Andrei Caldararu

RTG Graduate Summer School

Geometry of Quantum Fields and Strings

UPenn, Philadelphia

Notes by Orit Davidovich

June, 2009

1 Lecture 1

1.1 Keller’s Motivation.

Definition: Two complexes C•1 , C•2 are said to be quasi isomorphic if there

exists a third complex C• and maps

C•

f gC•1 C•2

such that H•(f), H•(g) are isomorphisms. We will denote quasi-isomorphism

by C•1q∼= C•2 .

Basic Question: When are two complexes quasi isomorphic? This question issimilar in nature to the question of when two topological spaces are homotopyequivalent. Two complexes C•1 , C

•2 are not quasi isomorphic if Hi(C•1 ) 6∼= Hi(C•2 )

for some i. But the converse is not true, Hi(C•1 ) ∼= Hi(C•2 ) for all i does not im-

ply C•1q∼= C•2 . This is a subtle point, C•1 is quasi-isomorphic to C•2 if there exists

an actual chain map that induces an isomorphism on the level of cohomologies.

Counter Example: Let A = k[x, y] and consider the following two complexesof A-modules.

C•1 : 0→ A⊕2 → A→ 0

C•2 : 0→ A0→ k → 0

The map A⊕2 → A is defined to be (f, g) 7→ xf + yg. The cohomology groupsof these two complexes are isomorphic. We will show that they are not quasi

1

isomorphic. But using the basic definition is not going to work. Instead we usethe following characterization.

Claim: Every complex C• which only has two adjacent non-vanishing coho-mology groups, say, in degrees 0 and −1, is determined by H0(C•), H−1(C•)and a class α ∈ Ext2(H0(C•), H−1(C•)).

Exercise: Show that in the above claim α is a quasi-isomorphism invariant.

Note that every complex C• determines a class in Ext2(H0(C•), H−1(C•)) givenby the short exact sequence

0→ H−1(C•)→ C−1/Imd−2 d−1

−−→ ker d0 → H0(C•)→ 0

where di : Ci → Ci+1 is the differential. By the above claim this class fullycharacterizes C• up to quasi-isomorphism.

In this example both complexes have adjacent cohomology groups H0(C•i ) ∼= kand H−1(C•i ) ∼= A and each is determined by a class in Ext2(k,A) given by

α1 : 0→ A→ A⊕2 → A→ k → 0

α2 : 0→ Aid→ A

0→ kid→ k → 0

Geometrically speaking, A = OA2 and k = O(0,0) is the structure sheaf of thepoint (0, 0) ∈ A2. Two-step extensions can be computed using Serre Duality

Ext2A2(O(0,0),OA2) ∼= Ext0

A2(OA2 ,O(0,0))∨ ∼= k

It can be shown that α2 is trivial while α1 is not1. This implies C•1 6q∼= C•2 .

Detour: Let C• be a complex and assume Hn(C•) = 0 for all n > N . Wedefine the right-truncated complex τ≤NC• to be

· · · → CN−2 → CN−1 → ZN → 0→ · · ·

We have a map of complexes τ≤NC• → C• which induces an isomorphism onthe level of cohomology. Similarly, when Hm(C•) = 0 for all m < M we candefine the left-truncated complex τ≥MC•. We have a map C• → τ≥MC

• whichinduces isomorphism on the level of cohomology. In addition, we have maps ofcomplexes τ≤NC• → HN (C•)[−N ] and HM (C•)[−M ]→ τ≥MC

•.

Let C• be a complex with two adjacent non-vanishing cohomology groups, sayin degrees −1 and 0 and consider f : H−1(C•)[1] → τ≥−1C

• as above. Takingthe cone of f we have an exact triangle in the homotopy category

H−1(C•)[1] → τ≥−1C•

6 Cone(f)

1α1 is the Kozul resolution of k over A.

2

Explicitly, the cone of f is the complex

0→ H−1(C•)→ C−1/Imd−2 → C0 → C1 → · · ·

In particular, we have a map H0(C•)[0] → Cone(f) which induces an isomor-phism on cohomology. In the derived category we have an exact triangle

H−1(C•)[1] → C•

6 H0(C•)[0]

The morphismH0(C•)[0] 6→ H−1(C•)[1] gives a class in Ext2(H0(C•), H−1(C•)).

Moral of Story: Showing that two complexes are not quasi-isomorphic whenthey have isomorphic cohomology groups requires the use of more subtle in-variants and the one we just used cannot be pushed further to a more generalsetting. An A∞ structure, on the other hand, provides a complete set of invari-ants.

Claim: If we regard A as an A∞−algebra and C•1 , C•2 as A∞−modules over A

then C•1q∼= C•2 iff they are isomorphic as A∞−modules.

1.2 Kontsevich’s Motivation.

Theorem: [Bondal-van den Bergh; Toen - Vaquie] Let X be a complex mani-fold. Then X is algebraic iff Db(X) admits a split generator.

Fact: If E is a split generator for Db(X) then the derived category can berecovered from the A∞−algebra Ext•(E,E).

The analogy to keep in mind is: affine varieties to commutative rings are like al-gebraic varieties to A∞−rings. In the former case the correspondence is unique,in the later case it is not. Db(X) can have several split generators and X cannot be fully recovered from Db(X). Still, we think of any algebraic variety as“affine” in the A∞ world.

1.3 Stasheff’s Motivation.

Let X be a topological space with a base point ∗ and consider the space Ω(X, ∗)of loops based at ∗, that is, the space of maps γ : [0, 1] → X such thatγ(0) = ∗ = γ(1). We wish to consider Ω(X, ∗) as a group with multiplica-tion given by concatenating two loops and tracing each at twice the speed.

This multiplication is clearly not associative; the maps (γ1 γ2) γ3 and γ1 (γ2 γ3) are not identical. But they are homotopic, and we can prescribe a

3

γ2

γ1

γ3

γ1 γ2 γ3

γ1 γ2 γ3t = 0

t = 1

6

Figure 1: Composition of loops.

homotopy h(3) between them independent of γi. See for example fig. 1. Givenfour loops, we can homotope ((γ1 γ2) γ3) γ4 to γ1 (γ2 (γ3 γ4)) using h(3)

in more than one way, and we can prescribe a homotopy h(4) between those,which is again independent of γi. And so on and so forth, every homotopy weprescribe will satisfy higher homotopy associativity relations. This collection ofoperations m,h(3), h(4), . . . characterizes Ω(X, ∗) up to homotopy equivalence.

Theorem: [Stasheff] Yh∼= Ω(X, ∗) iff Y can be equipped with such collection

of operations.

An A∞−algebra is the algebraic analogue of this structure.

1.4 A∞ Algebras.

Almost Definition: A (flat) A∞ algebra is a vector space together with op-erations

mk : A⊗k → A , k ≥ 1

satisfying an infinite sequence of equations

Assocn :∑±ma(x1, . . . , xl,mb(xl+1, . . . , xl+b), xl+b+1, . . . , xn) = 0

where we sum over all a + b = n + 1 and l < a. On the level of an “almost”definition we ignore the issue of signs. We can represent each element in thesum by a tree with n leaves, one root and one internal edge.We write down the first few associativity equations.

