Knot Categorification and Mirror Symmetry - math.ksu.edu · Along the way, we will discover...

and Mirror Symmetry

Mina Aganagic UC Berkeley

Knot Categorification

to the problem of categorifying

link invariants.

In this talk, I will describe an application of mirror symmetry

Along the way,

we will discover examples of

where structures that are usually out of reach

homological mirror symmetry,

become tractable,

in part thanks to deep relation to representation theory.

depends on a choice of a Lie algebra,

of its strands by representations of .

A quantum invariant of a link

and a coloring

The link invariant,

in addition to the choice of the Lie algebra

depends on one parameter

and its representations,

Quantum invariants of a link with this data

can be thought of as originating from conformal field theory with

Lie algebra symmetry.

from geometry.

We will start by recalling how this comes about.

We will then go on to rediscover

the relevant structures

To eventually get invariants of knots in or 3,

which is a complex plane with punctures.

xx

x

we want to start with a Riemann surface

x

xx

xxx

It is equivalent, but better for our purpose,

to be a punctured infinite cylinder.

to take

x

To punctures at finite point

xxx

x

we will associate finite dimensional representations

of .

of infinite dimensional, Verma module representations of

To punctures at the two ends at infinity,

we will associate a pair

xxx

x

,

x x x

x

To a 3-punctured sphere

a chiral vertex operator

which acts as intertwiner between pairs of Verma module representations.

conformal field theory associates

``conformal block”

obtained by sewing chiral vertex operators.

To a Riemann surface with punctures

it associates a

xxx

x

we get to make choices of intermediate Verma module representations,

xxx

x

so conformal blocks

In sewing chiral vertex operators

we get in this way live in a vector space.

in terms of vertex operators and sewing,

one can describe them as solutions to a differential equation.

x xx

x

Rather than characterizing conformal blocks

is the equation discovered by Knizhnik and Zamolodchikov in ’84:

xxx

x

conformal blocks of The equation solved by

on

The equation is known as the Knizhnik-Zamolodchikov equation

of trigonometric type since

xxxx

we are thinking of the Riemann surface

as the infinite cylinder.

The coefficients on its right hand side

are the classical r-matrices of :

where

in the standard Lie theory notation.

By varying the positions of vertex operators on as a function of

we get a colored braid in three dimensional space, which is

“time”

which is to analytically continue

This leads to a monodromy problem,

the fundamental solution to the Knizhnik-Zamolodchikov equation

along the path described by the braid.

Monodromy along a path depends only on its homotopy type,

so the resulting monodromy matrix

is an invariant of the colored braid.

The monodromy problem of the Knizhnik-Zamolodchikov equation

was solved by Tsuchia and Kanie in ’88 and by Drinfeld and Kohno in ’89.

corresponding to

is an R-matrix of the quantum group

They showed that monodromy matrix that reorders

a neighboring pair of vertex operators

Action by monodromies

turns the space of conformal blocks into a module for the

quantum group in representation,

The representation is viewed here as a representation of ,

and not of , but we will denote by the same letter.

The monodromy action

is irreducible only in the subspace of

x xx

x

of fixed

corresponding to conformal blocks of the form

quantum invariants of not only braids

This perspective leads to

but knots and links as well.

Any link K can be represented as a

a closure of some braid.

=

of the braiding matrix,

taken between a pair of conformal blocks

The corresponding quantum link invariant is the matrix element

which correspond to the top and the bottom of the picture.

The conformal blocks

which describe pairs of vertex operators,

we need are very special solutions to KZ equations

which come together and “fuse” to disappear.

colored by complex conjugate representations

both braiding and fusion of conformal field theory

play an important role in the story.

This way,

To categorify quantum knot invariants,

one would like to associate

to the space conformal blocks one obtains at a fixed time slice

a bi-graded category,

and to each conformal block an object of the category.

x x x

To braids,

one would like to associate

functors between the categories

corresponding to the

top and the bottom.

Moreover,

we would like to do that in the way that

recovers the quantum knot invariants upon

de-categorification.

