Knot Categorification from Geometry via String Theory · 2019-01-21 · Knot Categorification...
Transcript of Knot Categorification from Geometry via String Theory · 2019-01-21 · Knot Categorification...
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Knot Categorification
from Geometry,
via String Theory
Mina Aganagic UC Berkeley
Based on joint work to appear with Andrei Okounkov, and further work with Dimitri Galakhov.
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This talk is about
categorification of knot invariants.
I will explain how
two geometric approaches to the problem,
and the relationship between them,
follow from string theory.
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The approaches we will find,
have the same flavor as those in the works of
Kamnitzer and Cautis,
Seidel and Smith,
Thomas and Smith,
Webster
although details differ.
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There is another approach
with the same string theory origin,
due to Witten.
We will see that string theory leads to a unified framework
for knot categorification.
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To begin with, it is useful to recall
some well known aspects of knot invariants.
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To get a quantum invariant of a link K
one starts with a Lie algebra,
and a coloring
of its strands by representation of .
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The link invariant,
in addition to the choice of a group
and
representations,
depends on one parameter
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Witten showed in his famous ’89 paper,
that the knot invariant
comes from
Chern-Simons theory with gauge group based on the Lie algebra
and (effective) Chern-Simons level
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In the same paper, he showed that
underlying Chern-Simons theory is a
two-dimensional conformal field theory,
with
affine current algebra symmetry.
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The space conformal blocks of
on a Riemann surface with punctures
is the Hilbert space of Chern-Simons theory on
x x x
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To eventually get invariants of knots in or 3,
we want to take
to be a complex plane with punctures,
xxx
or equivalently, a punctured infinite cylinder.
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The corresponding conformal blocks
are correlators of chiral vertex operators:
xx
x
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Punctures at and ,
are labeled by a pair of highest weight vectors
of Verma module representations of the algebra.
x xx
x
and
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x x x x
associated to a finite dimensional representation
A chiral vertex operator
of adds a puncture at a finite point on
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It acts as an intertwiner
between a pair of Verma module representations,
x x x x
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The space of conformal blocks corresponding to
is a vector space.
Its dimension is that of a subspace of representation
when one sews the
whose weight is
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The finite dimensional space of conformal blocks
comes from choices of intermediate Verma module representations
when one sews the chiral vertex operators together.
x x x
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The Chern-Simons path integral on
in the presence of a braid
gives the corresponding
quantum braid invariant.
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The braid invariant one gets
is a matrix that transports
the space of conformal blocks,
along the braid .
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To understand what it means to the transport the space of conformal blocks
along a path corresponding to the braid
a different perspective on
conformal blocks is helpful.
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Instead of characterizing conformal blocks
in terms of vertex operators and sewing,
we can equivalently describe them as solutions to a differential equation.
x xx
x
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The equation solved by conformal blocks of ,
is the Knizhnik-Zamolodchikov equation:
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The quantum invariant of the braid
is the monodromy matrix of the Knizhnik-Zamolodchikov equation,
along the path in the parameter space corresponding to
the braid .
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The monodromy problem of the Knizhnik-Zamolodchikov equation
They showed that its monodromy matrices are given in terms of the
R-matrices of the quantum group
was solved by Drinfeld and Kohno in ’89.
corresponding to
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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
of fixed weight
which in our setting equals to
x xx
x
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This perspective leads to
quantum invariants of not only braids
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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The corresponding quantum link invariant is the matrix element
of the braiding matrix,
taken between a pair of conformal blocks
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correspond to the top and the bottom of the picture.
The pair of conformal blocks
that define the matrix element
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The conformal blocks
are specific solutions to KZ equations
which describe pairwise fusing vertex operators
into copies of trivial representation.
Necessarily they correspond to subspace of ,
of weight .
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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 aimed to categorify.
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In the two of the approaches
we are about to describe,
the second step is automatic.
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The starting point for us is
a geometric realization of conformal blocks,
with origin in supersymmetric quantum field theory.
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This is not readily available.
We will eventually find not
one but two such interpretations.
However, to explain how they come about,
and to find a relation between them,
it is useful to ask a slightly different question first.
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q-conformal blocks of
Namely, we will first ask for a
geometric interpretation of
the quantum affine algebra that is a q-deformation of
the affine Lie algebra.
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are q-deformations of conformal blocks of ,
The q-conformal blocks of
which I. Frenkel and Reshetikhin
discovered in the ’80’s.