Assoc1 : m1(m1(x0)) = 0Assoc2 : m1(m2(x, y))±m2(m1(x), y)±m2(x,m1(y)) = 0Assoc3 : m2(x,m2(y, z))±m2(m2(x, y), z)±m1(m3(x, y, z))

±m3(m1(x), y, z)±m3(x,m1(y), z)±m3(x, y,m1(z)) = 0

4

. . . . . .. .

ma

mb

x1 xl+1 xl+b xn

Figure 2: Tree diagram.

Remarks:• m1 is a differential on A so we will often denote d := m1 (Assoc1 is justd2 = 0). Also, we will often write xy for m2(x, y).• Assoc2 is the Leibnitz Rule d(xy) = (dx)y + x(dy).• By Assoc3 we have

(xy)z − x(yz) = dm3(x, y, 3)±m3(dx, y, z)±m3(x, dy, z)±m3(x, y, dz)

This means m2 descends to an associative multiplication on H•(A, d).

Immidiate Examples:• Any associative algebra is an A∞ algebra with m2 the only non-vanishingoperation.• A dg-algebra is an A∞ algebra with mk = 0 for all k ≥ 3.

Interesting Example: Let A be an associative algebra. Let α be a Hochschildk-cochain, meaning a linear map α : Ak → A, and k ≥ 3. Assume α is closed.This means

x1α(x2, . . . , xk+1)− α(x1x2, x3, . . . , xk+1)+α(x1, x2x3, . . . , xk+1)− . . .± α(x1, . . . , xk)xk+1 = 0

We construct an A∞ algebra, Aα, with the same underlying vector space andm2 = ·, mk = α. The fact that α is closed implies Assock (all other associativityequations are obviously satisfied).

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Curved A∞ Algebras: To get a curved A∞ algebra we add a map

m0 : k → A

Let W := m0(1). We have an additional associativity constraint

Assoc0 : dW = 0

The other associativity constraints need to be readjusted. In particular

Assoc1 : d2(x)± xW ±Wx = 0

This means we no longer have a differential. An immediate example is givenby an associative algebra A with an element W ∈ Z(A) and multiplication asbefore.

Grading: There are two types of graded A∞ algebras that appear in the lit-erature. One is a Z-graded and the other is a Z/2Z-graded vector space. Theoperations are defined to be

mk : A⊗k → A[2− k]

Note that m2 does not shift. The differential shifts by 1, as expected.

1.5 Co-free Co-Algebra.

Let V be a vector space. Consider the free algebra on V to be TV = ⊕k≥0V⊗k

with multiplication given by

(x1| · · · |xk) · (xk+1| · · · |xn) = (x1| · · · |xn)

We also have a co-free co-algebra on V . It is given by the same underlyingvector space CV = ⊕k≥0V

⊗k with co-multiplication given by

∆(x1| · · · |xn) =n∑k=0

(x1| · · · |xk)⊗ (xk+1| · · · |xn)

Note this does not make the underlying vector space into a bi-algebra2. Forx ∈ V ⊗1 and y ∈ V ⊗1 we have

∆(x|y) = 1⊗ (x|y) + x⊗ y + (x|y)⊗ 1∆(x)∆(y) = (1⊗ x+ x⊗ 1)(1⊗ y + y ⊗ 1)

= 1⊗ (x|y) + 2x⊗ y + (x|y)⊗ 1

which shows ∆ is not a morphism of algebras.

2There are other products / co-products called the shuffle product / co-product which make⊕k≥0V

⊗k into a bi-algebra.

6

+ +f f f f

Figure 3: Defining f .

Theorem: There exists a natural isomorphism

HomVect(V, TV ) ∼−→ Der(TV,TV)

where Der stands for derivations.

This natural isomorphism is given by sending f : V → TV to a derivation

f : (x1| · · · |xn) 7→∑k

(x1| · · · |f(xk)| · · · |xn)

For example, a map f : V → V ⊗3 can represented as a tree with one branchand three roots. The corresponding f will be evaluated on V ⊗3 and take itsvalues in V ⊗5 as in fig. 3.

Checking f is a derivation means showing the following diagram commutes.

A⊗A ·−→ A

f ⊗ 1 + 1⊗f ↓ ↓ f

Af−−→ A

(1)

Theorem: There exists a natural isomorphism

HomVect(CV, V ) ∼−→ CoDer(CV,CV)

where CoDer stands for co-derivations.

Co-derivations can be defined by inverting all arrows in equation 1 and replacingmultiplication by co-multiplication. To define the natural isomorphism in thiscase we just turn fig. 3 up-side down.

Theorem: An A∞ structure on V is a square-zero co-derivation on CV .

Given a co-derivation on CV we get a map f : CV → V which translatesinto a collection of operations f (k) : V ⊗k → V . The requirement that 2 = 0

7

D0

HomVect(CV, V )

D2 = 0

D1

Figure 4: Deformation of A∞ structure.

translates into the Assocn equations. Taking 2 means we are considering treeswith exactly one internal edge.

Deformations of A∞ Algebras: Given the above theorem, we can think ofthe ‘space’ of A∞ structures on an underlying vector space V as a ‘subspace’ ofHomVect(CV, V ) ∼= HomCoDer(CV,CV ) given by the locus of points satisfyingD2 = 0.

We wish to understand “TA(D2 = 0)” − the tangent space to D2 = 0 at asquare-zero derivation D0 corresponding to an A∞−algebra A. A first orderdeformation of D0 is of the form D0 + ~D1 satisfying (D0 + ~D1)2 = 0 up tofirst order in ~. This implies the tangent space is given by

TA(D2 = 0) = D1 ∈ HomCoDer(CV,CV ) |D0D1 +D1D0 = 0

This is equivalent to the Hochschild complex of the A∞ algebra A.

2 Lecture 2

2.1 Derived Categories.

Motivation: Consider a simply connected manifold X with its de-Rham com-plex (Ω(X), d). Knowing the complex with a bit of extra data completely de-termines the rational homotopy type of X. On the other hand, just knowingH•DR(X) is not enough. The moral of this example is that working with chaincomplexes, rather than with their cohomology groups, retains more information.

Categories of Complexes: Let A be an abelian category. This could bethe category of modules over a ring or the category of coherent sheaves over ascheme. We construct the category dg(A). Its objects are complexes of objectsin A . Its Hom sets are differential graded vector spaces defined by

Homidg(A)(C

•, D•) :=∏j

HomA(Cj , Dj+i)

df• := dD• f − f dC•

dg(A) is a dg-category, meaning it is a category enriched over dgVect .

