Knot homology and KLR algebras I - Northeastern Universitymathserver.neu.edu/~bwebster/CRM-I.pdf · Knot homology and KLR algebras I Ben Webster University of Virginia June 26, 2013

Knot homology and KLR algebras I

Ben Webster

University of Virginia

June 26, 2013

Ben Webster (UVA) Knot homology and KLR algebras I June 26, 2013 1 / 27

The big picture

We’ve now had a number of talks about polynomial invariants of knotscoming from Chern-Simons theory. We’ve also had a number of talksabout homological (i.e. vector space valued) knot invariants.

What I’d like to do in these talks is to get these to mix. We’ve alreadyseen one overlap between these:

Khovanov homology categorifies the Jones polynomial, theinvariant attached to C2 as a representation of sl2.

So I’m going hunting for generalizations of Khovanov homology:1 In the first lecture, I’ll concentrate on the sl2 case; that is, give a

much more complicated construction of Khovanov homology.2 In the second, I’ll move on to the general case. The underlying

principles are the same, but a lot hides under the surface.3 In my third lecture, I’ll talk about a very different realization of the

same construction, using A-branes on quiver varieties.Ben Webster (UVA) Knot homology and KLR algebras I June 26, 2013 2 / 27

The big picture








The big picture








The big picture








The big picture








Knot invariants

Knot and 3-manifold invariants à la WittenChern-Simons theory defines a topological quantum field theory. Itassigns:

a number to a closed 3-manifold M (a path integral overconnections) anda vector space V(Σ) assigned to every 2-manifold Σ (the Hilbertspace obtained by taking sections of a line bundle), anda map HN : V(Σ1)→ V(Σ2) for a manifold with ∂N = Σ̄1 t Σ2.

It also has Wilson operators labeled with representations of a compactgroup G that sit on 1-manifolds L ⊂ M and 0-manifolds in Σ. These gettheir own numbers, vector spaces and maps respectively.

In particular, this assigns a number w(K ,V) to a knot K labeled with aG-representation V .


Knot invariants

Knot and 3-manifold invariants à la WittenChern-Simons theory defines a topological quantum field theory. Itassigns:

a number to a closed 3-manifold M (a path integral overconnections) anda vector space V(Σ) assigned to every 2-manifold Σ (the Hilbertspace obtained by taking sections of a line bundle), anda map HN : V(Σ1)→ V(Σ2) for a manifold with ∂N = Σ̄1 t Σ2.

It also has Wilson operators labeled with representations of a compactgroup G that sit on 1-manifolds L ⊂ M and 0-manifolds in Σ. These gettheir own numbers, vector spaces and maps respectively.

In particular, this assigns a number w(K ,V) to a knot K labeled with aG-representation V .


Gluing

Furthermore, everything in the picture satisfies a gluing formula.

That is, if we have 2 cobordisms N1 and N2, with ∂Ni = Σ̄i t Σi+1, then

HN2 ◦ HN1 = HN2∪Σ2 N1

Σ1 Σ2 Σ3

N1 N2

Atiyah: TQFT is a functor!


Knot invariants

OK, that’s lovely, but I don’t know how to evaluate path integrals.However, I can reformulate things in terms I can understand.

Let’s only worry about knots in S3. Then we can cut these knots upinto very simple pieces, by picking a height function X = S3

→ [0,1]with 2 critical points (a max and a min).

Let X[a,b] be the points of height between a and b. This is alwaysS2× I or D3.

For any 0 < a1 < · · · < an < 1, we can write the invariant w(K ,V) asthe gluing of the submanifolds X[0,a1],X[a1,a2], . . . ,X[an ,1], that is, asmaps between the vector spaces attached to a sphere with finitelymany labels.


Knot invariants

Associated to a surface decorated with Wilson operators with circleboundary, we have a representation of g.

trivial representation C.

λsimple representation Vλ.

λ1λ2 tensor product Vλ1 ⊗ Vλ2 .

A D2 carrying ` Wilson operators with labels Vλi has associatedrepresentation Vλ = Vλ1 ⊗ · · · ⊗ Vλ` .

An S2 carrying ` Wilson operators with labels Vλi corresponds to theinvariants Vgλ = (Vλ1 ⊗ · · · ⊗ Vλ`)

g.


Knot invariants

By picking enough ai ’s, we can reduce to understanding the trivialcobordism I × S2 decorated with

a pair of Wilson operators labeled with Vλ and Vλ∗ := V ∗λ beingborn, that is, a local minimum of the knot.

· · · · · ·

λλ∗

a pair of such operators dying, that is, a maximum of the knot.

· · · · · ·

λλ∗

a braid in the Wilson operators.

· · · · · ·

I actually prefer to take these in D2, rather than S2. It’s easier todescribe the maps between tensor products and then take invariants.


