knot categorification and mirror symmetry - math.ksu.edu · along the way, we will discover...
TRANSCRIPT
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and Mirror Symmetry
Mina Aganagic UC Berkeley
Knot Categorification
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to the problem of categorifying
link invariants.
In this talk, I will describe an application of mirror symmetry
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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.
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depends on a choice of a Lie algebra,
of its strands by representations of .
A quantum invariant of a link
and a coloring
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The link invariant,
in addition to the choice of the Lie algebra
depends on one parameter
and its representations,
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Quantum invariants of a link with this data
can be thought of as originating from conformal field theory with
Lie algebra symmetry.
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from geometry.
We will start by recalling how this comes about.
We will then go on to rediscover
the relevant structures
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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
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xxx
It is equivalent, but better for our purpose,
to be a punctured infinite cylinder.
to take
x
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To punctures at finite point
xxx
x
we will associate finite dimensional representations
of .
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of infinite dimensional, Verma module representations of
To punctures at the two ends at infinity,
we will associate a pair
xxx
x
,
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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
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``conformal block”
obtained by sewing chiral vertex operators.
To a Riemann surface with punctures
it associates a
xxx
x
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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.
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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
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is the equation discovered by Knizhnik and Zamolodchikov in ’84:
xxx
x
conformal blocks of The equation solved by
on
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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.
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The coefficients on its right hand side
are the classical r-matrices of :
where
in the standard Lie theory notation.
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By varying the positions of vertex operators on as a function of
we get a colored braid in three dimensional space, which is
“time”
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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.
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Monodromy along a path depends only on its homotopy type,
so the resulting monodromy matrix
is an invariant of the colored braid.
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The monodromy problem of the Knizhnik-Zamolodchikov equation
was solved by Tsuchia and Kanie in ’88 and by Drinfeld and Kohno in ’89.
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corresponding to
is an R-matrix of the quantum group
They showed that monodromy matrix that reorders
a neighboring pair of vertex operators
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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.
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The monodromy action
is irreducible only in the subspace of
x xx
x
of fixed
corresponding to conformal blocks of the form
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quantum invariants of not only braids
This perspective leads to
but knots and links as well.
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Any link K can be represented as a
a closure of some braid.
=
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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.
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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
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both braiding and fusion of conformal field theory
play an important role in the story.
This way,
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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
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To braids,
one would like to associate
functors between the categories
corresponding to the
top and the bottom.
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Moreover,
we would like to do that in the way that
recovers the quantum knot invariants upon
de-categorification.
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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.
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The virtue of the two approaches
I will describe in these lectures,
is that the second step is automatic.
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The starting point for us is
a geometric realization
Knizhnik-Zamolodchikov equation.
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we will find two such geometric realizations,
More precisely,
related by two dimensional equivariant mirror symmetry.
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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.
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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.
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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
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``A-model’’
Quantum multiplication on
is defined by Gromov-Witten theory,
or, the oftopological
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The first, term of the quantum multiplication
subsequent terms are quantum corrections.
is the classical product on :
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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
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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
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we need can be described in several different ways.
The manifold
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is as the Coulomb branch
of a certain
three dimensional quiver gauge theory
One description of
with N=4 supersymmetry.
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The corresponding quiver
based on the Dynkin diagram of .is
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attached to the nodes of the Dynkin diagram
The integers
and
encode the representation conformal blocks transform in:
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(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 .
,
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singular monopoles, with prescribed Dirac singularities, on
is as the moduli space of
Another description of
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The vector in
encodes the singular monopole charges
and the order in which they appear.
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The choice of
determines the total monopole charge,
including that of smooth monopoles.
in
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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.
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The manifold
is holomorphic symplectic, so it has hyper-Kahler structure.
The positions of singular monopoles on
are the moduli of its metric.
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The positions of singular monopoles on
are the real Kahler moduli of ,
and their positions on the complex structure moduli.
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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,
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We chose all the singular monopoles to be at the origin of
in
in order for this to be symmetry.
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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.
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All the ingredients in
have a geometric interpretation in terms of .
x xx
x
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we take a singular monopole
For every vertex operator
at the corresponding point on
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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 .
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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
.
,
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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
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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.
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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
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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.
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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
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in terms of
The geometric interpretation of conformal blocks of
has far more information than the conformal blocks themselves.
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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.
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The physical interpretation of
Gromov-Witten vertex function
with target on
is the partition function of the supersymmetric sigma model
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supersymmetry is preserved byIn the interior of ,
an A-type topological twist,
so the partition function is computed by Gromov-Witten theory of .
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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
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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
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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.
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The vertex function
in two different ways.
generalizes the central charge function of the brane
is a solution to the Knizhnik-Zamolodchikov equation which
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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 .
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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”
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The Pi stability central charge
is obtained from the equivariant central charge by
setting the T-equivariant parameters to zero.
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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
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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:
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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
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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
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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
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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
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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,
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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,
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a Bridgeland stability conditions,
Our setting should provide a model example of
whose variations generate braid group actions
on the derived category.
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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.
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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.
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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,
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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
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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 .
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To extract the monodromy matrix elements
cut the infinite cigar near its boundary, at
and insert a “complete set of branes”.