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Homotopy Category: We define the homotopy category Ho(A) to be thecategory whose objects are the same as those of dg(A) and

HomHo(A)(C•, D•) := H0(Homdg(A)(C•, D•))

To understand what the morphisms ofHo(A) are take a 0-cocycle f• in Hom0(C•, D•).It is a map f• : C• → D• such that df• = dD•f

• − f•dC• = 0, namely, f• is achain map. Modding out by 0-coboundaries amounts to modding out by homo-topy. Hence Ho(A) is the category whose objects are chain complexes of objectsin A and whose morphisms are homotopy classes of chain maps. Note that

Hn(Homdg(A)(C•, D•)) = H0(Homdg(A)(C•, D•[n]))

therefore knowing the shift functor allows us to compute all cohomology groupsof Hom spaces in dg(A).

Definition: A morphism in Ho(A) is called a quasi-isomorphism if H•(f•) isan isomorphism.

Example: Let A = Z-mod. Consider the following chain map.

0 → Z ×2→ Z → 0↓ ↓

0 → 0 → Z/2Z → 0

The map induces an isomorphism in cohomology. But there is no chain mapgoing in the other direction. The map is a quasi-isomorphism but is not invert-ible in Ho(A). We will construct a category of chain complexes in which thismap is invertible.

Localization: Let C be a category and S a collection of morphisms in C ).S−1C is a category together with a functor C → S−1C such that the followingholds.

• The image of any morphism in S is an isomorphism in S−1C

• S−1C satisfies the following universal property: for any other categoryD and a functor C → D with the above property, there exists a uniquefunctor S−1C → D making the appropriate diagram commute.

We can construct S−1C explicitly. Its object are the same as those of C. Mor-phisms from A to B are given by ‘zig-zag’ diagrams.

X1 X2 · · · Xn

A Y1 Y2 · · · Yn−1 B

where arrows pointing left are in S (the ones pointing right are arbitrary).

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The Derived Category: Consider the homotopy category Ho(A) and take Sto be the collection of all quasi-isomorphisms in Ho(A). We define the derivedcategory of A to be D(A) := S−1Ho(A).

Both Ho(A) and D(A) are additive categories, but they are no longer abelian,for example, we added new morphisms to D(A) but not necessarily their kernelsand co-kernels.

Triangulated Category: Both Ho(A) and D(A) are triangulated categories.Roughly speaking, a triangulated category is a category equipped with an auto-equivalence, a ‘shift’, denoted by “[1]”, and a distinguished collection of dia-grams called distinguished triangles of the form A→ B → C → A[1] satisfyingsome axioms. We often write down distinguished triangles as

A → B6

C

For any morphism Af−→ B there exists a distinguished triangle A

f−→ B →C → A[1] (C need not be unique), in particular, A id−→ A → 0 → A[1] is adistinguished triangle. Given a triplet of isomorphisms making all rectanglescommute

A → B → C → A[1]↓ ↓ ↓ ↓A′ → B′ → C ′ → A′[1]

the bottom line is also a distinguished triangle.

For any morphism class C•f•−→ D• in Ho(A), we can define the cone of f• to

be a complex (Conef•)i = Ci+1 ⊕Di with differential

diCone =(di+1C 0f i+1 diD

)There is a distinguished triangle in Ho(A) given by inclusion / projection ontoeach summand

C•f•−→ D•

6 Cone(f•)

We think of Cone(f•) as encoding the kernel and co-kernel of f•.

Given a short exact sequence 0→ A→ B → C → 0 inA we have a distinguishedtriangle in D(A)

A[0] → B[0]6

C[0]

10

arising from0 → 0 → C → 0↑ ↑ ↑ ↑0 → A → B → 0↓ ↓ ↓ ↓0 → A → 0 → 0

where the top arrow is a quasi-isomorphism. The above triangle is not a distin-guished triangle in Ho(A).

Enhancement: We wish to argue that D(A) has an enhancement, namely,that there exists a dg-category DG(A) such that Ho(DG(A)) ∼= D(A).

Theorem: If C• and D• are complexes of injectives in A , then a chain mapC• → D• is a quasi-isomorphism if and only if it is invertible up to homotopy.

We take DG(A) to be dg(Inj(A)).

2.2 Homological Perturbation Lemma.

Let A• be a (flat) A∞ algebra (A• is Z -graded). We denote d := m1 andconsider the complex (A•, d). The problem we are trying to solve is decompos-ing A• = B• ⊕ (something), such that B• is a smaller complex encoding thecohomology of A• . The motivation for this arises from Hodge theory where thecomplex of harmonic forms is a subcomplex of Ap,q(X) and encodes its coho-mology, namely, there is a unique harmonic form in every class.

From now on we will drop bullets from our notation; all objects and maps arein the category of complexes. Let π : A→ A be a chain map such that π2 = π.This decomposes A into A = Imπ ⊕ kerπ. We denote B = Imπ.

Assume π is homotopy equivalent to the identity, namely, there exists H : A→A[1] such that id−π = dH+Hd. Denote the inclusion i : B → A, the projectionp = i π, and we have π = i p.

We will define an A∞ structure on B. The differential dB will be given by therestriction of dA to B. The multiplication will be given by mB

2 := p mA2 (i⊗i).

To define higher multiplication, consider T a planar tree with n leaves and oneroot. All internal vertices are of valence at least 3. Define mT

n : B⊗n → B[2−n]by labeling all the leaves of T by the embedding i, the root of T by the pro-jection p, all internal vertices of T by mA

k (for the appropriate k), and internaledges by the homotopy H. An example of such labeling appears in fig. 5.

11

iii

p

Hm^A

m^A

i i i

p

Hm^A

m^A

i i i

p

m^A

Figure 5: Labeled trees mT3 .

Define the higher multiplication on B to be

mBn :=

∑T

±mTn

Note that mBn has the right degree. Let VT be the set of vertices of T , let ET

be the set of edges and EintT its subset of internal edges, then

deg(mTn ) = −|Eint

T |+∑v∈VT

2− (deg(v)− 1)

= −|EintT |+ 3|VT | − (2|Eint

T |+ n+ 1)

= 3(|VT | − |EintT |)− n− 1

= 2− n

Theorem: dB ,mBn form a A∞ structure on B.

Proof: The proof will be entirely combinatorial. We will demonstrate it formB

3 . We will ignore signs completely (signs should work out the way we expectthem to). Remember we need to show AssocB3 = 0. To make things clearer, wewill denote all operations in A by bullets, all H’s by circles, and all π’s by stars.Assoc3(B) is given in fig. 6 (operations in B are not marked). We expand

+_ +_ +_ +_ +_

(a) (b) (c) (d) (e) (f)

Figure 6: AssocB3 .

Assoc3(B) in terms of operations in A, π and H (see fig 7). We use the factthat π is homotopy equivalent to the identity. Pictorially we can represent it asin fig. 8. We plug in the homotopy into Assoc3(B). This introduces three terms

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* *(e)

(a)

(b)

(d)

(f)

(c)

&

Figure 7: Expanding AssocB3 in terms of mAn ,π and H.

replacing each (e),(f) in fig. 7. We find that the terms in the right most columnof fig. 7 and a couple of terms from (e),(f) cancel out thanks to Assoc3(A). Weare left with trees appearing in fig. 9. Note that now each black dot on a leafor a root is closest to one distinguished vertex. We re-group all trees accordingto such common distinguished vertex. But now, each row in fig. 9 vanishes dueto Assoc2(A). That finishes the proof of the vanishing of Assoc3(B). Similarconsiderations work for higher order Assoc.

