto its genus and orientability up to diffeomorphism.

Proof. The proof can be found in [12]. �

Theorem 5.LetΣ be a two-dimensional compact orientable manifold whoseboundary is a disjoint union of closed strings, thenΣ can be categorized ac-cording to its genus and the number of closed strings in its boundary. Thuswe say thatΣ is a (Σ, g, k) manifold (with genus g and number of closedstrings k).

Proof. Fix an orientation onΣ, a closed 2 manifold with genusg, i.e. a(Σ, g, 0) manifold. Then given a (Σ′, g, k) 2 manifold, formΣ′ from Σ byremovingk open disks.

If (Σ′, g, k) is a 2 manifold and (Σ, g, l) is a 2 manifold wherek , l, thenΣ′

andΣ are not equivalent as they have non-isomorphic fundamentalgroups.

Conversely, given an (Σ′, g, k) 2 manifold, by sewingk open disks to eachclosed string (essentially closing each boundary) we form an oriented closed2 manifoldΣ of genusg, which can be classified. �

Theorem 6. Every(Σ, g, k) manifold can be formed from sewing copies ofcups and pairs of pants.

Proof. Utilizing a cup and pair of pants, we proceed to construct any(Σ, g, k)manifold by induction ong and k. Observe that this construction is notunique.

• If g = 0, sewk − 2 pants as follows

If k < 3 sew the necessary cups.

• If g = 1, sew pants together in the following manner


then sew ag = 0 piece with the appropriate number ofk’s.• If g > 1 sew the appropriate number ofg = 1 pieces then sew a

g = 0 piece with the necessaryk.• If (Σ, g, k) = ∐n

i=1(Σi , gi, ki) where∑n

i=1 gi = g,∑n

i=1 ki = k thenconstruct (Σ, g, k) by constructing each (Σi , gi, ki).

�

Remark 5. Although a(Σ, g, k) manifold can be made in this manner, wehave seen for example that a(Σ, 0, 3) manifold can be realized as a left orright pair of pants. As cobordisms, they are not equivalent.Each can beformed by sewing the appropriate g= 0 pieces with the appropriate numberof k’s to either end of the formed surface with genus g.

Abrams [1] utilizes the left and right caps and pants as well as the cylinderas generators with the following relations to solidify the previous theoremin proposition 12. These generators and relations will allow the subsequenttheorem to be proven almost effortlessly.

18 GEILLAN ALY

Theorem 7. Every two-dimensional TQFT Z: 2COB|C → Complex Vec-tor Spaces is equivalent to a symmetric commutative Frobenius algebraAwhich is finite-dimensional overC.

The spirit of the proof shows that Abrams’ generators and relations that2COB|C adheres to are paralleled in a Frobenius Algebra. A two-dimensionalTQFT is a functor from 2COB to the category of Complex Vector Spaces.This theorem is being restricted to the subcategory 2COB|C. Pedantically,a TQFT over open/closed or just closed strings will be used to indicate theappropriate domain, 2COB or 2COB|C, respectively.

This proof is a sketch of the ideas of the equivalence of the categories. Theactual proof utilizes Morse theory and will not be explored in this paper.Lauda and Pfeiffer [9] utilize Morse theory to describe the generators andrelations that fully determine 2COB.

Proof. Part 1: We show that a TQFT gives rise to a commutative symmetricFrobenius algebraA in a natural way.

It suffices to simply consider a cap and pair of pants as these are the funda-mental manifolds which when sewn together can form any relevant (Σ, g, k)manifold. [1]

SinceS1 is the only closed connected oriented 1 manifold, define the algebraA as a vector space whereZ(S1) = A.

A right-capR is a cobordism fromS1 to ∅, so associate the cobordismRwith the trace mapθ : A → C, soZ(R) = θ : A→ C.

Aθ // C

A left-capL is a cobordism from∅ to S1, thus the unit mapC→A is asso-ciated to the elementZ(L) ∈ Hom (C,A) such that 17→ 1A. To prove thiswe observe that∅ ∐ S1

� S1, therefore the following objects are equivalent

in dCOB:


If φ(1) is the image of 1 of the left-cap and~v ∈ Z(S1), then

1 7→ φ(1)⊗ ~v 7→ φ(1) · ~v

These maps are identical to the identity map

~v 7→ ~v.

So we can concludeφ(1) = 1A.

A left-pair of pants considered as a cobordism fromS1 to (S1 ∐ S1) is theproductA⊗A → A.

A⊗A // A

A right-pair of pants considered as a cobordism fromS1 to (S1 ∐ S1) isassociated to comultiplicationα : ~v 7→ ~v⊗~v which can be defined as above.