One typically proceeds by coming up with a category,

and then one has to work to prove

that de-categorification gives

the quantum knot invariants one set out to categorify.

The virtue of the two approaches

I will describe in these lectures,

is that the second step is automatic.

The starting point for us is

a geometric realization

Knizhnik-Zamolodchikov equation.

we will find two such geometric realizations,

More precisely,

related by two dimensional equivariant mirror symmetry.

so are one of the following types:

We will specialize to be a simply laced Lie algebra

The generalization to non-simply laced Lie algebras

involves an extra step, which we will not have time to describe.

It turns out that Knizhnik-Zamolodchikov equation of

is the “quantum differential equation” of a

This result has been proven recently

certain holomorphic symplectic manifold.

by Ivan Danilenko, in his thesis.

Quantum differential equation of a Kahler manifold

The connection is defined in terms of “quantum multiplication” by divisors

over the complexified Kahler moduli space.

of a connection on a vector bundle

is an equation for flat sections

with fibers

``A-model’’

Quantum multiplication on

is defined by Gromov-Witten theory,

or, the oftopological

The first, term of the quantum multiplication

subsequent terms are quantum corrections.

is the classical product on :

algebraic geometry and in mirror symmetry.

Just as the Knizhnik-Zamolodchikov equation

is central for many questions in representation theory,

quantum differential equation

is central for many questions in

to coincide with the Knizhnik-Zamolodchikov equation

solved by conformal blocks of ,

one wants to take to be a very special manifold.

To get the quantum differential equation

we need can be described in several different ways.

The manifold

is as the Coulomb branch

of a certain

three dimensional quiver gauge theory

One description of

with N=4 supersymmetry.

The corresponding quiver

based on the Dynkin diagram of .is

attached to the nodes of the Dynkin diagram

The integers

and

encode the representation conformal blocks transform in:

(For knot theory purposes, the is always dominant.)

transversal slices in affine Grassmannian of G

The manifold also has a description as

a resolution of a certain intersection of

where is the Lie group of adjoint type with Lie algebra .

,

singular monopoles, with prescribed Dirac singularities, on

is as the moduli space of

Another description of

The vector in

encodes the singular monopole charges

and the order in which they appear.

The choice of

determines the total monopole charge,

including that of smooth monopoles.

in

positions of these smooth monopoles on

The monopole moduli space

can be thought of as parameterized, in part, by

keeping the positions of singular monopoles fixed.

The manifold

is holomorphic symplectic, so it has hyper-Kahler structure.

The positions of singular monopoles on

are the moduli of its metric.

The positions of singular monopoles on

are the real Kahler moduli of ,

and their positions on the complex structure moduli.

unless we work equivariantly with respect to a torus action

Since is holomorphic symplectic,

that scales the holomorphic symplectic form

its quantum cohomology is trivial,

We chose all the singular monopoles to be at the origin of

in

in order for this to be symmetry.

and the positions of singular monopoles on

If all the representations are minuscule,

is smooth. are generic,

A smooth holomorphic symplectic manifold with a

symmetry that scales its form is called

an equivariant symplectic resolution.

All the ingredients in

have a geometric interpretation in terms of .

x xx

x

we take a singular monopole

For every vertex operator

at the corresponding point on

with the highest weight of the representation

coloring the vertex operator.

is identified by Langlands correspondence

The charge of the singular monopole

Here, is the simply connected Lie group with Lie algebra .

To get the quantum differential equation to coincide

one needs to work equivariantly with respect to a larger torus of symmetries

with the Knizhnik-Zamolodchikov equation solved by

in addition to the action which scales

.

,

the highest weight vector

preserves the holomorphic symplectic form,

and comes from the maximal torus of .

Its equivariant parameters determine

The symmetry corresponding to

which enters:

and which is not fixed by

The fact that Knizhnik-Zamolodchikov equation solved by

has a geometric interpretation as the

quantum differential equation of

computed by equivariant Gromov-Witten theory,

implies the conformal blocks too have a geometric interpretation.

equivariant counts of holomorphic maps

Solutions of the quantum differential equation are

equivariant Gromov-Witten theory.

