Categorifying Reshetikhin-Turaev invariants
Ben Webster
U. of Oregon
June 17, 2013
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
1/1
Introduction
This talk will be online at
http://math.mit.edu/˜bwebster/faroIII.pdf
if I ever get the wireless to work.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
2/1
Introduction
The story thus far
Let me just remind you what we’ve done thus far.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
3/1
Introduction
The story thus far
Let me just remind you what we’ve done thus far.
(λ1 , . . . , λ` )
Ben Webster (U. of Oregon)
algebra T λ
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
3/1
Introduction
The story thus far
Let me just remind you what we’ve done thus far.
functor Bσ : D(T λ -mod) → D(T σ·λ -mod)
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
3/1
Introduction
The story thus far
Let me just remind you what we’ve done thus far.
functor Bσ : D(T λ -mod) → D(T σ·λ -mod)
Now, we have to be able to turn around.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
3/1
Introduction
Coevalution and quantum trace
We also need functors corresponding to the cups and caps in our theory.
First, consider the case where we have two highest weights λ and
−w0 λ = λ∗ . We must first define an isomorphism between Vλ∗ and Vλ∗ . That
is to say, a pairing Vλ × Vλ∗ → C(q).
We start with a chosen highest weight vector of both representations vλ , vλ∗
(this comes from the irrep in Tλλ -mod ∼
= k -mod). So, a pairing is fixed by a
choice of lowest weight vector.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
4/1
Introduction
Coevalution and quantum trace
We also need functors corresponding to the cups and caps in our theory.
First, consider the case where we have two highest weights λ and
−w0 λ = λ∗ . We must first define an isomorphism between Vλ∗ and Vλ∗ . That
is to say, a pairing Vλ × Vλ∗ → C(q).
We start with a chosen highest weight vector of both representations vλ , vλ∗
(this comes from the irrep in Tλλ -mod ∼
= k -mod). So, a pairing is fixed by a
choice of lowest weight vector.
Pick a reduced expression
w0 = s1 · · · sn with corresponding roots α1 , · · · , αn .
Then we have a lowest weight vector of the form
(α∨
n (sn−1 ···s1 λ))
vlow = Fin
∨
(α∨
2 (s1 λ)) (α1 (λ))
Fi1
vλ
· · · Fi2
We will always choose this one.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
4/1
Introduction
Invariants
We should look for a categorification of the unique invariant vector
c ∈ Vλ ⊗ Vλ∗ . We can actually guess quite easily what this should be.
The space of invariants is orthogonal under the Euler form to all projectives of
the form Fi M for any i. We know by counting arguments that all but one
indecomposable projective is a summand of a Fi M.
We actually know exactly what this remaining projective Pλ is; it corresponds
to the sequence of weights and roots
(α∨
1 (λ))
(λ, α1
Ben Webster (U. of Oregon)
(α∨
2 (s1 λ))
, α2
(α∨
n (sn−1 ···s1 λ))
, . . . , αn
Categorifying Reshetikhin-Turaev invariants
, λ∗ ).
June 17, 2013
5/1
Introduction
Invariants
We should look for a categorification of the unique invariant vector
c ∈ Vλ ⊗ Vλ∗ . We can actually guess quite easily what this should be.
The space of invariants is orthogonal under the Euler form to all projectives of
the form Fi M for any i. We know by counting arguments that all but one
indecomposable projective is a summand of a Fi M.
We actually know exactly what this remaining projective Pλ is; it corresponds
to the sequence of weights and roots
(α∨
1 (λ))
(λ, α1
(α∨
2 (s1 λ))
, α2
(α∨
n (sn−1 ···s1 λ))
, . . . , αn
, λ∗ ).
So, an element of invariants is given by the simple quotient of Pλ . Denote this
Lλ .
It’s pretty easy to check by hand that Lλ is killed by all Ei .
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
5/1
Introduction
Coevalution and evaluation
The coevaluation functor is categorified by the functor
∗
V∅ ∼
= Vect → V λ,λ sending C → Lλ .
The evaluation functor is categorified by
∗
RHom(Lλ , −)[2ρ∨ (λ)](2hλ, ρi) : V λ,λ → V ∅ ∼
= Dfd (Vect).
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
6/1
Introduction
Coevalution and evaluation
The coevaluation functor is categorified by the functor
∗
V∅ ∼
= Vect → V λ,λ sending C → Lλ .
The evaluation functor is categorified by
∗
RHom(Lλ , −)[2ρ∨ (λ)](2hλ, ρi) : V λ,λ → V ∅ ∼
= Dfd (Vect).