Knot invariants

By picking enough ai ’s, we can reduce to understanding the trivialcobordism I × S2 decorated with

a pair of Wilson operators labeled with Vλ and Vλ∗ := V ∗λ beingborn, that is, a local minimum of the knot.a pair of such operators dying, that is, a maximum of the knot.a braid in the Wilson operators.

I actually prefer to take these in D2, rather than S2. It’s easier todescribe the maps between tensor products and then take invariants.

Reshetikhin-TuraevThese maps between tensor products can be interpreted in terms ofquantum groups at a root of unity.


Categorification

So, what happens if we categorify this picture? We expect to assign:a vector space (or maybe complex) to a closed 3-manifold M withsome Wilson operators anda category C(Σ) to a closed 2-manifold Σ anda functor FN : C(Σ1)→ C(Σ2) when ∂N = Σ̄1 t Σ2.

Better yet, if we’re satisfied with knots in S3, we don’t have to definetoo many of these; we can assume that Σ � D2,S2.

DefinitionLet C(λ1, . . . , λ`) denote the category attached to the disk

λ1 λ`λ2 λ`−1· · ·


Categorification

We expect that this new TQFT will be compatible with CS theory in thesense we get the old theory by decategorifying.

We expect that the “dimension” (correctly understood) of thevector space is the CS invariant of M.The Grothendieck group of the category C(Σ) should be theHilbert space of Σ. Recall that the GG of C is the formal span overZ of [A ] for A ∈ Ob(C) modulo the relations [B] = [A ] + [C] for ashort exact sequence

0 A B C 0

We’ll actually want a grading on C(Σ), and thus aZ[q,q−1]-module, with q[A ] the grading shift of A .The functor FN should be exact, so the map [A ] 7→ [F(A)] iswell-defined and matches the map between Hilbert spaces.


Categorification

So, the central question is the category C(λ1, . . . , λ`). What is it?

Work of Khovanov, Lauda and Rouquier gives us a hint. They define anotion of a categorical g-action, that is, a category with an analogueof an action of g. We expect that C(λ1, . . . , λ`) should be such acategory.

For just one Wilson operator C(λ), this is enough to uniquely specifythe category; in the general case, it provides some serious guidance.The category C(λ1, . . . , λ`) has several different definitions:

(Axiomatic) C(λ1, . . . , λ`) is the unique category with a categoricalg-action, Grothendieck group isomorphic to Vλ, and certain otherconditions.(Diagrammatic) We can just write down generators and relations.(Geometric) It’s the category of A-branes with certain conditions at∞ on quiver varieties.Ben Webster (UVA) Knot homology and KLR algebras I June 26, 2013 10 / 27

Diagrams

For the first two talks, I’ll only consider the diagrammatic definition:

Let Tλ be the algebra generated by diagrams of red and black lines;red lines are labeled by weights, black by simple roots.

λ1 ij in

· · ·· · ·

deg = 〈αij , αij 〉

i1 ij ij+1 λ`

· · ·· · ·

deg = −〈αij , αij+1〉

i1 ij λk in

· · ·· · ·

deg = 〈αij , λk 〉

Reds never cross!


Relations

i i

=

i i

+

i i i j

=

ji

aij

j

aji

i

+

i i

= 0

ii j

=

ii j

+∑

a+b=aij−1

i

b

i

a

j

i λ

=α∨i (λ)

i λ ii λ

=

ii λ

+∑

a+b=α∨i (λ)−1

i

b

i

a

λ

λ i

=

λ

α∨i (λ)

i i

· · · = 0


Diagrams

Theorem (W.)

The Grothendieck group of the category C(λ) := D(Tλ -mod) is thetensor product Vλ = Vλ1 ⊗ · · · ⊗ Vλ` ; the categorical g-action inducesthe usual action of g on Vλ.

This construction clarifies a lot of interesting structures on tensorproducts. For example, Lusztig’s canonical basis matches the classesof the indecomposable projects.


Wilson operators

The real proof is if we can define the functors attached to tangles withthe Wilson operators. To normalize, one must take ribbon tangles.

Theorem

For every ribbon tangle τ in D2× I, we have a functor Fτ : C(λ)→ C(λ′)

between the categories associated to the top and bottom, such thatFτ′◦τ � Fτ′ ◦Fτ, which categorifies the Reshetikhin-Turaev construction.

These functors are defined as derived tensor product withdiagrammatically defined bimodules as well.

λj+1λjλj λj+1

· · · · · ·

µ µ∗

· · ·

µ µ∗

· · ·

v


Wilson operators

The real proof is if we can define the functors attached to tangles withthe Wilson operators. To normalize, one must take ribbon tangles.

Theorem

For every ribbon tangle τ in D2× I, we have a functor Fτ : C(λ)→ C(λ′)

between the categories associated to the top and bottom, such thatFτ′◦τ � Fτ′ ◦Fτ, which categorifies the Reshetikhin-Turaev construction.

These functors are defined as derived tensor product withdiagrammatically defined bimodules as well.