They are known as the Givental’s J-function

of all degrees computed by

or, “cohomological functions” of .

generating functions

The domain curve

is best thought of an infinite cigar with an boundary at infinity.

The boundary data is a choice of a K-theory class

Knizhnik-Zamolodchikov equationIt determines which solution of the

the vertex function computes.

The vertex function is a vector

due to insertions of classes at the origin of D.

of .

Geometric Satake correspondence,

with the weight subspace of

representation

identifies

in terms of

The geometric interpretation of conformal blocks of

has far more information than the conformal blocks themselves.

Underlying the Gromov-Witten theory of

with as a target space.

is a two-dimensional supersymmetric “sigma model”

The sigma model describes all maps

not only holomorphic ones.

The physical interpretation of

Gromov-Witten vertex function

with target on

is the partition function of the supersymmetric sigma model

supersymmetry is preserved byIn the interior of ,

an A-type topological twist,

so the partition function is computed by Gromov-Witten theory of .

Boundary conditions form a category, and

One places at infinity a “B-type” boundary condition.

working equivariantly with respect to is

the derived category of equivariant coherent sheaves on

the category of boundary conditions of the sigma model on ,

preserving a B-type supersymmetry and

Picking, as a boundary condition, an object

we get as the partition function.

only through its K-theory class

It depends on the choice of the brane

generalized central charge of the brane The vertex function is the

where the central charge that is being generalized is

the Pi stability central charge,

considered by Douglas in his work on stability conditions.

The vertex function

in two different ways.

generalizes the central charge function of the brane

is a solution to the Knizhnik-Zamolodchikov equation which

Firstly, the vertex function is a vector

due to insertions of classes at the origin of D.

and secondly it depends on equivariant parameters

of the -action on .

Undoing only the first generalization, by placing no insertion at the origin

we get a scalar analog of the vertex function

which is the “equivariant central charge function”

The Pi stability central charge

is obtained from the equivariant central charge by

setting the T-equivariant parameters to zero.

since the complexified Kahler moduli of

are the relative positions of vertex operators on

that avoids singularities,complexified Kahler moduli

A braid has a geometric interpretation as a path in

there are no quantum corrections with equivariant parameters turned off,

Since is holomorphic symplectic,

so the Pi stability central charge has an exact expression in classical geometry.

For example, for a brane supported on a holomorphic Lagrangian

with a vector bundle over it

the exact central charge is simply:

A geometric realization of the action of

on the space of conformal blocks is

along the path in its Kahler moduli corresponding to the braid.

monodromy of the quantum differential equation of

of the brane at the boundary at infinity

K-theory class a-priori comes from the action on the

The action of monodromy

so the quantum group acts on equivariant K-theory

since they differ by the data at the origin,

and monodromy acts at infinity.

the scalar functions and vector vertex functions

have exactly the same monodromy

and

A simple consequence is that

Since the sigma model needs the actual brane

to serve as the boundary condition,

an action on the brane itself,

the action of monodromy comes from

Along a path in Kahler moduli

the derived category stays the same, so we get a braid group action

that acts on its branes by

which is an auto-equivalence the derived category,

every object of the derived category, i.e. every B-type brane

direct sums, shifts and cones starting with semi-stable branes.

and undergoes monodromy

as we go around a closed loop in moduli space.

has a description in terms of

What happens is that,

This description depends on the choice of stability condition,

a Bridgeland stability conditions,

Our setting should provide a model example of

whose variations generate braid group actions

on the derived category.

The specific braid group action on the derived category

which one gets from the sigma model categorifies the action of

quantum group on equivariant K-theory

via monodromies of quantum differential equation,

This is a theorem, in our case, due to Bezrukavnikov and Okounkov.

From sigma model perspective, the monodromy problem arizes

according to the braid, in the neighborhood of the boundary at infinity.

by letting the moduli of the theory vary

The direction along the cigar coincides with the “time ” along the braid.

