The Fukaya category of the punctured 2-torus...Fukaya categories The symplectic topology of Liouville manifolds Symplectic manifolds A symplectic manifold is a smooth manifold M with

Fukaya categories

The Fukaya category of the punctured 2-torus

Tim Perutz

University of Texas at Austin

Rice University colloquium, January 20, 2011

T. Perutz Fukaya categories

Fukaya categories

Plan

Plan of the colloquium

1 The symplectic topology of Liouville manifolds

2 What are Fukaya categories?

3 Example: the punctured 2-torus (Lekili–P.).


Fukaya categories

The symplectic topology of Liouville manifolds

Symplectic manifolds

A symplectic manifold is a smooth manifold M with a2-form ω that is (i) non-degenerate as a skew bilinear formTxM × TxM → R at every point x ; and (ii) closed (dω = 0).

Condition (i) is controlled by bundle theory, and (ii) by deRham cohomology, but their combination is subtle.

Symplectic topology studies global questions about thestructure of symplectic manifolds.


Fukaya categories


The rise of ‘exact’ symplectic topology

The 1990s saw strides in understanding the topology of closed (i.e.compact, boundaryless) symplectic manifolds.Since 2000, there has been more progress on symplectic manifoldswith non-empty boundary. To get an interesting theory, one needsa boundary condition.

A Liouville domain is a compact manifold M2n with boundaryand a 1-form θ such that

(i) the exact 2-form ω := dθ is non-degenerate (so symplectic),(ii) the Liouville vector field λ, characterized by θ = ω(λ, ·), points

outwards along the boundary.

Example: Ln a Riemannian manifold, M = (T ∗L)≤1 thecotangent vectors of length ≤ 1, θ the ‘p dq’ 1-form.


Fukaya categories


Liouville manifolds

Liouville domains have continuous parameters, e.g.vol (M, θ) :=

∫M ωn.

By gluing R+ × ∂M to the boundary ∂M, one can ‘complete’

a Liouville domain to a boundaryless Liouville manifold (M, θ)with infinite volume. Here

(i) ω := d θ is non-degenerate, and

(ii) the Liouville vector field λ, defined by θ = ω(λ, ·), generates aflow φt for all times t such that

d

dtφ∗t ω

∣∣∣∣t=0

= ω.

E.g. complete (T ∗L)≤1 to T ∗L.

Any smooth complex affine variety X ⊂ Cn is a Liouvillemanifold (θ is pulled back from Cn = T ∗Rn).


Fukaya categories


Exotic Liouville manifolds

Liouville manifolds have no continuous parameters.Formally: in any 1-parameter family (Mt , θt), there arediffeomorphisms ft : Mt → M0 such that f ∗t θt − θ0 = dht forcompactly supported functions ht (so f ∗t ωt = ω0).But:

Abouzaid–Seidel, 2010: Let M be the completion of a simplyconnected Liouville domain M of dimension ≥ 12. Then thereare uncountably many Liouville manifolds diffeomorphic to M,no two of them isomorphic as symplectic manifolds.


Fukaya categories


Invariants

Like the large majority of results in symplectic topology sinceGromov’s seminal 1985 paper, Abouzaid–Seidel’s theorem isbased on pseudo-holomorphic curves.

Specifically, they use an invariant called symplecticcohomology to distinguish Liouville manifolds.

The proof involves constructing Liouville domains bysuccessive handle attachments, and tracking the change insymplectic cohomology via the Fukaya category.

The use of Fukaya categories as a ‘master invariant’ is arecent but growing trend in symplectic topology.


Fukaya categories

Fukaya categories: generalities

The Fukaya category: what is it?

What has a Fukaya category? A Liouville domain M.[Reference: P. Seidel, Fukaya categories and Picard–Lefschetz

theory, 2008]

What is it? An A∞-category F(M), linear over an arbitraryground field K.

What it’s not. In studying the Fukaya category, methods of‘pure’ category theory (adjoint functors, limits, etc.) are oflittle use. Need homological algebra instead.

What’s an A∞-category, roughly? You can extract from anyA∞-category a differential graded K-algebra (DGA), i.e., agraded associative algebra with a degree 1 endomorphism dsuch that d ◦ d = 0 and d(ab) = (da)b + (−1)deg aa(db). Youdon’t lose much by thinking of the Fukaya category as a DGA.


Fukaya categories


The Fukaya category: what’s it good for?

