Refined Chern-Simons Theory, Topological Strings and Knot Homology

Transcription

1 Refined Chern-Simons Theory, Topological Strings and Knot Homology Based on work with Shamil Shakirov, and followup work with Kevin Scheaffer arxiv: arxiv:

2 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 S 3.

3 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.

4 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.

5 Chern-Simons theory is solvable explicitly, so it provides a physical way to not only interpret knot polynomials, but to calculate them as well.

6 Chern-Simons theory gained new relevance when it emerged as a part of string theory and the topological string.

7 Topological string is counting holomorphic maps to a Calabi-Yau threefold X. φ : Σ g X φ = 0

8 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 : Σ g X φ = 0 Since X is a Calabi-Yau threefold, M is three dimensional.

9 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.

10 Open topological string in this case is the same as SU(N) Chern-Simons theory on M. Z CS (M,q) = Z Top (T * M,λ) The parameter that keeps track of the genus in topological string is related to q of Chern-Simons theory by 2πi k + q = e N = e λ 2g 2+# holes λ # edges # vertices λ

11 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 S 3 T * S 3 L K S 3 = K Now, we must allow boundaries on L_K as well.

12 As explained by Gukov, Schwarz and Vafa, categorified knot invariants are related to the refined topological string.

13 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.

14 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.

15 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.

16 Ordinary topological string on a Calabi-Yau X is related to M-theory on (X 2 S 1 ) q where as one goes around the 2 S 1 rotate by, the complex coordinates of

17 The supersymmetric partition function of M-theory in this geometry Z M (X,q) = Tr( 1) F q S 1 S 2 equals the closed topological string partition function of X Gopakumar, Vafa Dijkgraaf, Vafa,Verlinde Z M (X,q) = Z top (X,λ) Here, around the S 1,S 2 z 1, z 2 are generators of rotations planes, and q = e λ The M-theory trace corresponds to trace around the factor in S 1 (X 2 S 1 ) q

18 Open topological string, with boundaries on arizes by putting M5 branes on L X L S 1 in X 2 S 1

19 In particular, if we take X = T * M SU(N) Chern-Simons theory on M corresponds M-theory on (T * M 2 S 1 ) q with N M5 branes on where corresponds to one of the two factors in 2

20 The partition function of N M5 branes in this geometry Z M 5 (M,q) = Tr ( 1) F q S 1 S 2 Ooguri-Vafa equals the SU(N) Chern-Simons partition function on M Z M 5 (M,q) = Z CS (M ) S 1,S 2 2πi k + N Here q = e is related to the level k of Chern-Simons and are generators of rotations around the z 1,z 2 planes.

21 The refined topological string theory is defined via M-theory on a slightly more general background, Nekrasov Nekrasov, Okounkov... (X 2 S 1 ) q,t where as one goes around the, the complex coordinates of 2 S 1 rotate by

22 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

23 When the theory on N M5 branes on preserves supersymmetry, its partition function Z M 5 (M,q,t) = Tr ( 1) F q S 1 S R t S R S 2 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.

24 The extra U(1) symmetry exists when M is a Seifert manifold S 1 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. Rotations of the fiber of this is the U(1) we need. S 1

25 We can introduce knots in the theory by introducing, for each knot K an additional Lagrangian and M5 branes wrapping L K S 1. The L K preserves the U(1) symmetry if K wraps the S 1 fibers. L K

26 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.

27 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.

28 In refined Chern-Simons theory, only a subset of amplitudes generated by S and T enter, since only those preserve the U(1) symmetry.

29 S The 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

30 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. R i

31 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 and inside, R i R j up to the SL(2,Z) transformation of their boundaries.

32 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.

33 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 G k WZW model This allows one solve the theory explicitly, and in particular obtain the Jones and other knot polynomials.

34 Explicitly, for G=SU(N), the S and the T matrices are given by where s R (x) = Tr R x are Schur functions. For q = e 2πi k + N this generates a unitary representation of SL(2,Z) in the Hilbert space labeled by SU(N) representations at level k.

35 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 = q β

36 In the refined SU(N) Chern-Simons theory, t = q β S 00 = c N 1 n=0 i=1 (q n/2 t i/2 q n/2 t i/2 ) N i (q n/2 t (i+1)/2 q n/2 t (i+1)/2 ) N i where M R (x i,q,t) are the Macdonald polynomials. For q = e 2πi k +βn this gives a unitary representation of SL(2,Z) in the Hilbert space is labeled by SU(N) representations at level k.

37 Similar formulas hold for any ADE group where and where y is the dual Coxeter number of G.

38 The fact that these provide a unitary representation of SL(2,Z) was proven in the 90s by Ivan Cherednik.

39 The relation of refined topological string to knot homologies, leads us to conjecture the following.

40 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 = k H ijk 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.

41 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 S 1 where D is a disk. S,T and everything else can be recovered from this by gluing.

42 For separated M5 branes, the M-theory index Z M 5 (M,q,t) = Tr ( 1) F q S 1 S R t S R S 2 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

43 In the present case, all the holomorphic curves are annuli, which are isolated. The moduli of the flat bundle is an S 1 One gets precisely two cohomology classes. From the M-theory perspective, the simplicity of the spectrum is the key to solvability of the theory.