By asking how monodromy

Cecotti and Vafa,studied twenty years ago by

one gets a Berry phase type problem

acts on the quantum state produced at

by the path integral over the cigar,

The solution of the problem is the linear map

the monodromy of the quantum differential equation,

which acts on the K-theory class of the brane

The path integral depends only on the homotopy type path B.

so taking all the variation to happen near the boundary,

we get to keep the moduli constant over the entire cigar,

but produce a new boundary condition

which is the image of under the derived equivalence functor .

and whose K-theory class is .

To extract the monodromy matrix elements

cut the infinite cigar near its boundary, at

and insert a “complete set of branes”.

with the pair of B-branes at the boundary

We end up with the description of monodromy matrix element

are the vertex functions of the branes.

and

as computed by the path integral of the sigma model on the annulus,

where

annulus with a pair of B-type branes at the boundary

where we take the time to run around the ,

computes the index of a supercharge preserved by the two branes.

The path integral on the

is per definition the graded Hom space between the branes

computed in

The cohomology of the supercharge Q

Its Euler characteristic is per construction

of

the matrix element

of monodromy matrix of the Knizhnik-Zamolodchikov equation.

So far, we understood that

manifestly categorifies

braiding matrix elements.

The quantum invariants of links should also be categorified by

since they too can be expressed as matrix elements of the braiding matrix

between pairs of conformal blocks.

The first step is to find objects of

whose vertex functions are conformal blocks

in which pairs of vertex operators fuse to trivial representation.

For this, one makes use of a classic result in conformal field theory,

which is that

fusion diagonalizes braiding.

In looking for objects of

whose vertex functions are conformal blocks

we will discover that not only braiding,

but also fusion has a geometric interpretation in terms of

Braiding changes the order in which one sews the Riemann surface

from chiral vertex operators

xx xx

xx xx

and acts on the space of conformal blocks by R-matrices of

,

(which correspond to 3-punctured spheres)

sewing chiral vertex operators as follows:

it is more natural to first bring them together, and sew instead like this:

xx xx

x

xx

x

approach, one gets a new natural basis of conformal blocks.

Rather than using conformal blocks obtained by

As a pair of vertex operators

Conformal blocks in the fusion basis,

x

xx

x

are represented diagrammatically like this:

obtained by first bringing vertex operators together,

xx xx

unlike in the basis we started with

Both choices of basis span the space of solutions

to the Knizhnik-Zamolodchikov equation, but

braiding acts diagonally,in the fusion basis,

xxxx

in which the Riemann surface degenerates:

A basic, but key, result in conformal field theory is that one can

determine the eigenvectors and eigenvalues of braiding in the limit

which corresponds to replacing a pair of vertex operators by a single one

The possible choices of fusion products

that occur in the tensor product,

and where are conformal dimensions of vertex operators.

are labeled by representations

Replacing

in the conformal block,

gives a basis of solutions of the KZ equation

whose behavior as is,

where “finite” stands for terms that are non-vanishing and regular.

that solutions of the KZ equation obtained in this wayIt follows

with eigenvalue

which is read off from

are eigenvectors of braiding

is reflected in the behavior of its central charges.

One of the lessons from the very early days of mirror symmetry

is that the geometry of near a point in its moduli space

where it develops a singularity

For us, the central charges are close cousins of conformal blocks,

must be reflected in the geometry of .

so the behavior of conformal blocks we just found

develops a singularity due to a collection of cycles that vanish:

labeled by representations in the tensor product

One can show that, as ,

whose dimension is where

corresponding to a pair of vertex operators that approach each other,

These vanishing cycles give rise to objects of the derived category

as the conformal blocks in the fusion basis.

whose vertex functions have the same leading behavior near

the singularity at

of the derived category .

do not in general come from of actual objects

Conformal blocks which diagonalize the action of braiding

on which

are rare.

