Quantum R-matrices, classical integrability and knot invariants Andrei Mironov P.N.Lebedev Physics Institute and ITEP Integrability and Combinatorics, 2014
A.Mironov (LPI/ITEP) Knots and tau-functions 2014 2 / 25
Basic examples A generating function of the non-linear Schr¨ odinger equation (at infinite coupling) − → Refs classical non-linear Schr¨ odinger at finite A.Its, A.Izergin, V.Korepin, N.Slavnov (1990) coupling A.Zabrodin (A.Alexandrov, V.Kazakov, Master T-operator for inhomogeneous XXX S.Leurent, Z.Tsuboi) GL ( N ) spin chain with twisted boundary A.Izegin (1987) conditions − → Hirota blinear equation for mKP M.Jimbo, T.Miwa, M.Sato; T.T.Wu, B.M.McCoy, C.A.Tracy, E.Barouch; D.Bernard, Partition function of the 6-vertex model with A.LeClair; ... inhomogeneities (determinant formula) − → KP τ -function in Miwa variables . . . Goal for knot theory: from knots to equations Yet another approach: quantum A-polynomials (S.Garoufalidis, S.Gukov etc) A.Mironov (LPI/ITEP) Knots and tau-functions 2014 3 / 25
Plan HOMFLY-PT, generating functions and τ -functions 1 Rosso-Jones formula for torus knots/links 2 Tau-functions for torus knots 3 HOMFLY-PT via R-matrices 4 Deformations 5 Non-torus knots A.Mironov (LPI/ITEP) Knots and tau-functions 2014 4 / 25
HOMFLY-PT, generating functions and τ -functions HOMFLY-PT � � � � H K R ( q | A ) A = q N = Tr R P exp A � SU ( N ) K 3 d Chern-Simons theory with the gauge group SU ( N ) and the action � S = κ � A d A + 2 3 A 3 � 2 πi Tr q = exp 4 π κ + N d 3 x Skein relations for H K � ( q | A ) : H K R ( q | A ) is proportional to a Laurent polynomial in q and A ■ ❅ � ✒ ■ ❅ � ✒ { x } ≡ x − 1 � ❅ ❅ � A − 1 A − = { q } ✻✻ ❅ � x � ❅ � ❅ � ❅ A.Mironov (LPI/ITEP) Knots and tau-functions 2014 5 / 25
Color HOMFLY-PT: cabling Trefoil A.Mironov (LPI/ITEP) Knots and tau-functions 2014 6 / 25
Trefoil: simplest torus knot Examples ( q 2 + q − 2 ) A − A − 1 � { A } � H [2 , 3] = [1] { q } q 4 { A }{ Aq } � ( q 12 + q 6 + q 4 + 1) A 2 − q 8 − q 6 − q 2 − 1 + q 2 A − 2 � H [2 , 3] = [2] { q }{ q 2 } ( q − 12 + q − 6 + q − 4 + 1) A 2 − q − 8 − q − 6 − q − 2 − 1 + q − 2 A − 2 � { A }{ A/q } � H [2 , 3] [11] = q 4 { q }{ q 2 } A.Mironov (LPI/ITEP) Knots and tau-functions 2014 7 / 25
Ooguri-Vafa partition function as a τ -function � � �� � n 1 � � Z K ( q, A ) = H K Tr V n R ( q, A ) S R ( p ) = exp n Tr A dx K R n p k = Tr V k = � i v k i are external sources, S R ( p ) are Schur functions, Tr is taken over the fundamental representation. Ooguri-Vafa conjecture Z K ( q, A ) describes the topological string on the resolved conifold. When is a τ -function? Hopf link: Unknot: A.Mironov (LPI/ITEP) Knots and tau-functions 2014 8 / 25
