A non-torus link from topological vertex
aa r X i v : . [ h e p - t h ] A ug A non-torus link from topological vertex
Hidetoshi Awata a ∗ , Hiroaki Kanno a,b † , Andrei Mironov c,d,e ‡ , Alexei Morozov d,e § , and Andrey Morozov d,e,f ¶ FIAN/TD-08/18IITP/TH-10/18ITEP/TH-13/18 a Graduate School of Mathematics, Nagoya University, Nagoya, 464-8602, Japan b KMI, Nagoya University, Nagoya, 464-8602, Japan c Lebedev Physics Institute, Moscow 119991, Russia d ITEP, Moscow 117218, Russia e Institute for Information Transmission Problems, Moscow 127994, Russia f MIPT, Dolgoprudny, 141701, Russia
Abstract
The recently suggested tangle calculus for knot polynomials is intimately related to topological stringconsiderations and can help to build the HOMFLY-PT invariants from the topological vertices. We discussthis interplay in the simplest example of the Hopf link and link L n . It turns out that the resolved conifoldwith four different representations on the four external legs, on the topological string side, is describedby a special projection of the four-component link L n , which reduces to the Hopf link colored with twocomposite representations. Thus, this provides the first explicit example of non-torus link description throughtopological vertex. It is not a real breakthrough, because L n is just a cable of the Hopf link, still it canhelp to intensify the development of the formalism towards more interesting examples. Correlation functions of Wilson loops are the most interesting observables in gauge theories: these are thegauge invariant quantities needed to understand confinement and various phase transitions. Unfortunately, theyare much more difficult to calculate, even approximately. Therefore, of interest are particular cases, where Wilsonloop averages are exactly calculable, and one can hope to develop a formalism capturing and efficiently describingthe peculiar properties of these non-local observables. The best known example of such exactly solvable problemis 3 d Chern-Simons theory [1], which is topological and essentially Gaussian in particular gauges like A = 0,which allows one to formulate it in very different dual terms, for instance, as the Reshetikhin-Turaev (RT)lattice theory [2] on 4-valent graphs, which are the knot/link diagrams in projection from three to two spatialdimensions. In result, the correlators in simply-connected 3 d space-time are essentially rational functions withwell controlled denominators, naturally called knot/link polynomials [3,4], being investigated in knot theory foralready about a century.One of spectacular results of the RT approach in its modern Tanaka-Krein version [5] is a possibility ofdefining and exploiting “observables”, arising when the Wilson loops are cut in pieces, into Wilson lines. Anadvantage is that one can then construct the original correlators by ”gluing” them from much simpler buildingblocks. The problem is, however, that, in gauge theory, the open loops provide gauge-non-invariant quantitieswhich can not be ascribed any meaning, neither physical, nor mathematical. However, a reformulation in RTterms, where everything is automatically gauge invariant, allows one to bypass this difficulty and introduce“tangle blocks” [4, 6, 7], which has no clear definition in the original Chern-Simons theory, however, can beefficiently used in cut-and-glue procedures for constructing and evaluating link invariants [6,8]. A great capabilityof this approach was already demonstrated by developing the arborescent calculus [9], which is currently the ∗ [email protected] † [email protected] ‡ [email protected]; [email protected] § [email protected] ¶ [email protected] L n . See also [19], where theresolved conifold with four non-trivial representations was considered as a covering contribution to the Hopflink of SO/Sp
Chern-Simons theory. We demonstrate in this paper that the actual relation is less trivial: theresolved conifold stays related to the Hopf link, only composite representations get involved. As to L n , onlya piece of its invariant is reproduced in this way, while the entire expression is rather a sum of reduced conifoldcontributions over a set of representations at external legs. In fact, L n is a cable of the Hopf link, thus itis not a big surprise that these invariants are related this way, still this is the first example of a formula for anon-torus knot/link constructed from topological vertices.Thus, the main claim of the paper is that the resolved conifold with branes on the four external legs Z µ ,µ ; λ ,λ , which is a function of four representations λ , , µ , , describes a special (in a sense, maximal)projection of the colored HOMFLY invariant H L n of the link L n (in accordance with the Thistlethwaite linktable [20]) in the colored space, which ultimately reduces to the HOMFLY invariant of the Hopf link H Hopf with the two components colored by the composite representations ( λ , λ ) and ( µ , µ ): G L n µ × λ × µ × λ := Pr h H L n µ × λ × µ × λ i max = Z µ ,µ ; λ ,λ Z ∅ , ∅ : ∅ , ∅ = H Hopf( λ ,λ ) × ( µ ,µ ) (1)Note that this is an open question if there is a possibility of expressing generic knots and links through topologicalvertices within the network model framework. The only examples known so far are the torus knots/links, andtheir invariants are either constructed from the Chern-Simons S -matrix [12, 13, 21] which is the Hopf invariantand is associated with the topological vertex only in this way, or constructed [22] by basically substitutingthe Adams coefficients into the Rosso-Jones formula [23], with