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Knots and Mirror Symmetry
Mina Aganagic
UC Berkeley
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the basic question in knot theory:
When are two knots distinct?
Quantum physics has played a central role in answering
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Witten explained in ‘88,
that the Jones polynomial, one of the best known knot invariants,
is computed by the quantum SU(2) Chern-Simons theory on S3.
E. Witten, ’98
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In this talk, I will describe a conjecture relating
knot theory and mirror symmetry,
that provides a new way of thinking about knot invariants,
where classical geometry plays the central role.
The conjecture also provides a new perspective on mirror symmetry.
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The conjecture is a joint work
with Cumrun Vafa.
I will also discuss some work in progress with
Tobias Ekholm and Lenny Ng
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The conjecture is that, to every knot K, one can associate
a non-compact Calabi-Yau three-fold YK
that is a knot invariant.
or equivalently, a Riemann surface (which encodes all the same information)
YK : uv = FK (x, p;Q)
0 = FK (x, p;Q)
! !
4[u,v, x, p]
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Studying topological B-model on
together with a collection of B-branes
one can recover all the quantum SU(N) Chern-Simons
invariants of the knot K, for any N
YK
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Moreover, every Calabi-Yau manifold that appears
in this way is a distinct mirror of
We get not just a single mirror, but infinitely many:
one for each knot in S3.
This follows by a generalization of
Strominger-Yau-Zaslow mirror symmetry,
as applied to local Calabi-Yau manifolds.
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In the rest of the talk,
I will explain the motivation for the conjecture,
and its implication for mirror symmetry and knot theory.
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Strominger, Yau and Zaslow conjectured that
every compact Calabi-Yau X, with a mirror Y,
admits a family of special Lagrangian T3’s,
such that moduli space of a
special Lagrangian 3-brane wrapping the T3
is the mirror Calabi-Yau Y
- once one includes disk instanton corrections.
A. Strominger, S.T. Yau,
E. Zaslow ’96
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This way, the classical moduli space of a pointlike B-brane probing Y
and the disk instanton corrected moduli space
of a special Lagrangian 3-brane wrapping the T3 fiber
of X are the same.
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the mirror Y is the quantum-corrected geometry of X
as seen by the probe special Lagrangian T3 brane.
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Thus, for SYZ mirror pair of Calabi-Yau manifolds X and Y,
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On local toric Calabi-Yau manifolds.
the special Lagrangian T3 fibration
does not exist.
Instead, the mirror is obtained by considering a special Lagrangian
with topology of .
M.A, C. Vafa, ‘01
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M. Gross
R2x S1
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The moduli space of a special Lagrangian A-brane wrapping
is one complex dimensional:
It follows from a theorem by McLean which states that
the complex dimension of the moduli space of a special
Lagrangian three cycle L
together with a flat U(1) bundle on L is b1(L)
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R2x S1
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The mirror Y of any toric Calabi-Yau X turns out to be of the form
Y:
The mirror to the Lagrangian A-brane in X
is a B-brane in Y,
which wraps a curve given by setting v=0
and choosing a point on the Riemann surface
uv = F(x, p;Q)
0 = F(x, p;Q)
R2x S1
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The Riemann surface
0 = F(x, p;Qi )
which is the classical moduli space of the B-brane on Y
is also the disk instanton corrected moduli space
of the mirror A-brane on X.
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For a special Lagrangian A-brane
with
the Gromov-Witten potential on the disk
captures the corrections to the classical moduli space.
The classical moduli space is a cylinder
parameterized by x where
L ! X b1(L) = 1
W (x;Q) = rk ,mekx
k ,m!0
" Qm
!xW ! 0
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Defining
the exact moduli space of the A-brane is a Riemann surface
0 = F(x, p;Qi )
p = !xW (x;Q)
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Mirror symmetry relates the topological string amplitudes
of the A-model on X, and the B-model on Y.
In particular, it relates disk amplitudes
of the special lagrangian A-brane on X
and its mirror B-brane on Y.
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In the B-model on the mirror,
the classical geometry is exact,
and the disk amplitude can be read off from it.
