Knot invariants, A-polynomials, BPS states

Transcription

1 Knot invariants, A-polynomials, BPS states Satoshi Nawata Fudan University collaboration with Rama, Zodin, Gukov, Saberi, Stosic, Sulkowski

2 Chern-Simons theory Witten M A compact 3-manifold gauge field with semi-simple gauge group action is independent of metric of M S CS (M,k) = k Z Tr(A ^ da A ^ A ^ A) k 2 Z M is inverse of gauge coupling (CS level) The partition function is itself a 3-manifold invariant Z Z k (M) = DA e is CS(M,k) Witten-Reshetikhin-Turaev invariant Invariants can be evaluated algebraically by using surgery

3 Chern-Simons theory Witten Wilson loop along a knot colored by a representation I W R (K) =Tr R P exp( Expectation value is a knot invariant Z 1 J G,R (M,K) = DA W R (K)e is CS(M,k) Z k (M) This is indeed a Laurent polynomial of q 2 J( ) q 2 J( ) = (q q 1 )J( ) For example, Jones poly of the trefoil is K A) J( ) = q + q 3 + q 5 q 9 q =exp 2 i k+h _ M = S 3, G = SU(2), R =, then it is Jones polynomial R

4 Chern-Simons theory Witten Wilson loop along a knot colored by a representation I W R (K) =Tr R P exp( Expectation value is a knot invariant Z 1 J G,R (M,K) = DA W R (K)e is CS(M,k) Z k (M) This is indeed a Laurent polynomial of q =exp Two variable invariants M = S 3, G = SU(N) K A) 2 i k+h _ R colored HOMFLY polynomials: M = S 3, G = SO(N)/Sp(N) colored Kauffman polynomials: P (K; a, q) where a = q N F (K; a, q) where a = q N 1

5 Torus knots Torus knots can be evaluated by modular matrices T µ = q C 2( ) µ S µ = s (q )s µ (q + ) Moreover, using the Adams operation Rosso-Jones (S 0 ) m = X µ C µ (m)s 0µ In principle, one can determine the invariants for any rep. J (T m,n )= X µ C µ (m)q m n C 2(µ) S 0µ HOMFLY polynomial of torus knots can be expressed P (T m,n ; a, q) / 2 ' 1 (q 1 m,q 1 n ; q 1 m n ; q, a)

6 Non-torus knots Non-torus knots can be evaluated by braiding operations on WZW conformal blocks we need quantum j-symbols j 1 j 2 j 12 j j 3 = X j 23 j1 j 2 j 12 j 3 j j 23 j 1 j j 23 j 2 j 3 expression for SU(N) gauge group with symmetric rep S.N. Ramadevi, Zodinmawia = ( 1, 2, 12) ( 3,, 12) ( 1,, 23) ( 2, 3, 23)[N 1]! X n ( ) z [z + N 1]!C z ({ i }, 1 12, 23) 2 h , 1 _ + N _ 1i z! z2z h , 1 _ + N _ 1i z! 1 2 h , 1 _ + N _ 1i z! 1 z 2 h , 1 _ + N _ 1i! 1 z 2 h , 1 _ + N _ 1i! z 1 2 h , _ 1 + _ N 1i! z 1 2 h , _ 1 + _ N 1i! o 1

7 String theory U(N) Chern-Simons theory is realized in topological A-model A-model can be embedded in M-theory spacetime N M5 M5 Witten R T N T S 3 R D S 3 R D NK Gopakumar-Vafa At large N, there is geometric transition from deformed Ooguri-Vafa to resolved conifold T S 3! X = O( 1) spacetime O( 1)! CP1 R T N X R D LK BPS states associated to M2-M5 bound states can be identified Gukov-Schwarz-Vafa with knot homology Hknot = HBPS M5

8 3d/3d correspondence U(N) Chern-Simons theory is realized in topological A-model A-model can be embedded in M-theory Witten spacetime N M5 M5 R TN T S 3 R D S 3 R D N K N M5-branes on R D (S 3 \K) Gopakumar-Vafa compactify on S 3 \K localization on R D 3d N =2SCFTT N [K] on R D 3d/3d relation SL(N,C) Chern-Simons on S 3 \K Partition functions and moduli spaces are equivalent M vac (T N [K]) = M flat (S 3 \K, SL(N,C))