the second ingredient being eigenvalues of thecut-and-join (or Casimir) operator, which do not have any explicit meaning in topological calculus, and arejust introduced ”by hands”. Hence, even in the case of torus knots and links it remains unclear if they can beobtained directly via network configurations. Specifics of the Hopf link invariant is that, in this case, the Rosso-Jones formula, involving a Casimir-weighted sum of the Schur functions in various representations, accidentally(?) coincides with just a single Schur function at a peculiarly deformed topological locus, and thus acquires asimple representation theory meaning.The plan of the paper is as follows: in sections 2 and 3, we describe the HOMFLY Hopf invariant. Then, insection 4, the Hopf link is related with the conifold description, which naturally gives rise to link L n in thegeneral case. In fact, it turns out that, from the conifold approach, one obtains not the full colored HOMFLYinvariant for the L n link, but its projection, which is ultimately projection to composite representation.Hence, one needs to consider the Hopf link colored with composite representations. It is done in section 5.After this preliminary work, in section 6, we are able to formulate our main statement: that the conifoldconsideration through the topological vertex, indeed, gives rise to the projected L n link aka the Hopf link incomposite representations. This statement is proved in sections 7-8. Section 9 described symmetry propertiesof the HOMFLY Hopf invariant. At last, in section 10, we discuss relations of the description HOMFLY Hopfinvariant as the Schur function and the Rosso-Jones formula [23] and generalization to the superpolynomials.The Appendix contains illustrative examples of the main statement of this paper for first representations.2 Hopf recursion
The simplest relation provided by the tangle calculus of [8] arises when one connects the open ends of theHopf tangle ✲✲ λ λ µ ❲ and treats it in two different ways: ✲✲✛✛ λ µ ❲ ✲✲✛✛ λ λ µ ❲ The 3-component link at the right is composite, i.e. its reduced
HOMFLY “polynomial” is just a product oftwo, for the two constituent Hopf links. At the left, we treat the same configuration as a single Hopf link, wherereducible is the representation λ ⊗ λ , which can be decomposed into irreps, thus the HOMFLY polynomialis a sum of Hopf polynomials in different representations, taken with appropriate multiplicities N λλ λ (whichactually are all unities if either λ or λ is symmetric, or the both are rectangular). Therefore, we get a relationbetween the unreduced Hopf polynomials D µ · X λ ∈ λ ⊗ λ N λλ λ · H Hopf λ,µ = H Hopf λ ,µ · H Hopf λ ,µ (2)where D µ is the quantum dimension of the representation µ . This relation is correct for the Hopf polynomialin the standard, or canonical framing [8, 24–26], which we use throughout the paper.It can be used as a recursion providing efficient and explicit formulas for colored Hopf invariants [8], whichdo not involve any functions difficult to deal with, like the Schur polynomials. Its t -deformation changes thecoefficients in the sum over λ , still, it seems to persist for colored superpolynomials available from [27–30].This is non-trivial, because there is no reason why the Khovanov-Rozansky cohomological calculus, underlyingtoday’s theory of superpolynomials, should respect cut-and-gluing procedures of the effective lattice theory thatare behind the Reshetikhin-Turaev formalism. Indeed, it is not quite respected, because the coefficients aredeformed, but still it survives, since non-linear relations like (2) continue to exist. On the other hand, the topological vertex formalism [12–15, 17, 31] is associated with an expression for theHopf polynomials through characters, Schur or Macdonald functions, which is not very efficient in descriptionof knot polynomials but is very convenient for gluing via the Cauchy summation formulas. Namely, accordingto [6, 15, 17, 32], H Hopf λ,µ = q | λ || µ | N · Schur λ ( q − ρ ) · Schur µ ( q − λ − ρ ) (3)where ρ is the Weyl vector (half-sum of all positive roots) and λ , µ are the weights of representations, andSchur R ( x i ) is the Schur function associated with the Young diagram R , i.e. the character of the U ( N ) group The word “polynomial” should not lead to a confusion here: the colored HOMFLY invariants are, in fact, rational functions of q and A = q N with simple denominators proportional to quantum dimensions. Hereafter, we call them HOMFLY polynomials inorder to use the same term both for links and knots (when the reduced HOMFLY invariant is, indeed, a Laurent polynomial of A and q ). R [33]. Schur R ( x i ) is here a symmetric function of N variables x i , i = 1 ...N associated with components of the corresponding vectors in the Cartan plane. For instance, 2 ρ = ( N − , N − , . . . , − N + 3 , − N + 1) = { N − i + 1 } . In (3), we manifestly have taken into account the U (1)-factor q | λ || µ | N , [8, 24–26].The Hopf polynomial (3) depends on variables q and A = q N . The drawback of this presentation is that (3)is defined for a concrete N , and obtaining its A -dependence requires an analytic continuation. Instead, one canproceed with a presentation with the number of variables in the Schur function not related with the parameter A , [17]. To this end, one has to introduce the shifted Weyl vector 2 ρ = ( − , − , . . . ) = {− i + 1 } , and considerthe Schur functions as symmetric functions of two sets of variables: H Hopf λ,µ = q | λ || µ | N · Schur λ ( A − q − ρ , Aq ρ ) · Schur µ ( A − q − λ − ρ , Aq ρ ) (4)and now one may not restrict by N the number of variables in the Schur symmetric function.The Hopf polynomial is, in fact, symmetric under the permutation of λ and µ , i.e.Schur λ ( q − ρ ) · Schur µ ( q − λ − ρ ) = Schur µ ( q − ρ ) · Schur λ ( q − µ − ρ ) . (5)Then H Hopf λ ,µ · H Hopf λ ,µ = q | λ | + | λ | ) | µ | N · Schur µ ( q − ρ ) · Schur µ ( q − ρ ) · Schur λ ( q − µ − ρ ) · Schur λ ( q − µ − ρ ) . (6)Since we made a clever choice between the two versions in (5), the arguments of Schur λ and Schur λ dependon the same diagram µ , i.e. coincide, and one can use the multiplication ruleSchur λ · Schur λ = X λ ∈ λ ⊗ λ N λλ λ · Schur λ (7)where N λλ λ are integer-valued Littlewood-Richardson coefficients. This gives: H Hopf λ ,µ · H Hopf λ ,µ = X λ ∈ λ ⊗ λ N λλ λ · · D µ z }| { Schur µ ( q − ρ ) · H Hopf λ,µ z }| { q | λ | + | λ | ) | µ | N · Schur µ ( q − ρ ) · Schur λ ( q − µ − ρ )= D µ · X λ ∈ λ ⊗ λ N λλ λ · H Hopf λ,µ (8)i.e. reproduces (2).One can also use the time variables p k := P i x ki so that the quantum dimension is D µ = Schur µ { p ∗ } , p ∗ k := A k − A − k q k − q − k (9)Similarly, one can consider the time variables for q − λ − ρ , moreover, one can absorb the U (1)-factor (3) in theirdefinition so that p ∗ λk = q | λ | kN (cid:16) p ∗ k − A − k ( q k − q − k ) X i,j ∈ λ q k ( i − j ) (cid:17) = q | λ | kN (cid:16) p ∗ k + A − k X i q (2 i − k ( q − kλ i − (cid:17) (10)where λ i are lengths of lines of the Young diagram λ , | λ | = P i λ i and A = q N . In terms of these time variables H Hopf µλ = D λ · Schur µ { p ∗ λ } (11)For instance, H Hopf[1] , [1] D [1] = p ∗ [1]1 = q N A − A − ( q − q − ) q − q − (12)If λ is the adjoint representation, then the uniform Hopf polynomial [34] is associated with the time variables p ∗ adj k = ( q k − q − k ) p ∗ k (13)see [8, 35] for further details. 4 Conifold description
In topological vertex theory of [12–15, 17], the Hopf polynomial H Hopf λ,µ is associated with the brane pattern,symbolically depicted as (cid:0)(cid:0)(cid:0)
Qµ λ ←→ ✲✛ λ µ ❲ According to our logic in the present paper, λ can actually be a representation from the product of two. Infact, the same can be true for µ . In other words, we can consider four branes put on the external legs: (cid:0)(cid:0)(cid:0) µ λ µ λ ←→ ✲✲✛✛ λ λ µ µ ❲❲ and interpret this picture as the double sum P λ ∈ λ ⊗ λ P µ ∈ µ ⊗ µ N λλ λ · N µµ µ · (cid:0)(cid:0)(cid:0) µ λ ←→ ✲✛ λ µ ❲ Alternatively, one can drag two of these four branes through to the other external legs in order to get: (cid:0)(cid:0)(cid:0) µ λ µ λ S ←→ ✣✢✤✜✣✢✤✜✣✢✤✜✣✢✤✜ µ λ µ λ ←→ (cid:0)(cid:0)(cid:0) µ λ µ λ S The link on the r.h.s. is the 4-component L n , and it is obviously the same as the link in the second picturesof the present section. The identity reflecting the possibility of multiplying representations on one leg, or,alternatively, the possibility of brane-dragging states that H L n µ ,λ ,µ ,λ = X λ ∈ λ ⊗ ¯ λ µ ∈ µ ⊗ ¯ µ N λλ λ · N µµ µ · H Hopf λ,µ (14)and is immediately clear from the picture: 5 ❄✲✛ ❄✻✛✲ λ µ λ µ = ✲✛✛✲ ❄✻ ✻❄ λ µ λ µ Note that the apparent cyclic symmetry of the l.h.s. in (14) is hidden on its r.h.s.This is an interesting formula, because one can independently calculate both sides by the methods of [8].The l.h.s. is a necklace with H L n = Tr τ , where the lock-element τ , can be extracted from the knowledgeabout twist knots, where H twist k = Tr τ ¯ T k . The r.h.s. involves the Hopf polynomials, which we review in thenext section. An even more interesting option is to apply conifold formulas from [15,17], and we do this in sec.7below. Surprisingly or not, they do not reproduce (14), but pick up a single item H Hopf( λ ,λ ) × ( µ ,µ ) from the r.h.s.sum. We explain it in more details in the next sections. Colored HOMFLY polynomials for the Hopf link are in the intersection of application domains of verydifferent approaches: from the Rosso-Jones formula to conifold calculus. Moreover, while the Rosso-Jonesformula nicely describes “universal” representations, i.e. the representations of SU ( N ) that do not depend on N at large enough N , the conifold calculus deals with the composite (or rational, [36, 37]; or coupled, [38])representations, which manifestly depend on N , the simplest example of these being conjugate representations.First of all, we need to explain what is the composite representation, which is, in a sense, a “maximal”representation in the product of R and the conjugate ¯ P of P . The composite representation is the most generalfinite-dimensional irreducible highest weight representations of SU ( N ) [36–40], which are associated with theYoung diagram obtained by putting R atop of p lines of the lengths N − p ∨ i ( p ∨ i are length of lines of thetransposed Young diagram P ∨ ), i.e.( R, P ) = h r + p , . . . , r l R + p , p , . . . , p | {z } N − l R − l P , p − p lP , p − p l P − , . . . , p − p i or, pictorially, ......... . . .. . .. . .. . . ( R, P ) = ¯ PR ˇ P h P = l P ∨ = p Nl R l P . . . . . . . . . where l P is the number of lines in the Young diagram P . This ( R, P ) is the first (hence, the word “maximal”)representation contributing to the product R ⊗ ¯ P . It can be manifestly obtained from the tensor products (i.e. This independence on N means that the character is determined by some Young diagram(s) not involving N . The formaldefinition is related with the notion of universal character, see [36]. “First” means here the representation that is associated with the Young diagram obtained just by attaching the diagram R to¯ P atop.