λj+1λjλj λj+1

· · · · · ·

µ µ∗

· · ·

µ µ∗

· · ·

v


The case of sl2

To keep things simple, let’s do a detailed discussion for sl2, with λi = 1.Don’t need to label red and black strands, since there’s only onepossible label. Let T ` be the algebra with ` red strands.

= + = 0 =

= = +

= · · · = 0


The case of sl2

Just to get our head in place, let’s look at T2.no black strands: .

one black strand:

, , , , .two black strands:


A basis

There’s actually a combinatorial basis of T `. Each basis vector is builtout of 3 pieces.

a dotless diagram sweeping black strands right (they may cross)a diagram with no consecutive black strands, crossings or dotsa dotless diagram sweeping black strands right (they may cross)

Let’s reconsider the case of T2 with 1 strand:

, , , ,


A basis




, , , ,


A basis




, , , ,


A basis




, , , ,


Functors

The category T ` -mod has functors corresponding to the action of theChevalley generators E/F . These subtract/add a black line.

The action of F is “add a black line at the right,” and the action of E is“absorb a black line at the right.”

· · · · · ·

m

E

· · · · · ·

m

F


The case of sl2

T2 -mod should categorify the representation C2⊗ C2.

v1 ⊗ v1

q−1v1 ⊗ v−1 + v−1 ⊗ v1 f v−1 ⊗ v1 − q−1v1 ⊗ v−1

(q + q−1)v−1 ⊗ v−1

F

F

V3 V1

projectives simple


The case of sl2The cup (up to shift) is associated to derived tensor product with thebimodule:

Of course, we need to have some relations:

= 0 = 0 = 0

= = −

This is inserting two new tensor factors, and putting in the invariantmodule for those two factors.




= 0 = 0 = 0

= = −





= 0 = 0 = 0

= = −





= 0 = 0 = 0

= = −



The case of sl2So, for a circle, we get elements of the bimodule for each picture:

It’s easy to check that we can simplify so that the bubble is separate.

But we have to think a bit harder than this; the functor for a cup isn’texact! You need to use a projective resolution!

−

−1 0 1





−

−1 0 1





−

−1 0 1


The case of sl2We can evaluate compositions by noting that

= ∅ = 0 = 0

So, we can see the relations

== ∅ ⊕ ∅

−1 0 1



= ∅ = 0 = 0


== ∅ ⊕ ∅

−1 0 1



= ∅ = 0 = 0


== ∅ ⊕ ∅

−1 0 1



= ∅ = 0 = 0


=

= ∅ ⊕ ∅

−1 0 1



= ∅ = 0 = 0


== ∅ ⊕ ∅

−1 0 1



= ∅ = 0 = 0


=

= ∅ ⊕ ∅

−1 0 1


The action of cobordisms

Theorem (Chatav)There is an action of the cobordism 2-category on the categoriesDb(Tn -mod).

∅

∅ ∅

∅

∅

∅

∅

∅

In fact, one can easily check that this action satisfies the Bar-Natanrelations (S,T,G,NC).




∅

∅ ∅

∅

∅

∅

∅

∅





∅

∅ ∅

∅

∅

∅

∅

∅





∅

∅ ∅

∅

∅

∅

∅

∅





∅

∅ ∅

∅

∅

∅

∅

∅




Fun game: look at a construction in BN world, transport it to modulesover T `.

Dror tells us that

= Cone( )

The cone of an surjective map of bimodules over T ` is its kernel.

The kernel is the set of diagrams where we can see a pair of red lineswith no black between them.

To remind that we have to go around, snap the red strands together.



Fun game: look at a construction in BN world, transport it to modulesover T `.

Dror tells us that

= Cone( )

The cone of an surjective map of bimodules over T ` is its kernel.

The kernel is the set of diagrams where we can see a pair of red lineswith no black between them.

To remind that we have to go around, snap the red strands together.


Comparison to Khovanov homology

These bimodules define a functor Db(T `1 -mod)→ Db(T `2 -mod) forevery tangle connecting `1 points to `2; I should say “tangle projection”but we already know isotopy invariance!

TheoremThe resulting knot invariant is Khovanov homology.

Note that we can make functoriality work as usual using the action ofcobordisms. If you want to avoid sign problems, you need to use theMorrison-Walker disorientation scheme.

= Cone(

))

)


Jones-Wenzl projectors

Cooper and Krushkal constructed a categorified JW projector inBar-Natan’s category.

The image over T ` is surprisingly easy to describe.

· · ·

· · ·

w/ relations a = b

You can think of this as realizing T (n1,...,n` -mod as a quotient categoryof Tn1+···+n` -mod.

This allows us to transport their categorification of colored Jones to ourpicture. Actually, this is a very special case of the general construction,which I’ll discuss tomorrow.


Thanks for your attention.


Documents

Knot homology and KLR algebras I - Northeastern Universitymathserver.neu.edu/~bwebster/CRM-I.pdf · Knot homology and KLR algebras I Ben Webster University of Virginia June 26, 2013