Theorem: A and B are quasi-isomorphic as A∞ algebras3. In particular, if wetake B := H•(A) and choose a (not canonical) splitting A = B ⊕ (something),then A is quasi-isomorphic to H•(A).

3A map of A∞ algebras is more than a map of the underlying Z-graded vector spaces. Amap of A∞ algebras B → A corresponds to a map of co-algebras CB → CA. This map iscompletely determined by a collection of maps B⊗n → A[−] (satisfying some conditions).

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=+ + *

Figure 8: Homotopy.

Application: Given a space X we can construct a “big” dg-category DG(X)associated to it. We can then use the homological perturbation lemma to re-place DG(X) by a “small” A∞ category, A∞(X). We pay the price of addinghigher order multiplications.

Theorem: If E is a split generator of Db(X) := Db(Coh(X)) then the A∞algebra EndDb(X)(E) carries all information of A∞(X).

Split Generators: Roughly speaking, an object E in a triangulated categoryis a split generator if applying shifts, cones and direct summands to it giveseverything.

Example: A result due to Beilinson states that O,O[1], . . . ,O[n] generateDb(Pn). This is no longer true for an elliptic curve E. To see why, consider thesurjection

Db(E)→ K0(E)

C• →∑i

(−1)i[Ci]

This map has the following properties: a distinguished triangle A→ B → C →A[1] maps to [A] − [B] + [C] = 0 and a shift A[−1] maps to −[A]. If a finiteset of objects was to generate Db(E) their images would then generate K0(E).But K0(E) is not finitely generated.

On the other hand, we have a double cover π : E → P1 (branched over fourpoints). We can use the fact that π∗(C) is generated by O,O[1] in Db(P1), andthe fact that any C ∈ Db(E) is a direct summand of π∗π∗(C), to argue thatπ∗O, π∗O[1] split generate Db(E).

14

Figure 9: Using homotopy and re-grouping.

3 Lecture 3

3.1 Operads.

Definition: An operad O in a monoidal category C is a collection of objectsO(n) ∈ C for all n ≥ 1, that carry a Σn action4, together with compositionoperators

: O(k)⊗O(n1)⊗ · · · ⊗O(nk)→ O(n1 + . . .+ nk)

satisfying some axioms (associativity and identity). A map of two operads is acollection of morphisms that preserve the structure.

Associative Algebras: Denote the multiplication of an associative algebra bya binary tree with two inputs and one output. Multiplication of n elements canbe represented by a binary tree with n inputs and one output. This represen-tation depends on the order in which we choose to multiply.

4Σn is the permutation group on n letters.

15

= =

Figure 10: Associativity in trees.

For example, we have two binary trees with three inputs. The fact that multi-plication is associative tells us that the two trees are one and the same. Thisidentity is expressed in fig. 10.

The operad describing associative algebras is an operad in Sets. Each set O(n)contains trees with n inputs, one output, and one vertex of valence n+1, whoseleaves are labeled by 1, . . . , n (Σn acts on the labels). We denote this operadby Ass.

The operad describing commutative algebras is also an operad in Sets withO(n) = ∗ (the action of Σn on O(n) is trivial). We can think of O(n) ascontaining one tree with n inputs as above and no labeling. We denote thisoperad by Com.

Another important example arises from objects in monoidal categories. Let Vbe such object. Define an operad End(V ) to be

End(V )(n) := HomC(V ⊗n, V )

with composition given by

f, f1, . . . , fk 7→ f f1 ⊗ · · · ⊗ fk

Definition: Let O be an operad in C. An algebra over O is an object V in acategory enriched over C equipped with a map of operads O → End(V ).

An associative algebra is an algebra over Ass and a commutative algebra is analgebra over Com. The advantage of this terminology is that it allows us totalk about associative or commutative algebras in Sets, topological spaces andso forth.

More Examples: The free operad with one multiplication is an operad in Setswith

Free(n) := binary trees with n leaves and one root

Unlike in Ass, here we do distinguish between different orderings of multiplica-tion, for example, see fig. 11.

16

=

Figure 11: Distinguished orderings of multiplication.

The Lie operad, Lie, is given as a quotient of the free operad by the anti-commutativity and Jacobi identity relations. To make sense of these relations,we take Free to mean the vector space spanned by all binary trees with n leavesand one root. Hence Lie is an operad in vector spaces.

The Little Discs Operad: We define an operad Dk in topological spaces withDk(n) being space consisting of n k-dimensional discs embedded in the unit k-dimensional disc Dk (see fig. 12). Composition maps are given by rescaling discsand sticking them one into the other.

Figure 12: A point in D2(3).

If we think of each little disc as determined by its center and radius we canembed

Dk(n) ⊂ Dk × · · · × Dk︸ ︷︷ ︸n

× (0,∞)× · · · × (0,∞)︸ ︷︷ ︸n

where discs are not allowed to intersect. Up to homotopy, Dk(n) is the config-uration space of n distinct points in Dk

Dk(n) ∼=h Dk × · · · × Dk︸ ︷︷ ︸n

\all diagonals

Therefore π1(D2(n)) = Bn (the braid group on n strands).

Passing to Homology: Given any operad O in topological spaces we canconstruct a new operad H∗(O) in graded vector spaces or C∗(O) in differentialgraded vector spaces by taking homology or chains of each O(n) respectively.

17

Let us check what H∗(D1) should be. H∗(D1(n)) is concentrated in degree 0for all n and is a vector space of dimension n!. It is not hard to guess thatH∗(D1) = Ass when considering Ass as an operad over vector spaces by takingthe vector space generated by Ass(n) in each degree. If instead we take chainsthen C∗(D1) is the A∞ operad.

Let us go one dimension up. For n = 1, H∗(D2(1)) = H∗(D2) = k[0]. Thismeans an algebra over H∗(D2) carries a unary operation in degree 0. Now con-sider n = 2. To understand what D2(2) looks like note that any two pointsin D2 determine a line intersecting the boundary ∂D. We claim H∗(D2(2)) ∼=H∗(S1) ∼= k[0] ⊕ k[1]. This means an algebra over H∗(D2) carries two binaryoperations, one in degree 0 (multiplication) and the other in degree 1 (bracket).Note D2(n) is connected for all n. This implies our multiplication is associativeand commutative. The bracket is in fact a Lie bracket and is compatible withmultiplication in the sense that [xy, z] = x[y, z] + y[x, z].

Gerstenhaber Algebra: A Gerstenhaber algebra (G-algebra) is a graded vec-tor space, A, equipped with associative commutative multiplication of degree 0and a Lie bracket of degree −1 satisfying [xy, z] = x[y, z] + y[x, z]. A Gersten-haber algebra is an algebra over H∗(D2).