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They are defined as a correlation functions
of chiral vertex operators,
like in the conformal case, except all the operators are q-deformed.
x xx
x
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Just like conformal blocks of
the q-conformal blocks of
are solutions of the quantum Knizhnik-Zamolodchikov equation.
may be defined as solutions of the Knizhnik-Zamolodchikov equation,
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The quantum Knizhnik-Zamolodchikov (qKZ) equation
is a difference equation
which reduces to the Knizhnik-Zamolodchikov equation
in the conformal limit.
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It turns out that q-conformal blocks of
have a geometric realization,
and one that, most directly, originates from a
supersymmetric gauge theory.
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Let be a simply laced Lie algebra so in particular
and of the following types:
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The non-simply laced cases can be treated,
but I will not have time to describe this in this talk.
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three dimensional quiver gauge theory
The gauge theory we need
V1
W1
V2
W2
is a
with N=4 supersymmetry.
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defining the theory, is a collection of nodes and arrows between them,
based on the Dynkin diagram of
The quiver diagram
V1
W1
V2
W2
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The quiver encodes
the gauge symmetry and global symmetry groups
and representations of matter fields charged under it
V1
W1
V2
W2
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The ranks of the vector spaces
are determined by a pair of weights
of the representation ,
which the q-conformal blocks transform in:
Wa
Va
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of the three dimensional quiver gauge theory on
The solutions to qKZ equation
turn out to be supersymmetric partition functions
V1
W1
V2
W2
where
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To define the supersymmetric partition function we need,
one wants to use the
R-symmetry of the 3d N=4 theory.
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All the ingredients in the q-conformal block
have a gauge theory interpretation.
x xx
x
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of the qKZ equation is the parameter by which rotates,
To preserve supersymmetry,
this is accompanied by a holonomy around the ,
of the R- symmetry.
The step
as we go around the in
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The parameter in
is the holonomy around the in ,
of the off-diagonal
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The holonomies associated to masses of fundamental hypermultiplets
are the positions of vertex operators,
V1
W1
V2
W2
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is related to the holonomy of global symmetry
associated with Fayet-Iliopoulos parameters:
The highest weight vector of Verma module in
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The partition function
of the quiver gauge theory on
has a precise mathematical formulation,
in terms of
quantum K-theory
of the Nakajima quiver variety .
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The quiver variety ,
is the Higgs branch of the gauge theory:
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Quantum K-theory
of Nakajima quiver varieties,
generalizing their Gromov-Witten theory
was developed a few years ago
by Okounkov (with Maulik and Smirnov).
The theory is a variant of the theory
Givental put forward earlier, but uses crucially the fact
that these are holomorphic-symplectic varieties.
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The supersymmetric partition function of the gauge theory on
for is the K-theoretic “vertex function”
This is the generating function of equivariant, K-theoretic counts
of “quasi-maps” (i.e. vortices)
of all degrees.
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One works equivariantly with respect to:
is the maximal torus of rotations of
that preserve the symplectic form, and scales it,
and with respect to
which rotates the domain curve.
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A key result of the theory,
due to Okounkov,
is that the K-theoretic vertex function of X
solves the quantum Knizhnik-Zamolodchikov equation
corresponding to
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Which solution of the qKZ equation
computes depends on the choice of data
at infinity of
This choice means vertex functions should be thought of as valued in
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While
solve the quantum Knizhnik-Zamolodchikov equation,
they are not the q-conformal blocks of .
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q-conformal blocks are the solutions of the qKZ equation
which are holomorphic in a chamber such as
corresponding to choice of ordering of vertex operators in
x xx
x
This is a choice of equivariant parameters of ,
or mass parameters of the gauge theory.
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Instead,
are holomorphic in a chamber of Kahler moduli of X,
and Fayet-Iliopoulus parameters of the gauge theory.
x xx
x
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So, this does not give an answer to the
question we are after,
namely to find a geometric interpretation of conformal blocks,
even after q-deformation.
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It turns out that
is not the only geometry that underlies solutions to
the qKZ equation corresponding to our problem.
There is a second one,
which turns out to the relevant one.
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There are two natural holomorphic symplectic
varieties one can associate to
the 3d quiver gauge theory
with quiver
V1
W1
V2
W2
and N=4 supersymmetry.
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One such variety is
the Nakajima quiver variety
.