Commutativity can be verified when considering the topologyof a cobor-dism

S1 ∐ S1 → S1 = S1 ∐ S1 → S1.

Without regard to the ambient space, there exists a homeomorphism from apair of pants onto itself which gives the necessary relations.

The Frobenius property can again be verified by considering that the topol-ogy of the manifold does not change. The base case is seen below wherewe have the cobordism

(S1 ∐ S1) ∐ S1 → S1 = S1 ∐ (S1 ∐ S1)→ S1.

20 GEILLAN ALY

Part 2: A symmetric Frobenius algebra gives rise to a TQFT: As part 1is bijective, the axioms and properties of a symmetric Frobenius algebramanifest themselves in a TQFT. Using left and right-pairs ofpants, left andright-caps and cylinders one can generate each element in 2COB in the spiritof theorem 6. The relations given above by Abrams, complete the sketch ofthe proof showing that two diffeomorphic manifolds are the manifestationof isomorphic symmetric Frobenius algebras.

�

5.2. Properties of 2-dimensional TQFT’s Over Closed Strings.

Property 5.2.1. Let T be a manifold defined by a torus with one outgoingboundary circle. Letα ∈ A be an element such thatZ(T) = α. Then IfK isa (K, g, 0) surface,Z(K) = θ(αg).

Proof. This can be done by induction.Let g = 1 thenK is composed ofT sewn to a left-capX1. We can nowfollow the formulations given to associate a symmetric Frobenius algebrawith a 2-dimensional space. Therefore

Z(K) ∈ Hom(

Z(∅),Z(S1),Z(S1)∗,Z(∅))

= θ(Z(T)) = θ(α).

Let g ∈ N thenK is composed of (Kg−1∐ T) sewn to a left-pair of pantsX1,sewn to a left-capX2. Then

Z(K) = Hom(

Z(Kg−1 ∐ T),Z(X1),Z(X2))

= Hom(

(

Z(Kg−1) ⊗ Z(T))

,Z(X1),Z(X2))

= θ(αg−1 × α)= θ(αg)

�

Property 5.2.2. If {ei} is a vector space basis forA, and{e∗i } is the dualbasis, thenα(1) =

∑

eiλ−1(e∗i ) whereλ : A → A∗ andλ−1 : A∗ → A.


Proof. We begin by decomposingT into a left-cap sewn to a right-pair ofpants sewn to a left-pair of pants.

Following the cobordisms and the associated TQFT, we get

Z(∅) // Z(S1) // Z(S1 ∐ S1) // Z(S1)

which is associated algebraically to

C // A∗ // (A⊗A) // A .

Sewing the left cap to the right-pair of pants, we are left with the followingmaps:

C // (A⊗A) // A .

If we follow where the element 1 maps to:

α(1) = (λ−1 ⊗ λ−1) ◦ β∗ ◦ λ(1)= (λ−1 ⊗ λ−1)θ= (λ−1 ⊗ λ−1) ◦ Σ(e∗i ⊗ λ(ei))=∑

λ−1(e∗i ) ⊗ ei

Our remaining map

A⊗A −→ A∑

ei ⊗ λ−1(e∗i ) 7−→

∑

eiλ−1(e∗i )

completes the proof. �

Property 5.2.3. For π : A → End (A), the left regular representation,θ(aα) = tr(π(a)) for a ∈ A, tr the trace map.

Proof. The following set of maps

Aπ // End (A) � A⊗A∗

IA⊗λ−1// A⊗A

defines a coproduct mapΨ ofA. The composition tr◦ π is equal to

AΨ // A⊗A

〈−,−〉 // C.

Since〈−,−〉 = θ ◦ µ, tr(π(a)) = θ ◦ µΨ)(a),

we can rewrite this sequence as

(θ ◦ µ ◦ α(⊗idA))(a) = θ(aα),

thus proving the property. �

22 GEILLAN ALY

Property 5.2.4.A is semi-simple, i.e., a sum of copies ofC, if and only ifα is invertible.

Proof. Sinceθ(a, b) is non-degenerate fora, b ∈ A, then the trace function(a, b) 7→ trπ(a)trπ(b) is non-degenerate if and only ifα is invertible sincefor non-zeroa, b ∈ A we have

0 , θ(a, b) = θ(abαα−1) = tr(abα−1),

which implies that the finite-dimensional algebra is semi-simple by nonde-generacy. �

Property 5.2.5. If the characteristic polynomial ofα is χ(t) =∏

(t − λi)thenΨg =

∑

λg−1i for g ≥ 1.

Proof.