Now, we know that if we want quantum trace, we should compromise between
Lλ [2ρ∨ (λ)](2hλ, ρi)
and
Lλ [−2ρ∨ (λ)](−2hλ, ρi)
Definition
The positive ribbon twist acts on the category by [2ρ∨ (λ)](2hλ, ρi).
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
6/1
Introduction
Ribbon structure
∨
So this decategorifies to (−1)2ρ (λ) q2hλ,ρi . Note: this is a strange ribbon
element! (It appeared in work of Snyder and Tingley on half-twist elements.)
For each ribbon element, there is a notion of “quantum dimension,” and in this
∨
picture, qdimV|q=1 = (−1)2ρ (λ) dim V. For example, in sl2 ,
qdimVn = (−1)n
qn+1 − q−n−1
.
q − q−1
From now on, all my knots are ribbon knots (in the blackboard framing), and
I’ll really get invariants of ribbon knots (but twists just give grading shifts).
=
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
7/1
Introduction
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext• (Lλ , Lλ )[2ρ∨ (λ)](2hλ, ρi)
has graded Euler characteristic given by the quantum dimension of Vλ .
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
8/1
Introduction
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext• (Lλ , Lλ )[2ρ∨ (λ)](2hλ, ρi)
has graded Euler characteristic given by the quantum dimension of Vλ .
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ is
really the dimension of Vλ . In particular, if λ = ωi for g = sln , then
Aλ ∼
= H ∗ (Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼
= H ∗ (Grλ ).
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
8/1
Introduction
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext• (Lλ , Lλ )[2ρ∨ (λ)](2hλ, ρi)
has graded Euler characteristic given by the quantum dimension of Vλ .
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ is
really the dimension of Vλ . In particular, if λ = ωi for g = sln , then
Aλ ∼
= H ∗ (Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼
= H ∗ (Grλ ).
On the other hand, if λ is not miniscule, things blow up. For example, if
g = sl2 and λ = 2, then
j
j
i,j (−t) dimq Aλ
P
Ben Webster (U. of Oregon)
6= q−2 t2 + 1 + q2 t−2
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
8/1
Introduction
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext• (Lλ , Lλ )[2ρ∨ (λ)](2hλ, ρi)
has graded Euler characteristic given by the quantum dimension of Vλ .
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ is
really the dimension of Vλ . In particular, if λ = ωi for g = sln , then
Aλ ∼
= H ∗ (Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼
= H ∗ (Grλ ).
On the other hand, if λ is not miniscule, things blow up. For example, if
g = sl2 and λ = 2, then
j
j
i,j (−t) dimq Aλ
P
Ben Webster (U. of Oregon)
= q−2 t2 + 1 + q2 t−2 +
Categorifying Reshetikhin-Turaev invariants
q2 −q2 t
1−t2 q4
June 17, 2013
8/1
Introduction
Coevalution and quantum trace
In particular, the algebra (which is the invariant of the circle)
Aλ = Ext• (Lλ , Lλ )[2ρ∨ (λ)](2hλ, ρi)
has graded Euler characteristic given by the quantum dimension of Vλ .
If Vλ is miniscule, then everything works beautifully. The dimension of Aλ is
really the dimension of Vλ . In particular, if λ = ωi for g = sln , then
Aλ ∼
= H ∗ (Grass(i, n)).
Conjecture
If λ is miniscule, Aλ ∼
= H ∗ (Grλ ).
On the other hand, if λ is not miniscule, things blow up. For example, if
g = sl2 and λ = 2, then
j
j
i,j (−1) dimq Aλ
P
Ben Webster (U. of Oregon)
= q−2 + 1 + q2
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
8/1
Introduction
Coevalution and quantum trace
To do this in general, you can construct natural bimodules Kµ . This is given
by the picture.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
9/1
Introduction
Coevalution and quantum trace
To do this in general, you can construct natural bimodules Kµ . This is given
by the picture.
αi∨1 (µ)
αi∨n (sin−1 · · · si1 µ)
λ`
λ1
···
λ1
Ben Webster (U. of Oregon)
µiµ1n∗
Lµ
Categorifying Reshetikhin-Turaev invariants
···
λ`
June 17, 2013
9/1
Introduction
Coevalution and quantum trace
There’s exactly one interesting relation here, which says that
···
µiµ1n∗
···
···
µiµ1n∗
···
=
Lµ
Lµ
Fi v ⊗ cλ = Fi (v ⊗ cλ ).
Theorem
Tensor product with this bimodule categorifies coevaluation/quantum cotrace,
and Hom with it categorifies evaluation/quantum trace.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
10 / 1
Introduction
Knot invariants
Now, we start with a picture of our knot (in red), cut it up into these
elementary pieces, and compose these functors in the order the elementary
pieces fit together.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
11 / 1
Introduction
Knot invariants
Now, we start with a picture of our knot (in red), cut it up into these
elementary pieces, and compose these functors in the order the elementary
pieces fit together.
