Topological Strings and Knot Homologies Sergei Gukov Outline

Topological Strings and Knot Homologies Sergei Gukov

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

Perturbative Topological String X Calabi-Yau 3 -fold map from a Riemann surface 3 -fold X is characterized by • genus g of • to Calabi-Yau

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

Perturbative Topological String B-model: symplectic basis of 3 -cycles holomorphic Ray-Singer torsion

Holomorphic Anomaly [M. Bershadsky, S. Cecotti, H. Ooguri, C. Vafa] (determines up to holomorphic ambiguity)

Wave Function Interpretation quantization of symplectic structure Wave Function depends on (choice of polarization) [E. Witten]

Mirror Symmetry

Applications • Physical Applications • compute F-terms in string theory on X [H. Ooguri, A. Strominger, C. Vafa], … • Black Hole physics • dynamics of SUSY gauge theory [R. Dijkgraaf, C. Vafa], … • Mathematical Applications • • • Enumerative geometry Homological algebra Low-dimensional topology Representation theory Gauge theory

D-branes Open topological strings A-model: Lagrangian submanifolds in X (+ coisotropic branes) B-model: Holomorphic cycles in X

Open String Field Theory N D-branes [E. Witten] A-model: U(N) Chern-Simons gauge theory B-model: 6 d: holomorphic Chern-Simons [R. Dijkgraaf, C. Vafa] 2 d: BF theory 0 d: Matrix Model

Homological Mirror Symmetry A-branes: objects in the Fukaya category Fuk (X) B-branes: objects in the derived category of coherent sheaves homological mirror symmetry: Fuk (X) [M. Kontsevich]

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)

Large N Duality [R. Gopakumar, C. Vafa]

Large N Duality [R. Gopakumar, C. Vafa] N D-branes U(N) Chern-Simons theory on Closed topological string on resolved conifold

Counting BPS states: 5 d [R. Gopakumar, C. Vafa] M-theory on M 2 -brane on number of BPS states with charge Example (conifold): and spin g

Counting BPS states: 4 d Type II string theory on X [H. Ooguri, A. Strominger, C. Vafa] number of BPS states of 4 d black hole with electric charge q and magnetic charge p evaluated at , the attractor value

A-model Open Closed 3 d Chern-Simons theory Gromov-Witten theory holomorphic Chern. Kodaira-Spencer Simons theory B-model theory Matrix model

Computing non-compact (toric) holomorphic anomaly relative Gromov-Witten large N duality heterotic/type IIA duality gauge theory compact small g (ambiguity) (in practice only small g) ? partial results for all g ?

Gromov-Witten Invariants via Gauge Theory X symplectic 4 -manifold [C. Taubes] topological twist of N=2 abelian gauge theory with a hypermultiplet

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)

Enumerative Invariants Rational (maps) Closed Open Refinement Integer (gauge theory, embeddings) GW DT/GV Equivariant (stable maps) (ideal sheaves) open GW Knot BPS invariants (relative Homologies stable maps)

Polynomial Knot Invariants • In Chern-Simons theory Wilson loop operator polynomial in q • Quantum groups & R-matrix [E. Witten]

Polynomial Knot Invariants • Jones polynomial: unknot Example:

Polynomial Knot Invariants • Quantum sl(N) invariant: unknot

Polynomial Knot Invariants • HOMFLY polynomial: unknot

Polynomial Knot Invariants • HOMFLY polynomial: unknot Example:

Polynomial Knot Invariants • Alexander polynomial: unknot Example:

• Question (M. Atiyah): Why integer coefficients?