Problems in symplectic topology. As an invariant, it encodes ahuge amount of information.

Homological mirror symmetry. The Fukaya categories of somesmooth complex algebraic varieties are equivalent tocategories built from the coherent sheaves on a ‘mirror’variety. When this is proven there is two-way information flowbetween symplectic and algebraic geometry.

Low-dimensional topology. Some invariants of 3-manifolds orknots, notably Heegaard Floer cohomology and Khovanovcohomology, have Fukaya-categorical structures.

Geometric representation theory (D-modules, etc.).


Fukaya categories


First approximation: the cohomology category HF(M)

Objects of HF(M): oriented exact Lagrangian submanifolds,i.e., closed embedded submanifolds of middle dimension,Ln ⊂ M2n, such that θ|L = d(some function).

Morphisms L0 → L1: Floer cohomology HF (L0, L1), aZ/2-graded K-vector space.

Composition: the ‘triangle product’HF (L1, L2)⊗ HF (L0, L1)→ HF (L0, L2).


Fukaya categories

Fukaya categories: Floer cohomology

Floer cohomology as a black box

(Let K = Z/2 to simplify statements.)

Floer cohomology assigns a finite-dimensional (Z/2-graded)vector space HF (L0, L1) to any two closed (oriented) exactLagrangian submanifolds of a Liouville manifold.

HF (L, L) ∼= H∗(L; K).

Continuity: in a 1-parameter family of oriented exactLagrangians {Λt}t∈[0,1], one has HF (Λt , L) ∼= HF (Λ0, L).

If L0 t L1 then dim HF (L0, L1) ≤ #geom(L0 ∩ L1).


Fukaya categories


Digression: a theorem of Gromov

The existence of Floer cohomology implies that R2n has nocompact exact Lagrangian submanifolds (so HF(R2n) is empty):If Λ ⊂ R2n were a compact exact Lagrangian then

HF (Λ,Λ) ∼= H∗(Λ; K) 6= 0.

Translations Λt := tv + Λ are again exact Lagrangians. Bycontinuity, HF (Λt ,Λ) ∼= HF (Λ,Λ) 6= 0.

For t � 0, Λt ∩ Λ = ∅, sodim HF (Λt ,Λ) ≤ #geom(Λt ∩ Λ) = 0.

So HF (Λt ,Λ) is both zero and non-zero. #


Fukaya categories


Constructing Floer cohomology

To construct HF (L0, L1), first move L0 to ensuretransversality: L0 t L1.

The aim is to build a Z/2-graded cochain complex CF (L0, L1)and define HF (L0, L1) = H∗CF (L0, L1).

Define CF (L0, L1) = KL0∩L1 . (Basis vector 〈x〉, wherex ∈ L0 ∩ L1, has mod 2 degree given by the intersection sign.)

Need differential µ1 ∈ End1CF (L0, L1) with µ1 ◦ µ1 = 0.Write this as µ1〈x〉 =

∑y∈L0∩L1

n(x , y)〈y〉.n(x , y) is a count of pseudo-holomorphic bigons with certainboundary conditions.


Fukaya categories


The matrix coefficients

On a symplectic manifold M, one can make TM a family ofcomplex vector spaces (contractible choice).n(x , y) is then a count of pseudo-holomorphic bigons:u : D2 \ {i ,−i} → M with complex-linear derivativeDu : TD2 → u∗TM, subject to boundary conditions pictured.[Only count the bigons that come in 0-dimensional families.]

x

y

LL 01

uD2

M


Fukaya categories

Fukaya categories: pseudo-holomorphic polygons

Generalization: pseudo-holomorphic polygons

n(x1, . . . , xd ; y): count of pseudo-holomorphic (d + 1)-gons.

x

y

x

x

LL0

1

L

L

3

2

1

2

3

uD2

M

These are structure coefficients for a d-fold composition map

µd : CF (Ld−1, Ld)⊗ · · · ⊗ CF (L1, L2)⊗ CF (L0, L1)→ CF (L0, Ld),

µd(xd , . . . , x1) =∑

y∈L0∩Ld

n(x1, . . . , xd ; y)y .


Fukaya categories


The A∞-relation

If one is careful about the precise way the composition maps µd areset up, one finds that they satisfy the A∞ “associativity” equation∑j ,k

(−1)•µd−j+1(xd , . . . , xj+k+1, µk(xj+k , . . . , xj+1), xj , . . . , x1) = 0.

xxx x xx 123456

These relations hold true because of the ways in which a disc withboundary punctures can degenerate.