Eigensheaves of braiding

the braiding functor acts as

implies that the derived category analogue of the fact that

fusion diagonalizes braiding in conformal field theory

is existence of a filtration

near the singularity whose terms are labeled by the fusion products,

by the order of vanishing of the central charge

Mirror symmetry,

The m-th term in the filtration

and where the order of vanishing increases as decreases:

generated by objects whose central charges vanish at least as fast as

where

is a subcategory of the derived category

the dimension of the vanishing cycle

vanishes at least as fast aswhose central charge

Here a subcategory of

starting with -semi-stable objects of

obtained by

This filtration is induced from an analogous filtration of

abelian heart of

and taking all possible extensions.

One gets such a filtration near each

wall in Kahler moduli,

togehter.

xxxx

corresponding to a pair of vertex operators coming

Exchanging the order of a pair of vertex operators:

corresponds to a generalized flop that trades

where

encode two different orderings of monopoles :

there is an analogous filtration on the other side of the wall

which comes from the filtration on its abelian heart subcategories

Objects of are obtained from those in

by taking direct sums, degree shifts and cones.

By analyticity of the central charge,

teaches us that braiding always mixes up branes

of a given order of vanishing of the central charge,

Mirror symmetry,

with those that vanish faster,

but never the other way around.

This means while braiding cannot be diagonalized,

it preserves the filtrations.

Picking a path in Kahler moduli

the variation of the central charge

gives a functor

to

that takes

by mixing up objects at a given order of the filtration

with those from lower orders.

In particular, on the quotient category

in which one treats all objects of coming from as zero

the functor acts at most by degree shifts

which one can read off from

Derived equivalences of this type are called perverse equivalences

Rouquier and Chuang.

by

We are able to explicitly characterize them for any

and all of its walls by predicting the shift functors

This reproduces the known results for the generalized flop of

cotangent to Grassmannians

and

by thinking about them as

and

due to Kamnitzer, Cautis and Licata,

the -th and the -th anti-symmetric representation.

where and are given by its for

We can now characterize branes of

whose vertex functions are conformal blocks

describing cups and caps.

The cap colored by a representation

which approach each other and fuse to the identity.

colored by conjugate representationscomes from a pair of vertex operators,

The vanishing cycle,

where is a maximal parabolic subgroup of

which are known as minuscule Grassmannians.

associated to the minuscule representation

associated to identity representation in the tensor product

belong to the lowest term of the filtration

so they are necessarily eigensheaves the braiding functor

for the same reason the identity representation is special.

Even then, they are extremely special ones,

The objects corresponding to such conformal blocks

corresponding to bringing and together

When a collection of vertex operators come together in pairs of

our manifold has a local neighborhood where we can approximate it as

where

minuscule representations and their conjugates

is a product of minuscule Grassmannians:

We get a very special B-type brane

which is the structure sheaf of this vanishing cycle,

Among other things, the vertex function of this brane is the conformal block

which we will denote by

For any derived auto-equivalence

coming from our action of braiding on …

the graded homology group

categorifies the corresponding link invariant.

Namely, by the theorem of Bezrukavnikov and Okounkov

is a braid invariant whose Euler characteristic

is the matrix element

of the corresponding braiding matrix .

For our brane

the vertex function

is the conformal block described by

……so the Euler characteristic of the homology theory

the link invariant.

is automatically

In fact a stronger statement is true,

is itself be a link invariant.

namely,

For this, one needs to show that additional relations hold.

1. A version of ``pitchfork’’ identity:

2. Reidermeister 0 or “S-move”:

3. Framed Reidermeister I move:

For all of these, in conformal field theory, one wants to view a cap

as a map between the space of conformal blocks of the form

which come from

and the space of conformal blocks obtained by pair creation

which come from and look like

on the derived category

and their relation to conformal field theory

these relations hold in the derived category.

The existence of perverse filtrations

provides an apriori way to understand why

For example,

states that we get a derived equivalence

where

are the cap functors on the left and the right and

corresponds to braiding with

We can identify

and

as the bottom most parts of double filtrations of

which one gets near the intersection of a pair of walls

and

where three vertex operators come together.

The functor acts on the bottom parts of

These turn out to be trivial in our case

or otherwise the relation we are trying to prove would not hold

even in conformal field theory.