Example: Let X be a smooth manifold. Let A = ⊕iΓ(ΛiTX) be the al-gebra of poly-vector fields. A is a G-algebra. The multiplication is givenby the wedge product and the bracket is the Lie bracket extended to A via[v1 ∧ v2, v3] = v1 ∧ [v2, v3] + v2 ∧ [v1, v3].

Deligne Conjecture: Let A be an associative algebra, then HH∗(A) is a Ger-stenhaber algebra. In other words, if A an algebra over H∗(D1) then HH∗(A)is an algebra over H∗(D2).

In fact, a more general statement is true.

Theorem:[Kontsevich - Lurie] If A is an algebra over H∗(Dn), then HH∗(A)is an algebra over H∗(Dn+1). This statement is also true at chain level, namely,if A is an algebra over C∗(Dn), then C∗(A) is an algebra over C∗(Dn+1), whereC∗(A) is the Hochschild cochain complex of A, appropriately defined.

3.2 Hochschild Cohomology.

We already encountered Hochschild cohomology in section 1.4 where we con-structed an example of an A∞−algebra using a Hochschild cocycle.

Let A be an associative algebra over a field k. We define the vector space ofp-chains to be

Cp(A) := Homk(A⊗p, A)

18

We define a differential b : Cp(A)→ Cp+1 to be

(bf)(a1, . . . , ap+1) = a1 · f(a2, . . . , ap+1)− f(a1 · a2, . . . , ap+1) + · · ·· · · ± f(a1, . . . , ap) · ap+1

We claim (C•(A), b) is a chain complex. We denote HH•(A) := ker b/Imb.

Low Degree Comology: Let us consider the first couple of terms in HH•(A).Let a ∈ Homk(k,A) be a 0-cochain. We identify it with a(1) and b(a) : x 7→xa−ax. A 0-cocycle is therefore an element of the center and HH0(A) = Z(A).Let f ∈ Homk(A,A) be a 1-cochain and (bf)(x, y) = xf(y)− f(xy) + f(x)y. A1-cocycle is therefore a derivation of A. A 1-coboundary is given by the naturalmap [ , ] : A→ Der(A) taking a to [−, a]. We have HH1(A) = Der(A)/[−, A].

When A is commutative HH0(A) = A and HH1(A) = Der(A). A prototypicalexample is A = C∞(X) and HH0(A) = A = Γ(Λ0TX), HH1(A) = Der(A) =Γ(Λ1TX). In higher degrees we have HHm(A) = Γ(ΛmTX). We already arguedthat in this example HH•(A) has a structure of a G-algebra.

Deformations: Given an algebra A, we wish to consider a family of algebrasA~, with the same underlying vector space, such that A ∼= A0. In other words,we wish to deform the multiplication on A to

x ∗ y = xy + ~c2(x, y) + ~2c3(x, y) + . . .

Up to first order, the associativity of x ∗ y is equivalent to dc2 = 0. It is nothard to show that deformations via 2-coboundaries are trivial up to first order.In other words, HH2(A) controls deformations of the multiplication on A up tofirst order.

Multiplication: We define multiplication in C•(A)

Cp(A)⊗ Cq(A)→ Cp+q(A)(f · g)(x1, . . . , xp+q) := f(x1, . . . , xp)g(xp+1, . . . , xp+q)

We haveb(f · g) = (bf) · g + (−1)pf · (bg)

We spell out the definition for b(f · g) in fig. 13. The top row represents (bf) · gand the bottom row represents f · (bg). As a result this multiplication descendsto a multiplication in HH•(A)5.

Two path components inD1(2) map into two operations in End(C•(A)), namely,f · g and g · f . When embedded into D2(2), they belong to the same path com-ponent. Up to higher homotopy, they are connected by two paths h and h′, one

5HH•(A) is a (graded) commutative algebra as a consequence of Gerstenhaber’s theorembelow.

19

....

f g

m

....

f g

m

....

f g

m

....

f gm

....

f gm

.. ..

f g

m

b(fg)= + ...

...+_ +_

Figure 13: Leibnitz rule in C•(A).

going clockwise and the other going counter-clockwise. By abuse of notation wedenote their image in HomdgVect(C•(A)⊗2, C•(A)[−1]) by the same letters.

Explicitly, given f ∈ Cp(A) and g ∈ Cq(A) we define hf,g ∈ Cp+q−1(A) to be

hf,g(a1, . . . , ap+q−1) :=∑

1≤i≤p

(−1)(i−1)qf(a1, . . . , g(ai, . . . , ai+q−1), . . . , ap+q−1)

and h′f,g := hg,f . A theorem due to Gerstenhaber states that

hf,bg − bhf,g + (−1)q−1hbf,g = (−1)q−1 (g · f − (−1)pqf · g)

Bracket: We define a bracket in C•(A) by

[f, g] := hf,g − (−1)pqhg,f

descending to a Lie bracket on cohomology.

3.3 Batalin-Vilkovisky Algebras.

Definition: A Batalin-Vilkovisky algebra (BV-algebra) is a graded vector space,A, equipped with associative commutative multiplication of degree 0 and a sec-ond order differential operator ∆ of degree −1, such that the associated bracketis a Lie bracket.

A second order differential operator is an operator such that the associatedbracket

[x, y] := ∆(xy)−∆(x)y − x∆(y)

is a derivation in each variable.

A BV-algebra is a G-algebra in which the Lie bracket comes from a second orderdifferential operator and that operator is part of the data.

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H∗(D1) Associative algebraC∗(D1) A∞ algebraH∗(Dfr

1 ) Frobenius algebraC∗(Dfr

1 ) Cyclic A∞ algebraH∗(D2) Gerstenhaber algebraC∗(D2) Homotopy Gerstenhaber algebraH∗(Dfr

2 ) Batalin-Vilkovisky algebraC∗(Dfr

2 ) Homotopy Batalin-Vilkovisky algebra

Table 1: Dictionary of Operads and their Algebras.

Example continued: Let X be a smooth manifold and A = ⊕iΓ(ΛiTX) thealgebra of poly-vector fields. As discussed in section 3.1, A is a G-algebra. IfX is orientable, and we choose a no-where vanishing section of ΛtopT ∗X , weget an identification Γ(ΛiTX) ∼−→ Γ(Λn−iT ∗X). The operator ∆ is given by thecomposition

Γ(ΛiTX) ∼−→ Γ(Λn−iT ∗X) d−→ Γ(Λn−i+1T ∗X) ∼−→ Γ(Λi−1TX)

making ∆ an enhancement of the Lie bracket.

Framed Little Discs Operad: This operad is the same as the little discsoperad only we add a point on the boundary of every boundary circle and re-quire that points match when we compose. We denote this operad by Dfr

2 .H∗(Dfr

2 (1)) = H∗(S1) = k[0] ⊕ k[1]. An algebra over H∗(Dfr2 ) is equipped with

a unary operation ∆ in degree −1. It is not hard to guess ∆ is an enhancementof the Lie bracket.

