Volume Conjecture: Refined and Categorified

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1 Volume Conjecture: Refined and Categorified Sergei Gukov based on: hep-th/ (generalized volume conjecture) with T.Dimofte, arxiv: (review/survey) with H.Fuji and P.Sulkowski, arxiv: (new!)

2 Kashaev s observation knot K [R. Kashaev, 1996] invariant <K> 2 n labeled by a positive integer n lim 1 log n/n n defined via R-matrix very hard to compute lim log <K> = Vol ( n 3 Vol (S \ K) ( volume conjecture )

3 A first step to understanding the Volume Conjecture <K> = J (q) colored Jones polynomial n n with q = exp(2pi/n) Hitoshi Murakami Jun Murakami (1999)

4 Colored Jones polynomial J (q) = J (q) = Jones polynomial 2 In Chern-Simons gauge theory [E.Witten] Wilson loop operator R polynomial in q 2-dimn l representation of SU(2)

5 Colored Jones polynomial J (q) = J (q) = Jones polynomial 2 Skein relations: unknot Example:

6 Colored Jones polynomial knot K n-colored Jones polynomial: Cabling formula : J (K;q) n R = n-dimn l representation of SU(2)

7 Colored Jones polynomial knot K n-colored Jones polynomial: J (K;q) n R = n-dimn l representation of SU(2)

8 Volume Conjecture Murakami & Murakami: <K> = n quantum group invariants (combinatorics, representation theory) classical hyperbolic geometry

9 Physical interpretation of the Volume Conjecture [S.G., 2003] analytic continuation of SU(2) is SL(2, ) classical SL(2, ) Chern-Simons theory = classical 3d gravity (hyperbolic geometry)

10 Physical interpretation of the Volume Conjecture [S.G., 2003] constant negative curvature metric on M flat SL(2, ) connection 3 on M = S \ K classical SL(2, ) Chern-Simons theory = classical 3d gravity (hyperbolic geometry)

11 Physical interpretation of the Volume Conjecture [S.G., 2003] constant negative curvature metric on M flat SL(2, ) connection 3 on M = S \ K classical solution in 3D gravity with negative cosmological constant ( ) classical solution in CS gauge theory

12 Example: unknot = BTZ black hole K = unknot M = solid torus Euclidean BTZ black hole:

13 Physical interpretation of the Moral: Volume Conjecture lim ( ) x n/n SU(2) Chern- Simons Classical limit q = exp(2pi/n) / 1 [S.G., 2003] SL(2,C) Chern- Simons leads to many generalizations:

14 Physical interpretation of the Volume Conjecture Generalization 1: t Hooft limit: partition function of SL(2, ) Chern-Simons theory

15 Generalized Volume Conjecture = classical action of SL(2, ) Chern-Simons theory : space of solutions 3 on S \ K A-polynomial of a knot K

16 Algebraic curves and knots x = e u y = e v A-polynomial of a knot K

17 Physical interpretation of the Volume Conjecture [S.G., 2003] Generalization 2: n-colored Jones polynomial: knot K J (K;q) n recursion relation: rational functions

18 Quantum Volume Conjecture using we can write this recursion relation as: [S.G., 2003] where so that

19 Quantum Volume Conjecture [S.G., 2003] In the classical limit q/1 the operator becomes A(x,y) and the way it comes about is that in the mathematical literature was independently proposed around the same time, and is know as the AJ-conjecture [S.Garoufalidis, 2003]

20 Quantization and B-model Bergman kernel simple formula that turns classical curves A(x,y) = 0 into quantum operators [S.G., P.Sulkowski]

21 Quantization and B-model [S.G., P.Sulkowski]

22 Quantization and B-model [B.Eynard, N.Orantin] [V.Bouchard, A.Klemm, M.Marino, S.Pasquetti] [A.S.Alexandrov, A.Mironov, A.Morozov] [R.Dijkgraaf, H.Fuji, M.Manabe] : g x(p) and y(p) W Z ^ A(x,y) ^ ^ n ( ^ recall: A(x,y) ^ ^ Z(M) = 0 )

23 Refinement / Categorification Knots / SL(2,C) CS Matrix models Algebraic curve A-polynomial spectral curve Refinement Homological invariants P (q,t) n b-deformation 4d gauge theories 3d superconformal indices Topological strings (B-model) Seiberg-Witten curve A-polynomial mirror Calabi-Yau geometry A(x,y) = zw refinement

24 Deformation vs Quantization

25 Knot Homologies Khovanov homology: [M.Khovanov] Example: i j

26 (Large) Color Behavior of Knot Homologies similarly, is the n-colored sl(2) knot homology: categorify n-colored Jones polynomials:

27 (Large) Color Behavior of Knot Homologies similarly, is the n-colored sl(2) knot homology: satisfy recursion relations exhibit beautiful large-n asymptotic behavior Both controlled by a refined algebraic curve

28 Refinement / Categorification Generalized Volume Conjecture: [S.G., H.Fuji, P.Sulkowski] where

29 Refined Algebraic Curves Example: refinement: where

30 Refinement / Categorification Quantum Volume Conjecture: [S.G., H.Fuji, P.Sulkowski] Example:

31 Refinement / Categorification Quantum Volume Conjecture: [S.G., H.Fuji, P.Sulkowski] Example:

32 Deformation vs Quantization

33 Powerful Calculations (3,4) torus knot Colored HOMFLY homology has dimension 8,170,288,749,315

35 A-polynomial

36 Mirror Symmetry for Knots

Knot Homology from Refined Chern-Simons Theory

Knot Homology from Refined Chern-Simons Theory Mina Aganagic UC Berkeley Based on work with Shamil Shakirov arxiv: 1105.5117 1 the knot invariant Witten explained in 88 that J(K, q) constructed by Jones