For a link L, we get a functor FL : V ∅ ∼
= D(Vect) → V ∅ ∼
= D(Vect). So
FL (C) is a complex of vector spaces (actually graded vector spaces).
Theorem
The cohomology of FL (C) is a knot invariant, and finite-dimensional in each
homological and each graded degree. The graded Euler characteristic of this
complex is JV,L (q).
As usual, we can take a generating series of FL (C). This will not be a
polynomial, but it should be a rational function.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
11 / 1
Introduction
Knot invariants
Start with C.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
12 / 1
Introduction
Knot invariants
V
V∗
A1 = C ⊗ K1,2
V
Replace with projective
resolution B1
Start with C.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
12 / 1
Introduction
Knot invariants
V
V∗
V
V∗
A2 = B1 ⊗ K1,2
V
Replace with injective
resolution B2
V
V∗
A1 = C ⊗ K1,2
V
Replace with projective
resolution B1
Start with C.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
12 / 1
Introduction
Knot invariants
Replace with projective
resolution B3
V
V
V∗
V ∗ A3 = RHom(Bi , B2 )
V
V∗
V
V∗
A2 = B1 ⊗ K1,2
V
Replace with injective
resolution B2
V
V∗
A1 = C ⊗ K1,2
V
Replace with projective
resolution B1
Start with C.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
12 / 1
Introduction
Knot invariants
Replace with projective
resolution B4
V
V
V∗
V∗
V
V
V∗
V ∗ A3 = RHom(Bi , B2 )
V
V∗
V
V∗
A2 = B1 ⊗ K1,2
V
Replace with injective
resolution B2
V
V∗
A1 = C ⊗ K1,2
V
Replace with projective
resolution B1
A 4 = B 3 ⊗ B1
Replace with projective
resolution B3
Start with C.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
12 / 1
Introduction
Knot invariants
V
V
V∗
V∗
A 5 = B 4 ⊗ B3
Replace with injective
resolution B5
V
V
V∗
V∗
A 4 = B 3 ⊗ B1
Replace with projective
resolution B4
V
V
V∗
V ∗ A3 = RHom(Bi , B2 )
V
V∗
V
V∗
A2 = B1 ⊗ K1,2
V
Replace with injective
resolution B2
V
V∗
A1 = C ⊗ K1,2
V
Replace with projective
resolution B1
Replace with projective
resolution B3
Start with C.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
12 / 1
Introduction
Knot invariants
V
V ∗ A6 = RHom(K2,3
V , B5 )
Replace with injective
resolution B6
V
V
V∗
V∗
A 5 = B 4 ⊗ B3
Replace with injective
resolution B5
V
V
V∗
V∗
A 4 = B 3 ⊗ B1
Replace with projective
resolution B4
V
V
V∗
V ∗ A3 = RHom(Bi , B2 )
V
V∗
V
V∗
A2 = B1 ⊗ K1,2
V
Replace with injective
resolution B2
V
V∗
A1 = C ⊗ K1,2
V
Replace with projective
resolution B1
Replace with projective
resolution B3
Start with C.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
12 / 1
Introduction
Knot invariants
A7 = RHom(K1,2
V , B6 )
V
Knot homology!
V ∗ A6 = RHom(K2,3
V , B5 )
Replace with injective
resolution B6
V
V
V∗
V∗
A 5 = B 4 ⊗ B3
Replace with injective
resolution B5
V
V
V∗
V∗
A 4 = B 3 ⊗ B1
Replace with projective
resolution B4
V
V
V∗
V ∗ A3 = RHom(Bi , B2 )
V
V∗
V
V∗
A2 = B1 ⊗ K1,2
V
Replace with injective
resolution B2
V
V∗
A1 = C ⊗ K1,2
V
Replace with projective
resolution B1
Replace with projective
resolution B3
Start with C.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
12 / 1
Introduction
Comparison to other knot homologies
So, one of the things I had promised you is that this homology would be a
generalization of previous known ones. How does one check a thing like that?
Well, it helps that this whole construction is actually exactly the same as the
one done by Stroppel to interpret Khovanov homology in terms of parabolic
category O.
Thus, we just have to make sure that the categories and functors we feed into
this construction are the same.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
13 / 1
Introduction
Comparison to other knot homologies
So, one of the things I had promised you is that this homology would be a
generalization of previous known ones. How does one check a thing like that?
Well, it helps that this whole construction is actually exactly the same as the
one done by Stroppel to interpret Khovanov homology in terms of parabolic
category O.