9 String theory The large N duality can be seen at the level of partition functions where f µ (K, a, q) contains information of BPS degeneracies the parameters are identified by q = e g s, a = e t Since counts the number of M2-M5 bound states, they should be integers. This can be written in infinite product formula X P (K; a, q)s (x) = Y Y µ i,j k=0 Labastida-Marino-Ooguri-Vafa 1Y (1 a i q j+k x µ ) N µ,i,j(k)

10 String theory The M2-brane realization of knot in topological strings Witten spacetime N M5 M2 R TN T S 3 R D S 3 R {0} K After geometric transition, we are left with spacetime M2 R TN X R {0} K Open problem M2-brane (D2-brane) configurations of representation R More precise formulation in this setting and understand differences between M5-branes and M2-branes

11 Refined Chern-Simons theory From perspective of refined topological string, the refined modular S, T matrices are defined T µ = q q 1 2 t 2 2 q N 2 2 q 1 2N 2 1 µ, S µ = M (q 1 )M µ(q 1 q 2 ) Aganagic-Shakirov Using refined modular matrices, one can evaluate torus knot invariants that This is equivalent to Poincare polynomial of HOMFLY homology for torus knots/links µ

12 Modular transformations in 3d/3d relation HOMFLY polynomials of Hopf link Effect of S-surgery in CS theory S-transformation in 3d N=2 theory Witten Same correspondence for T-transformation 3d/3d correspondence identifies modular transformations of pure Chern-Simons and 3d N=2 gauge theory

13 Geometric engineering Gorsky-Negut By geometric engineering, this is equivalent to 5d U(1) Nekrasov partition function with surface operator associated to a knot K P(T n,m : a, q, t) = X µ`n e n eg µ SYT X of shape µ Q n i=1 1 S m/n (i) i (1 a i )(q i t) q 2 q t n t n 1 Y 1applei<japplen ( j q i )(t j i ) ( j i )(t j q i ) where i denotes the q,t-content of the box labeled by i in the tableau, and S m/n (i) = eg µ = Y 2 im n (i 1)m n (1 q a( ) t l( )+1 ) Y 2, e = (1 q a( ) 1 t l( ) ) (t 1)(q 1) q(t q) the expression admits the following integral representation P(T n,m : a, q, t) = Z Q n i=1 zs m/n(i) i 1 az i z i 1 1 qt z 2 z qt z n z n 1 Y 1applei<japplen! zi z j dz1 2 iz 1... dz n 2 iz n

14 geometric transition p=2 ( 1, 3 ) ( 1, 0 ) ( 1, 1) p= ( 1, 2 ) ( 0, 1 ) Chern-Simon theory on Lens space S 3 /Z p ( 1, 1) ( 1, 1 ) ( 1, 2 ) ( 1, 1 ) ( 1, 1 ) ( 1, 1 ) topological string on toric CY ( 1, 1 ) ( 1, 1 ) ( 1, 2 ) ( 1, 1 ) ( 1, 2 ) p=3 ( 1, 3 ) Open problem Evaluate (refined) invariants of an unknot in Lens space by using (refined) topological vertex Perform SL(2,Z) transformation and obtain invariants of torus knots

15 knot homology Categorifications of quantum knot invariants lead to doubly-graded homology Khovanov, Khovanov-Rozansky, Webster, J g (K; q) = X i,j ( 1) j q i dim H g (K) i,j Poincare polynomials are more powerful than quantum invariants. Even uncolored sl(2) homology distinguish some mutant pair P g (K; q, t) = X i,j q i t j dim H g (K) i,j There is a triply-graded homology behind HOMFLY polynomials Khovanov-Rozansky P (K; a, q) = X i,j,k( 1) k a i q j dim H (K) i,j,k

16 knot homology HOMFLY homology P (K; a, q, t) = X a i q j t k dim H (K) i,j,k i,j,k t = 1 HOMFLY polynomial P (K; a, q) sl(n) homology J sl(n) (K; q, t) = X i,j,k q i t j dim H sl(n) (K) i,j a = q N t = 1 sl(n) quantum invariant J sl(n) (K; q)