For a B-brane wrapping a holomorphic curve C
it is given in terms of the period
of the holomorphic three form
on a 3-chain B(C, C*) with boundary on the B-brane.
!3,0
B(C ,C* )
"
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In the present case, for a Calabi-Yau of the form,
where the B- brane wraps and a point on the Riemann surface
the B-model disk amplitude becomes simply
uv = F(x, p;Q)
!3,0
=du
u" dp" dx
with
v = 0
W = pdxx*
x
!
0 = F(x, p;Q)
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In particular, in the B-model on the mirror,
the disk potential W can be read off from the classical geometry.
Alternatively, W, computed by the A-model of
X determines the geometry of the mirror Y.
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Now, suppose we can find more than one
special Lagrangian A-brane on X
with the topology of R2 x S1.
Then would get more than one mirror,
as the branes would generically see the geometry of X differently.
Namely, one would get different disk amplitudes W for different branes.
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As I will now explain,
there is an infinite family of such special Lagrangian branes
of topology R2xS1 on
There is one such brane for every knot in S3.
Each such brane leads to a new mirror of X,
depending on the knot.
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A way to construct branes is using a
geometric transition
R2xS1
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On X* =T*S3, for every knot K in S3,
we get a special Lagrangian LK of the topology of R2x S1
by taking the knot together with its conormal bundle
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X* =T*S3
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The special Lagrangian LK in X*
is related, by geometric transition,
geometric transition
to a special Lagrangian LK on X,
also of the topology of R2xS1
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Thus, for every knot K in S3
we get a Lagrangian LK in X, of the topology of R2x S1,
and with it a new mirror of X.
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FK (ex,e
p,Q) = 0
For each knot K,
the Gromov-Witten potential
of X together with an A-brane on
encodes the classical geometry of the mirror Calabi-Yau,
viewed as the disk instanton corrected moduli space of the brane.
LK
WK(x,Q)
p = !xW
K
This is the mirror Riemann surface, a set of points x, p related by
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There is a way to sum up disk instanton corrections and determine
for any K, and with it the mirror Calabi-Yau,
WK(x,Q)
using a conjecture that relates Gromov-Witten theory of
to a solvable theory.
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with N A-branes on the S3
The geometric transition is now a duality -- the large N duality.
Gopakumar and Vafa conjectured in ’98 that
large N duality relates topological A-model string on
with of size
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N!
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The A-model on with N A-branes on the S3
is the same as the SU(N) Chern-Simons theory on S3
The relation to Chern-Simons theory means the A-model is solvable in this
case.
E. Witten, ’93
X*
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A-model string amplitudes receive contributions
only from holomorphic maps into a Calabi-Yau. In
all such maps are degenerate.
Witten showed in ’93 that they degenerate to
Feynman graphs of SU(N) Chern-Simons theory on S3
E. Witten, ’93
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Introducing in additional
Lagrangian branes LK which are conormal to the knot K,
corresponds simply to studying invariants of knot K in Chern-Simons theory.
H. Ooguri, C.Vafa ’99
CS
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X*= T
*S3
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Geometric transition/large N duality relates
this to the Lagrangian on X,
we called LK as well,
of the topology of R2xS1.
Large N
geometric transition
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Thus, Large N duality relates
Chern-Simons invariants of a knot K in S3 ,
with Gromov Witten invariants
of a Lagrangian LK in X
The rank of Chern-Simons gauge group
is encoded in the parameter Q
keeping track of the degree of maps to the , by
Q= e!N"
P1
!Here, is the effective coupling constant which is the same
in Chern-Simons theory and the topological string.
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Considering Chern-Simons HOMFLY knot invariants
colored by the rank n symmetric representation
gives the exact partition function of a single A-brane on LK:
in X.
HRn
Rn
! (K ,",Q)e#nx
H. Ooguri, C.Vafa ’99
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!K(x,",Q) =
HRn(K;Q)
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All genus A-model amplitudes on X with LK
!(x,Q,") = exp( Fg,hg,h
# (x,Q) "2g$2+h
)
HRn
Rn
! (K ,",Q)e#nx
!K(x,",Q) =
arise as expansion of this in powers of : !
by summing all Chern-Simons
ribbon diagrams of fixed topology
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= exp(WK(x,Q) / ! + ......)