44 The partition function on the solid torus, D S 1 is a wave-function, depending on the holonomy around the S 1. x I 0 x = Δ q,t (x) = 1 I < J N n=1 (q n/2 e (x J x I )/2 q n/2 e (x I x J )/2 ) (q n/2 t 1/2 e (x J x I )/2 q n/2 t 1/2 e (x I x J )/2 )

45 This implies that line operators inserting wilson loops in a solid torus x R i = x O Ri 0 = Δ q,t (x)m Ri (e x ) R i and giving a basis for are given by Macdonald polynomials. In this basis, the identity operator acting on has the natural matrix elements corresponding to Z(S 2 S 1, R i, R j ) R i R j = d N x Δ q, t (x) 2 M Ri (e x )M Rj (e x ) = 1 ij This equals 1 or 0 depending on whether R i, j are equal or not.

46 Obtaining the S and T matrices from solving the theory on two solid tori M L = D S 1, 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.

47 One can give the S 1 fibration over S 2 first Chern class, by multiplying one of two wave functions by exp( ptr x 2 / 2ε 1 ) before gluing. p For example, viewed this way, is the Hopf fibration, with. S 3 p = 1

48 The M-theory partition function on is computing the TST matrix element corresponding to refined Chern-Simons theory on a framed S^3. R i TST R j = d N From M-theory, we obtain T * S 3 The fact that this takes the explicit form we gave before 1 x Δ q,t (x) 2 e Tr x 2 2ε 1 M Ri (e x )M Rj (e x ) was proven in the 90 s in the context of proving Macdonald Conjectures Cherednik, Kirillov, Etingof

49 The theory also generalizes Verlinde algebra. The algebra of line operators k O Ri O Rj = N ij k has coefficients O Rk N ijk =

50 They satisfy the Verlinde formula, as required by three-dimensional topological invariance.

51 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 S 1 inside T * M 2 S 1, but for a general three manifold M, for which the refined index need not exist.

52 The index Z M 5 (M,q,t) = Tr H ( 1) F q S 1 S R t S R S 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 S R associated to it.

53 By contrast, the Poincare polynomial of the theory of N M5 branes wrapping M in, X = T * M Tr H Q q S 1 t S 2 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 the supersymmetry operator Q. H Q of

54 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.

55 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) P 1

56 The large N duality is a geometric transition that shrinks the S 3 and grows the P 1 while eliminating boundaries SU(N) SU(N) Chern-Simons theory on S 3, or open topological string on Gromov-Witten theory on X V = O( 1) O( 1) P 1 X = T * S 3 with P 1 of size Nλ

57 Ooguri-Vafa Adding a knot on K S 3 corresponds to adding a Lagrangian in, and geometric transition relates this to a Lagrangian in. L K X L K X V X = T * S 3 X V = O( 1) O( 1) P 1

58 The large N duality of Chern-Simons theory implies topological string on Ooguri-Vafa 99 X V = O( 1) O( 1) P 1 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.

59 The large N duality extends to the refined case: e.g. the partition function of the refined SU(N) Chern-Simons theory on equals the partition function of the refined topological string on S 3 X Z CS (q,t, N) = Z rtop (X,q,t,a) a = t N t 1/2 q 1/2

60 The duality relates computing knot invariants of the refined Chern-Simons theory on to computing the refined index Z M 5 (L K, X,q,t) = Tr ( 1) F q S 1 S R t S R S 2 S 3 of M5 branes wrapping (L K S 1 ) q,t after the transition, in M-theory on (X 2 S 1 ) q,t

61 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.

62 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.

63 While refined Chern-Simons theory is computing an index, in many cases this index agrees with the Poincare polynomial of the theory DGR constructed.

64 The invariant of an (n, m) torus knot in S 3 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

65 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 q = t, t = q / t and a = q N / t

66 There also seems to persist, for all representations colored by knots in symmetric or anti-symmetric representations.

67 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 X = O( 1) O( 1) P 1 L K graded by the spins S 1,S and their charge in H 2 2 (X, ).

68 Some structural properties of the answer follow from its M-theory definition.

69 Topological string is computing a particular combination of Chern-Simons knot observables det(1 U V ) 1 = Tr SU (N ) R (U) q Tr SU (N ) R (V ) R

70 In the refined topological string, the story is richer we have to specify which branes we are studying: whether they wrap or plane q t 1 in M-theory on (X 2 S 1 ) q,t We will call these the q- and the t-branes.

71 Depending on the kinds of branes, the topological string is computing different refined Chern-Simons amplitudes. n=0 det(1 q n U V ) det(1 tq n U V ) det(1 U V ) SU ( N ) t 1 = M R (U) SU (N )q M R (V ) R G R = ˆM R (U) ( 1) R M SU ( N ) R T (V ) t q q t q R q q

72 a = t N t / q a = t N t / q 1 M R (U) SU (N )q M R (V ) R G R ˆM R T (U) ( 1) R M SU ( N ) R (V ) t R a However, at large N, the branes on the hence we get a prediction S 3 disappear, Z R1 R n (K 1, K n,a,q,t) / G R1 G Rn = ( 1) R 1+ +R n Z R T 1 R T n (K 1, K n,a,t 1,q 1 )

73 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) P 1 Cozacs, Iqbal It was proven recently that the refined topological vertex predictions for the Hopf-link colored by arbitrary representations, S ij ee with the S-matrix amplitude of the refined Chern-Simons theory, at large N, as predicted by large N duality.

74 One can show that similarly, the refined topological vertex Aganagic, Shakirov 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

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