Ψg = θ(αg) = θ(αg−1α) = tr(αg−1) =

∑

λg−1i

sinceα can be written as a diagonal matrix of its eigenvalues. �

Property 5.2.6. If g = 1 thenΨg = Ψ1 = θ(α) = θ(∑

eiλ−1(e∗i )) = dim(A).

This property can be visualized as the identification of the ends of a cylinderto form a torus. Once formed, the torus can be decomposed intoa left-cap,right-pair of pants, two cylinders, a left-pair of pants anda right-cap.

5.3. Two-Dimensional TQFTs with Open Strings. In a parallel develop-ment, we expand the results of studying 2-dimensional TQFTsdefined on2COB|C to all of 2COB. We begin with a simpler construction and thenintroduce boundary conditions on the boundaries of the openstrings. Ifboundary conditions are not considered (i.e. all boundary conditions aretrivial), then certain properties that will be illustratedwill be straightfor-ward. If the reader prefers, they may ignore the boundary conditions of theopen strings when they are introduced. The absences of boundary condi-tions provide a simpler model for the open string case.

We begin by expanding the definition of a cobordism.

Definition 14. A cobordismis a 4-tuple (Σ,X,Y, ∂cstrΣ) such that the bound-ary ofΣ is the disjoint union ofX andY, whereX indicates the orientationreversal ofX. ∂cstrΣ is theconstrained boundary, a cobordism from∂X to∂Y. In the case where∂X = ∂Y = ∅ as before, we will only refer to the3-tuple (Σ,X,Y).

Similar to a (Σ,X,Y) cobordism, in a (Σ,X,Y, ∂cstrΣ), the direction in whichthe cobordism propagates will be determined by the orientation of Σ.


Example 13. (Σ, 1 1∐1 1, 1 1, 1∐1) is a cobordism from two line segmentsto one. So the reader does not confuse the lack of boundary conditions witha cobordism whose boundaries are closed strings, the identity element 1 isconsidered the boundary. Again, it is necessary to emphasize the order inwhich the disjoint union of 1-dimensional manifolds appear.

The introduction to Moore and Segal’s paper [13] is concerned with a gen-eralization of an open string theory, where the constrainedparts of theboundaries of morphisms are equipped with boundary conditions. The setof boundary conditions is itself a linear category, where the compositioncorresponds to

c

b

a

b c

a

a

d

c

d

c

b

a

b

a

d

c

d

The introduction to Moore and Segal’s paper is written more in the style of aphysics paper, in that it does not include details in the introductory sectionsthat discuss this material - this omission could possibly befor the sake ofbrevity. Here is our attempt to mathematically capture the meaning of thissection.

We now introduce boundary conditions, whereB0 is a fixed linear category.

Definition 15. a. A B0 decorated string is an oriented compact 1-manifold(with or without boundary) with a labelling of the boundary componentsby elements ofB0.

b. A B0 decorated morphism is a cobordism as in definition 14 with thelabeling of the connected components of the constrained boundary byelements ofB0.

24 GEILLAN ALY

Definition 16. B0-decorated 2COB [2COB with boundary conditions] isthe category whose objects consist ofB0-decorated strings (up to isomor-phism), and the morphisms consist of equivalence classes ofB0-decoratedmorphisms, where the labelling of the morphism is consistent with the la-belling of the unconstrained part of the boundary (up to isomorphism).

Example 14. (Σ, ab∐ bc, ac, a∐ c) is a cobordism from the line segmentsab∐ bc to the line segmentac. Observe thata andc are the constrainedboundaries - cobordisms from∂(ab∐ bc) to ∂(ac). a, b andc will also bereferred to as boundary conditions. It is particularly important to heed theorientation of the line segments. A line segmentab is a cobordism fromb toa, whereas a lineba is a cobordism froma to b. Moreover, it is necessary toemphasize the order in which the disjoint union of 1-dimensional manifoldsappear.

c

b

a

b c

a

Henceforth, consider surfaces whose boundaries are both open and closedstrings, i.e. objects and morphisms in the category 2COB. Since this sectionsucceeds the results 2-dimensional TQFT’s with closed strings, many of theresults will be straightforward or will echo previous developments.

Theorem 8. Expanding on theorem 5, letΣ be a two-dimensional compactorientable manifold whose boundary is a disjoint union of closed and openstrings, thenΣ can be categorized according to its genus and the number ofclosed and open strings in the boundary. Thus we say thatΣ is a (Σ, g, k, l)manifold (with genus g, number of closed strings k and numberof openstrings l).

Again, in the spirit of remark 5, two different cobordisms can be formedfrom a (Σ, g, k, l) manifold.