The disk amplitude is the leading piece:
!(x,Q,!)
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Thus, geometric transition allows one
not only to construct the Lagrangians
but to compute their amplitudes as well,
in particular
Large N
geometric transition
in terms of Chern-Simons invariants of knots on S3
WK(x,Q)
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The simplest example of this is the case when K is a unknot.
Then, the Lagrangian one gets is the one that leads to the canonical
mirror of the conifold,
derived for example by Hori and Vafa.
K. Hori, C. Vafa, ’00
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Taking the trefoil knot instead,
the mirror looks more complicated.......
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For the figure 8 knot......
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The mirror conjecture implies that
closed B-model
on all the Calabi-Yau manifolds is the same.
So far, the equivalence has been proven at genus zero,
for torus knots.
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The mirror Riemann surfaces that arize in this way
generalize the classical A-polynomial of the knot. Setting
Q = 1
the mirror Riemann surface contains the SL(2,C) character variety
as a factor. This is a consequence of the analytic continuation that
relates SU(N) and SL(N,C) Chern-Simons theory,
and the fact that, at any finite N
Q = e!N"
#1
as goes to zero.!
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The way we found the new mirrors is roundabout,
using large N duality and Chern-Simons theory.
There is a more direct way.
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There is a way to count holomorphic disks ending on
the Lagrangians of this type,
in the framework of
knot contact homology.
The closed string version of this was pioneered by
Eliashberg, Givental and others.
The open version, knot contact homology, was developed by first by
Lenny Ng.
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At infinity of both X and X* approach
V ! R
where is a contact manifold. V = S3! S
2
Moreover, approaches LK
LK! !
K" R
where is a torus, which is Legendrian in .!K
V
This is a setting where knot contact homology can be used.
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To a Legendrian in one associates
a differential graded algebra (DGA).
The algebra is associative but not commutative,
with coefficients in
is a torus bounding the knot,
so the latter is generated by three elements
corresponding to the longitude of the knot, to the meridian,
and the class in
The coefficients themselves count the number of holomorphic disks
ending on in the corresponding relative homology class.
!K V
H2(V ,!
K)
!K
ex,e
p,Q
!K
H2(V )
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The DGA that arises in this way, starting with the conormal Lagrangian
in X is an invariant of the knot K in
A small byproduct of the algebra is
the augmentation variety
For every point in this variety, the DGA has a trivial,
one dimensional representation.
LK
AK (x, p,Q) = 0
S3
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Since knot contact homology is counting holomorphic
disks, it is natural to expect the mirror curves
obtained via Chern-Simons theory
and the augmentation varieties of the knot are in fact the same.
The fact that this is the case in all known examples
is a direct confirmation of large N duality.
AK (x, p,Q) = FK (x, p,Q)
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So far we focused on mirror symmetry implications.
The conjecture has no less interesting consequences
for Chern-Simons theory.
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We have translated the problem of computing arbitrary
Chern-Simons invariants associated to the knot K
to a computation in the topological B-model on Calabi-Yau YK
with appropriate branes.
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To get arbitrary Chern-Simons invariants of the knot K
we simply need to study the
mirror of not just with a single brane on LK, but
with arbitrary many of them.
k branes k {
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The quantum invariants of the B-model on
are determined essentially uniquely
from the classical Riemann surface
uv = F(x, p;Qi )
0 = F(x, p;Qi )
at least assuming the Riemann surface is smooth.
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More precisely, defining the quantum topological string on the
local Calabi-Yau
requires a choice of periods
(related to holomorphic anomaly of the topological string)
and a choice of a point on the Riemann surface, related
to choosing the moduli of the brane.
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uv = F(x, p;Qi )
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This means that topological string provides
a new classical knot invariant:
the mirror Calabi-Yau YK itself
Together with the finite data to define the quantum B-model
this should be as good at distinguishing
knots as all of SU(N) Chern-Simons theory.
