Topological%20Strings%20and%20Knot%20Homologies

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Topological StringsandKnot Homologies

• Sergei Gukov

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Outline

• Introduction to Topological String Theory

• Relation to Knot Homologies

based on
S.G., A.Schwarz, C.Vafa, hep-th/0412243
N.Dunfield, S.G., J.Rasmussen, math.GT/0505662
S.G., J.Walcher, hep-th/0512298
joint work with E.Witten
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Perturbative Topological String
X Calabi-Yau 3-fold
map from a Riemann surface to Calabi-Yau
3-fold X is characterized by

• genus g of

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Perturbative Topological String
Topological string partition function
A-model
Kahler moduli
number of holomorphic maps of genus
g curves to X which land in class
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Perturbative Topological String
B-model
symplectic basis of 3-cycles
holomorphic Ray-Singer torsion
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Holomorphic Anomaly
M.Bershadsky, S.Cecotti, H.Ooguri, C.Vafa
(determines up to holomorphic ambiguity)
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Wave Function Interpretation
E.Witten
quantization of
symplectic structure
Wave Function
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Mirror Symmetry
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Applications

• Physical Applications
• compute F-terms in string theory on X
• Black Hole physics
• dynamics of SUSY gauge theory
• Mathematical Applications
• Enumerative geometry
• Homological algebra
• Low-dimensional topology
• Representation theory
• Gauge theory

H.Ooguri, A.Strominger, C.Vafa,
R.Dijkgraaf, C.Vafa,
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D-branes
Open topological strings
A-model Lagrangian submanifolds in X
( coisotropic branes)
B-model Holomorphic cycles in X
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Open String Field Theory
N D-branes
E.Witten
A-model U(N) Chern-Simons gauge theory
B-model 6d holomorphic Chern-Simons
R.Dijkgraaf, C.Vafa
2d BF theory 0d Matrix Model
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Homological Mirror Symmetry
A-branes objects in the Fukaya category
Fuk (X)
homological mirror symmetry
M.Kontsevich
Fuk (X)
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Matrix Factorizations
B-branes at Landau-Ginzburg point are described
by matrix factorizations
Topological Landau-Ginzburg model with
superpotential W
CY-LG correspondence
MF (W)
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Large N Duality
R.Gopakumar, C.Vafa
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Large N Duality
R.Gopakumar, C.Vafa
N D-branes
Closed topological string on resolved conifold
U(N) Chern-Simons theory on
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Counting BPS states 5d
R.Gopakumar, C.Vafa
M-theory on
M2-brane on
Example (conifold)
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Counting BPS states 4d
H.Ooguri, A.Strominger, C.Vafa
Type II string theory on X
number of BPS states of 4d black hole with
electric charge q and magnetic charge p
evaluated at , the attractor value
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Large N Dualities
Open Closed
A-model 3d Chern-Simons theory Gromov-Witten theory
B-model holomorphic Chern-Simons theory Matrix model Kodaira-Spencer theory
Mirror Symmetry
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Computing
non-compact (toric) compact
holomorphic anomaly small g (ambiguity)
relative Gromov-Witten (in practice only small g)
large N duality ?
heterotic/type IIA duality partial results for all g
gauge theory ?
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Gromov-Witten Invariants via Gauge Theory
X symplectic 4-manifold
C.Taubes
topological twist of N2 abelian gauge theory
with a hypermultiplet
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Gromov-Witten Invariants via Gauge Theory
D.Maulik, N.Nekrasov, A.Okounkov,
R.Pandharipande
X Calabi-Yau 3-fold
topological twist of abelian gauge theory in six
dimensions localizes on singular U(1) instantons
(ideal sheaves)
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Enumerative Invariants
Rational (maps) Integer (gauge theory, embeddings) Refinement
Closed GW (stable maps) DT/GV (ideal sheaves) Equivariant
Open open GW (relative stable maps) BPS invariants Knot Homologies
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Polynomial Knot Invariants

• In Chern-Simons theory

E.Witten
Wilson loop operator
polynomial in q

• Quantum groups R-matrix

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Polynomial Knot Invariants

• Jones polynomial

unknot
Example
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Polynomial Knot Invariants

• Quantum sl(N) invariant

unknot
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Polynomial Knot Invariants

• HOMFLY polynomial

unknot
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Polynomial Knot Invariants

• HOMFLY polynomial

unknot
Example
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Polynomial Knot Invariants

• Alexander polynomial

unknot
Example
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• Question (M.Atiyah)