This is the
Higgs branch of vacua
of the 3d gauge theory.
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The other is the Coulomb branch,
which we will denote by
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The Coulomb branch
of our gauge theory is a certain intersection of slices in
the (thick) affine Grassmanian of G.
Here, G is the adjoint form of a Lie group with Lie algebra g.
Braverman, Finkelberg, Nakajima
Bullimore, Dimofte, Gaiotto
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where are the pair of weights defining the quiver ,
with
is an intersection of a pair of orbits
in the affine Grassmanian of complex dimension
The Coulomb branch
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The two holomorphic symplectic varieties
in general live in different dimensions,
and between them,
the roles of Kahler (Fayet-Illiopolus) and equivariant (mass) parameters,
get exchanged:
,
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and the Kahler (FI) parameter of
The positions of vertex operators on
x xx
x
are equivariant (mass) parameters of
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Another way to think about
is as the moduli space of
-monopoles,
on
where is the charge of singular monopoles,
and the total monopole charge.
Hanany, Witten
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In this language, the positions of vertex operators on
x xx
x
are the (complexified) positions of singular monopoles on
x x x x
in .
(all monopoles sit at the origin of to preserve the symmetry)
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Three dimensional mirror symmetry
leads to relations between certain computations on
and
This is referred to as
“symplectic duality”,
in some of the literature that explores it.
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Physically, one expects to be able to compute the supersymmetric
partition function of the gauge theory on
by starting with the sigma model of either
or
with suitable boundary conditions at infinity.
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We prove that, whenever it is defined,
the K-theoretic vertex function of
solves the same qKZ equation
as
the K-theoretic vertex function of .
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From perspective of
the qKZ equation
is the quantum difference equation since the -variables
are the Kahler variables of .
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The “quantum difference equation"
is the K-theory analogue of
of the quantum differential equation of Gromov-Witten theory.
Here “quantum” refers to the
quantum cohomology cup product on
used to define it.
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While
and
solve the same qKZ equation,
they provide two different basis of its solutions.
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While
does the opposite.
leads to solutions of qKZ which are analytic in
-variables, but not in -variables,
Kahler for
and
equivariant for
Kahler for
and
equivariant for
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Now we can return to
our main interest,
which is obtaining a geometric realization of
conformal blocks.
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The conformal limit is the limit which takes
and the qKZ equation to the corresponding KZ equation.
It is amounts to
This corresponds to keeping the data of the conformal block fixed.
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very differently,
since it treats the
Kahler for
Kahler for
differently:
The conformal limit treats
and
- and the -variables,
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The conformal limit,
is not a geometric limit from perspective of the Higgs branch
since the -variables
which go to in the limit are its Kahler variables.
The limit results in a badly singular space,
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By contrast, from perspective of the Coulomb branch,
the limit is perfectly geometric.
Its Kahler variables are the -variables,
x xx
x
the positions of vertex operators,
which are kept fixed.
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the conformal limit,
is the cohomological limit
taking:
From perspective of ,
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The Knizhnik-Zamolodchikov equation
we get in the conformal limit
becomes the quantum differential equation
of .
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It follows that conformal blocks of
have a geometric interpretation as
cohomological vertex functions
computed by equivariant Gromov-Witten theory of
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The cohomological vertex function counts holomorphic maps
equivariantly with respect to
where one scales the holomorphic symplectic form of by
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The domain curve
is best thought of as an infinite cigar with an boundary at infinity.
The boundary data is a choice of a K-theory class
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The K-theory class
inserted at infinity of determines which solution of
the quantum differential equation, and thus which conformal block,
computes.
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Underlying
the Gromov-Witten theory of
,
is a two-dimensional supersymmetric sigma model with
as a target space.
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The geometric interpretation of
conformal blocks of
in terms of the supersymmetric sigma model to
has far more information than the conformal blocks themselves.
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The physical meaning of
is the partition function of the supersymmetric sigma model
with target
on
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one has A-type twist in the interior of
and at infinity, one places a B-type boundary condition,
corresponding to the choice of
To get
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The B-type boundary condition
is a B-type brane on . The brane we need to get
is an object
of the derived category of -equivariant coherent sheaves on
whose K-theory class is
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conformal block of .
The choice of a B-type brane
at infinity of determines which
computes.
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From perspective of ,
the action of braiding
on the space of conformal blocks
is the monodromy of the quantum differential equation,
along the path in its Kahler moduli.