6s a projector from R ⊗ ¯ P ) by formula [36]( R, P ) = X Y,Y ,Y ( − l Y N RY Y N PY ∨ Y Y ⊗ Y (15)where ∨ denotes the transposition of the Young diagram.The adjoint representation in this notation is adj = ([1] , [1]), while the conjugate of the representation R is R = ( ∅ , R ). The product [ m ] ⊗ [ m ] = m X k =0 ([ k ] , [ k ]) (16)where ( ∅ , ∅ ) SU ( N ) ∼ = ∅ , and the other items are diagrams with 2 k lines, k of length N − k of length one.Similarly, [1 m ] ⊗ [1 m ] = m X k =0 ([1 k ] , [1 k ]) (17)where contributing are the diagrams with just two lines of lengths N − k and k .The expression for the Hopf polynomial in arbitrary composite representations can be again written in theform H ( R,P ) × ( T,S ) = D ( R,P ) · Schur ( T,S ) { p ∗ ( R,P ) } (18)Moreover, the main ingredients in this formula can be presented in a universal form that involves N (if at all)as a simple parameter, however, the explicit expressions for them are far more involved: p ∗ ( R,P ) k implied by (10)for the composite Young diagram ( R, P ) are [6, 35] p ∗ ( R,P ) k = q | R |−| P | N k p ∗ k + 1 A k · l R X j =1 q (2 j − k · ( q − kr j −
1) + A k · l P X i =1 q (1 − i ) k · ( q kp i − (19)the quantum dimensions are [35] D ( R,P ) = D R ( N − l P ) D P ( N − l R ) Q l R i =1 [ N − l P − i ]! Q l P i ′ =1 [ N − l R − i ′ ]! Q l R + l P i =1 [ N − i ]! l R Y i =1 l P Y i ′ =1 [ N + r i + p i ′ + 1 − i − i ′ ] (20)where [ ... ] denotes quantum numbers, and the corresponding Schur functions are [35–37]Schur ( R,P ) { p ∗ ( T,S ) } = X η ∈ R ∩ P ∨ ( − ) | η | · Schur
R/η { p ∗ ( T,S ) } · Schur
P/η ∨ { p ∗ ( T,S ) ( A − , q − ) } (21)where Schur R/η denotes the skew Schur function. Note that the “mirror-reflecting” substitute (
A, q ) → ( A − , q − ) in (21) could be replaced just by the transposition of Young diagrams except for the U (1)-factor q | R |−| P | N k that have to be accordingly changed. L n C ξµλ ( q ) = q κ ( λ ) · Schur µ ( q ρ ) X η Schur ξ/η ( q µ + ρ ) · Schur λ ∨ /η ( q µ ∨ + ρ ) , (22) One can make the change q → /q so that the Schur polynomials would contain exponentials q − ρ like in formulas of sec.3.This would require the identification Q = A − and reversing arrows in the figure below, and interchanging representations in thecomposite representation: ( R, P ) → ( P, R ). κ ( λ ) = 2 P i,j ∈ λ ( j − i ), and the time variables are p ( µ ) k = p k ( q µ + ρ ) = ∞ X j =1 q (2 µ j − j +1) k = 1 q k − q − k + ∞ X j =1 q (1 − j ) k ( q µ j k −
1) = 1 q k − q − k + ( q k − q − k ) X i,j ∈ µ q k ( j − i ) , (23)Note that they are independent of A and different from p ∗ µk in (10). In topological string theory, the parameter q is associated with the string coupling (genus expansion parameter) g s by q = e − g s / . Since η in the skew-characters should be a sub-diagram of λ , the sum over η is actually finite. Though it is not manifest in (22), C ξ ∨ µλ ∨ ( q ) · q − κ ( µ ) is symmetric under the cyclic permutation of ( ξ, µ, λ ). ❄(cid:0)(cid:0)(cid:0)✲ (cid:0)(cid:0)(cid:0)✠ ✛✻ ξ λ λ µ µ “Four point function” on the resolved conifold geometry is given by gluing two topological vertices: Z µ ,µ ; λ ,λ = X ξ ( − Q ) | ξ | C ξµ ∨ λ ( q ) C ξ ∨ µ ∨ λ ( q ) , (24)where Q = e t and t is the K¨ahler parameter of the rational curve P represented by the internal edge. Theprescription of large N duality tells that the ’t Hooft coupling is t = N g s = πiNN + k , and we may identify Q = q N = A . In the precise formulation, the diagram is equipped with arrows which distinguish between therepresentations and their transposed. In writing down (24), we set the clockwise order of edges at each vertex.If the edge is outgoing, we have to use the transpose of the Young diagram attached to the edge. To make thesummation over ξ easy, we put ξ to the first position by using the cyclic symmetry of the topological vertex.In the toric diagram, the slope of the edge corresponds to a cycle of T fibration, in particular, the horizontal(vertical) line corresponds to (1 , , µ , and λ , are linkedwith linking number ±
1, while µ ( λ ) and µ ( λ ) are parallel to each other. Note that the notation is adjustedto purposes of the present paper, where the cyclic ordering is less important and can be made obscure.Our main claims in this paper are: • the normalized sum (24) is equal toˆ G L n µ × λ × µ × λ = Z µ ,µ ; λ ,λ Z ∅ , ∅ : ∅ , ∅ = ( − A ) | µ | + | µ | q κ ( λ )+ κ ( λ )+ κ ( µ )+ κ ( µ ) · D ( µ ,µ ) ×× P σ,η ,η ( − A ) | η | + | η |−| σ | · Schur λ ∨ /η { p ( µ ∨ ) } · Schur λ ∨ /η { p ( µ ∨ ) } · Schur η ∨ /σ { p ( µ ) } · Schur η ∨ /σ ∨ { p ( µ ) } (25)where the sums run over η ⊂ λ ∨ , η ⊂ λ ∨ and σ ⊂ η ∩ η . The quantum dimension of the compositerepresentation, D ( µ ,µ ) is given in (20). • instead of the full unreduced HOMFLY of link