17 knot homology HOMFLY homology P (K; a, q, t) = X i,j,k a i q j t k dim H (K) i,j,k t = 1 d N di erential HOMFLY polynomial P (K; a, q) sl(n) homology J sl(n) (K; q, t) = X i,j,k q i t j dim H sl(n) (K) i,j a = q N t = 1 sl(n) quantum invariant J sl(n) (K; q)

18 d N knot homology homology w.r.t. differentail is isomorphic to sl(n) homology For example, deg d N =( 2, 2N, 1) Dunfield-Gukov-Rassmusen P(T 3, )=a 10 t 8 + a 8 (q t 3 + q 2 t 5 + t 5 + q 2 t 7 + q t 7 ) + a (q t 0 + q 2 t 2 + t + q 2 t + q t ) taking homology with respect to kills six generators d 2 P sl(2) (T 3, )=q t 0 + q 10 t 2 + q 12 t 3 + q 12 t + q 1 t 5

19 Properties of HOMFLY homology sl(n) differential colored differential H (H,d N ) = H sl(n) d [r]![k] Gukov-Stosic Gorsky-Gukov-Stosic mirror symmetry H (H [r],d [r]![k] ) = H [k] (H ) i,j,k = (H T ) i, j, exponential growth property dim H [r] =[dimh [1] ] r moreover, we have P [r] (K; a, q =1,t)=[P [1] (K; a, q =1,t)] r

20 Properties of HOMFLY homology 8 a H [1,1] (T 2,3 ) q 8 a 0 0 H [2] (T 2,3 ) q There are two-ways to assign homological gradings Using these two gradings, we can introduce auxiliary Q-grading Q = q + t r t c

21 Properties of HOMFLY homology 0 a q a q H [2] (T 2,3 ) H [1,1] (T 2,3 ) a Q a Q

22 Properties of HOMFLY homology With Q-grading, the properties of HOMFLY homology become more transparent Self-symmetry ( e H [r] ) i,j,k,` = ( e H [r] ) i, j,k j,` rj refined exp growth prop ep [r] (K; a, Q, t r,t c = 1) = [ e P [1] (K; a, Q, t r,t c = 1)] r a a Q Q

23 Properties of HOMFLY homology With Q-grading, the properties of HOMFLY homology become more transparent Self-symmetry ( e H [r] ) i,j,k,` = ( e H [r] ) i, j,k j,` rj refined exp growth prop ep [r] (K; a, Q, t r,t c = 1) = [ e P [1] (K; a, Q, t r,t c = 1)] r Open problem Understand physics of two homological gradings as well as auxiliary Q-grading properties of HOMFLY homology colored by hook-like Young diagrams

24 Examples (2s-1,1,2t-1)-pretzel knot rx ep [r] (K s,t ; a, Q, t r,t c )=( t r t c ) r(sgn(s)+sgn(t))/2 $ s,i (a, t r,t c )$ t,i (a, t r,t c ) i=0 where apple ( a 2 2(r 1) tc ) i ( a 2 Q 2 t r t c ; t 2 c) i ( a 2 Q 2 t 3 rt 1+2r c ; t 2 r c) i i t 2 c $ p,i (a, t r,t c )= a i i(i 1)/2 tc ix ( 1) j (at r t c ) 2pj t j=0 (2p+1)j(j 1) c 1 a 2 t 2 rt j c (a 2 t 2 rt 2j c ; t 2 c) i+1 apple i j t 2 c

25 Volume Conjecture Volume conjecture relates quantum knot invariants and classical geometry of knot complement lim n!1 2 n log J n(k; q = e 2 i n One can take double scaling limit J n (K; q) r!1 ~!0 exp 1 ~ S 1(x) 3 2 log ~ + X k ) = Vol(S 3 \K) ~ k S k (x)! Kashaev Murakami,Murakami x = q n :fixed the leading term is given by S 1 (x) = Z A(K;x,y(x))=0 log y dx x Gukov Gukov-Murakami The integral is over an algebraic curve determined by A-polynomial that describe moduli space of SL(2,C) flat connections M flat (T 2,SL(2, C) = C C Z 2 {A(x.y) =0} = M flat (S 3 \K, SL(2, C)