Ooguri-Vafa partition function as a τ -function � � �� � n 1 � � Z K ( q, A ) = H K Tr V n R ( q, A ) S R ( p ) = exp n Tr A dx K R n p k = Tr V k = � i v k i are external sources, S R ( p ) are Schur functions, Tr is taken over the fundamental representation. Ooguri-Vafa conjecture Z K ( q, A ) describes the topological string on the resolved conifold. When is a τ -function? Hopf link: Unknot: A.Mironov (LPI/ITEP) Knots and tau-functions 2014 8 / 25
Ooguri-Vafa partition function as a τ -function � � �� � n 1 � � Z K ( q, A ) = H K Tr V n R ( q, A ) S R ( p ) = exp n Tr A dx K R n p k = Tr V k = � i v k i are external sources, S R ( p ) are Schur functions, Tr is taken over the fundamental representation. Ooguri-Vafa conjecture Z K ( q, A ) describes the topological string on the resolved conifold. When is a τ -function? Hopf link: Unknot: A.Mironov (LPI/ITEP) Knots and tau-functions 2014 8 / 25
Ooguri-Vafa partition function as a τ -function � � �� � n 1 � � Z K ( q, A ) = H K Tr V n R ( q, A ) S R ( p ) = exp n Tr A dx K R n p k = Tr V k = � i v k i are external sources, S R ( p ) are Schur functions, Tr is taken over the fundamental representation. Ooguri-Vafa conjecture Z K ( q, A ) describes the topological string on the resolved conifold. When is a τ -function? Hopf link: Unknot: A.Mironov (LPI/ITEP) Knots and tau-functions 2014 8 / 25
τ -function as sum over characters � τ ( t ) = g R S R ( p ) t k = p k /k R What are the conditions for g R ? g [22] g [0] − g [21] g [1] + g [2] g [11] = 0 These are the Pl¨ ucker relations. Their general g [32] g [0] − g [31] g [1] + g [3] g [11] = 0 solution is: c R = det ij M R i − i,j g [221] g [0] − g [211] g [1] + g [2] g [111] = 0 . . . A particular example � � � � ( R i − i + 1 / 2) k − ( − i + 1 / 2) k � � � S R (¯ p ) S R ( p ) exp ξ k C k ( R ) C k = R k i is a τ -function of both the KP and Toda lattice hierarchy. A.Mironov (LPI/ITEP) Knots and tau-functions 2014 9 / 25
τ -function as sum over characters � τ ( t ) = g R S R ( p ) t k = p k /k R What are the conditions for g R ? g [22] g [0] − g [21] g [1] + g [2] g [11] = 0 These are the Pl¨ ucker relations. Their general g [32] g [0] − g [31] g [1] + g [3] g [11] = 0 solution is: c R = det ij M R i − i,j g [221] g [0] − g [211] g [1] + g [2] g [111] = 0 . . . A particular example � � � � ( R i − i + 1 / 2) k − ( − i + 1 / 2) k � � � S R (¯ p ) S R ( p ) exp ξ k C k ( R ) C k = R k i is a τ -function of both the KP and Toda lattice hierarchy. A.Mironov (LPI/ITEP) Knots and tau-functions 2014 9 / 25
τ -function as sum over characters � τ ( t ) = g R S R ( p ) t k = p k /k R What are the conditions for g R ? g [22] g [0] − g [21] g [1] + g [2] g [11] = 0 These are the Pl¨ ucker relations. Their general g [32] g [0] − g [31] g [1] + g [3] g [11] = 0 solution is: c R = det ij M R i − i,j g [221] g [0] − g [211] g [1] + g [2] g [111] = 0 . . . A particular example � � � � ( R i − i + 1 / 2) k − ( − i + 1 / 2) k � � � S R (¯ p ) S R ( p ) exp ξ k C k ( R ) C k = R k i is a τ -function of both the KP and Toda lattice hierarchy. A.Mironov (LPI/ITEP) Knots and tau-functions 2014 9 / 25