Claim: An algebra over H∗(Dfr2 ) is a BV-algebra.

Question: By Deligne’s conjecture, HH•(A) is a G-algebra for every asso-ciative algebra A. What restriction should we put on A for HH•(A) to be aBV-algebra?

For our prototypical example, we needed X to orientable, i.e., we needed a triv-ialization of ωX = ΛtopT ∗X . In the world of algebraic geometry, a trivializationof ωX is the Calabi-Yau condition.

Theorem: If A is Frobenius algebra then HH•(A) is a BV-algebra. Moreover,if A is a cyclic A∞ algebra, then C•(A) is an algebra over C∗(Dfr

2 ).

We will discuss cyclic A∞ algebras in lecture 5 in connection with open 2-dimensional topological conformal field theories. A dictionary of operads andtheir associated algebras appears in table 1.

21

4 Lecture 4

4.1 Topological Field Theory.

Fix a collection of labels Λ whose elements are referred to as branes.

Definition: We define the category of strings, denoted as Strng. The objects ofStrng are finite ordered sets of oriented circles and intervals IA,B with endpointslabeled by elements A,B ∈ Λ. The circles are referred to as closed strings, andthe intervals are referred to as open strings.

A morphism between an ‘incoming’ ordered set and an ‘outgoing’ one is thefollowing data: an oriented surface with boundary split into two componentsmarked ‘in’ and ‘out’ (equipped with induced orientation), an orientation re-versing embedding of the ‘incoming’ set of strings into the ‘in’ component, andan orientation preserving embedding of the ‘outgoing’ set of strings into the‘out’ component. All this is defined up to diffeomorphism of the surface relativeto the boundary.

Those parts of the boundary that are not in the image of the above embeddingsare referred to as free boundaries. The free boundary components are labeledby elements of Λ.An example of a morphism in Strng appears in fig. 14. It has two bound-ary components with embedded incoming closed string and embedded outgoingopen string. The outgoing open string starts and end at the same brane (freeboundary segment). This imposes the same ’boundary condition’ at the end-points of the open string. One can think of the morphism in fig. 14 as describingthe time evolution of a closed string into an open string. At time t = t′ thetransition from closed to open occurs.

A A

A A

A

t=0 t=Tt=t'

Figure 14: A morphism in Strng

Composition in Strng is given by gluing. Strng is a symmetric monoidal cate-gory where tensor product is given by disjoint union. It is also rigid in the sensethat objects have duals, namely, the same manifold with reversed orientation.

Claim: Every morphism in Strng is a composition of morphisms in fig. 15 andtheir duals.

22

(a) Pc (b) Dc

(c) Ic

A

B

C

(d) Po

A

B

(e) Io

A

A

A

(f) T

Figure 15: Generators of morphisms in Strng

Fig. 15(a): Pc is a genus 0 surface with 2 incoming and 1 outgoing closedstrings.

Fig. 15(b): Dc is a disc with 1 incoming closed string and no outgoing strings.

Fig. 15(c): Ic is a genus 0 surface with 1 incoming and 1 outgoing closed string.

Fig. 15(d): Po is a disc with 2 incoming and 1 outgoing open strings.

Fig. 15(e): Io is a disc with 1 incoming and 1 outgoing closed string.

Fig. 15(f): T was discussed in fig 14.

Definition: The category of open strings, Strngo, is the full subcategory ofStrng with objects open strings only. The category of closed strings, Strngc, isthe full subcategory with objects closed strings only.

23

Definition: A 2-dimensional open-closed topological field theory (TFT) is amonoidal functor

Ψ : Strng→ Vectk

A 2-dimensional open TFT is a monoidal functor with Strngo as its domain.The same goes for a 2-dimensional closed TFT.

Let Ψ be a 2-dimensional open-closed TFT. From now on, when we say TFT wemean open-closed TFT. Let us see what this entails. Denote V := Ψ(S1). Wehave Ψ(Pc) : V ⊗V → V a multiplication on V . We also have Ψ(Dc) : V → k atrace on V . The composition Ψ(Dc Pc) endows V with a non-degenerate innerproduct. This makes V into a commutative Frobenius algebra.

Claim: A 2-dimensional closed TFT determines a commutative Frobenius al-gebra and vice versa.

For A,B ∈ Λ we have Ψ(IA,B) ∈ Vect. This makes Λ into a k-linear categorywith

HomΛ(A,B) := Ψ(IA,B)

Composition in Λ is given by

Ψ(Po(A,B,C)) : HomΛ(A,B)⊗HomΛ(B,C)→ HomΛ(A,C)

For every A ∈ Λ we have a trace map

HomΛ(A,A)T (A)tS1Dc−−−−−−−−→ k

and a perfect pairing

HomΛ(A,B)⊗HomΛ(B,A)Po(A,B,A)−−−−−−−→ HomΛ(A,A) −→ k

This implies HomΛ(B,A) ∼−→ HomΛ(A,B)∨. A k-linear category with such iden-tification is known as a Calabi-Yau category.

Claim: A 2-dimensional open TFT determines a Calabi-Yau category and viceversa.

When we combine the above two claims together we get:

Claim: A 2-dimensional TFT determines a pair, (V,Λ), where V is a commu-tative Frobenius algebra and Λ is a Calabi-Yau category.

For the converse to hold we need some compatibility relations between V andΛ. The whistle diagrams T (A) and T (A)∨ (see fig. 15) relate open and closedstrings. This means we need to have a pair of maps for every A ∈ Λ

T (A) : EndΛ(A)→ V

T (A)∨ : V → EndΛ(A)

24

The two maps are actually dual when we use the identifications V ∼−→ V ∨ andEndΛ(A) ∼−→ EndΛ(A)∨ given by the non-degenerate pairings in V and Λ. (Weabuse notation when denoting the surface T (A) and its corresponding linearmap by the same letters). These maps need to satisfy the following:

• The map T (A)∨ : V → EndΛ(A) is a ring homomorphism, and its imageis in the center of EndΛ(A).

• The Cardy condition holds. See fig. 16.

=

Figure 16: The Cardy condition.

Example: Let G be a finite group. We can construct a 2-dimensional TFTfrom G with values in VectC. Its associated commutative Frobenius algebra isV = C[G]G. Elements of V are group algebra elements fixed by the action G onitself by conjugation. These can be thought of as class functions with productgiven by convolution. The trace map is given by f 7→ 1

|G|f(e). The associatedCalabi-Yau category is Λ = RepCG. The two are related by

C[G]G ∼= End(idRepG)

that is, V is identified with endomorphisms of the identity functor of Λ. Acommon notation for the endomorphisms of the identity is Z(Λ) known as thecenter of Λ.

The maps arising from T (A) and its dual are given by

EndRepG(A)→ C[G]G

idA 7→ χA

C[G]G → EndRepG(A)η → ηA

where η ∈ C[G]G is considered as an endomorphism of the identity functor andηA is its restriction to a representation A.

