Refined Chern-Simons Theory,
Topological Strings and
Knot Homology
Based on work with Shamil Shakirov, arXiv: 1105.5117
and followup work with Kevin Scheaffer arXiv: 1202.4456
Chern-Simons theory played a central role in mathematics and physics ever since
Witten’s discovery in ’88 that
the knot invariant J K (q)
constructed by Jones
is computed by SU(2) Chern-Simons theory on S3.
Chern-Simons theory with gauge group G on a three-manifold M
is given in terms of the path integral
where
is a connection valued in the Lie algebra of G
is an integer, and the action is
The theory is independent of the metric on M, so is automatically
topologically invariant.
The natural observable in the theory is the Wilson loop,
associated to a knot K in M
taken in various representations
of the gauge group.
To any three manifold with a collection of knots in it,
the Chern-Simons path integral
associates a knot and three manifold invariant.
Chern-Simons theory is solvable explicitly,
so it provides a physical way to not only interpret
knot polynomials,
but to calculate them as well.
Chern-Simons theory gained new relevance
when it emerged as a part of string theory
and the topological string.
Topological string is counting holomorphic maps to a Calabi-Yau
threefold X.
φ:
Σg
→
∂φ=0
X
This can be extended to open topological string theory,
counting maps
from Riemann surfaces with boundaries to X, where the boundaries
fall on a Lagrangian submanifold M of X.
M
φ:
Σg
→
X
∂φ=0
Since X is a Calabi-Yau threefold, M is three dimensional.
Contact with knot theory emerges if we take open topological string on
X = T *M
Witten ’92
where the boundaries map to the zero section. We take N copies of M,
and distinguish which copy a boundary maps to.
Open topological string in this case is the same as
SU(N) Chern-Simons theory on M.
Z CS (M ,q) = ZTop (T * M , λ )
The parameter that keeps track of the genus in topological string is
related to q of Chern-Simons theory by
q=e
2π i
k+N
= eλ
→
λ
2 g−2+# holes
λ # edges − # vertices
We can introduce knots in the theory by
adding Lagrangian branes to X.
For a knot K in S^3, we associate a Lagrangian L_K
K
S3
T *S 3
LK ∩ S 3 = K
Now, we must allow boundaries on L_K as well.
As explained by Gukov, Schwarz and Vafa,
categorified knot invariants are related to the
refined topological string.
Categorification program, initiated by Khovanov in ’99,
interprets Chern-Simons knot invariants as Euler characteristics
of a knot homology theory.
Categorifying the Jones polynomial, one associates to a
knot K bi-graded cohomology groups
in such a way that the Jones polynomial
arises as the Euler characteristic, taken with respect
to one of the gradings.
In this way, one makes integrality of the knot polynomials
manifest.
This also leads to refinement, where one computes
Poincare polynomials of the knot homology theory instead.
String theory allows one to relate topological string,
and hence also Chern-Simons partition functions,
to computing certain indices in M-theory on the Calabi-Yau.
In the M-theory formulation,
contact with knot homology can be made.
Ordinary topological string on a Calabi-Yau X
is related to M-theory on
(X ×  2 × S1 )q
1
where as one goes around the S , the complex coordinates of
 2 rotate by
The supersymmetric partition function of M-theory in this geometry
Z M (X,q) = Tr ( −1) q S1 −S2
F
equals the closed topological string partition function of X
Z M (X,q) = Z top (X, λ )
Gopakumar,Vafa
Dijkgraaf,Vafa,Verlinde
Here,
around the
S1 , S2
are generators of rotations
z1 , z2 planes, and
q = eλ
The M-theory trace corresponds to trace around the S1
factor in
(X ×  2 × S1 )q
Open topological string, with boundaries on
arizes by putting M5 branes on
L ×  × S1
in
X ×  2 × S1
L ∈X
In particular, if we take X = T * M
SU(N) Chern-Simons theory on M
corresponds M-theory on
(T * M ×  2 × S1 )q
with N M5 branes on
where  corresponds to one of the two factors in  2
The partition function of N M5 branes in this geometry
Z M 5 (M , q) = Tr ( −1) q S1 − S2
F
equals the SU(N ) Chern-Simons partition function on M
Z M 5 (M , q) = Z CS (M )
Ooguri-Vafa
Here q = e
2π i
k+N
is related to the level k of Chern-Simons
and
S1 , S2 are generators of rotations around the
z1 ,z 2 planes.
The refined topological string theory is defined
via M-theory on a slightly more general background,
Nekrasov
Nekrasov, Okounkov
......