Thus, we just have to make sure that the categories and functors we feed into
this construction are the same.
Theorem
They are!
For the standard representation of sln , this invariant is the same as the
homology given by Mazorchuk-Stroppel and Sussan.
For n = 2, 3, it’s known that this is Khovanov’s original homology. For n > 3,
it’s conjectured to be Khovanov-Rozansky, but it isn’t proven.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
13 / 1
Introduction
Functoriality?
It’s not known at the moment if this is functorial in cobordisms between
knots. What sort of structure would we expect to see if it were? Well, for one
thing, the invariant of a circle would be a Frobenius algebra, with the
Frobenius action coming from cobordisms between flat tangles.
Theorem
For λ miniscule, Aλ is a finite-dimensional homogeneous Frobenius algebra.
I conjectured earlier that this is cohomology of a smooth finite dimensional
manifold. You can regard this as some evidence for that.
What if λ isn’t miniscule? Wildly false; has a really nice fake proof.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
14 / 1
Introduction
Functoriality?
In non-miniscule weights, can Humpty-Dumpty be put back together again?
Well, maybe. You’ll recall that a couple of days ago, I killed some perfectly
innocent fake bubbles for no particularly good reason.
What if I had kept them? Then the algebra would have finite global
dimension, so one could replay the past 3 talks with these more “bubbly”
algebras, and get a finite dimensional answer!
Except maybe there was a good reason to kill the bubbles: this isn’t a knot
invariant. Reidemeister I is all wrong now. On the other hand, the invariant of
a circle is a finite dimensional Frobenius algebra.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
15 / 1
Introduction
Infinite-dimensional algebras
On the other hand, maybe it’s not so bad to be free from the tyranny of
finite-dimensionality.
Work in progress: Just as you can define highest weight representations, you
can also define lowest weight representations. Of course, if g is
finite-dimensional, these are equivalent, but if g is a Kac-Moody algebra, they
are different.
You can define tensor products of these modules, and braidings as before.
Now it’s easier to define cups and caps, since you should only define them for
pairs of highest and lowest weight reps.
You can now define knot homologies for Kac-Moody algebras, and they will
be finite in each homological degree!
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
16 / 1
Introduction
4d TQFT
One of the inspirations for studying categorifications is the connections
between higher categories and quantum field theory.
The quantum knot invariants arise from a 3-d TQFT: Chern-Simons theory.
You can think of this as built up from attaching the category of Uq (g)
representation to a circle and building the 2-and 3-dimensional layers from
that.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
17 / 1
Introduction
4d TQFT
One of the inspirations for studying categorifications is the connections
between higher categories and quantum field theory.
The quantum knot invariants arise from a 3-d TQFT: Chern-Simons theory.
You can think of this as built up from attaching the category of Uq (g)
representation to a circle and building the 2-and 3-dimensional layers from
that.
Can one make a 4-dimensional TQFT of some kind out the category of
2-representations of this categorified quantum group?
Gukov and other physicists have done work on this, but as far as I know,
nothing mathematically rigorous has appeared.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
17 / 1
Introduction
Quiver varieties
So, you might wonder what my picture has to do with the one Sabin talked
about. The relationship is a little strange, but ultimately will hopefully lead
somewhere interesting.
The rough answer is that my picture is the deformation quantization of theirs.
Nakajima quiver varieties are symplectic varieties, which means that
functions on them have a Poisson bracket {−, −}. Thus, we can imagine (and
in fact find) a star-product on the sheaf of algebraic functions such that
f ?~ g = fg + ~{f , g} + · · ·
Theorem
The algebra T λ is the Ext-algebra of a particular sheaf of modules over
(O[[h]], ?~ ).
Taking associated graded sends you back into Sabin’s picture.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
18 / 1
Introduction
Quiver varieties
Maybe it would make people happier to take Fukaya categories.
I don’t have the technical chops to make this precise, but sheaves of modules
over deformation quantizations are supposed to be a “first approximation” to a
Fukaya category.
The correspondences Sabin talked about are actually Lagrangian, so they
induce functors between quilted Fukaya categories.
Conjecture
These correspondences induce a categorical g-action on quilted Fukaya
categories. The algebra T λ is the Ext-algebra of a particular object in the
quilted Fukaya category of a Nakajima quiver variety (with non-compact
Lagrangians interpreted suitably).
Unfortunately, at the moment, I don’t know how to describe the functors for
braiding, cups and caps in terms of geometry.
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
19 / 1
Introduction
That’s all, folks!
Ben Webster (U. of Oregon)
Categorifying Reshetikhin-Turaev invariants
June 17, 2013
20 / 1