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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
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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
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is per definition the graded Hom space between the branes
computed in
The cohomology of the supercharge Q
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Its Euler characteristic is per construction
of
the matrix element
of monodromy matrix of the Knizhnik-Zamolodchikov equation.
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So far, we understood that
manifestly categorifies
braiding matrix elements.
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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.
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The first step is to find objects of
whose vertex functions are conformal blocks
in which pairs of vertex operators fuse to trivial representation.
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For this, one makes use of a classic result in conformal field theory,
which is that
fusion diagonalizes braiding.
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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
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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)
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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
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Conformal blocks in the fusion basis,
x
xx
x
are represented diagrammatically like this:
obtained by first bringing vertex operators together,
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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
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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
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The possible choices of fusion products
that occur in the tensor product,
and where are conformal dimensions of vertex operators.
are labeled by representations
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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.
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that solutions of the KZ equation obtained in this wayIt follows
with eigenvalue
which is read off from
are eigenvectors of braiding
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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
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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
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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,
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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
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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
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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,
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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
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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.
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One gets such a filtration near each
wall in Kahler moduli,
togehter.
xxxx
corresponding to a pair of vertex operators coming
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Exchanging the order of a pair of vertex operators:
corresponds to a generalized flop that trades
where
encode two different orderings of monopoles :
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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,
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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.
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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.
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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
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Derived equivalences of this type are called perverse equivalences
Rouquier and Chuang.
by
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We are able to explicitly characterize them for any
and all of its walls by predicting the shift functors
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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
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We can now characterize branes of
whose vertex functions are conformal blocks
describing cups and caps.
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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,
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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
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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
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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:
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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
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For any derived auto-equivalence
coming from our action of braiding on …
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the graded homology group
categorifies the corresponding link invariant.
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Namely, by the theorem of Bezrukavnikov and Okounkov
is a braid invariant whose Euler characteristic
is the matrix element
of the corresponding braiding matrix .
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For our brane
the vertex function
is the conformal block described by
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……so the Euler characteristic of the homology theory
the link invariant.
is automatically
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In fact a stronger statement is true,
is itself be a link invariant.
namely,
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For this, one needs to show that additional relations hold.
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1. A version of ``pitchfork’’ identity:
2. Reidermeister 0 or “S-move”:
3. Framed Reidermeister I move:
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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
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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
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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
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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.
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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
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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
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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 .
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of both sides of
and using
one finds
Taking the Euler characteristic
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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.
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Let me briefly also describe some aspects of mirror symmetry,
which we used repeatedly.
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The ordinary, non-equivariant mirror of
is a hyper-Kahler manifold
which is, to a first approximation,
given by a hyper-Kahler rotation of
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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
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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
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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
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There is an alternative symplectic geometry approach,
where the dependence of the theory
instead of being mysterious,
is manifest.
on ,
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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.
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The fixed locus of the action on ,
is a holomorphic Lagrangian in since it is mid-dimensional and
We will call the core of .
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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
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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.
,
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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
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While embeds into
fibers over with holomorphic Lagrangian fibers
holomorphic Lagrangian submanifold of dimension
as a
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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.
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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,
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The ordinary mirror of which is an surface,
is which is a “multiplicative”
with a potential which we will not need.
surface,
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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 .
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They project to Lagrangians in that begin and end at the
punctures.
There are Lagrangian spheres in
which are mirror to vanishing ’s in .
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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.
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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
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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
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and of
Projecting to the common SYZ base of
is the same as projecting ,
the moduli space of singular monopoles on
to
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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
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The potential is a sum of three terms,
which one should think of as associated to the nodes and arrows
of the quiver.
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Think of the Riemann surface
as a complex plane with
deleted.finite punctures at and
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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
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From perspective of conformal blocks
were….
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…..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
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In this setting,
the Knizhnik-Zamolodchikov equation
which the conformal blocks solve, arizes as the
quantum differential equation.
,
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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
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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.
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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
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the Landau-Ginzburg integral solves
is also the quantum differential equation of ….
The fact that the Knizhnik-Zamolodchikov equation which
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…..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
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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
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For us, the coordinates are the positions of vertex operators on ,
and the equations solved are the Knizhnik-Zamolodchikov equations.
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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 ,
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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
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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.
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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.
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Defining a theory of
A-branes on a non-compact manifold such as
requires work, to cure the non-compactness.
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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.
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there are no issues with non-compactness of .
For such Lagrangians,
and the superpotential would have played no role either,
were it single valued.
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What the non-single valued potential does
is to provide additional gradings on
the Floer cohomology groups.
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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.
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Varying the complex structure along the braid
give rise to derived equivalences of
which come from variation of the central charge function
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The fact that
provides a stability condition on
follows from a theorem by Thomas, in the special case
which corresponds to an surface .of
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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
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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
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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
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Finally, the equivariant central charge:
tracks the equivariant degrees as well as Maslov degrees.
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Equivariant mirror symmetry
equivariant mirror
mirror
mirror
helps us understand exactly which questions we need to ask
to recover homological knot invariants from .
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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.
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the first construction of Khovanov homology,
This leads to, for example,
which is entirely based on symplectic geometry.
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* ** *
leads to the construction of Jones polynomial due to Bigelow,
Taking the Euler characteristic of the theory