(X ×  2 × S1 )q,t
where as one goes around the S1, the complex coordinates
of  2 rotate by
To preserve supersymmetry of the trace,
in the refined theory
we need an additional U(1) symmetry
that acts on the supersymmetry generators Q
defining the trace
When the theory on N M5 branes on
preserves supersymmetry, its partition function
Z M 5 (M , q,t) = Tr ( −1) q S1 − SR t SR − S2
F
defines the partition function of
the refined SU(N) Chern-Simons theory on M.
For q=t, this reduces to the index
that computes ordinary Chern-Simons partition function.
The extra U(1) symmetry exists when
M is a Seifert manifold
S1 → M → Σ
The extra U(1) symmetry arizes as follows:
consider a rank 2 sub-bundle of T^*M,
corresponding to cotangent vectors orthogonal to the S1 .
Rotations of the fiber of this is the U(1) we need.
We can introduce knots in the theory by introducing,
for each knot K an additional
Lagrangian LK
The LK
and M5 branes wrapping LK ×  × S1 .
preserves the U(1) symmetry if K wraps the
S1
fibers.
Thus, when M is a Seifert three manifold,
with Seifert knots,
we can define refined Chern-Simons theory
as a supersymmetric partition function
of M5 brane theory in Nekrasov background.
Moreover, just like ordinary Chern-Simons theory,
the refined Chern-Simons theory is solvable exactly.
Recall that, in a topological theory in three dimensions,
all amplitudes, corresponding to any three-manifold and collection of
knots in it, can be written in terms of three building blocks,
the S and T matrices and the braiding matrix B.
In refined Chern-Simons theory, only a subset of amplitudes generated by
S and T enter,
since only those preserve the U(1) symmetry.
The S
and
T
matrices
of a 3d topological QFT,
provide a unitary representation of the action of the modular
group SL(2, Z)
on the Hilbert space
of the theory on T.2
A basis of Hilbert space
is obtained by considering
the path integral on a solid torus,
obtained by shrinking the (1,0) cycle of the boundary,
with Wilson loops in representations
inserted in its interior, along the (0,1) cycle of the torus.
Ri
An element
of SL(2,Z) acting on the boundary of the torus,
permutes the states in
The matrix element
has interpretation as the path integral on a three manifold obtained by
gluing two solid tori, with knots colored by Ri and R j inside,
up to the SL(2,Z) transformation of their boundaries.
From S and T, by cutting and gluing rules of a 3d TQFT
one can compute any amplitude for arbitrary
Seifert knots in Seifert manifolds.
In Chern-Simons theory with gauge group G and level k,
the Hilbert space is the space of conformal blocks
of the corresponding WZW model.
The S, T are the S, T matrices of the Gk WZW model
This allows one solve the theory explicitly, and in particular
obtain the Jones and other knot polynomials.
Explicitly, for G=SU(N), the S and the T matrices are given by
where
sR (x) = TrR x are Schur functions. For
q=e
2π i
k+N
this generates a unitary representation of SL(2,Z) in the
Hilbert space labeled by SU(N) representations at level k.
The S and T matrices of refined Chern-Simons theory
are computable explicitly from M-theory,
for any A, D, E gauge group.
They depend on q and one additional parameter,
t = qβ
In the refined SU(N) Chern-Simons theory,
t = qβ
(q − n/2t −i/2 − q n/2t i/2 )N −i
S00 = c∏ ∏ − n/2 −(i+1)/2
n/2 (i+1)/2 N −i
(q
t
−
q
t
)
n=0 i=1
∞
N −1
where M R (xi ,q,t) are the Macdonald polynomials. For
q=e
2π i
k+βN
this gives a unitary representation of SL(2,Z) in
the Hilbert space is labeled by SU(N) representations at level k.
Similar formulas hold for any ADE group
where
and where y is the dual Coxeter number of G.
The fact that these provide a unitary representation
of SL(2,Z) was proven in the ‘90s by
Ivan Cherednik.
The relation of refined topological string to knot homologies,
leads us to conjecture the following.
Let M be a Seifert three-manifold
with a collection of (linked) knots wrapping the S^1 fiber.
1) For any G=A,D, E, the colored knot homology should admit and additional grade
H ij = ⊕ H ijk
k
which allows one to define a refined index
∑
(−1)k q i t j−k dim H ijk
i, j,k
such that setting t = −1 one gets Chern-Simons invariants back.
2) The index is computable explicitly in terms of S and T matrices
of the refined Chern-Simons theory with the corresponding group.
The idea of the derivation is to use the M theory definition and topological
invariance to solve the theory on the simplest piece,
the solid torus
M L = D × S1
where D is a disk.
S,T and everything else can be recovered from this by gluing.