The functor all of whose degree shifts are identity acts trivially

the double filtration only by degree shifts.

so one finds

Finally, a very elementary, but nice consequence

is a geometric explanation for

mirror symmetry of link invariants

which states that the invariants of

are related by

a link and its mirror image

For us this follows from Serre duality

with branes and at the two ends

and -cohomology obtained by a reflection that exchanges the endpoints.

which is an isomorphism of -cohomology

The shift in the equivariant degree comes from the fact that, while

is trivial its unique holomorphic section is not invariant under .

of both sides of

and using

one finds

Taking the Euler characteristic

The fact that Serre duality implies mirror symmetry

is not an accident since the directions along the interval

and along the link, which get reflected, coincide.

Let me briefly also describe some aspects of mirror symmetry,

which we used repeatedly.

The ordinary, non-equivariant mirror of

is a hyper-Kahler manifold

which is, to a first approximation,

given by a hyper-Kahler rotation of

has only complex but no Kahler moduli turned on.

As has only Kahler but not complex moduli,

due to the equivariance we impose,

to be at the origin of in )

( since we took all the singular monopoles

A description based on

would give a symplectic geometry approach to the categorification problem,

with

replaced by its homological mirror,

Lagrangian branes on an appropriate derived category of

such as those in the works of Seidel and Smith, for Khovanov homology.

“symplectic” homological link invariants,

which only capture the theory at

At the moment, one only knows how to obtain from

There is an alternative symplectic geometry approach,

where the dependence of the theory

instead of being mysterious,

is manifest.

on ,

action on

since we want to work equivariantly with respect to the

which scales its symplectic formholomorphic

We will take advantage of the fact that,

all the relevant information about the geometry of

is contained in the fixed locus of this action.

The fixed locus of the action on ,

is a holomorphic Lagrangian in since it is mid-dimensional and

We will call the core of .

Viewing as the moduli space of monopoles on

its core is a locus in the moduli space where all the monopoles,

singular or not, are at the origin of and at points in

Instead of working with

one can work with the the core and the core’s mirror

and its mirror

mirror

mirror

Working equivariantly with respect to action on

the bottom row has as much information about the geometry

as the top.

,

we will call the equivariant mirror of .

mirror

mirror

Since the bottom row has as much information about the geometry as the top,

equivariant mirror

While embeds into

fibers over with holomorphic Lagrangian fibers

holomorphic Lagrangian submanifold of dimension

as a

A model example to keep in mind is

which is an surface.

Its core looks like

it is a collection of ’s with a pair of infinite discs attached.

This example comes from taking

with representation theoretic data encoded in the quiver

in presence of singular ones.

is the moduli space of a single smooth monopole,

The ordinary mirror of which is an surface,

is which is a “multiplicative”

with a potential which we will not need.

surface,

is a -fibration over

The “multiplicative” surface ,

is an infinite cylinder with marked points in the interior.

At the marked points, the fibers of degenerate .

They project to Lagrangians in that begin and end at the

punctures.

There are Lagrangian spheres in

which are mirror to vanishing ’s in .

is a single copy of the Riemann surface

where the fibration degenerates.

where the conformal blocks live:

The positions of vertex operator correspond to the marked points

This is not an accident.

By SYZ mirror symmetry,

the mirror pair

share a common base,

on in presence of singular ones.

which is the moduli space of one smooth monopole

More generally,

and the ordinary mirror of its core ,

the equivariant mirror of

corresponding to

is

and is our Riemann surface with punctures.

where is the total number of smooth monopoles

and of

Projecting to the common SYZ base of

is the same as projecting ,

the moduli space of singular monopoles on

to

Including an equivariant action

corresponds to adding to the sigma model on

a specific potential,

which is a multi-valued complex function on .

on and on

The potential is a sum of three terms,

which one should think of as associated to the nodes and arrows

of the quiver.

Think of the Riemann surface

as a complex plane with

deleted.finite punctures at and

which comes from the -action

and the two terms which come from the action by

,

Then the term in the potential

on is given by

From perspective of conformal blocks

were….