4.2 Universal TFT.

Question: Every TFT gives rise to an open TFT Ψo by restriction to Strngo.This restriction is known as the open string sector of that TFT. The same goesfor the closed string sector Ψc. Given an open TFT, is there an open-closed

25

TFT with that given open TFT as its open sector?

We have two embeddings

io : Strngo → Strngic : Strngc → Strng

with their associated pullbacks, given by restriction of an open-closed TFT toits open or closed sectors. To answer the above question we would like to con-struct a left adjoint, (Lio)∗, to (io)∗. Given Ψo an open TFT, (Lio)∗(Ψo) willbe the universal open-closed TFT with open sector Ψo.

Theorem: If Λ is a 2-dimensional open TFT given by a Calabi-Yau cate-gory, then there exists a universal open-closed TFT whose open sector is givenby Λ and whose closed sector is given by (ic)∗(Lio)∗(Λ) := HH∗(Λ), whereHH∗(Λ) := Z(Λ) = EndΛ(idΛ).

Symmetric Frobenius Algebra: We consider the above theorem in the sim-plest case where Λ is a Calabi-Yau category with one object. In this caselabellings are redundant. A monoidal category with one object is simply a ring.We will denote it by the same letter Λ := EndΛ(•). Λ is a Frobenius algebra,not necessarily commutative. It is endowed with a symmetric pairing comingfrom the Calabi-Yau structure.

If we want to construct an open-closed TFT compatible with Λ we will needto define a vector space V and a map T (•) : Λ → V . In fact, for any diagramwith some number of open inputs and a closed output we need to construct suchmap. The most ‘universal’ thing we can do is define

V = Spank

all open-closed diagrams withopen inputs labeled by elementsλ ∈ Λ and one closed output

/ ∼

where we mod out by some relations. Given an open-closed diagram, we candefine its corresponding map, going from Λ⊗n to V , by taking an elementλ1 ⊗ · · · ⊗ λn to a vector in V given by the same diagram with n incomingopen strings labeled λ1 to λn.

The Cardy condition allows us to turn any open-closed diagram into a dia-gram with an open part glued to a whistle. The Cardy condition needs to beincorporated into V , hence we have

V = Spank

all open diagrams with open in-puts labeled by elements λ ∈ Λ& one whistle

/ ∼

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S T

Figure 17: A morphism with open part glued to a whistle.

But now, let us consider an open diagram glued to a whistle, for example, seefig. 17. In any open-closed TFT compatible with Λ, the map associated withT S in has to factor through Λ.

λ1 ⊗ λ2 ⊗ λ3TS−−−→ diagram ST with incoming open

strings labeled by λ1, λ2, λ3

S ↓ ||

λT−→ whistle diagram with incoming

open string labeled by λ

where λ is completely determined by operations in Λ. This implies the two vec-tors on the right-hand column are equal. Such identities need to be incorporatedinto V , therefore we have

V = Spank

all whistle diagrams with openstring labeled by elements λ ∈ Λ

/ ∼

Now V carries a multiplication induced from Λ. But the multiplication in Vhas to be commutative. We should have a surjection

Λ/[Λ,Λ]→ V

We argue that the universal TFT, with Λ its open sector, is given by the pair(Λ,Λ/[Λ,Λ]).

Note that the trace defined on Λ/[Λ,Λ] is not induced from Λ. The trace inV is defined by the diagram in fig. 18. Let us compute it explicitly. Given theperfect pairing in Λ we can choose an orthonormal basis ei. We can write

27

Figure 18: Trace in V

down the map associated to fig. 18 using the Cardy relation.

λ⊗ λ′ 7→∑i

λ⊗ ei ⊗ λ′ ⊗ ei

7→∑i

λei ⊗ λ′ei

7→∑i

trΛ(λeiλ′ei)

=∑i

〈λeiλ′, ei〉Λ

This is exactly the trace of the map x 7→ λxλ′.

Conclusion: Let Λ be a symmetric Frobenius algebra and assume 〈λ, λ′〉 :=Tr(x 7→ λxλ′) is a non-degenerate pairing on V := Λ/[Λ,Λ]. Then (Λ, V ) is theuniversal open-closed TFT with open sector Λ.

Note that HH0(Λ) = Λ/[Λ,Λ]. The Calabi-Yau condition allows us to identifyHochschild homology with co-homology. In particular, Z(Λ) ∼= Λ/[Λ,Λ]. Thisagrees with the theorem we previously stated.

For a general Calabi-Yau category we HHi(C) ∼= HHn−i(C). In fact, the Calabi-Yau condition gives an isomorphism on the level of chain complexes.

28

5 Lecture 5

5.1 Moduli Spaces of Curves.

Let Mg,n denote the moduli space of genus g stable curves with n markedpoints6. We should think of stable curves as surfaces that admit a metric withconstant curvature −1, i.e., hyperbolic. We have the following

M0,3 = ptM0,4 = P1\3 ptsM0,5 = P1 × P1\7 linesM1,1 = A1 (the j-line)

The Deline-Mumford compactification of the above is

Mo,3 = ptM0,4 = P1

M0,5 = P2 blown up at 4 pointsM1,1 = P1

The Deligne-Mumford compactification is constructed by adding degenerations.For example, when two punctures on a sphere collide we get a bubbling phe-nomena as in fig. 19. In the case of M1,1 we add a nodal curve given byx2 = y2(y − 1).

Figure 19: Bubbling.

5.2 Ribbon Graphs.

Definition: A Ribbon Graph (RG), Γ, is a graph with a cyclic ordering of halfedges around each vertex. We assume the valency at each vertex is at least 3.A few examples are given in figure 20.

6In what follows marked points will sometimes be labeled and sometimes not, this distinc-tion will be important when we discuss operads.

29

Figure 20: Ribbon Graphs.

Note that a ribbon graph is the same data as two permutations σ and τ inΣ

2×edges such that σ2 = id and has no fixed points. σ encodes how half edgesare associated. τ encodes the cyclic data at the vertices. For example, considera ribbon graph as in fig. 21.

1 2

3

4

5

6

Figure 21: Ribbon Graph Permutations.

The permutations associated to it are

σ =(

1 2 3 4 5 62 1 4 3 6 5

), τ =

(1 2 3 4 5 65 4 2 3 6 1

)We can identify

Vertices V of Γ = cycles of τ = (1, 5, 6)(2, 4, 3)Edges E of Γ = cycles of σ = (1, 2)(3, 4)(5, 6)Faces F of Γ = cycles of τσ = (1, 6, 2, 3)(4)(5)

We define the genus of Γ to be the number g such that 2 − 2g = V − E + F .The genus of the first three ribbon graphs in fig. 20 is zero, the genus of thelast one is 1. Note g is the smallest genus surface on which Γ can be drawnwithout intersecting edges. We also define the degree of Γ to be deg(Γ) =∑v∈V (valence(v) − 3). The degree of the first two ribbon graphs in fig. 20 is

zero, the degree of the third is 1.