For separated M5 branes, the M-theory index
Z M 5 (M , q,t) = Tr ( −1) q S1 − SR t SR − S2
F
can be computed by counting
cohomologies of the moduli of holomorphic curves embedded in Y, endowed
with a flat U(1) bundle,
and with boundaries on the M5 branes.
Gopakumar and Vafa
Ooguri and Vafa
In the present case, all the holomorphic curves are annuli, which are isolated.
1
The moduli of the flat bundle is an S
One gets precisely two cohomology classes.
From the M-theory perspective, the simplicity of the spectrum
is the key to solvability of the theory.
D × S1 is a wave-function,
S1.
depending on the holonomy xI around the
The partition function on the solid torus,
0 x = Δ q,t (x) =
∏
1≤ I < J ≤ N
(q − n /2 e( xJ − xI )/2 − q n /2 e( xI − xJ )/2 )
∏
− n /2 −1/2 ( x J − x I )/2
t e
− q n /2t 1/2 e( xI − xJ )/2 )
n =1 (q
∞
This implies that line operators inserting wilson loops in a solid torus
x
x Ri = x OR 0 = Δq ,t (x )M R (e )
i
and giving a basis for
Ri
i
are given by Macdonald polynomials.
In this basis, the identity operator acting on
has the natural matrix elements corresponding to Z(S 2 × S1, Ri , R j )
Ri | R j = ∫ d N x | Δ q, t (x) |2 M Ri (e− x )M R j (e x )
This equals 1 or 0 depending on whether
= 1ij
Ri, j are equal or not.
Obtaining the S and T matrices from solving the theory on
two solid tori M L = D × S1 , and gluing the pieces back together,
lifts in M-theory to solving the theory on
T * M L =  2 × *
with N M5 branes on M L and gluing.
One can give the S1 fibration over S 2 first Chern class
by multiplying one of two wave functions by
,
p
exp(− pTr x 2 / 2ε1 )
before gluing.
For example, viewed this way, S 3 is the Hopf fibration, with p = 1 .
* 3
The M-theory partition function on T S
is computing the TST matrix element
corresponding to refined Chern-Simons theory on a framed S^3.
From M-theory, we obtain
Ri TST R j = ∫ d N x | Δ q,t (x) |2 e
−
1
Tr x 2
2 ε1
M Ri (e− x )M R j (e x )
The fact that this takes the explicit form we gave before
was proven in the ’90’s in the context of proving Macdonald Conjectures
Cherednik, Kirillov, Etingof
The theory also generalizes Verlinde algebra. The
algebra of line operators
ORi OR j = ∑ N ijkORk
k
has coefficients
N ijk =
They satisfy the Verlinde formula,
as required by three-dimensional topological invariance.
×
×
×
The refined Chern-Simons theory connects
to the recent work of Witten
on obtaining knot homologies from string theory.
Witten studies M5 branes in M-theory on the same geometry,
M ×  × S1 inside T * M ×  2 × S1 ,
but for a general three manifold M ,
for which the refined index need not exist.
The index
Z M 5 (M ,q,t) = TrH
( −1)F q S −S t S −S
1
R
R
2
is simple to compute, when it exists:
as a partition function,
it is computable by cutting and gluing.
This is the case when the theory has an extra
U(1) symmetry, and the
Hilbert space has an extra grade
SR associated to it.
By contrast, the Poincare polynomial of the theory
of N M5 branes wrapping M
in,
X = T *M
TrH Q q S1 t − S2
is defined for any three manifold, and any collection of knots,
but it is hard to compute in general,
since one needs to know the cohomology H
the supersymmetry operator Q.
Q
of
The refined SU(N) Chern-Simons theory has a
“large N dual” description
in terms of the refined topological string on a different manifold.
To begin with, recall how this works in the
case of the ordinary Chern-Simons theory.
The
SU(N ) Chern-Simons theory on
S 3,
or equivalently, the ordinary open topological string on
X = T *S 3
is dual to closed topological string theory on
X V = O(−1) ⊕ O(−1) → P1
The large N duality is a geometric transition that shrinks
the S 3 and grows the P1 while eliminating boundaries
SU(N )
SU(N ) Chern-Simons theory
3
on S ,
or open topological string on
X = T *S 3
Gromov-Witten theory on
X V = O(−1) ⊕ O(−1) → P1
with P1 of size N λ
Ooguri-Vafa
Adding a knot K on S 3
corresponds to adding a Lagrangian LK in X ,
and geometric transition relates this to a Lagrangian LK in X V.
X =T S
* 3
X V = O(−1) ⊕ O(−1) → P1
The large N duality of Chern-Simons theory implies
topological string on
Ooguri-Vafa ’99
X V = O(−1) ⊕ O(−1) → P1
with boundaries on L_K computes colored HOMFLY
polynomial. Relating this to M-theory on this space and the M-theory index
explains the integrality of the HOMFLY polynomial.