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Monodromy of the quantum differential equation
acts on
via its action on K-theory classes
inserted at the boundary at infinity of
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A theorem of Bezrukavnikov and Okounkov says that,
the action of braiding matrix on
via the monodromy of the quantum differential equation
lifts to
a derived auto-equivalence functor of
for any smooth holomorphic symplectic variety .
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This implies that, for a smooth
derived auto-equivalence functors of
categorify the
action of the R-matrices
on conformal blocks of .
,
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=
This also implies that from
we get categorification of quantum invariants of links
since they can be expressed as matrix elements of the braiding matrix
between pairs of conformal blocks.
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Denote by
the branes that give rise to conformal blocks
the image of under the braiding functor.
and by
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The matrix element
is the partition function of the B-twisted sigma model to on
with the pair of B-branes at the boundary.
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The corresponding categorified link invariant
is the graded Hom between the branes
computed in
In addition to the homological grade, there is a second grade,
coming from the -action,
that scales the holomorphic symplectic form on ,
with weight
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The three dimensional gauge theory
we started with
in addition, leads to a
second description
of the categorified knot invariants.
V1
W1
V2
W2
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It leads to a description in terms of a
two-dimensional equivariant mirror of
The mirror description of is also
a new result.
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The mirror
is a Landau-Ginzburg theory with target ,
and potential
.
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The Landau-Ginzburg potential
and the target
can be derived from the 3d gauge theory.
V1
W1
V2
W2
The potential is a limit of the three dimensional effective
superpotential, given as a sum of contributions
associated to its nodes and its arrows.
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Recall that, in the conformal limit,
has no geometric interpretation in terms of X.
One instructive, if roundabout, way to
discover the mirror description,
is as follows.
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While its conformal limit is not given in terms of ,
it must exist.
To find it one wants to make use the integral formulation of
which one can derive by thinking of maps to
in geometric invariant theory terms.
Aganagic, Frenkel, Okounkov
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The integrals
come from studying quasi-maps to the pre-quotient and,
projecting to gauge invariant configurations,
leads to integration over the maximal torus of
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The conformal limit of has the form:
It gives integral solutions to the Knizhnik-Zamolodchikov equation
corresponding to
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The function that enters
is the Landau-Ginzburg potential,
and is a top holomorphic form on .
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The potential is a sum over three types of terms
V1
W1
V2
W2
one of which comes from the nodes
and two from the arrows.
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The integration in
is over a Lagrangian cycle in
the target space of the Landau-Ginzburg model.
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the integral representations of conformal blocks of
They are very well known,
and go back to work of Feigin and E.Frenkel in the ’80’s
and Schechtman and Varchenko.
We are re-discovering
from
geometry and supersymmetric gauge theory,
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The fact that the Knizhnik-Zamolodchikov equation which
the Landau-Ginzburg integral solves
is also the quantum differential equation of ….
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…..gives a Givental type proof of 2d mirror symmetry
at genus zero,
relating
equivariant A-model on ,
to
B-model on with superpotential .
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The Landau-Ginzburg origin of
conformal blocks
automatically
leads to categorification of the corresponding
braid and link invariants.
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From the Landau-Ginzburg perspective
the conformal block
is the partition function of the B-twisted theory on ,
with A-type boundary condition at infinity, corresponding to the
Lagrangian in .
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Thus, corresponding to a solution to the
Knizhnik-Zamolodchikov equation
is an A-brane at the boundary of at infinity,
the Fukaya-Seidel category of A-branes on with potential .
The brane is an object of
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The categorified link invariant arizes as the Floer cohomology group
where the second grade is
is the winding number,
associated to the
non-single valued potential.
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We get a 2d mirror description of categorified knot invariants
based on
the Fukaya-Seidel category of A-branes on ,
the target of the Landau-Ginzburg model,
with potential .
This description is
developed in a work with Dimitrii Galakhov.
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one based on Fukaya-Seidel category of A-branes on with potential .
the category of equivariant B-branes on
and the other based on
The two geometric descriptions of knot homology groups,
are related by equivariant mirror symmetry
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It turns out that there is
a third approach to categorification
which is related to the other two,
though less tractable.
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It is important to understand the connection,
in particular because what will emerge
is a unified picture of
the knot categorification problem,
and its solutions.
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This will also demystify an aspect of the story so far
which seems strange:
What do three dimensional supersymmetric gauge theories
have to do with knot invariants?