L n , which in this case is given by (14) and contains manyterms: H L n λ × µ × λ × µ = X λ ∈ λ ⊗ ¯ λ µ ∈ µ ⊗ ¯ µ N λλ λ · N µµ µ · H Hopf λ × µ (26)the sum (25) provides the contribution of only the maximal representations λ max ∈ λ ⊗ ¯ λ and µ max ∈ µ ⊗ ¯ µ so that G L n λ × µ × λ × µ = H Hopf( λ ,λ ) × ( µ ,µ ) = D ( µ ,µ ) · Schur ( λ ,λ ) { p ∗ ( µ ,µ ) } (27)8ere G differs from ˆ G in (25) by changing the framing to the standard one and taking into account the U (1)-factor: G := ( − | λ | + | λ | + | µ | + | µ | · q − C ( λ ) − C ( λ ) − C ( µ ) − C ( µ ) · q | λ |−| λ | )( | µ |−| µ | ) N · ˆ G (28)Here C ( R ) = κ ( R ) + N | R | is the eigenvalue of the second Casimir operator in the representation R , andwe took into account that, for the composite representations, C (( R, S )) = C ( R ) + C ( S ), [39].In the next two sections, we explain how to get formulas (25), (27), and, in the Appendix, how the identity (27)works. Here we present summation formulas that lead to formula (25).We start with the simplest example: when a diagram at external line is empty, say, λ = ∅ , then the sumover η in (22) is restricted to η = ∅ and C ξµ ∅ = Schur µ ( q ρ ) · Schur ξ ( q µ + ρ ) (29)Then (24) implies: Z µ ,µ ; ∅ , ∅ ∼ X ξ ( − Q ) | ξ | · Schur ξ ( q µ + ρ ) · Schur ξ ∨ ( q µ + ρ ) = exp − X k Q k p ( µ ) k p ( µ ) k k ! (30)Substituting (23), we get for the µ -dependent factorexp − X k Q k p ( µ ) k p ( µ ) k k ! ∼ exp − X k Q k k X j q µ j k − q − µ j k q k − q − k · q ( µ j − j +1) k + X j q µ j k − q − µ j k q k − q − k · q ( µ j − j +1) k ++ X j ,j ( q µ j k − q µ j k − q − j + j − k (31)For particular diagrams µ and µ , the ratios in this formula become polynomials, and the entire expression isa product of factors (1 − Qq n ) with some µ -dependent powers n . For example, if µ = µ = [1], there are justtwo factors:exp − X k Q k p ([1]) k p ([1]) k k ! ∼ exp − X k Q k k (cid:16) q k + q − k z }| { q k − q − k (cid:17) = (32)= (1 − Qq )(1 − Qq − ) = A (cid:16) Aq − ( Aq ) − (cid:17)(cid:16) Aq − − A − q (cid:17) = A ( q − q − ) D adj Generally, restoring the µ -independent factorexp − X k k · Q k ( q k − q − k ) ! = ∞ Y i =1 (1 − Qq − i ) i (33)one obtains exp − X k Q k p ( µ ) k p ( µ ) k k ! = ( − A ) | µ | + | µ | q κ ( µ κ ( µ h µ h µ D ( µ ,µ ) ∞ Y i =1 (1 − Qq − i ) i (35) Alternatively, one can derive this formula using instead of p ( µ ) k the components of q µ + ρ = q µ j − j +1 in the Cartan plane,(23). Then, the l.h.s. of (35) is a productexp − X k Q k p ( µ ) k p ( µ ) k k ! = Y i,j ≥ (cid:16) − Qq µ i + µ j − i − j +1) (cid:17) (34)which straightforwardly gives the r.h.s. of (35), see [42] for further details. h µ := Q i,j ∈ µ ( q h i,j − q − h i,j ), h i,j is length of the hook ( i, j ), and the quantum dimension D ( µ ,µ ) is givenin (20).Let us now note that Schur µ ( q ρ ) = q κ ( µ )2 h µ (36)Hence, we finally obtain Z µ ,µ ; ∅ , ∅ Z ∅ , ∅ ; ∅ , ∅ = ( − A ) | µ | + | µ | q κ ( µ )+ κ ( µ ) D ( µ ,µ ) (37)which is a particular case of (25) and, being proportional to the quantum dimension of ( µ , µ ), is consistentwith (27) at λ = λ = ∅ .When diagrams λ are also non-trivial, non-trivial can also be η and summation over the intermediaterepresentation ξ can be performed by the Cauchy formula for skew Schur functions X ξ ( − Q ) | ξ | · Schur ξ/η { p } · Schur ξ ∨ /η { p ′ } = (38)= exp − X k Q k p k p ′ k k ! · X σ ( − Q ) | η | + | η |−| σ | · Schur η ∨ /σ { p ′ } · Schur η ∨ /σ ∨ { p } Now formulas (24), (22), along with (35) and (38), give rise to (25).
Now let us prove that the triple sum (27) reduces to the Hopf polynomial in the composite representationin the form (21). To this end, we use the formulaSchur
R/T { p (1) + p (2) } = X P Schur
R/P { p (1) } · Schur
P/T { p (2) } (39)Let us start studying formula (25) with the case of λ = µ = ∅ . In this case, the sum in (25) reduces to X η ( − A ) | η | · Schur λ ∨ /η { p ( µ ∨ ) } · Schur η ∨ { p ( ∅ ) } = X η A | η | · Schur λ /η ∨ {− p ( µ ∨ ) } · Schur η ∨ { p ( ∅ ) } == X η A | λ | · Schur λ /η ∨ {− A − k p ( µ ∨ ) k } · Schur η ∨ (cid:26) A k q k − q − k (cid:27) = A | λ | · Schur λ (cid:26) A k q k − q − k − A − k p ( µ ∨ ) (cid:27) == A | λ | · Schur λ ( p ∗ k + A − k X i q (2 i − k ( q − kµ i − ) = q − | λ || µ | N A | λ | · Schur λ { p ∗ µ } (40)where we used thatSchur R ∨ /T ∨ { p k } = ( − | R | + | T | Schur
