Cabling Procedure for the HOMFLY Polynomials
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1 Cabling Procedure for the HOMFLY Polynomials Andrey Morozov In collaboration with A. Anokhina ITEP, MSU, Moscow 1 July, 2013, Erice Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 1 / 28
2 Chern-Simons theory Current knowledge of the gauge eld theories 2-dimensional theories - e.g CFT 4-dimensional theories - e.g. SW Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 2 / 28
3 Chern-Simons theory Current knowledge of the gauge eld theories 2-dimensional theories - e.g CFT 4-dimensional theories - e.g. SW AdS/CFT AGT Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 2 / 28
4 Chern-Simons theory Current knowledge of the gauge eld theories 2-dimensional theories - e.g CFT 3-dimensional theories - Chern-Simons theory 4-dimensional theories - e.g. SW AdS/CFT AGT Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 2 / 28
5 Chern-Simons theory Current knowledge of the gauge eld theories 2-dimensional theories - e.g CFT 3-dimensional theories - Chern-Simons theory 4-dimensional theories - e.g. SW 5-dimensional theories 2 AdS/CFT AGT 5d AGT Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 2 / 28
6 Chern-Simons theory Current knowledge of the gauge eld theories 2-dimensional theories - e.g CFT 3-dimensional theories - Chern-Simons theory 4-dimensional theories - e.g. SW 5-dimensional theories 2 AdS/CFT AGT 5d AGT In the last few years many connections between CS and the other theories such as Integrable Systems, Topological Strings Theories and Conformal Field Theories has been found. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 2 / 28
7 Chern-Simons theory Chern-Simons theory 3-dimensional topological gauge theory - Chern-Simons theory S CS = k 4π d 3 x ( AdA A A A) Wilson-loop averages: ( ) W K Q = tr Q Pexp Adx K CS(N,q) Q is a representation of the group SU(N) K is a contour (i.e. a knot) q = exp( 2πi k+n ) 1 E.Witten, Comm.Math.Phys. 121 (1989) 351 Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 3 / 28
8 Chern-Simons theory Chern-Simons theory 3-dimensional topological gauge theory - Chern-Simons theory S CS = k 4π d 3 x ( AdA A A A) Wilson-loop averages: ( ) W K Q = tr Q Pexp Adx K CS(N,q) Q is a representation of the group SU(N) K is a contour (i.e. a knot) q = exp( 2πi k+n ) In 1989 E.Witten 1 suggested that the Wilson-loop averages of the CS theory are equal to the knot polynomials: H K = Q W K Q CS(N,q) 1 E.Witten, Comm.Math.Phys. 121 (1989) 351 Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 3 / 28
9 Knot theory HOMFLY polynomials HOMFLY polynomials are topological invariants and the Laurent polynomials in variables A and q. The fundamental HOMFLY can be evaluated using the skein relations A K ( ) A 1 = q q 1 K K A H K A 1 H K In the Chern-Simons theory q = e 2πi k+n = ( q q 1) H K and A = q N Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 4 / 28
10 Knot theory Color in the knot theory How can the color (i.e. representation Q) be introduced? Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 5 / 28
11 Knot theory Color in the knot theory How can the color (i.e. representation Q) be introduced? Colored skein relations For the representation [2]: q6 q A 2 q 4 +q 4 A 6 = q6 q A 4 q 6 Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 5 / 28
12 Knot theory Color in the knot theory How can the color (i.e. representation Q) be introduced? Colored skein relations For the representation [2]: q6 q A 2 q 4 +q 4 A 6 = q6 q A 4 q 6 Cabling procedure To dene it we will rst provide more details about the other methods of evaluating the knot polynomials. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 5 / 28
13 Knot theory Braid representation To describe the knot it is convenient to use its braid representation: Then in the Reshetikhin-Turaev formalism 2 for each crossing the R-matrix should be inserted and the answer for the HOMFLY polynomials then is given by the trace over the product of all the R-matrices: H K Q = Tr Q 2 i 2 N.Yu.Reshetikhin and V.G.Turaev, Comm. Math. Phys. 127 (1990) 1-26 Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 6 / 28 R i
14 Knot theory Current knowledge about the R-matrices R-matrices depend on many parameters: Representations of the group SU(N) placed on the dierent strands Number of the strands in the braid The pair of the crossing strands The exact matrix form has been constructed for any R-matrix in the fundamental representation. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 7 / 28
15 Cabling procedure Cabling procedure The main idea behind the cabling procedure is that H K Q m = H Km Q K m is a knot K were the strand is replaced with the m parallel strands: Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 8 / 28
16 Cabling procedure Projectors The cabling of the knot gives the answer in the reducible representation Q m therefore to nd the answer in some other irreducible or reducible representation T the operator called projector P Q m T constructed. should be These projectors can be described as a combination of the added crossings between the m strands in the cable. In other words the knot polynomials (Wilson-loop average) in the representation T can be represented as a linear combination of the knot polynomials in the fundamental representations but for the knots with T times more strands. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 9 / 28
17 Cabling procedure Answers for the projectors The general answer for the projectors is known in the matrix form. The answer which describes the connections between the colored and the fundamental knot polynomials is known only in the several simplest examples. P 2 = R 1 + q 1 q + q 1 P 11 = R 1 q q + q 1 There is also a suggestion of the recursive procedure which in principle should be able to construct any projector. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 10 / 28