Why integer coefficients?
Two recent developments - Categorification -
Integer BPS invariants
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Categorification
categorification
categorification
Number
Vector Space
Category
dimension
Grothendieck group
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Categorification
categorification
categorification
Number
Vector Space
Category
dimension
Grothendieck group
Example
Category of branes on the flag variety
N!
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Categorification
categorification
categorification
Number
Vector Space
Category
dimension
Grothendieck group

• Knot homology

Euler characteristic polynomial knot invariant
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Knot Homologies

• Knot Floer homology

P.Ozsvath, Z.Szabo J.Rasmussen
Example
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Knot Homologies

• Khovanov homology

M.Khovanov
Example
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Knot Homologies

• sl(N) knot homology

N3 foams (web cobordisms)
M.Khovanov
Ngt2 matrix factorizations
M.Khovanov, L.Rozansky
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A general picture of knot homologies
G Knot Polynomial Knot Homology
U(11) Alexander knot Floer homology .
SU(1) Lees deformed theory .
SU(2) Jones Khovanov homology .
SU(N) sl(N) homology .
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sl(N) knot homology

• is a functor (from knots and cobordisms to
  bigraded abelian groups and homomorphisms)
• is stronger than
• is hard to compute (only sl(2) up to crossings)
• cries out for a physical interpretation!

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Physical Interpretation
S.G., A.Schwarz, C.Vafa
space of BPS states
M-theory on
(conifold)
Lagrangian
M5-brane on
Earlier work H.Ooguri, C.Vafa J.Labastida,
M.Marino, C.Vafa
BPS state membrane ending on the Lagrangian
five-brane
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• Surprisingly, this physical interpretation leads
  to a rich theory, which unifies all the existing
  knot homologies

N.Dunfield, S.G., J.Rasmussen
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Families of Differentials

• differentials

• cohomology

sl(N) knot homology N gt 2 Lees theory
N1 knot Floer homology N0
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Matrix Factorizations, Deformations, and
Differentials
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a
q
Non-zero differentials for the trefoil knot.
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Differentials for 8 . The bottom row of
dots has a-grading 6. The leftmost dot on that
row has q-grading -6.
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Differentials for 10 . The bottom row of
dots has a-grading -2.
153
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Whats Next?

• Generalization to other groups and
  representations
• The role of matrix factorizations
• Finite N (stringy exclusion principle)
• Realization in topological gauge theory

S.G., J.Walcher

• Boundaries, corners,
• Surface operators
• Braid group actions on D-branes

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Gauge Theory and Categorification
gauge theory on a 4-manifold X
number Z(X)
(partition function)
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Gauge Theory and Categorification
gauge theory on a 4-manifold X
number Z(X)
(partition function)
gauge theory on
vector space
(Hilbert space)
3-manifold
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Gauge Theory and Categorification
gauge theory on a 4-manifold X
number Z(X)
(partition function)
gauge theory on
vector space
(Hilbert space)
3-manifold
gauge theory on
category of branes
(boundary conditions)
surface
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gauge theory on X
Z(X) counts solutions
self-duality equations
0
gauge theory on
monopole equations
F MM 0
A
topological A-model/B-model
gauge theory on
vortex equations
0
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Gauge Theory with Boundaries
In three-dimensional topological gauge theory
vector
vector
vector space
Z
Z
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Gauge Theory with Boundaries
In three-dimensional topological gauge theory
vector
vector
vector space
Z
Z
Z
Z
Z
Y
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Gauge Theory with Corners
In four-dimensional topological gauge theory
time
brane
brane
category of branes on
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Gauge Theory with Corners
In four-dimensional topological gauge theory
time
brane
brane
category of branes on
A-model
(Atiyah-Floer conjecture)
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From Lines to Surfaces

• A line operator lifts to an operator in 4D gauge
  theory localized on the surface
  where the gauge field A has a prescribed
  singularity

Hol (A) C
fixed conjugacy class in G
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Braid Group Actions on D-branes

• Any four-dimensional topological gauge theory
  which admits supersymmetric surface operators
  provides (new) examples of braid group actions on
  D-branes

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Moduli space
complex surface with three singularities

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a-brane
a-brane corresponds to the static configuration
of surface operators below (time direction
not shown)
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s(a)-brane
s(a)-brane corresponds to the static
configuration of surface operators with a
half-twist
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s (a)-brane corresponds to the static
configuration of surface operators with
three half-twists
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Topological Twists of SUSY Gauge Theory

• N2 twisted gauge theory

• N4 twisted SYM (adjoint non-Abelian monopoles)

• Partial twist of 5D super-Yang-Mills

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The End
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