R/T {− p k } , p ( µ ∨ ) k = ( − k +1 p ( µ ) k (cid:12)(cid:12)(cid:12) q →− /q (41)and formula (39) with T = ∅ . Taking into account all the factors in (25) and (28), we immediately obtain from(27) formula (11) for the Hopf link.In complete analogy, in the general case, using (39) we can evaluate in (25) the sum over η X η ( − A ) | η | · Schur λ ∨ /η { p ( µ ∨ ) } · Schur η ∨ /σ { p ( µ ) } = A | σ | ( − A ) | λ | · X η Schur λ /η ∨ {− A − k p ( µ ∨ ) k } · Schur η ∨ /σ { A k p ( µ ) k } == A | σ | ( − A ) | λ | · Schur λ /σ { A k p ( µ ) k − A − k p ( µ ∨ ) k } = q − | λ |−| σ | )( | µ |−| µ | ) N A | σ | ( − A ) | λ | · Schur λ /σ { p ∗ ( µ ,µ ) } (42)and, similarly, we evaluate the sum over η : X η ( − A ) | η | · Schur λ ∨ /η { p ( µ ∨ ) } · Schur η ∨ /σ ∨ { p ( µ ) } = q | λ |−| σ | )( | µ |−| µ | ) N A | σ | ( − A ) | λ | · Schur λ /σ ∨ { p ∗ ( µ ,µ ) ( A − , q − ) } (43)10ence, we are remaining with( − A ) | λ | + | λ | q | λ |−| λ | )( | µ |−| µ | ) N X σ ( − | σ | · Schur λ /σ { p ∗ ( µ ,µ ) } · Schur λ /σ ∨ { p ∗ ( µ ,µ ) ( A − , q − ) } (44)which is exactly the sum in (21). Now, taking into account all the factors in (25), (28), we obtain (27). The main tools of Hopf calculus in [8] are the recursion (2), which is a characteristic feature of charactersand thus implies (3), and a peculiar property of the Hopf link, H Hopf R × ¯ R ( A, q ) = H Hopf R × R ( A − , q − ) (45)which is obvious from the picture ❄ ✻ ¯ R R = ✻ ✻ R R supplemented by the property of R -matrix, that its inversion is equivalent to inversion of A and q . Note thatthe equality (45) specifically holds for the Hopf link despite R R × ¯ R ( A, q ) = R R × R ( A − , q − ).Other symmetry properties for the Hopf polynomials in the case of composite representations are H Hopf(
R,P ) × S = H Hopf(
P,R ) × S ( A − , q − ) (46)which follows from the relation ( R, P ) = (
P, R ), and H Hopf(
R,P ) × ( S,T ) = ( − | R | + | P | + | S | + | T | q | R |−| P | )( | S |−| T | ) N H Hopf( R ∨ ,P ∨ ) × ( S ∨ ,T ∨ ) ( A, q − ) (47)which is a Hopf version of the standard level-rank duality, [43] H { R i } = H { R ∨ i } ( A, q − )the latter valid for any HOMFLY invariant being a fact from group representation theory. Formula (47) followsfrom (18) and (21) using (19) and (41).From the three identities (45)-(47), it follows that H Hopf(
R,P ) × [1] = ( − | P | + | R | +1 q | P |−| R | ) N H Hopf( P ∨ ,R ∨ ) × [1] ( A − , q ) (48)In fact, the change of variable A → A − is natural from the viewpoint of conifold geometry. Under the flopoperator of conifold, which flips the sign of the K¨ahler parameter and hence A → A − , the representationattached with λ = [1] changes from µ to µ . Hence, this identity.While formulas of the type (3) and their generalizations to the composite representations [35, 37] as wellas the Rosso-Jones formula are convenient for general studies of the Hopf link, the most efficient for obtaininggeneral explicit formulas is at the moment the tangle calculus of [8]. Making use of these identities and oneadditional calculation for H Hopfadj × [ s ] , the old knowledge of Hopf polynomials in two symmetric representations H Hopf[ r ] × [ s ] can be extended to the case of one arbitrary diagram and one symmetric.Explicit formulas for the Hopf polynomials that we use in the Appendix are all picked up from [8].
10 Rosso-Jones formula and superpolynomials
There is another, basically very different formula for the Hopf polynomial based on the paper by M. Rossoand V. F. R. Jones [23] H Hopf λ × µ = q | λ || µ | N q κ λ + κ µ X η ∈ λ ⊗ µ N ηλµ · q − κ η · D η (49)11n this paper, we explained how the representation of the Hopf polynomial as a specialization of the character,(3) is directly related with the topological vertices. However, any direct relation either of these two to the Rosso-Jones formula (49) is not clear, to our best knowledge. All known proofs of equivalencies with this relation areprovided by an additional averaging procedure, either in Chern-Simons theory, [44], or via (related) a multipleintegral representation [45], or using a scalar product of two characters defined in some other way [46]. Thereis, certainly, a derivation based on the modular categories [47], i.e. basically on the Verlinde formula [48],however, it is not straightforward as well. As a particular manifestation of this problem, generalizing the Rosso-Jones formula for other knots and links, one can not obtain a counterpart of the character specialization (see,however, [49]).Thus, we have a diagram of the typeSchur representation (3) ←−−−−− Averaging procedure տ ր xy
Hopf polynomial y ւ ց