Fukaya categories


The associahedron

Stasheff’s associahedron Kd+1 compactifies the space of d bluepunctures and 1 red puncture on ∂D2, mod Aut(D2) = PSU(1, 1).It’s a polyhedron of dimension (d + 1)− 3.Its codimension 1 faces are products Kj+1 × Kd−j+1. These areresponsible for the A∞-equation.

vertex = disc

leaf = blue puncture

root = red puncture

interior edge = node

Planar rooted trees


Fukaya categories


The Fukaya category

The cohomological category HF(M) has oriented exactLagrangians as objects and Floer cohomology groupsHF (L0, L1) as morphism spaces.

The Fukaya category F(M) has the same objects, but itsmorphism spaces are the cochain complexes

hom(L0, L1) = CF (L0, L1).

These have all the composition maps µd : the differential µ1,the composition µ2 which satisfies the Leibniz rule but is onlyassociative up to homotopies defined by µ3, and so on. Thisstructure is called an A∞-category.


Fukaya categories


The Fukaya category as an invariant

HF(M) is a hopeless invariant, because it’s usually notfeasible to find all the objects.

F(M) is worse because it has the same objects but morestructure.

But you can use homological algebra to formally enlargeF(M) to a triangulated (not A∞) category DF(M), called thederived Fukaya category. This is sometimes computable inpractice, when studied up to equivalence.


Fukaya categories

The punctured torus

Example: the punctured 2-torus T0 (Lekili–P.)

The objects are easy to understand—they are curves L suchthat

∫L θ = 0. Up to isomorphism in F(T0), there’s one object

Lq for each slope q ∈ Q ∪ {∞}.The morphism spaces, generated by the intersection points,are easy too.

When the relevant curves are transverse, the compositionmaps µd are count immersed polygons with convex corners.

The difficulty is how to extract the essential information fromthis infinite data set. What characterizes the A∞-structure?


Fukaya categories

The punctured torus

Generators

The two objects L0 and L∞ generate the derived Fukayacategory DF(T0)—they ‘know’ everything.

Let A be the A∞-subcategory with objects L0 and L∞,morphisms and compositions inherited from F(T0). Let A bethe cohomological category HA.

Consider pairs (A∞-category B, iso. i : HB→ A). Theorem:If 6 6= 0 ∈ K then there are invariants (m,m′) ∈ K2 whichcompletely classify such pairs.


Fukaya categories

The punctured torus

Homological mirror symmetry

Consider the Weierstrass cubic curveC (a, b) = {y2z = 4x3 − axz2 − bz3} ⊂ P2(K).

One can use the algebraic geometry of sheaves on C (a, b) tocook up an A∞-category B = B(a, b) and an isomorphismi : HB→ A.

One has (m(B, i),m′(B, i)) = (a, b). That is, Weierstrasscubics parametrize A∞-structures on A.

So DF(T0) is equivalent to DB for some cubic—actually, anodal cubic.


Fukaya categories

The punctured torus

Homological mirror symmetry

trivial A -structure

A -structure of Fukayacategory of punctured torus

8

8

Over complex numbers:A -structure of Fukayacategory of closed torus (Polishchuk-Zaslow)

8

Discriminant curvea =27b3 2

y = 4x - ax - b2 3

K2 parametrizes all possible A∞-structures on A. To (a, b)corresponds the A∞-category B(a, b) for the cubic C (a, b).


Fukaya categories

The punctured torus

3-manifolds with torus boundary

We (Lekili–P.) have constructed a new extension ofOzsvath–Szabo’s Heegaard Floer theory from closed3-manifolds to compact 3-manifolds with boundary.

To a 3-manifold Y bounding a torus, we attach a module overF(T0). We studied F(T0) in order to understand what thisinvariant means.

We still don’t understand its content, but at least we nowunderstand F(T0).

Strange but inevitable conjecture: Heegaard Floer homologyhas a ‘mirror’ theory: an algebro-geometric interpretationwhich attaches to a 3-manifold with torus boundary acomplex of vector bundles over a nodal cubic curve.


Documents

The Fukaya category of the punctured 2-torus...Fukaya categories The symplectic topology of Liouville manifolds Symplectic manifolds A symplectic manifold is a smooth manifold M with