The large N duality extends to the refined case:
e.g. the partition function of
the refined SU(N) Chern-Simons theory on S 3
equals the partition function of the refined topological string on
Z CS (q,t, N ) = Z rtop (X,q,t,a)
a = t N t 1/2 q −1/2
X
The duality relates computing knot invariants of the
3
refined Chern-Simons theory on S
to computing the refined index
Z M 5 (LK , X, q,t) = Tr ( −1) q S1 − SR t SR − S2
F
of M5 branes wrapping
(LK ×  × S1 )q,t
after the transition, in M-theory on
(X ×  2 × S1 )q,t
Gukov,Vafa and Schwarz proposed that
keeping track of the two gradings S_1, S_2 in M-theory
instead, corresponds to categorification of the colored HOMFLY polynomial.
A triply graded knot homology theory
categorifying the HOMFLY polynomial
was constructed by Dunfield, Rassmussen and Gukov,
(the colored versions were studied recently by Gukov et. al.)
The Poincare polynomial of this theory is the superpolynomial
Setting t=-1, this reduces to the HOMFLY polynomial H(K,q,a).
Taking the cohomology with respect to a certain differential d_N
and setting a=q^N, this reduces
to knot homology theory of Khovanov and Rozansky,
categorifying the SL(N) knot invariants.
While refined Chern-Simons theory is computing an index,
in many cases this index agrees with the
Poincare polynomial of the theory DGR constructed.
The invariant of an
(n, m)
3
torus knot in S
is computed from refined Chern-Simons
by the TQFT rules
where
K represents an element of SL(2,) that takes
the (0,1) cycle into (n, m) cycle
There is strong evidence that, for all torus knots K ,
the refined SU(N ) Chern-Simons invariant
colored by the fundamental representation, computes the
superpolynomial
of the knot homology theory categorifying the HOMFLY polynomial
when written in terms of
and
q = t,
a = −q N / t
t = − q /t
There also seems to persist, for all representations
colored by knots in symmetric or anti-symmetric
representations.
Our results provide support for the conjecture
of Gukov,Vafa and Schwarz, that
the knot homologies
of a triply graded homology theory categorifying the HOMFLY polynomial
are, up to regrading, the spaces of supersymmetric ground states
in M-theory, with M5 branes on
LK
X = O(−1) ⊕ O(−1) → P1
graded by the spins S1 , S2 and their charge in H 2 (X, ) .
Some structural properties of the answer follow from its M-theory
definition.
Topological string is computing a
particular combination of Chern-Simons knot observables
det(1− U × V )−1
SU ( N )q
= ∑ TrR (U )
R
SU ( N )
TrR (V )
In the refined topological string, the story is richer
we have to specify which branes we are studying:
whether they wrap  q or  −1 plane
in M-theory on
t
(X ×  2 × S1 )q,t
We will call these the q- and the t-branes.
Depending on the kinds of branes, the topological string is computing
different refined Chern-Simons amplitudes.
det(1− q nU × V )
∏
n
n=0 det(1− tq U × V )
∞
= ∑ M R (U )
R
SU ( N )q
q
1
M R (V )
GR
q
q
q
det(1− U × V )
= ∑ M̂ R (U )
R
SU ( N )t
SU ( N )t
(−1)R M RT (V )
t
q
t
q
a = tN t / q
a = tN t / q
∑
R
M R (U )
SU ( N )q
1
M R (V )
GR
∑
R
M̂ RT (U )
SU ( N )t
(−1)R M R (V )
a
However, at large N, the branes on the S 3 disappear,
hence we get a prediction
Z R1Rn (K1 ,K n ,a,q,t) / GR1 GRn = (−1)R1 ++Rn Z RT
T
1R n
(K1 ,K n ,a,t −1 ,q −1 )
The refined topological string
on a large class of toric Calabi-Yau manifolds
is captured by the combinatorics of the refined topological vertex,
analogously to the ordinary topological string.
Cozacs, Iqbal,Vafa
Gukov, Iqbal,Vafa
X = O(−1) ⊕ O(−1) → P1
Cozacs, Iqbal
It was proven recently that the refined topological vertex
predictions for the Hopf-link colored by arbitrary representations,
agree with the S-matrix amplitude Sij of the refined Chern-Simons theory, at large N,
as predicted by large N duality.
One can show that similarly,
Aganagic, Shakirov
the refined topological vertex
follows from refined Chern-Simons theory,
just as the original derivation of the ordinary topological vertex
came from Chern-Simons theory,
using large N dualities.
Aganagic, Klemm, Marino,Vafa ’03