…..partition functions of the sigma model with target

with A-twist in the interior and a B-type boundary condition at infinity,

on which is an infinitely long cigar

which is an object of

In this setting,

the Knizhnik-Zamolodchikov equation

which the conformal blocks solve, arizes as the

quantum differential equation.

,

the partition function of the B-twisted theory on ,

with an A-type boundary condition at infinity, corresponding to a

Lagrangian in .

one expects to get the conformal blocks as

exchanges the A and the B-modelsMirror symmetry

so in the theory based on

with potential

where is the top holomorphic form on ,

Such amplitudes have the following form

and where ’s come from insertions of

point observables at the origin.

We are rediscovering here, from mirror symmetry,

conformal blocks

which goes back to work of Feigin and E.Frenkel in the ’80’s

and Schechtman and Varchenko.

the integral formulation of the

the Landau-Ginzburg integral solves

is also the quantum differential equation of ….

The fact that the Knizhnik-Zamolodchikov equation which

…..gives a Givental type proof of 2d mirror symmetry

the T-equivariant A-model on ,

to

B-model on with potential .

at genus zero, relating

One of the harder problems in

in terms of which the amplitudes satisfy a simple set of equations.

with some target and potential

is identifying the “flat coordinates” on the moduli space

a Landau-Ginsburg B-model

where is the matrix of multiplication by

For us, the coordinates are the positions of vertex operators on ,

and the equations solved are the Knizhnik-Zamolodchikov equations.

are expected to be equivalent to all genus.

Thus, the B-twisted the Landau-Ginsburg model

and A-twisted sigma model on

the resolution of slices in the affine Grassmannian of

working equivariantly with respect to ,

Knizhnik-Zamolodchikov equation

is an A-brane at the boundary of at infinity,

the derived Fukaya-Seidel category of A-branes on with potential .

The brane is an object of

Corresponding to a solution of the

The objects of are graded Lagrangians

where the grading is the Maslov grading,

together with additional grades which come from

the non-single valued potential.

The additional grades are defined

analogously to the way the lift of the phase of

by lifting the phase of

to a real valued function on the Lagrangian,

is used used to define the cohomological, Maslov grading.

Defining a theory of

A-branes on a non-compact manifold such as

requires work, to cure the non-compactness.

In the present case, we are after a symplectic-geometry based

The Lagrangians we need are for this purpose are all compact,

since they are related by mirror symmetry

to compact vanishing cycles on .

approach to knot homology.

there are no issues with non-compactness of .

For such Lagrangians,

and the superpotential would have played no role either,

were it single valued.

What the non-single valued potential does

is to provide additional gradings on

the Floer cohomology groups.

In the mirror language,

a braid in is the path in complex structure moduli of

Since is just the moduli space

of points on the monodromy on the moduli space comes

becomes completely geometric.

Varying the complex structure along the braid

give rise to derived equivalences of

which come from variation of the central charge function

The fact that

provides a stability condition on

follows from a theorem by Thomas, in the special case

which corresponds to an surface .of

with central charge

The stable branes of

are just special Lagrangians on

stable branes map to straight lines on

Since is a product and is flat

x

An brane at the m-th step in the filtration as come together

may look something like this:

where is the dimension of the vanishing cycle

xxx

x

xx

x

Derived equivalences act by generalized Dehn twists,

with those that vanish faster:

x

xxx

which mix cycles with a given order of vanishing,

by dimension of the vanishing cycle.

manifestly preserving the filtration

Finally, the equivariant central charge:

tracks the equivariant degrees as well as Maslov degrees.

Equivariant mirror symmetry

equivariant mirror

mirror

mirror

helps us understand exactly which questions we need to ask

to recover homological knot invariants from .

Since is an ordinary mirror of ,

the key step is to understanding how to recover

homological knot invariants from , instead of

equivariant mirror

mirror

mirror

The rest follows by ordinary mirror symmetry.

the first construction of Khovanov homology,

This leads to, for example,

which is entirely based on symplectic geometry.

* ** *

leads to the construction of Jones polynomial due to Bigelow,

Taking the Euler characteristic of the theory