Topological vertex approach Rosso-Jones formulaIt would be interesting to restore the missing arrows.What is, however, important, the Rosso-Jones formula admits very immediate and simple generalizationto any other torus knots and links [23], and also to the superpolynomials [27], while other ingredients at thediagram are only easily generalizable to the superpolynomials [21, 31, 46]. Indeed, the Rosso-Jones formula forthe Hopf superpolynomial looks like (hereafter in this section, we omit the U (1)-factor) P Hopf λ,µ = q − ν λ − ν µ t ν ′ λ + ν ′ µ X η ∈ λ ⊗ µ N ηλµ · q ν η t − ν ′ η · M η (50)where ν λ := 2 P i ( i − λ i , ν ′ λ := ν λ ∨ so that κ λ = ν ′ λ − ν λ , and M η is the Macdonald dimension of η , i.e.the specialization of the Macdonald symmetric function M η { q, t | p } at the topological locus (in time variables) p k = p ∗ k [27]: M η := M η { q, t | p ∗ } , p ∗ k = A k − A − k t k − t − k (51)where we use the Gothic letters in order to stress the superpolynomial deformation. In particular, the coefficients N ηλµ are defined now as coefficients of expansion of the products of Macdonald polynomials M λ { q, t | p } · M µ { q, t | p } = X η ∈ λ ⊗ µ N ηλµ · M η { q, t | p } (52)In contrast with the Littlewood-Richardson coefficients, N ηλµ are not obligatory integer.A counterpart of formula (3) in the superpolynomial form looks as P Hopf λ,µ = M λ ( t − ρ ) · M µ ( q − λ t − ρ ) = M λ · M µ ( q − λ t − ρ ) (53)These symmetric functions of the components of vectors in the Cartan plane can be again rewritten in termsof the time variables p ∗ λk = p ∗ k − A − k ( q k − q − k ) X i,j ∈ λ t k (2 i − q k (1 − j ) = p ∗ k + A − k X i t (2 i − k ( q − kλ i −
1) (54)where, as above, the Gothic letter refers to the superpolynomial deformation: M µ ( q − λ t − ρ ) = M µ { q, t | p ∗ λk } (55)Thus, (53) can be written in the form P Hopf λ,µ = M λ · M µ { q, t | p ∗ λk } (56)The equivalence of the two representations for the Hopf superpolynomials, (50) and (56) is again proved throughan intermediate averaging representation [21, 31, 46].Now one can repeat the machinery developed in the present paper, in this superpolynomial case, [50].12 In this letter, we considered an elementary example of the tangle calculus of [8]: the quadratic recursionformula (2) for Hopf polynomials, which immediately follows from pictorial gluing of free ends of the Hopf tangle.We explained the relation to the traditional description of Hopf polynomials, which provides an algebraic proofof the recursion identity from multiplication of Schur functions. This proof is easily lifted to superpolynomials,and emerging identity suggests that tangle calculus can be extended to this area, despite the Reshetikhin-Turaevformalism, of which it is a simple corollary, is no longer applicable in this case in any known way. This addsa new evidence to the comparably surprising results of [27] and [51] about survival of evolution property forthe torus superpolynomials and even for the torus Khovanov-Rozansky polynomials at finite N . As anotherapplication, we described the relation of the link polynomial for L n and of the Hopf polynomial.To summarize this application, the link L n , which one could naturally associate with the 4-point toricdiagram, in knot theory is distinguished by existence of two dual descriptions: as a closed necklace made from4 unknots and as a Hopf link in reducible representations ( µ ⊗ µ ) × ( λ ⊗ λ ). ✻ ❄✲✛ ❄✻✛✲ λ µ λ µ The possibility of two different descriptions allows one to double-check the formulas, see [8], but here we usedonly the Hopf-related expressions. Our claim (27) was that the sum (24) provides the contribution of thecomposite representations G L n λ × µ × λ × µ = H Hopf( µ ,µ ) × ( λ ,λ ) (57)rather than the full unreduced HOMFLY polynomial.It is an interesting task to extend all the components of our discussion: tangle calculus, character calculus,conifold calculus and their refinement beyond the Hopf link example. Some parts of this extension are alreadylong-standing problems, however, the interplay between these different approaches seems to provide new toolsto finally solve them and make new steps towards building a powerful and efficient calculational technique. Acknowledgements
Our work is supported in part by Grants-in-Aid for Scientific Research ( ppendix. Explicit examples of formula (27)
Here we give a series of examples that illustrate how formula (27) works. First of all, we will need a seriesof specializations of the Schur functions, some of them being an illustration of formula (36):Schur [1] ( q ρ ) = p ( q ρ ) = ( q − q − ) − , Schur [2] ( q ρ ) = 12 (cid:16) ( p ( q ρ )) + p ( q ρ ) (cid:17) = ( q − q − ) − (1 + q − ) − , Schur [1 ] ( q ρ ) = 12 (cid:16) ( p ( q ρ )) − p ( q ρ ) (cid:17) = ( q − q − ) − (1 + q ) − . (58)Other answers that we will need in examples areSchur [1] ( q [1]+ ρ ) = ( q − q − ) + Schur [1] ( q ρ ) , Schur [1] ( q [2]+ ρ ) = (1 + q )( q − q − ) + Schur [1] ( q ρ ) , Schur [1] ( q [1 ]+ ρ ) = (1 + q − )( q − q − ) + Schur [1] ( q ρ ) , (59) λ = λ = ∅ This case was already considered in (37). Using formula (20), we can compare E ([1] , [1]) ∼ exp ( − X k Q k k (cid:16) q k + q − k (cid:17)) = (1 − Qq )(1 − Qq − ) = A ( q − q − ) [ N + 1][ N − G L n [1] × ∅ × [1] × ∅ = [ N + 1][ N −