Proof. Following theorem 5, form a (Σ, g, k + l) manifold. Then gluel ofthe following pieces tol closed strings.

a

a

�


Definition 17. A B0-decorated topological quantum field theory is a func-tor from fromB0-decorated 2COB to the category of linear algebra, wheredisjoint unions correspond to tensor products.

In the two-dimensional case, by property 4.0.1, a cobordismwith openstrings in the boundary gives a linear map:

c

b

a

b c

a

(1) Oab ⊗ Obc// Oac

Observe thatab is a cobordism fromb to a. Thus, we consider the vectorspaceZ(ab) = Hom (b, a) = Oab to be the set of morphisms fromb to a.We will always assume that the theories are reduced as in Remark 3.

Theorem 9. In the undecorated category 2COB, every two-dimensionalTQFT Z: 2COB→ Complex Vector Spaces is equivalent to the followingalgebraic structure:

• A finite-dimensional, symmetric and commutative Frobeniusalge-bra overC with a non-degenerate trace mapθA : A→ C.• A finite-dimensional, symmetric, not necessarily commutative Frobe-

nius algebra overC with a non-degenerate trace mapθZ(I) : Z(I ) →C.• A homomorphismı : A → Z(I ) such thatı(1A) = 1Z(I) and the

image ofA is in the center of Z(I ).

The algebraic consequences and properties of the theorem are outlined insection 5.4. The sketch of the proof is provided below. As there is no gener-alization of Morse Theory to manifolds with corners, Lauda and Pfeiffer’sproof [9] of the equivalence of these categories is illuminating and shouldbe consulted for a greater understanding of the subject. Their account of thenecessity and sufficiency of the generators and relations in the open/ closedcase nicely generalizes the closed string case. The reader should note thatthe conditions of the algebraic structure are defined by Lauda and Pfeifferto be aknowledgeable Frobenius Algebra. The term knowledgeable comesfrom the mapping ofA into the center ofZ(I ) indicating thatA gives someinformation on the structure ofZ(I ).

Proof. Part 1: Again, we show that a TQFT gives rise to a noncommutativesymmetric Frobenius algebra in a natural way. We consider the open string

26 GEILLAN ALY

manifolds analogous to the closed string cap and pants. The only argumentthat is not analogous is the noncommutativity of the symmetric Frobeniusalgebra.

Consider the “flat right-cap” as a cobordism fromaa to ∅, associate to it thenondegenerate traceθa : Oaa→ C onOaa.

a

a

Oaa// C

Noncommutativity is clear when one realizes that due to the boundary con-ditions, there is no homeomorphism from a manifoldM to itself that willgive the necessary conditions. An attempt to imitate the closed string con-dition does not yield the desired result. If a TQFT over open strings werecommutative, then forψ ∈ Oba andφ ∈ Oab,

θb(ψφ) = θa(φψ)

a priori, there is no reason for the above statement to hold true in all cases,and thus the Frobenius algebra is said to be not necessarily commutative.

a

b

a

b a

a

b

a

b

a b

b

Part 2: Due to the complicated nature of the objects and morphisms in2COB, it is difficult to confirm that a list of generators and relations is com-plete. Lauda and Pfeiffer accomplish this goal by describing a normal formfor a cobordism in 2COB (similar to the categorization of theorem 8). Nextthey construct moves that change a given cobordism to its normal form.Thus the relations found below along with those from the closed string casegenerate all objects in 2COB.


c

b

a

b

a

d

c

d

c

b

a

b

a

d

c

d

c

b

a

b

a

d

c

d

c

b

a

b

a

d

c

d

a

a

a

a

a a

bb

a

b

a

b

a

a

a

a

a a

bb

b

ba

a

b

b a

a

c

c

b

a

a

b

c

c

b

a

a

b

c

c

b

a

a

b

c

c

b

a

a

b

c

c

b

a

a

b

c

c

b

a

a

b

c

c

b

a

a

b

c

c

b

a

a

b

c

c

b

a

a

b

c

c

b

a

a

b

c

b

a

b

c

b

a

b

a

a

d

c

d

bb

c

a

a

a

a

a

a

a

a

a

a

a

a

c

aa

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

a

b

b

a b

b

ba

a

b

b

ba

a

�

5.4. Properties of 2-dimensional TQFT’s Over Open Strings.Defininga TQFT for decorated open strings is trickier. Considerations need to bemade since the objects in 2COB are also cobordisms. In a manner of speak-ing, a TQFT for decorated strings needs to be defined inductively to ensurethat for an objectba ∈ 2COB, which is a one-dimensional cobordism, theproperties of a TQFT are also taken into account. Although wecan nowspecify the categoryB. Objects are the class of boundary conditions. Theset of morphisms fromb to a is the vector spaceOab. Composition of mor-phisms is given by the relation above in equation (1). For thefollowingproperties, we will generalize to decorated open strings sothe reader canappreciate where the boundaries need to be compatible. The relations givenearlier are given alongside the algebraic information.