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The explanation comes from string theory.
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More precisely, it comes from the six dimensional
labeled by a simply laced Lie algebra
little string theory
with (2,0) supersymmetry.
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The six dimensional string theory is
obtained by taking a limit of IIB string theory on
In the limit, one keeps only the degrees of freedom
supported at the singularity of and decouples the 10d bulk.
the ADE surface singularity of type
,
.
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The q-conformal blocks of the
can be understood
as the supersymmetric partition functions of
the -type little string theory,
with defects that lead to knots.
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One wants to study the six dimensional (2,0) little string theory on
where
x xx
x
is the Riemann surface where the conformal blocks live:
and is the domain curve of the 2d theories we had so far.
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The vertex operators on the Riemann surface
x xx
x
come from a collection of defects in the little string theory,
which are inherited from D-branes of the ten dimensional string.
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The D-branes needed are
two dimensional defects of the six dimensional theory on
x xx
x
supported on and the origin of
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The choice of which conformal
blocks we want to study
translates into choices of defects
x xx
x
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The theory on the defects is the quiver gauge theory
that arose earlier in the talk.
V1
W1
V2
W2
This is a consequence of the familiar description of
D-branes on ADE singularities
due to Douglas and Moore in ’96.
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The theory on the defects supported on ,
is a three dimensional quiver gauge theory on
rather than a two dimensional theory on ,
due to a stringy effect.
V1
W1
V2
W2
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x
These turn the theory on the defects supported on ,
to a three dimensional quiver gauge theory on
where the is the dual of the circle in .
In a string theory,
one has to include the winding modes of strings around C.
xxx
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The conformal limit of the algebras
coincides with the point particle limit of little string theory
in which it becomes the six dimensional conformal field theory
of type (with (2,0) supersymmetry)
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In the point particle limit,
the winding modes that made the theory
on the defects three dimensional instead of two,
become infinitely heavy.
xxx
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As a result, in the conformal limit,
the theory on the defects
becomes a two dimensional theory on
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The two dimensional theory on the defects
of the six dimensional (2,0) theory was sought previously.
It is not a gauge theory,
but it has two other descriptions,
I described earlier in the talk.
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One description
is based on the supersymmetric sigma model
describing maps
The other is in terms of the mirror
Landau-Ginzburg model on with potential
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The approaches to
categorification
started with the theories on the
defects.
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There is a third description,
due to Witten.
It is obtained from
the perspective of the 6d theory in the bulk.
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Compactified on a very small circle,
the six dimensional -type (2,0) conformal theory
with no classical description,
becomes a -type gauge theory
in one dimension less.
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To get a good 5d gauge theory description of the problem,
the circle one shrinks corresponds to in
so from a six dimensional theory on
one gets a five-dimensional gauge theory on a manifold with a boundary
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The five dimensional gauge theory is supported on
where
It has gauge group
which is the adjoint form of a Lie group with lie algebra .
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The two dimensional defects are monopoles
of the 5d gauge theory on
supported on and at points on,
along its boundary.
,
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Witten shows that the five dimensional theory on
can be viewed as a gauged
Landau-Ginzburg model on with potential
on an infinite dimensional target space ,
corresponding to connections on
with suitable boundary conditions (depending on the knots).
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To obtain knot homology groups in this approach,
one would end up counting solutions to
certain five dimensional equations.
The equations arise in
constructing the Floor cohomology groups
of the five dimensional Landau-Ginzburg theory.
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Thus, we end up with three different approaches
to the knot categorification problem,
all of which have the same
six dimensional origin.
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They all describe the same physics
starting in six dimensions.
The two geometric approaches,
describe the physics from perspective of the defects.
The approach based on the 5d gauge theory,
describe it from perspective of the bulk.
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In general,
theories on defects
capture only the local physics of the defect.
In this case,
they should capture all of the relevant physics,
due to a version of supersymmetric localization:
in the absence of defects,
the bulk theory is trivial.
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The approach based on
should be equivalent to that
of Kamnitzer and Cautis in type A.
The 2d mirror approach based on
should be related,
by Calabi-Yau/Landau-Ginzburg correspondence
to the Fukaya categories approach by Abouzaid, Smith and Seidel.
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The idea that the 5d Landau-Ginzburg theory
could have a 2d Landau-Ginzburg counterpart
was suggested in the works of
Gaiotto, Witten and Moore.