1] = D adj = H Hopf([1] , [1]) × ∅ ,E ([2] , [1]) ∼ exp ( − X k Q k k (cid:16) q k + 1 + q − k (cid:17)) = (1 − Qq )(1 − Q )(1 − Qq − ) == A ( q − q − ) [ N + 2][ N ][ N − G L n [2] × ∅ × [1] × ∅ = 1[2] [ N + 2][ N ][ N −
1] = D ([2] , [1]) = H Hopf([2] , [1]) × ∅ ,E ([1 , , [1]) ∼ exp ( − X k Q k k (cid:16) q k + 1 + q − k (cid:17)) = (1 − Qq )(1 − Q )(1 − Qq − ) == A ( q − q − ) [ N + 1][ N ][ N − G L n [1 , × ∅ × [1] × ∅ = 1[2] [ N + 1][ N ][ N −
2] = D ([1 , , [1]) = H Hopf([1 , , [1]) × ∅ ,E ([2] , [2]) ∼ exp ( − X k Q k k (cid:16) q k + 2 + q − k (cid:17)) = (1 − Qq )(1 − Q ) (1 − Qq − ) == A ( q − q − ) [ N + 3][ N ] [ N − G L n [2] × ∅ × [2] × ∅ = 1[2] [ N + 3][ N ] [ N −
1] = D ([2] , [2]) = H Hopf([2] , [2]) × ∅ ,E ([1 , , [1 , ∼ exp ( − X k Q k k (cid:16) q k + 2 + q − k (cid:17)) = (1 − Qq )(1 − Q ) (1 − Qq − ) == A ( q − q − ) [ N + 1][ N ] [ N − G L n [1 , × ∅ × [1 , × ∅ = 1[2] [ N + 1][ N ] [ N −
3] = D ([1 , , [1 , = H Hopf([1 , , [1 , × ∅ ,E ([2] , [1 , ∼ exp ( − X k Q k k (cid:16) q k + q k + q − k + q − k (cid:17)) = (1 − Qq )(1 − Qq )(1 − Qq − )(1 − Qq − ) == A ( q − q − ) [ N + 2][ N + 1][ N − N − G L n [2] × ∅ × [1 , × ∅ = 1[2] [ N + 2][ N + 1][ N − N −
2] = D ([2] , [1 , = H Hopf([2] , [1 , × ∅ , (60)where we used the notation exp (cid:18) − P k Q k p ( µ ) k p ( ν ) k k (cid:19) := E ( µ, ν ).14 = [1] , λ = ∅ In this case we have Z µ ,µ ;[1] , ∅ = Schur µ ( q ρ ) Schur µ ( q ρ ) X ξ ( − Q ) | ξ | X τ Schur ξ ∨ /τ ( q µ + ρ ) Schur [1] /τ ( q µ ∨ + ρ ) Schur ξ ( q µ + ρ ) , (61)where we can write down the summation over τ explicitly; X τ Schur ξ ∨ /τ ( q µ + ρ ) Schur [1] /τ ( q µ ∨ + ρ ) = Schur ξ ∨ ( q µ + ρ ) Schur [1] ( q µ ∨ + ρ ) + Schur ξ ∨ / [1] ( q µ + ρ ) . (62)Performing the summation over ξ by the Cauchy formula, we obtain Z µ ,µ ;[1] , ∅ = Z µ ,µ ; ∅ , ∅ (cid:16) Schur [1] ( q µ ∨ + ρ ) − Q · Schur [1] ( q µ + ρ ) (cid:17) . (63)Now, using (59), we obtain G L n [1] × [1] × [1] × ∅ ( A, q ) = [ N + 1][ N ][ N − q − q − ) = H Hopf([1] , [1]) × [1] G L n [2] × [1] × [1] × ∅ ( A, q ) = q /N · [ N + 2][ N ][ N − (cid:16) [3][ N ] − A − q − ( q − q − ) (cid:17) = H Hopf([2] , [1]) × [1] G L n [2] × [1] × [2] × ∅ ( A, q ) = [ N + 3][ N ] [ N − (cid:16) [3][ N ] + [ N + 2]( q − q − ) (cid:17) = H Hopf([2] , [2]) × [1] G L n [2] × [1] × [1 , × ∅ ( A, q ) = [ N + 2][ N + 1][ N ][ N − N − ( q − q − ) = H Hopf([2] , [1 , × [1] and all other cases of the Young diagrams up to the level 2 are obtained from these using formulas (46)-(48). λ = λ = [1] In this case we have Z µ ,µ ;[1] , [1] = Schur µ ( q ρ ) Schur µ ( q ρ ) X ξ ( − Q ) | ξ | X τ Schur ξ ∨ /τ ( q µ + ρ ) Schur [1] /τ ( q µ ∨ + ρ ) × X σ Schur ξ/σ ( q µ + ρ ) Schur [1] /σ ( q µ ∨ + ρ ) . (64)Using (62), we find G L n µ × [1] × µ × [1] = G L n µ × ∅ × µ × ∅ · z ( µ , µ ) , (65)where z ( µ , µ ) := Schur [1] ( q µ ∨ + ρ ) Schur [1] ( q µ ∨ + ρ ) − Q (1 + Schur [1] ( q µ + ρ ) Schur [1] ( q µ ∨ + ρ )+ Schur [1] ( q µ + ρ ) Schur [1] ( q µ ∨ + ρ )) + Q · Schur [1] ( q µ + ρ ) Schur [1] ( q µ + ρ )) , (66)which is symmetric under µ ↔ µ . Thus we obtain G L n [1] × [1] × [1] × [1] = [ N + 1][ N − (cid:16) − [ N ] (cid:17) = H Hopfadj × adj G L n [2] × [1] × [1] × [1] = [3][2] [ N + 2][ N + 1][ N ][ N − (cid:16) ( q + q − )[ N ] − [ N + 1] (cid:17) = H Hopf([2] , [1]) × adj , G L n [2] × [1] × [2] × [1] = [ N + 3][ N + 1][ N ] [ N − (cid:16) ( q + q − ) [ N + 1] − q + q − )[ N + 2] + [ N + 3] (cid:17) = H Hopf([2] , [2]) × adj , G L n [2] × [1] × [1 , × [1] = [ N + 2][ N + 1][ N − N − (cid:16) ( q + q − )[ N ] − [ N + 1] (cid:17)(cid:16) ( q + q − )[ N ] − [ N − (cid:17) = H Hopf([2] , [1 , × adj (67)15 = [2] or [1 , , λ = ∅ For λ = [2], we have Z µ ,µ ;[1] , ∅ = Schur µ ( q ρ ) Schur µ ( q ρ ) X ξ ( − Q ) | ξ | X τ Schur ξ ∨ /τ ( q µ + ρ ) Schur [2] /τ ( q µ ∨ + ρ ) Schur ξ ( q µ + ρ ) , (68)where X τ Schur ξ ∨ /τ ( q µ + ρ ) Schur [2] /τ ( q µ ∨ + ρ )= Schur ξ ∨ ( q µ + ρ ) Schur [2] ( q µ ∨ + ρ ) + Schur ξ ∨ / [1] ( q µ + ρ ) Schur [1] ( q µ ∨ + ρ ) + Schur ξ ∨ / [2] ( q µ + ρ ) . (69)The Cauchy formula gives G L n µ × [2] × µ × ∅ = G L n µ × ∅ × µ × ∅ (cid:16) Schur [2] ( q µ ∨ + ρ ) − Q · Schur [1] ( q µ ∨ + ρ ) Schur [1] ( q µ + ρ ) + Q · Schur [1 , ( q µ + ρ ) (cid:17) . (70)When µ = µ = [1], we obtain G L n [1] × [2] × [1] × ∅ = A [ N + 1][ N − (cid:16) s [2] ( q [1]+ ρ ) − Qs [1] ( q [1]+ ρ ) + Q s [1 , ( q [1]+ ρ )) (cid:17) == [ N + 1][ N ][ N − (cid:16) ( q + q − )[ N + 2] − [ N + 3] (cid:17) = H Hopfadj × [2] (71)On the other hand, if λ = [1 , [2] and Schur [1 , switch places everywhere, and the parallel compu-tation gives A · G L n [1] × [1 , × [1] × ∅ ( A, q ) = A · G L n [1] × [2] × [1] × ∅ ( A, q − ) = H Hopfadj × [1 , . (72) References [1] S. Chern, J. Simons, Proc.Nat.Acad.Sci. 68 (1971) 791-794; Annals of Math. 99 (1974) 48–69A.S. Schwarz,
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