28 GEILLAN ALY

Property 5.4.1. To limit ourselves to a reduced TQFT in the spirit of Re-mark 3, we see that comparable to the cylinder for closed strings, a cobor-dism (Σ,Oba,Oba, a∐ b) acts as an identity element onOba.

a a

bb

Open strings and closed strings are necessary to define a TQFTover an openstring. Thus we relate them using the following properties.

Property 5.4.2. There exists a linear map

ıa : Oaa→ A

a

a

Property 5.4.3. There also exists a linear map

ıa : A→ Oaa

a

a

ıa has the following properties:

Property 5.4.4. ıa is an algebra homomorphism since forφi ∈ A,

ıa(φ1φ2) = ıa(φ1)ıa(φ2).

a

a

a

a

a

a

a

a

Property 5.4.5. ıa preserves the identity in the sense thatıa(1A) = 1a.

a

a

a

a

Property 5.4.6. ıa is central in that it maps into the center ofOaa. ıa(φ)ψ =ψıa(φ) for all φ ∈ A, ψ ∈ Oaa.


a

aa

aa

a

a

a

a

a

a

a

a

a

a

a

Property 5.4.7. ıa andıa are adjointsθ(ıa(ψ)φ) = θa(ψıa(φ)) for all ψ ∈ Oaa,φ ∈ A.

a

a

a

a

a

a

a

a

a

a

Property 5.4.8. The Cardy Condition: For a spaceOab with basisψµ, thenOba is it’s dual space with basisψν. Let πa

b : Oaa → Obb be defined in thefollowing manner.

πab(ψ) =

∑

p

ψµψψν.

This implies the Cardy condition

πab = ıb ◦ ı

a.

We can understand the Cardy condition with the use of the following dia-gram:

a

a

a

b

b

a b

b

ba

a

b

b

ba

ab

ba

a

The associated algebra is as follows:

Oaa// Oab ⊗ Oba

// Oba⊗ Oab// Obb

so that an elementψ gets mapped as follows:

ψ � // cµνψµ ⊗ ψν � // cµνψν ⊗ ψµ � // cµνψνψµ

To answer the question of whatcµν is, we follow the following diagram:

30 GEILLAN ALY

b

a

a

a

a

bb

a

a

ba

b

ba

a

b

b

a

a

a

The diagram on the left is a map fromOab⊗Oba⊗Oaa toC. Apply the mapto basis elementsψν0 ⊗ ψµ0 ⊗ ψ for some fixed basis elementsψν0, ψµ0 andψ ∈ Oaa. Since the diagram on the right is equivalent to the diagram on theleft, the triple (φ, ψν0, ψµ0) maps toθb(ψµ0ψψ

ν0). We can therefore concludethatcµ0ν0 = θb(ψµ0ψψ

ν0).

We can validate this idea by checking thatθb(ψµ0ψψν0)ψν equalsψµψ. Begin

by settingψµψ = ψ ∈ Oba. Then we haveθb(φψν)ψν = φ but we know thatφ =∑

φνψν soφν = θb(φψν)

It is clear that one can consider a TQFT over closed strings asa restrictionof a TQFT over open strings. What is unclear is whether the converse istrue. Moore and Segal explore this question and use K-theoryto illustrateresults related to this question in [13].

6. Example: The Dijkgraaf-Witten Toy Model

There is an example of ann-dimensional TQFT associated to a finite groupG, the Dijkgraaf-Witten Toy model.

Definition 18. Let M be a topological space,K a field. Then anK vectorbundleof rankk overM is a topological spaceE with a surjective continu-ous mapπ : E→ M such that

• for eachx ∈ M, the setEx = π−1(x) ⊂ E, the fiber of E over x,is endowed with the structure of ak-dimensionalK vector space sothatE =

⋃

x∈M Ex.• for eachx ∈ M, there exist a neighborhoodU of x in M and a

homeomorphismΦ : π−1(U) → U ×Kk (called alocal trivializationof E overU) such that the following diagram commutes:

π−1(U) ⊂ EΦ //

π

��

U × Kk

π1xxpppppppppppp

U

whereπ1 is projection on the first factor such that for eachq ∈ Uthe restriction ofΦ to Eq to {q} × Kk

� Kk [11].