τ -function as sum over characters � τ ( t ) = g R S R ( p ) t k = p k /k R What are the conditions for g R ? g [22] g [0] − g [21] g [1] + g [2] g [11] = 0 These are the Pl¨ ucker relations. Their general g [32] g [0] − g [31] g [1] + g [3] g [11] = 0 solution is: c R = det ij M R i − i,j g [221] g [0] − g [211] g [1] + g [2] g [111] = 0 . . . A particular example � � � � ( R i − i + 1 / 2) k − ( − i + 1 / 2) k � � � S R (¯ p ) S R ( p ) exp ξ k C k ( R ) C k = R k i is a τ -function of both the KP and Toda lattice hierarchy. A.Mironov (LPI/ITEP) Knots and tau-functions 2014 9 / 25
Torus knots/links and braid representation Closed braid: If m and n are mutually prime, it is torus. Otherwise, it is a link with l components if l is the largest common divisor of m and n . A.Mironov (LPI/ITEP) Knots and tau-functions 2014 10 / 25
Adams operation. Pletism Adams operation: step 1 p [ m ] → v m = p mk v i − i k Adams operation: step 2 For links with l components: For knots: � l S R ( p [ m ] ) = C Q � � R S Q ( p ) C Q S R a ( p [ m ] ) = R 1 ...R l S Q ( p ) Q ⊢ m | R | a =1 Q ⊢ m | R | A.Mironov (LPI/ITEP) Knots and tau-functions 2014 11 / 25
Adams operation. Pletism Adams operation: step 1 p [ m ] → v m = p mk v i − i k Adams operation: step 2 For links with l components: For knots: � l S R ( p [ m ] ) = C Q � � R S Q ( p ) C Q S R a ( p [ m ] ) = R 1 ...R l S Q ( p ) Q ⊢ m | R | a =1 Q ⊢ m | R | A.Mironov (LPI/ITEP) Knots and tau-functions 2014 11 / 25
Adams operation. Pletism Adams operation: step 1 p [ m ] → v m = p mk v i − i k Adams operation: step 2 For links with l components: For knots: � l S R ( p [ m ] ) = C Q � � R S Q ( p ) C Q S R a ( p [ m ] ) = R 1 ...R l S Q ( p ) Q ⊢ m | R | a =1 Q ⊢ m | R | A.Mironov (LPI/ITEP) Knots and tau-functions 2014 11 / 25
Rosso-Jones formula For knots: � H m,n 2 m C 2 ( Q ) C Q n R S ∗ ( q | A ) = q For links with l components: R Q Q ⊢ m | R | � n H m,n m C 2 ( Q ) C Q R 1 ...R l S ∗ R 1 ...R l ( q | A ) = q Q Here Q ⊢ m | R | p k = { A k } S ∗ Q ≡ S Q ( p ∗ ) { q k } S ∗ Q is the HOMFLY-PT for unknot and is equal to the quantum dimension of Q : { Aq i − j } [ N + i − j ] q � A = q N � S ∗ Q = − → { q h i,j } [ h i,j ] q ( i,j ) ∈ Q ( i,j ) ∈ Q h i,j is the hook length. A.Mironov (LPI/ITEP) Knots and tau-functions 2014 12 / 25
Rosso-Jones formula For knots: � H m,n 2 m C 2 ( Q ) C Q n R S ∗ ( q | A ) = q For links with l components: R Q Q ⊢ m | R | � n H m,n m C 2 ( Q ) C Q R 1 ...R l S ∗ R 1 ...R l ( q | A ) = q Q Here Q ⊢ m | R | p k = { A k } S ∗ Q ≡ S Q ( p ∗ ) { q k } S ∗ Q is the HOMFLY-PT for unknot and is equal to the quantum dimension of Q : { Aq i − j } [ N + i − j ] q � A = q N � S ∗ Q = − → { q h i,j } [ h i,j ] q ( i,j ) ∈ Q ( i,j ) ∈ Q h i,j is the hook length. A.Mironov (LPI/ITEP) Knots and tau-functions 2014 12 / 25
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