Categorification Number categorification Vector Space dimension categorification Category Grothendieck group

Categorification Number categorification Vector Space dimension categorification Category Grothendieck group Example: N! Category of branes on the flag variety

Categorification Number categorification Vector Space dimension categorification Category Grothendieck group • Knot homology Euler characteristic = polynomial knot invariant

Knot Homologies • Knot Floer homology: Example: [P. Ozsvath, Z. Szabo] [J. Rasmussen]

Knot Homologies • Khovanov homology: [M. Khovanov] Example: 3 2 1 0 i 1 3 5 7 9 j

Knot Homologies • sl(N) knot homology: N=3: “foams” (web cobordisms) [M. Khovanov] N>2: matrix factorizations [M. Khovanov, L. Rozansky]

A general picture of knot homologies G Knot Polynomial U(1|1) Alexander “SU(1)” SU(2) SU(N) Jones Knot Homology knot Floer homology . Lee’s deformed theory . Khovanov homology . sl(N) homology .

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!

Physical Interpretation space of BPS states M-theory on (conifold) M 5 -brane on Lagrangian [S. G. , A. Schwarz, C. Vafa] Earlier work: [H. Ooguri, C. Vafa] [J. Labastida, M. Marino, C. Vafa] BPS state: membrane ending on the Lagrangian five-brane

• Surprisingly, this physical interpretation leads to a rich theory, which unifies all the existing knot homologies [N. Dunfield, S. G. , J. Rasmussen] graded by charge , , and membrane

Families of Differentials • differentials • cohomology sl(N) knot homology N>2 Lee’s theory N=1 knot Floer homology N=0

Matrix Factorizations, Deformations, and Differentials

a q Non-zero differentials for the trefoil knot.

Differentials for 8 19. The bottom row of dots has a-grading 6. The leftmost dot on that row has q-grading -6.

Differentials for 10 153. The bottom row of dots has a-grading -2.

What’s Next? • Generalization to other groups and representations [S. G. , J. Walcher] • The role of matrix factorizations • Finite N (stringy exclusion principle) • Realization in topological gauge theory • Boundaries, corners, … • Surface operators • Braid group actions on D-branes

Gauge Theory and Categorification gauge theory on a 4 -manifold X number Z(X) (partition function)

Gauge Theory and Categorification gauge theory on a 4 -manifold X number Z(X) (partition function) gauge theory on = 3 -manifold vector space (Hilbert space)

Gauge Theory and Categorification gauge theory on a 4 -manifold X number Z(X) (partition function) gauge theory on = 3 -manifold vector space (Hilbert space) gauge theory on = surface category of branes (boundary conditions)

gauge theory on X self-duality equations: Z(X) counts solutions +…=0 gauge theory on monopole equations: FA + MM + … = 0 gauge theory on vortex equations: +…=0 =H( ) = moduli space topological A-model/B-model = Fuk b D

Gauge Theory with Boundaries In three-dimensional topological gauge theory: vector ZY 1 vector space vector ZY 2

Gauge Theory with Boundaries In three-dimensional topological gauge theory: vector ZY vector space ZY 2 1 ZY = ZY ZY 1 2

Gauge Theory with Corners In four-dimensional topological gauge theory: “time” corner brane Y 1 category of branes on brane Y 2

Gauge Theory with Corners In four-dimensional topological gauge theory: “time” brane category of branes on Y 1 A-model: Y 2 = HF * symp (“Atiyah-Floer conjecture”) Y 1 Y 2

From Lines to Surfaces • A line operator lifts to an operator in 4 D gauge theory localized on the surface S = where the gauge field A has a prescribed singularity Hol (A) C fixed conjugacy class in G

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 • Example: topological twist (GL twist) of N=4 super. Yang-Mills on with 4 surface operators =

Moduli space: = complex surface with three singularities =

a-br ane a-brane corresponds to the static configuration of surface operators below (“time” direction not shown)

e s(a ) an r b - s(a)-brane corresponds to the static configuration of surface operators with a half-twist

e bran 3 s(a) 3 s (a)-brane corresponds to the static configuration of surface operators with three half-twists

Closing the braid gives 3 = space of a - s (a) strings = Casson-like invariant for knots

Topological Twists of SUSY Gauge Theory • N=2 twisted gauge theory: = D (q) • Alexander polynomial N=4 twisted SYM (adjoint non-Abelian monopoles): doubly-graded knot homology • Partial twist of 5 D super-Yang-Mills: = ZVafa-Witten =H( instanton )