Definition 19. Let G be a fixed finite group andP a fiber bundle over amanifoldM. If G acts onP from the right, thenP is aprincipal G bundleifthe following properties hold:

• Any fiberπ−1(U) � G

•P×G −→ P(p, g) 7−→ pg

• p(g1g2) = (pg1)g2

• If pg1 = pg2 for somep ∈ P theng1 = g2

In the toy model, forΣ a closedd manifold,ΨΣ is the sum of isomorphismclasses of principalG bundles overΣ with weight 1/|Aut P|, where AutPis the group of auomorphisms ofP covering idΣ. Define

ΨΣ ≔∑

[P]

1|Aut P|

where [P] are the isomorphism classes of principalG bundles overΣ.

Reducing to the case whereΣ is connected, there exists the following cor-respondence.

Lemma 2. There is a bijection between the isomorphism classes of princi-pal G bundles[P] overΣ and homomorphismsρ ∈ Hom

(

π1(Σ),G)

moduloconjugacy.

Proof. Let Σ be the universal cover ofΣ, ρ : π1(Σ) → G a homomorphism,and p: P → Σ a surjective, continuous map. AsΣ is the universal cover,π1(Σ) acts onΣ. Consider the spaceΣ×ρ G � Σ×G/ ∼ρ where ifα ∈ π1(Σ),(n, g) ∼ (αn, ρ(α)g). DefineP � Σ ×ρ G.

It remains to show thatP is a principal bundle.G acts onP from the rightsince forh ∈ G,

[(αn, ρ(α)g)] · h ∼ [(n, g)] · h = [(n, gh)] ∼ [(αn, ρ(α)gh)].

Finally, we show thatP/G � Σ by first considering elements inΣ × GmoduloG. We have that [(n, g)] ∼ [(n, 1)]. Takingρ into account, [(n, 1)] ∼[(αn, ρ(α))] ∼ [(αn, 1)]. This shows that if [(m, 1)] ∼ [(n, 1)] thenm andn differ by a multiple of an element ofπ1(Σ). Thus we see thatP/G �Σ/π1(Σ) � Σ.

Thus,P is a principal bundle associated toρ. �

Lemma 3. The group Aut(P) is identical to CG(ρ), the subgroup of ele-ments of G which commute with the image ofρ.

32 GEILLAN ALY

Proof. First we choose an elementgρ ∈ G that commutes with the image ofρ and define the map

gρ : Σ ×ρ G −→ Σ ×ρ G[(n, h)] 7−→ [(n, gρh)][(αn, ρ(α)h)] 7−→ [(αn, gρρ(α)h)] = [(αn, ρ(α)gρh)]

Since [(n, h)] ∼ [(αn, ρ(α)h)] and [(n, gρh)] ∼ [(αn, ρ(α)gρh)], we have awell defined action on equivalence classes. Therefore,gρ is an element ofAut (P).

SinceΣ × G is free overG, chooseψ ∈ Aut P. Theψ’s are the same asπ1(Σ) equivariant mapsΣ → Σ ×G. Then we get (n, 1) 7→ (n′, ψ) for ψ ∈ Gwhereπ(n) = π(n′). Sincen andn′ are in the same fiber, they are related bya deck transformation, so we can rewrite this as a map (n, 1) 7→ (n, ψ(n)) forsomeψ(n) ∈ G.

The equivariance ofψ means thatψ commutes withρ(α), for all α ∈ π1(Σ),then the set of all suchψ lie in the centralizer ofρ(π1(Σ)). �

Consequently, the isomorphism class ofP is isomorphic toG/CG(ρ(π1(Σ))).Therefore

∑

[P]

|G||Cg(ρ(π1(Σ)))|

= | Hom (π1(Σ),G)|

|G|∑

[P]

1| Aut (P)|

= | Hom (π1,G)|

Therefore

ψY =| Hom (π1(Y),G)|

|G|

We now describe the vector spaces associated to an (d − 1)-dimensionalmanifoldX and the maps associated to cobordisms.

Definition 20. Two principalG-bundlesπ1 : P1 → M, π2 : P2 → M areequivalentif there exists a homeomorphismH : P1→ P2 such that

P1H //

π1 AAA

AAAA

AP2

π2~~}}}}

}}}}

M

commutes andH(p · g) = H(p) · g for all g ∈ G.


Let PX be the set of isomorphism classes of principalG bundles onX andZ(X) ≔ CPX. LetΣ be a manifold where∂Σ = X. And takeψΣ(1) = Z(X) =C

PX to be determined as follows:

Let P→ X be aG bundle overX. Then

ψΣ(1)(P) =∑

[Q]

1| Aut (Q)|

where [Q] ranges over isomorphism classes ofG bundles overΣ such thatQ|∂Σ = P and where the isomorphisms fixQ|∂Σ.