18 Conclusion Conclusion The exact matrix form of the R-matrices in the fundamental representation has been constructed. This allows in principle to nd the HOMFLY polynomial of any knot or link in the fundamental representation. The matrix form of the projectors has been constructed. This allows in principle to nd the HOMFLY polynomial of any knot or link in any representation. The method to nd the projectors which describes the connections between the fundamental and the colored HOMFLY polynomials has been suggested. The collected data should be analyzed to nd the properties of the Wilson averages and the connections between the dierent knot polynomials. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 11 / 28
19 Conclusion THANK YOU FOR YOUR ATTENTION! Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 12 / 28
20 R-matrices and Character Expansion Yang-Baxter Equation HOMFLY polynomials satisfy three Reidemeister moves The third Reidemeister move is the Yang-Baxter equation which solutions are R-matrices. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 13 / 28
21 R-matrices and Character Expansion Character Expansion The eigenvectors of these R-matrices with the dierent eigenvalues correspond to the dierent irreducible representations from the decomposition of the Q m. Thus instead of using the R-matrices acting on the vectors one could use one for the irreducible representations. Then the answer for the Wilson-loop average (Knot polynomial) will be an expansion into the Schur-functions (characters of dierent irreducible representations of the studied group). The coecients of this expansion are not topologically invariant, but the whole sums are: H K Q (A q) = W K Q = T h T Q S T (A, q) Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 14 / 28
22 R-matrices R-matrices For the braid with m strands there are m 1 dierent R matrices corresponding to the crossings between dierent pairs of strands. R 2 R 1 R 2 R 1 R 1 = R 0 I R 2 = I R 0 All of them have the same eigenvalues but they are not the same ones. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 15 / 28
23 R-matrices Diagonal R-matrices For the crossing between representations T 1 and T 2 the eigenvalues are described by the irreducible representations in the decomposition Q i T 1 T 2. λ i = q κ Q i, κ Qi = 1 (i j) 2 {i,j} Q i If there are more than two strands then the following decomposition should be studied: T 1 T 2.. T m = ( i Q i ).. T m = j Q j The eigenvalues for the Qj are the same ones as for the corresponding Q i Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 16 / 28
24 R-matrices Non-diagonal R-matrices The general formula is known only for the fundamental representations. The matrix is block-diagonal with blocks 1 1 equal to q or q 1 and 2 2 equal to the b j = 1 q j [j] q [j+1]q[j 1] q [j] q [j+1]q[j 1] q [j] q q j [j] q Where [j] q is the quantum j: [j] q qj q j q q 1. Then the paths on the tree of the diagrams should be considered Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 17 / 28
25 R-matrices Tree of the diagrams Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 18 / 28
26 R-matrices Non-diagonal R-matrices R k 1 is described by the level k in the tree. The paths can go as singlets or doublets. In doublets two paths dier only on the level k q 1 in R 2 31 b 3 block in R q in R b 3 block in R b 4 block in R If the length of a hook, connecting the two added cells is equal to j then block b j 1 should be used Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 19 / 28
27 R-matrices Non-diagonal R-matrices R = 1 [2]q[4] q [3] qq 3 [3] q [2]q[4] q q 3 [2] q [3] q q q 1 1 [2]q[4] q [3] qq 3 [3] q [2]q[4] q [3] q q 3 [3] q Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 20 / 28
28 Projectors Path Projectors Projectors can be constructed n the same manner as the R-matrices. For the P Q in the tree only the paths which pass through the Q should remain. P = Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 21 / 28
29 Projectors Representation tree Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 22 / 28
30 Projectors Characteristic equations The denition using the paths is very constructive but does not provide the connections between the colored HOMFLY and the fundamental ones. To nd these connections the denition which uses the R-matrices should be constructed. This can be done using the operator properties of the R-matrices: The projector on eigenvalue λ i of the operator with the characteristic n equation (ˆR λ i ) = 0 is i=1 ˆR λ i ˆP λj = i j λ j λ i Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 23 / 28
31 Characteristic equation Characteristic equations for the R-matrices For the fundamental R-matrices the characteristic equation is known from the mathematics and is described by the skein relations: R R 1 ( q q 1) ( = (R q) R + 1 ) = 0 q ) = (q q 1 Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 24 / 28
32 Characteristic equation Colored skein relations From the characteristic equation the analogue of the skein relations for the colored R-matrices can be constructed though it is not very useful: ( R2 2 q 6) ( R q 2) (R 2 2 1) = 0 ( q 8 q 6 + q 2) +q 8 ( ) = q 6 q Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 25 / 28
33 Characteristic equation The projectors on [2] and [11] From the characteristic equation the projectors can be constructed ( (R q) R + 1 ) = 0 q P 2 = R 1 + q 1 q + q 1 P 11 = R 1 q q + q 1 Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 26 / 28
34 Characteristic equation The projectors on [21] There are also additional characteristic equations for the combinations of the R-matrices: ( ) (R 1 R 2 ) (R 1 R 2 ) 2 (q q 2 ) = 0 Together with the characteristic equations on each of the R-matrices this gives P 21 = (R 1 R 2 ) 2 q q 2 Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 27 / 28
35 Characteristic equation Recursive formula for the projector Q B Q = R i Q i=1 j=1 R Q +1 j The projector on the representation Q made from T with the addition of the cell {k, l} is equal to P Q=T (k,l) = i B Q q 2j 2i (i,j) (k,l) q 2l 2k q 2j 2iP T The product is over all possible additions of cells {i, j}. Andrey Morozov (ITEP, MSU, Moscow) Cabling procedure 1 July, 2013, Erice 28 / 28
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