RG Complex: We construct a complex with underlying complex vector spacegiven by the span of all ribbon graphs modulo the appropriate notion of isomor-phism. This vector space can be graded by the degree. We denote this graded

30

vector space by RG and define a map

d : RGn → RGn−1

dΓ =∑

Γ′/e=Γ

±Γ′

where we sum over all pairs (Γ′, e ∈ E(Γ′)) such that collapsing e in Γ′ to a pointproduces Γ. We claim that, given the right choice of signs, this is a differential.We denote this complex by (RG, d) and refer to it as the ribbon graph complex.The differential we just defined does not change the genus of Γ or the numberof faces (contracting an edge is a local operation). Hence we can fix g and Fand consider (RG(g, F ), d). The significance of ribbon graphs is apparent in thefollowing theorem.

Theorem: [Strebbel & Penner] The ribbon graphs complex (RG(g, n), d) isquasi isomorphic to C∗(Mg,n) for n ≥ 1.

Let us consider the case g = 0, F = 3. We have V − E = −1. Comparingedges to vertices we have 3V ≤ 2E. This implies V ≤ 27. We find two ribbongraphs in degree zero, and one ribbon graph in degree 1. The complex is givenby C → C2 → 0. The differential is shown in fig. 22. It is clear H∗(M0,3) ∼=H∗(RG(0, 3), d), though perhaps not a very convincing demonstration of thetheorem.

_d

Figure 22: Differential for RG(0, 3).

Metric RG: A metric ribbon graph is a ribbon graph with an assignment ofpositive length to each of its edges. We denote by Mcomb

g,n the moduli space of

metric ribbon graphs. We topologize it by gluing pieces of R#E(Γ)+ assigned to

each Γ ∈Mg,n via contraction of edges.

For every face f ∈ F (Γ) of a metric ribbon graph Γ we define its diameter

diam(f) :=∑e∈∂f

length(e)

7A similar computation shows RG(g, F ) is finite dimensional at each degree for all g, F .

31

We have a homeomorphism Mcomb0,3

∼−→ R3+ given by the diameter of each face.

Theorem: [Mumford, Penner, Strebbel] We have a homeomorphism

Mcombg,n

∼=Mg,n × Rn+

5.3 Topological Conformal Field Theory.

Definition: Fix a collection of labels Λ whose elements are referred to as branes.Let Strngcx denote the category with objects the same as Strng and morphismsare equipped with conformal structure (for the definition of Strng see section4.1). The category C∗(Strngcx) has the same objects as Strng and morphismsare given by

C∗(Strngcx)(X,Y) := C∗(Strngcx(X,Y))

Note that Strng = H0(C∗(Strngcx)), in other words, the path components ofStrngcx(X,Y) are determined by the topological type.

Definition: A conformal field theory (CFT) is a monoidal functor from Strngcx

to some monoidal target category.

We have no example of such a theory to date. Instead we consider theorieswhich are half way between CFT and TFT.

Definition: We define a 2-dimensional topological conformal field theory (TCFT)to be a monoidal functor of dg-categories C∗(Strngcx)→ dgVect.

Given A,B ∈ Λ, a TCFT associates to each object IA,B a differential gradedvector space which we denote by

Hom•Λ(A,B) := Ψ(IA,B)

As in the case of the open-closed TFT we consider Po in fig. 15(d) for ournotion of composition in Λ (see section 4.1). LetMPo

denote the moduli spaceof Riemann surfaces of topological type Po. Applying Ψ we get a map of dg-vector spaces

C∗(MPo)⊗Hom•(A,B)⊗Hom•(B,C)→ Hom•(A,C)

We no longer have a composition of morphisms in Λ but rather a differentialgraded space worth of them.

We will dedicate the rest of this lecture to a discussion of a theorem due toCostello which is a spruced up version of a theorem by the same author we en-countered earlier on in section 4.2. Replacing TFT by TCFT requires passingto the A∞ world. The adjoint functor to (io)∗ should now be derived and weneed a notion of Hochschild chain complex for this setting.

32

Theorem: [Costello] A 2-dimensional open TCFT determines a cyclic A∞ cat-egory Λ and vice versa. Given a cyclic A∞ category Λ there exists a universalopen-closed TFT whose open sector is given by Λ and whose closed string sectoris given by C∗(Λ), the Hochschild cochain complex of Λ.

Cyclic A∞ Algebra: We consider the above theorem in the simplest casewhere Λ is a cyclic A∞ category with one object. In this case the endomor-phisms of that single object form a cyclic A∞ algebra.

A cyclic A∞ algebra is an A∞ algebra, A, with a non-degenerate pairing 〈, 〉 :A⊗A→ k such that the following maps

cn+1 : A⊗(n+1) → k

x1 ⊗ · · · ⊗ xn+1 7→ 〈mn(x1, . . . , xn), x(n+1)〉

are invariant under cyclic permutation (up to signs depending on degrees). Forexample, a cyclic A∞ algebra with only m2 is a symmetric Frobenius algebra.

Invariants of Ribbon Graphs: A cyclic A∞ algebra, A, gives rise to a numer-ical invariant, ev(Γ,A), associated to every ribbon graph Γ. We will demonstratethe construction with the following example. Consider a ribbon graph Γ with amorsification as in fig. 23.

Figure 23: Morsification of a ribbon graph Γ.

We define ev(Γ,A) to be the composition of the following maps

C 〈,〉∗−−→ A⊗A µ∗⊗id−−−−→ A⊗A⊗A id⊗id⊗µ∗−−−−−−→ A⊗A⊗A⊗Aid⊗µ⊗id−−−−−→ A⊗A⊗A µ⊗id−−−→ A⊗A 〈,〉−→ C

We claim ev(Γ,A) does not depend on the choice of morsification. Moreover, thiscalculation can be generalized to ribbon graphs with vertices of valency greaterthan 3. In that case we use ck : Ak → A to get a map A⊗i → A⊗(k−i+1). A hasa non-degenerate pairing hence a finite basis ei. The above definition agrees

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1 2 3

4

1 2 3

4

d = _1 2

4 3

Figure 24: Vertex Expansion.

with the following.

ev(Γ,A) =∑

labeling ofhalf edgesof Γ byelements ofbasis

∏edges withhalveslabeled byei, ej

±〈ei, ej〉∏

vertices withadjacent 1/2-edges labeledei1 , · · · , eik inthis cyclic order

ck(ei1 , · · · , eik)

We ignored the issue of signs in this expression. We claim this expression doesnot depend on choice of basis.

Theorem: ev(dΓ,A) = 0. This means every cyclic A∞ algebra defines a co-cycle on (RG, d). In particular, it defines a class in H∗(Mg,n).

Taking the differential at a vertex of Γ of valence n+ 1 we get all trees with nleaves, one root, and one internal edge. For example, when Γ contains a vertexof valence 4, see fig. 24. The bracket in the picture is meant to remind us thatwe are looking at a part of Γ containing that vertex. Evaluation at an expandedvertex corresponds exactly to Assocn and hence equals zero.

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