6.1. The Two-Dimensional Case.We use Dijkgraaf-Witten to calculateZ(S1) andZ(Oba).to show that the toy model manifested in two dimensions in thecommu-tative symmetric Frobenius algebra (A, θS1) whereA is the center of thegroup ringC[G] and we define

θ(∑

λgg)

≔1|G|λ1.

6.1.1. Z(S1) � Z(C[G]), the center ofC[G]. Fix a basepointx ∈ S1 anda finite groupG. Let P � R ×ρ G be a principalG-bundle onS1. A G-bundle is determined up to isomorphism by the mapρ : Z→ G. By mapping1 7→ g for any g ∈ G, | Hom (Z,G)| = |G|. Hence, we can associate anymapρ with an elementg ∈ G. To determine isomorphism classes ofGbundles onS1, we simply consider what happens to the basepointp fortwo equivalentG-bundles. For a giveng associated withρ, g changes abasepointp 7→ p · g. This change in basepoint induces a change in the mapρ. Thus twoG-bundles are equivalent if two group elements acting on thespace are conjugate.

ThusZ(S1) is the space of complex valued functions on isomorphism classesof G-bundles ofS1. Namely, since twoG-bundles are isomorphic if they areinduced by conjugate elements inG, Z(S1) is the center of the group ringC[G].

6.1.2. Z(ba) � C[G]. Again, we follow the algorithm to determineZ(ba).Fix a basepointx ∈ ba. We consider the principalG-bundle onba. Thereis only one principal bundle onba sinceπ1(ba) = e, the identity element.Thus the only principal bundle is defined by the identity map.Hence, thereare no non-trivial isomorphism classes ofG-bundles to be concerned withandZ(ba) = C[G].

34 GEILLAN ALY

As we have boundary conditions, the categoryB is the category of finite-dimensional complex representationsT of G. As Oaa is the set of mor-phismsψ : a → a for some boundary conditiona, the traceθa : Oaa → C

takesψ ∈ End (T) 7→ 1/|G| × trace (ψ).

Theorem 10.Specific to the Dijkgraaf-Witten model, we have the followingproperties.

Property 6.1.1. The group ringC[G] =⊕

End (V) whereV runs throughthe irreducible representations ofG. Consider the set

eχ =χ(1)|G|

∑

g∈G

χ(g) · g

as a basis forC[G]:Sinceeχ is a function onG with complex coefficients constant over theconjugacy classes ofG, eχ ∈ C[G]. Secondeχi · eχ j = δi j .

Property 6.1.2. θ(1V) = λ−1V where 1V is the identity element in End (V)

andλV =|G|2

dim(V)2.

Proof.θ(1V0) = θ(χV0)

=1|G|2

χV0(1)

=(dimV0)2

|G|2

�

Property 6.1.3. α =∑

λV1V.

Proof. We know thatα =∑

eiΦ∗(e∗1) and for basis elementsei and e∗i ,

θ(eiΦ∗(e∗j )) = δi j .

Consider 1V to be a basis for End (V), we show thatΦ∗(e∗i ) = λV1V.Then we see that

θ(1ViλV j 1V j ) = λV jθ(1Vi 1V j ) = δi j

by the previous property. �

Property 6.1.4. ΨΣ = |G|2g−2∑ 1

(dimV)2g−2for Σ a closed manifold of

genusg.


Proof.ΨΣ = θ(αg)= θ((

∑

λV1V)g)=∑

λVθ(1V)g

= |G|2g−2∑ 1(dimV)2g−2

�

7. Appendix: 2-Categories

Definition 21. [7]A 2-categoryC is composed of the following:

• A class of objects|C|• For each pair ofX, Y ∈ |C|, a category Hom (X,Y).

– Objects in Hom (X,Y) are1-morphisms, i.e. morphisms be-tween objects inC

– Morphisms in Hom (X,Y) are 2-morphisms, i.e. morphismsbetween the morphisms between objects inC.

• Horizontal composition of 1-morphisms and 2-morphisms: For eachtriple X,Y,Z ∈ |C| a functor

µX,Y,Z : Hom (X,Y) × Hom (Y,Z)→ Hom (X,Z)

with the following conditions:– (Identity 1-morphism): For each objectX ∈ |C|, there exists an

object idX ∈ Hom (X,X) such that

µX,X,Y(IX, ·) = µX,Y,Y(·, IY) = I Hom (X,Y),

where idHom (X,Y) is the identity functor in Hom (X,Y).– (Associativity of horizontal compositions): For each quadruple

X,Y,Z,T ∈ obC,

µX,Z,T ◦ (µX,Y,Z × I Hom (Z,T)) = µX,Y,T ◦ (I Hom (X,Y) × µY,Z,T)

This definition can be visualized with the use of diagrams.