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uv = FK (x, p,Q)
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For the purposes of
distinguishing knots, the quantum invariants
of the B-model on carry
no additional information,
once the quantization procedure is defined.
YK
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Disclaimer: an important property of the new mirrors
is that they are all singular.
For the singular Riemann surfaces at hand,
one has to work to define what one means by
quantization.
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This gives a very different perspective on
knot invariants than what one is used to:
here, the classical data
defining the mirror curve
plays the central role.
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It is natural to ask how do links fit into this picture?
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In the case of a link with k components,
the augmentation variety
is a submanifold (in general reducible)
AK ,! (xi , p j ,Q) = 0 ! = 1,...,k
M
k= (!
*)2kof the phase space with coordinates
xi, p
j
corresponding to the fact that the Legendrian torus of the link
is now a union of k two dimensional tori, one for each knot
component.
i, j = 1,...,k
VK
VK:
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The augmentation variety of a link
can be related to Gromov-Witten invariants and
the HOMFLY polynomial of the link
in a similar, but somewhat more subtle way,
as in the knot case.
VK
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The variety is a union of components,
corresponding to different ways to fill
the Legendrian torus at infinity
VK= V
K(F)
F
!
!K
into a Lagrangian submanifold of X.LK(F)
Different fillings also correspond to different
saddle points of the HOMFLY invariants of the link, colored by totally
symmetric representations.
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Each of the irreducible components of the augmentation variety is
in fact Lagrangian with respect to the holomorphic symplectic structure
! = dpii
" # dxi
It comes from a potential
WK(F)(x
1,..., x
k,Q)
pi = !iWK (F)(x1,..., xk ,Q)as points on the variety satisfy
VK(F)
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There is a way to quantize the augmentation variety
of a link that parallels the case of a single knot.
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This uses the fact that the B-model on a Calabi-Yau
has a novel reformulation. It is equivalent to a certain
A-model on Mk=1
due to Kapustin and Witten, and studied by Dijkgraaf, Vafa and others.
together with an A-brane supported on
uv = F(x, p,Q)
0 = F(x, p,Q)
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For any k, in fact, the phase space is hyperkahlerMk
with three kahler forms
! = dpii
" # dxi =$ J + i$K
and
!I, !
J, !
K where
k = i (dxii
! " dxi + dpi " dpi ) =# I
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The A-model on such a hyperkaher manifold
admits a so called
“canonical coisotropic brane” which wraps the entire manifold,
and supports flux
This brane is coisotropic brane with respect to
complex structures I and K, but holomoprhic in complex structure J
The canonical coisotropic brane is thus an (A,B, A) brane
with respect to complex strures (I,J,K)
F =! J = Re( dpii
" # dxi )
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We add to this a brane wrapping the augmentation variety
of a K-component link.
In terms of the three complex structures, I,J, K,
the augmentation variety is holomorphic in the first,
and lagrangian in the last two:
it is a (B,A,A)-brane.
VK! M
k
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We conjecture that the topological A-model on
M
k
together with the canonical coisotropic brane on
and a (B,A,A) brane wrapping
VK! M
k
quantizes the augmentation variety
and gives rise to quantum Chern-Simons invariants of a link K.
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This description is closely related to the fact that HOMFLY polynomials
of a link are q-holonomic.
They define a D-module for the Weyl algebra generated
by
xj, p
i= ! "
i
The elements of the Weyl algebra are string states beginning and
ending on the coisotropic brane.
The D-module for these are the strings between
the (A,B,A) and (B,A,A) branes.
Garoufalidis 2012
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I described two conjectures that come from combining
(generalized) SYZ mirror symmetry and
large N duality of topological string:
- For the resolved conifold X,
the should be an infinite ambiguity as to what
the mirror of a Calabi-Yau is: there are as many mirrors as knots.
- The mirror geometry YK is a knot invariant.
The quantum invariants of the B-model on YK
should contain all Chern-Simons invariants of the knot.
Summary
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