26 Quantizablity Condition In order for a classical A-polynomial A(K; x, y) to be quantizable, periods of the imaginary and real parts of along a path γ is required to obey the Bohr-Sommerfeld condition (K2 condition) The necessary condition is that all roots of all face polynomials of its Newton polygon are roots of unity 5 y NW N NE 3 2 W E 1 SW S SE 12 x

27 Quantum curves O(1 N ) Quantum'Curves' quantization N F0 x y O(1 N ) N 2 2 g N finite q y x = 0 Fg b A(K; x, y)! A(K; x, y, q) g 0 non-commutative deformation of Darboux coordinates finite leads to operator that annihilates Jones polynomial N Gukov Garoufalidis b A(K; x, y, q)jn (K; q) = 0 x Jn (K; q) = q n Jn (K; q) y Jn (K; q) = Jn+1 (K; q) Recursion relations of colored Jones polynomials amounts to quantum A-polynomial N g g

28 knot homology HOMFLY homology P (K; a, q, t) = X i,j,k a i q j t k dim H (K) i,j,k t = 1 d N di erential HOMFLY polynomial P (K; a, q) sl(n) homology J sl(n) (K; q, t) = X i,j,k q i t j dim H sl(n) (K) i,j a = q N t = 1 sl(n) quantum invariant J sl(n) (K; q)

29 Deformation of A-polynomials super-a-polynomial Fuji-Gukov-Sulkowski A super (K; x, y, a, t) t = 1 a =1 Q-deformed A-polynomial refined A-polynomial A Q-def (K; x, y, a) A ref (K; x, y, t) a =1 t = 1 A-polynomial A(K; x, y)

30 Deformation of A-polynomials Q-deformed A-polynomial A Q-def (K; x, y, a) Aganagic-Vafa Agangic-Ekholm-Ng-Vafa = Augmentation polynomial of knot contact homology

31 Mirror symmetry Mirror symmetry is duality b/w A-model and B-model Kahler moduli in A-model is complex structure moduli in B-model Given a knot K, there is a corresponding mirror geometry Brini-Eynard-Marino wz = H K (x, y, a) Mirror geometry of torus knot brane is described by SL(2,Z) transformation of unknot curve Aganagic-Vafa Mirror geometry for a knot K is described by Q-deformed A-polynomial

32 J n (K; q) A-polynomial and topological recursion r!1 ~!0 exp 1 ~ S 1(x) 3 2 log ~ + X k ~ k S k (x)! x = q n :fixed Subleading terms can be obtained by applying topological recursion to character variety Need non-perturbative corrections with theta-functions Asymptotic expansion of HOMFLY polynomials colored by symmetric rep can be obtained by applying topological recursion to Q-deformed A-polynomial Gu-Jockers-Klemm-Soroush Modified Kernel is constructed by Brini-Eynard-Marino curve

33 A-polynomial and topological recursion Asymptotic expansion of HOMFLY polynomials colored by symmetric rep can be obtained by applying topological recursion to Q-deformed A-polynomial Gu-Jockers-Klemm-Soroush Modified kernel is constructed by Brini-Eynard-Marino curve Open problem Find a way to construct modified kernel for non-torus knot Matrix model construction of colored HOMFLY polynomials for non-torus knots Find spectral curve for torus knot in Lens space

34 Refined open BPS states There are two Cauchy formulas for Macdonald polynomials

35 Refined open BPS states Correspondingly, there are two formulations of geometric transition at refined level Reformulated invariants the following structure Remarkably are non-negative integers for all colors after appropriate change of variables

36 String theory The large N duality can be seen at the level of partition functions where f µ (K, a, q) contains information of BPS degeneracies the parameters are identified by q = e g s, a = e t Since counts the number of M2-M5 bound states, they should be integers. Labastida-Marino-Ooguri-Vafa This can be written in infinite product formula X P (K; a, q)s (x) = Y µ Y i,j 1Y (1 a i q j+k x µ ) N µ,i,j(k) k=0

37 Refined open BPS states Reformulated invariants the following structure Remarkably are non-negative integers for all colors after appropriate change of variables Open problem Is this formulation valid for non-torus knots? Provide formulations for other toric CY3s Precise mathematical formulations of refined open BPS states