Fix a 2-categoryC with objectsX,Y,Z andT. An object f in the categoryHom (X,Y) is a 1-morphism ofC:

Xf // Y .

A morphismα of the category Hom (X,Y) is a 2-morphism ofC:

α

��X

f

##

f ′

;; Y

36 GEILLAN ALY

Vertical composition of 2-morphisms is associative:

[γ ◦ (β ◦ α) = (γ ◦ β) ◦ α].

Given the diagram

α��X

f

!!

g

==h //β��

Y there exists β◦α

��X

f

##

g

;; Y

Horizontal compositions of 2-morphism is associative:

[(γ ∗ β)α = γ ∗ (β ∗ α)].

Given the diagram

α

��β

��X

f

##

f ′

;; Y

g

##

g′

;; Zthere exists β∗α

��X

g◦ f

##

g′◦ f ′

;; Z

There exists an identity for 2-morphisms. ForX, Y, Z ∈ objC, f , g: X→ Yandh: Y → Z all 1-morphisms. For each 1-morphismf , there exists a 2-morphismI f such thatα◦ Ig = I f ◦α = α for someα : g⇒ f . Compositionalso respects the identity in thatIh ∗ I f = Ih◦ f .

Compatibility of horizontal and vertical compositions of 2-morphisms. Fora given diagram

α�� β��X

f

!!

f ′′

==f ′ //α′��

Y

g

!!

g′′

==g′ //β′��

Z

then (β′ ◦ β) ∗ (α′ ◦ α) = (β′ ∗ α′) ◦ (β ∗ α).

References

[1] Lowell Abrams,Two dimensional topological quantum field theories and frobeniusalgebras, J. Knot Theory and its Ramifications5 (1996), 569–587.

[2] Michael F. Atiyah,Topological quantum field theory, Publication mathematiques del’I.H. E.S.68 (1988), 175–186.

[3] David A. Cox and Sheldon Katz,Mirror symmetry and algebraic geometry, AmericanMathematical Society, Rhode Island, 1999.


[4] Charles W. Curtis and Irving Reiner,Representation theory of finite groups and asso-ciative algebras, Interscience Publishers, New York, 1962.

[5] Daniel S. Freed,Quantum groups from path integrals, Particles and Fields (Gor-don W. Semenoff and Luc Vinet, eds.), Springer-Verlag, New York, 1998.

[6] Peter B. Gilkey,The geometry of spherical space form groups, World Scientific Pub-lishing Company, 1989.

[7] Tomas L. Gomez,Algebraic stacks, arXiv:math/9911199v1, November 1999.[8] Joachim Kock,Frobenius algebras and 2d topological quantum field theories, Cam-

bridge University Press, New York, 2004.[9] Aaron D. Lauda and Hendryk Pfeiffer, Open-closed strings: Two-dimensional ex-

tended tqfts and frobenius algebras, (2008).[10] Ruth Lawrence,An introduction to topological field theory, Proc. Symp. Appl. Math.

51 (1996), 89–128.[11] John Lee,Introduction to smooth manifolds, Springer-Verlag, New York, 2002.[12] William S. Massey,A basic course in algebraic topology, Springer-Verlag, New York,

1991.[13] Gregory W. Moore and Graeme Segal,D-branes and k-theory in 2d topological field

theory, arXiv:hep-th/0609042v1, 2006.[14] Graeme Segal, Lecture 1, topological field theories,

http://www.cgtp.duke.edu/ITP99/, July-August 1999.[15] , The definition of conformal field theory, Topology, Geometry and Quantum

Field Theory; Preceedings of the 2002 Oxford Symposium in Honour of the 60thBirthday of Graeme Segal (Ulrike Tillmann, ed.), CambridgeUniversity Press, 2004.

[16] Stanford University,Math 283: Topological field theories (class notes), Winter 2008.[17] Edward Witten,Topological quantum field theory, Communications in Mathematical

Physics117(1988), 353–386.

Documents

CAPS AND PANTS: TOPOLOGICAL QUANTUM FIELDgeillan/research/thesis.pdfCAPS AND PANTS: TOPOLOGICAL QUANTUM FIELD THEORIES IN DIMENSION 2 1 Abstract. A two-dimensional Topological Quantum